The harness's method zoo: 36 computational methods for quantum many-body problems, each with its accuracy class, an M1–M14 property table (tasks, regime, cost, scaling, failure modes), and verified benchmarks. Every row is generated from the method's knowledge-base card.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Full spectrum · all eigenvectors · level statistics · ETH diagnostics · thermodynamics · entanglement spectrum · order parameters · exact finite-cluster dynamics | Only method that natively returns the full spectrum; thermodynamics via exact canonical sum. |
| M2 regime | T=0 and finite-T (all temperatures from a single diagonalization) | Full spectrum covers all regimes at once. |
| M3 accuracy class | Numerically exact, deterministic | Errors are only floating-point rounding; no statistical error, no truncation bias. |
| M4 dimension fit (A1) | Any dimension / geometry | Applicable to any lattice; total site count N is the only limit, not geometry. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion / any | Local dim d enters as D_H = d^N; large d (Hubbard d=4, soft-core bosons) drastically shrinks reachable N. |
| M6 entanglement regime (B5) | Volume-law tolerated | No entanglement restriction; full Hilbert-space representation requires O(D_blk²) memory. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; no sign problem. |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂/parity block-diagonalize H; translation (k) and point-group reduce D_blk by ≈N and |G| respectively | Decisive cost lever: D_blk = D_H / (reduction factor). With full symmetry (Sᶻ + momentum + point group + parity), a spin-½ chain of N≈20 is trivial dense; dense ED reaches ~28–30 sites maximum (see M12). The ~40-site and 48–50-site numbers belong to iterative (Lanczos) ED — see ed-lanczos card. |
| M9 time complexity | O(D_blk³) | Dense eigensolve (LAPACK dsyevd); dominant cost for useful system sizes. |
| M10 memory | O(D_blk²) | Must store the full dense matrix and all eigenvectors; the first wall hit in practice. |
| M11 control knob | None (exact) — the only knob is which symmetry sectors to diagonalize | No convergence parameter: result is exact within machine precision. |
| M12 scale frontier | ~20 sites trivial dense; ~28–30 sites maximum for dense ED with full symmetry (N=28 full-symmetry D_blk~1.8×10⁵ → ~250–520 GB (real vs complex double); N=32 → ~44 TB, infeasible) | Dense O(D_blk²) memory is the binding wall. Iterative (Lanczos) ED reaches ~40 sites routine and 48–50 at the supercomputer frontier (Wietek–Läuchli 2018, arXiv:1804.05028) — see ed-lanczos card. |
| M13 primary approximation / bias | None — numerically exact | Only approximation is the finite cluster (open/periodic BC); bulk extrapolation requires finite-size scaling. |
| M14 hard blocker / failure mode | Hilbert-space d^N wall: dense O(D_blk²) memory walls by N ≈ 30 (spin-½) or N ≈ 16 (Hubbard d=4) | No workaround within ED full; switch to ED Lanczos for larger N or for a few eigenstates. |
O(D_blk³) — LAPACK dense eigensolveO(D_blk²) — stores the full matrix and all eigenvectorsD_blkN; primary validation target for DMRG, QMC, VMC, iPEPSN≈20, Hubbard up to N≈12–16 sitesmethod-property-map.md (ED profile): applicable when D_H ≲ 10⁷–10⁸ after symmetry reductionN=20: Hilbert-space dim D_H = 2^20 ≈ 10^6; with U(1) largest block C(20,10) ≈ 184,756; dense diag trivial on a laptop sandvik_2010_computational.d=4, N=16: D_H = 4^16 ≈ 4×10^9 (full Fock space, all fillings; at half-filling D_H = C(16,8)² ≈ 1.66×10^8); requires full symmetry reduction to be tractable [method-survey.md §1.1].Key reference: sandvik_2010_computational — comprehensive tutorial covering the full diagonalization approach, symmetry reduction (momentum sectors, point group, spin inversion), thermodynamic observables via exact canonical sums, and benchmarks for Heisenberg and Hubbard models.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Ground-state energy · ground-state wavefunction · few excited states & gaps · dynamical S(q,ω) (continued-fraction) · real-time dynamics (Krylov exponentiation) · entanglement entropy · order parameters | Level statistics and full thermodynamics require the full spectrum (ED full). |
| M2 regime | T=0 (a few low eigenstates); real-time dynamics on finite clusters | Finite-T via FTLM/TPQ (separate method) uses the same matrix-free machinery. |
| M3 accuracy class | Numerically exact, deterministic | The Lanczos recursion converges to machine precision for the targeted eigenpairs; no stochastic error. |
| M4 dimension fit (A1) | Any dimension / geometry | Same as ED full: geometry only affects N_nnz and the sparsity pattern, not the algorithm's applicability. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion / any | D_H = d^N; large d shrinks reachable N, same as ED full, but Lanczos reaches about 2× larger N than full diag. |
| M6 entanglement regime (B5) | Volume-law tolerated | No entanglement restriction; the ground-state wavefunction is stored as a full vector of length D_blk. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; no sign problem. |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂/parity block-diagonalize H; translation (k) and point-group reduce D_blk by ≈N and |G| | The decisive cost lever: symmetry reduction is the primary route to larger system sizes. With full symmetry, N≈48–50 is accessible (Wietek–Läuchli 2018). |
| M9 time complexity | O(n_iter · N_nnz) where N_nnz ≈ N · D_blk for short-range (A4) Hamiltonians | n_iter ≈ tens for gapped (B6) systems; hundreds for critical or near-degenerate systems; much higher for shift-invert (interior states). |
| M10 memory | O(D_blk) | Stores only a few Lanczos vectors of length D_blk; the key advantage over ED full. |
| M11 control knob | n_iter (number of Lanczos steps) — controls convergence of the targeted eigenpairs | Convergence criterion: residual norm ‖H|ψ⟩ − E|ψ⟩‖ < ε. For the GS, convergence is typically fast (tens of iterations); for excited states or S(q,ω), more iterations needed. |
| M12 scale frontier | ~40 sites routine with full symmetry; 48–50 sites at the supercomputer frontier (Wietek–Läuchli 2018, arXiv:1804.05028, sublattice-coding with translation + point group + parity) | Frontier requires distributed-memory storage of D_blk-length vectors across many nodes. |
| M13 primary approximation / bias | None — numerically exact for the targeted eigenpairs | The finite cluster approximation (open/PBC) is external; bulk extrapolation requires finite-size scaling. |
| M14 hard blocker / failure mode | d^N wall (same as ED full, but shifted to larger N): D_blk vector storage requires distributed memory beyond ~40 sites; near-degenerate or critical systems (B6) require many iterations (n_iter → ∞ at a quantum critical point) | Long-range interactions (A4) raise N_nnz and slow matrix-vector products. |
O(n_iter · N_nnz) with N_nnz ≈ N · D_blk for short-range models; n_iter ~ tens (gapped) to hundreds (critical)O(D_blk) — stores only a few Krylov vectors (the key advantage over ED full)n_iter (Lanczos iterations) — controls eigenpair convergence to machine precisionN≈40; the benchmark oracle for DMRG, QMC, VMC at these sizesS(q,ω) on small clusters: frustrated magnets (B8, C12 sign-ful) where QMC cannot reach real-frequency data without analytic continuationmethod-property-map.md (ED profile): applicable when D_H ≲ 10⁷–10⁸ after symmetry reduction; preferred over ED full whenever the full spectrum is not requiredmethod-survey.md §1.2].< 10⁻¹² in < 200 Lanczos steps for gapped 1D Heisenberg (N=36) sandvik_2010_computational.S(q,ω) via continued-fraction: n_CF ≈ 200 Lanczos steps per q point for Heisenberg chain N=24 gagliano_1987_dynamical.n_iter ~ 50 (gapped) vs ~ 500 (critical, near-degenerate) — B6 dependence verified in method-survey.md §1.2.Key reference: sandvik_2010_computational — covers the Lanczos algorithm, symmetry reduction, continued-fraction computation of S(q,ω), and benchmarks for Heisenberg chains and ladders.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Finite-T thermodynamics (free energy, internal energy, specific heat C_v, susceptibility χ) · finite-T dynamical S(q,ω) (FTLM) · canonical and grand-canonical ensembles (canonical TPQ) | Does NOT natively give the full spectrum or level statistics (need ED full for those). |
| M2 regime | Finite-T (all temperatures including T→0 limit via large β) | T=0 ground-state properties accessible via large-β limit; in practice ED Lanczos is cheaper for T=0. |
| M3 accuracy class | Stochastic (statistical error ∝ 1/√(R·D_H)); controlled-bias (improvable by increasing R) | Error shrinks with system size (typicality) — a key practical advantage at N ≳ 20. Deterministic per random vector; stochastic only across the average. |
| M4 dimension fit (A1) | Any dimension / geometry | Same as ED full/Lanczos: geometry enters only through N_nnz and the symmetry block structure. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion / any | D_H = d^N; memory O(D_blk) means the accessible N matches ED Lanczos (not ED full). |
| M6 entanglement regime (B5) | Volume-law tolerated | Operates in the full Hilbert space; no entanglement restriction. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; random vectors in Hilbert space, not field-space sampling. |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂/parity block-diagonalize; translation and point-group reduce D_blk | Same symmetry reduction as ED Lanczos; random vectors drawn within the symmetry sector. |
| M9 time complexity | O(R · n_iter · N_nnz) where R = number of random vectors, n_iter = Lanczos steps per vector, N_nnz ≈ N · D_blk | At high T, R can be small (1–10); at low T, thermal fluctuations are larger and R must grow. |
| M10 memory | O(D_blk) | Stores only the current Lanczos basis vectors; same footprint as ED Lanczos. |
| M11 control knob | R (number of random vectors) — controls statistical error 1/√(R·D_H) | At large N, even R=1 can give < 1% error (typicality); at small N, need R ≈ 10–100. Standard canonical TPQ (Sugiura–Shimizu 2012/2013) applies e^{-βH/2}|r⟩ via Krylov/Chebyshev — exact to machine precision with no Trotter error; Δτ is a knob only in Trotter-based TPQ implementations. |
| M12 scale frontier | Same as ED Lanczos: ~40 sites routine; frontier ~48–50 sites with full symmetry | Typicality means error improves with system size; FTLM/TPQ is most powerful for N ≳ 20 where it beats ED full on memory while giving accurate finite-T results. |
| M13 primary approximation / bias | Stochastic trace approximation with R random vectors; error ∝ 1/√(R·D_H) (controlled) | Finite cluster approximation for bulk thermodynamics; bulk limit requires finite-size scaling. |
| M14 hard blocker / failure mode | Very low T (large β): thermal fluctuations grow → R must increase to maintain accuracy; long imaginary-time Lanczos evolution becomes numerically delicate | D_H wall same as ED; sign-immune so no QMC sign problem. At very low T, ED Lanczos (T=0) or DMRG (finite-T purification) may be preferred. |
O(R · n_iter · N_nnz) with N_nnz ≈ N · D_blk for short-range modelsO(D_blk) — same as ED Lanczos; no full matrix or all-eigenvector storageR (random vectors) controlling statistical error 1/√(R · D_H); standard Krylov-TPQ has no Δτ (exact imaginary-time evolution); Δτ (error O(Δτ²)) applies only in Trotter-based TPQ implementationsN ≈ 20–40: where FTLM/TPQ beats ED full (memory O(D_blk) vs O(D_blk²)) while giving accurate finite-T resultsS(q,ω): dynamical neutron-scattering structure factor at finite T on finite clustersmethod-property-map.md: falls under the ED harness skill profile; preferred for D13 finite-T when QMC sign is present and LTRG/XTRG is unavailable∝ 1/√(R · D_H): for N=20 spin-½ (D_H ≈ 10^6), R=10 gives error ≈ 3×10^{-4} per observable sugiura_2012_thermal.N=28: agrees with Bethe ansatz TBA to < 0.5% for T ≥ J/10 with R=20 jaklic_1994_lanczos.n_iter ≈ 100–200 Lanczos steps for convergence at intermediate T [method-survey.md §1.3].Key reference: jaklic_1994_lanczos — original FTLM paper by Jaklič and Prelovšek; establishes the random-vector Lanczos approach for finite-T thermodynamics and dynamical functions in correlated systems.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Density of states (DOS) · spectral function A(ω) · dynamical structure factor S(q,ω) · optical / dynamical conductivity · finite-T traces via Chebyshev expansion of e^{-βH} | Does NOT provide individual eigenstates, the GS wavefunction, or entanglement data (these require Lanczos). |
| M2 regime | T=0 spectral densities · finite-T thermodynamic traces (stochastic Chebyshev) | Ground-state GS energy not native (no variational principle); spectral densities at all ω. |
| M3 accuracy class | Stochastic-trace + controlled-bias; deterministic per random vector, stochastic across R vectors | Bias: Gibbs ripples from truncating the Chebyshev series at order M — controlled by Jackson/Lorentz kernel; statistical error ∝ 1/√(R · D_H) from the stochastic trace. Resolution ≈ bandwidth/M improves with M. |
| M4 dimension fit (A1) | Any dimension / geometry — wins for large sparse Hamiltonians in any d | Sparsity (A4) is the key gate; short-range interactions → small N_nnz per site → cheap matrix-vector products. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion / any | D_H = d^N; memory O(D_H) (stores one or a few state vectors of length D_H). |
| M6 entanglement regime (B5) | Volume-law tolerated | No entanglement restriction; KPM never constructs a wavefunction ansatz — it works with the full D_H-dimensional Hilbert space. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; random vectors sampled in Hilbert space, not field space. |
| M8 symmetry exploitation (C9/C10) | Block structure can reduce D_H (same as ED Lanczos); random vectors drawn within a symmetry sector for sector-resolved DOS/S(q,ω) | Symmetry is less critical than for ED (no need to diagonalize): KPM already reaches full D_H scale cheaply. |
| M9 time complexity | O(M · R · N_nnz) where M = #Chebyshev moments, R = #random vectors, N_nnz = #nonzeros in H | Linear in M and R; no D_H³ or D_H^2 factor. Scales to very large sparse Hamiltonians inaccessible to Lanczos. |
| M10 memory | O(D_H) — stores one or a few state vectors of length D_H | No matrix storage (matrix-free); no eigenvalue array; just the state vectors and the recursion coefficients. |
| M11 control knob | M (#Chebyshev moments) — controls spectral resolution ≈ bandwidth/M and Gibbs-ripple suppression | Larger M → finer spectral resolution → higher cost (linearly). R controls statistical error 1/√(R · D_H). Both knobs are independent and can be tuned separately. |
| M12 scale frontier | Largest system sizes accessible among the exact methods: D_H ≈ 10^{9}–10^{10} feasible on a single workstation since only O(D_H) memory is needed for the vectors | Without the O(D_H²) or O(D_H³) wall, KPM reaches much larger D_H than ED full/Lanczos for the tasks it computes (spectral densities). |
| M13 primary approximation / bias | Truncated Chebyshev expansion at order M — spectral resolution limited to ≈ bandwidth/M; Gibbs oscillations damped by kernel (controlled) | The kernel choice (Jackson, Lorentz) determines the trade-off between resolution and Gibbs suppression; this is a controlled, improvable approximation. |
| M14 hard blocker / failure mode | O(D_H) memory wall — must store state vectors of length D_H; not a route to individual eigenstates or the GS wavefunction; poor for sharp spectral features if M is limited | Long-range interactions (A4) raise N_nnz and slow the matrix-vector products; dense Hamiltonians break the sparse advantage. |
O(M · R · N_nnz) — linear in M and R; N_nnz ≈ N · D_blk for short-range modelsO(D_H) — only a few state vectors; no eigenvalue storageM (#moments) → spectral resolution ≈ bandwidth/M; R (#random vectors) → statistical error 1/√(R · D_H)D_H ≈ 10^9–10^{10} feasible (memory-limited only by the state vector size)D_H is too large for Lanczos eigenstates but O(D_H) memory is feasibleR random vectors serve dual purpose as disorder-average samplesmethod-property-map.md (ED profile, extended): a matrix-free spectral-density method that extends the ED harness to larger D_H for spectral tasksO(M · R · N_nnz) — verified scaling in weisse_2006_kernel §III; resolution ≈ bandwidth/M confirmed for the 1D Heisenberg chain DOS.O(D_H) — storing two state vectors of length D_H; verified in [method-survey.md §1.4].∝ 1/√(R · D_H) — for D_H = 10^8 and R = 10, error ≈ 3×10^{-5} per spectral bin weisse_2006_kernel §II.D.M = 2000 gives resolution ≈ bandwidth/2000; Lanczos continued-fraction achieves similar resolution at comparable cost but requires a starting vector near the target state weisse_2006_kernel §IV.Key reference: weisse_2006_kernel — authoritative Reviews of Modern Physics paper by Weisse, Wellein, Alvermann, and Fehske; defines the Chebyshev expansion, kernel choices (Jackson, Lorentz, Dirichlet), stochastic-trace evaluation, and applications to DOS, dynamical conductivity, and spectral functions. The primary reference for all KPM implementations.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Exact ground-state energy and wavefunction · few excited states (EOM-FCI / excited-state CI) · one- and two-particle reduced density matrices (1-RDM, 2-RDM) · benchmark for approximate quantum-chemistry methods (CCSD, CASPT2, DMRG-CI) | FCI is the benchmark target; other methods are benchmarked against it. |
| M2 regime | T=0 ground state (primary); few excited states via sector targeting | Finite-T FCI possible (canonical-ensemble ED) but rarely used for chemistry; more relevant in lattice-model context. |
| M3 accuracy class | Numerically exact, deterministic (exact FCI); controlled-bias stochastic (FCIQMC, selected-CI) | Exact FCI = numerically exact within the chosen single-particle basis. FCIQMC = stochastic, error ∝ 1/√N_w·N_steps; selected-CI = variational upper bound with improvable selection threshold. |
| M4 dimension fit (A1) | Molecules and lattice clusters in any geometry (A1 irrelevant — FCI is basis-set, not lattice, driven) | Applicable to any molecular or cluster geometry; cost set by #orbitals, not spatial dimension. |
| M5 statistics & local dim (A3) | Fermions only; local dimension d = 4 per spatial orbital (empty/up/down/doubly-occupied) | d=4 per orbital is a factor of 4 harder than spin-½ per site; D_H = C(2·N_orb, N_e) grows exponentially in N_orb. |
| M6 entanglement regime (B5) | Volume-law tolerated | No entanglement restriction; exact FCI stores the full coefficient vector of length D_H. |
| M7 sign-problem dependence (C12) | Sign-immune (exact FCI, selected-CI) · stochastic with sign (FCIQMC, controlled by annihilation) | Exact FCI: sign-immune (direct matrix diagonalization). FCIQMC: fermionic sign problem controlled by the annihilation step, giving unbiased results in principle booth_2009_fermion. |
| M8 symmetry exploitation (C9/C10) | Sᶻ (N_↑, N_↓) + S² (spin symmetry) + spatial point-group (irreducible representations) reduce D_H | Point-group symmetry (C₂v, D₂h, etc.) of the molecule reduces the Hamiltonian block size; key cost lever analogous to translation symmetry in lattice ED. |
| M9 time complexity | Exponential in N_orb: O(D_H³) for direct dense diagonalization (LAPACK); O(N_orb² · D_H) per Davidson/Lanczos iteration (dominant two-electron sigma-vector step) | In practice, the Davidson iterative solver is used (analogous to Lanczos); memory O(D_H) per vector. Exponential wall prevents scaling beyond ~20 orbitals for exact FCI. |
| M10 memory | O(D_H) for iterative (Davidson/Lanczos FCI); O(D_H²) for dense diagonalization | Davidson FCI: stores a few vectors of length D_H; dense FCI (LAPACK): stores the full matrix D_H × D_H. |
| M11 control knob | N_orb (number of active orbitals) — the primary convergence parameter; for selected-CI: selection threshold ε₁ (controls which determinants are included) | Increasing N_orb → exponential cost increase. Selected-CI threshold ε₁ → 0 → exact FCI. FCIQMC: N_w (number of walkers) controls statistical error. |
| M12 scale frontier | ~18–20 spatial orbitals in routine codes (Davidson FCI); ~10¹² determinants at the distributed-memory frontier (trillion-determinant FCI, JCTC 2024); selected-CI extends effective reach via sparsity | The 10¹² frontier uses specialized bit-string algorithms and distributed memory; routine use is ~N_orb ≈ 16 (CASSCF active space). |
| M13 primary approximation / bias | Basis-set incompleteness (choice of single-particle orbitals); none within the chosen orbital space — FCI is exact in the given basis | Extrapolation to the complete-basis-set (CBS) limit is the main source of remaining error. For selected-CI: selection threshold bias (variational, controlled). |
| M14 hard blocker / failure mode | Exponential scaling in N_orb: D_H = C(2·N_orb, N_e) grows as ~4^{N_orb} (at half-filling) — the d^N wall translated to orbital language; standard FCI is limited to active spaces of ~20 orbitals | FCIQMC and selected-CI push the frontier but cannot escape the exponential wall for strongly correlated systems with many active orbitals. |
N_orb; Davidson iterative FCI ≈ O(N_orb² · D_H) per iteration (dominant two-electron sigma-vector); dense diag O(D_H³)O(D_H) for iterative (Davidson); O(D_H^2) for dense; D_H = C(2·N_orb, N_e)N_orb (active orbitals, exponential cost); selected-CI threshold ε₁ (improvable bias); FCIQMC walker count N_w (statistical error 1/√N_w)N_orb ≈ 20d=4 Hubbard); selected-CI (FCIQMC, CIPSI) pushes to larger N_orbN_orb ≈ 30–100 with near-FCI accuracy via stochastic sampling of the dominant determinantsmethod-property-map.md (ED profile): FCI is the quantum-chemistry arm of the ED harness; /method-ed skill; fermionic, any geometryN_w ≈ 10^8 walkers booth_2009_fermion.N_orb ≈ 30–50 with selected-CI threshold ε₁ ≈ 10^{-5} [method-survey.md §1.5].Key reference: booth_2009_fermion — original FCIQMC paper by Booth, Thom, and Alavi; introduces the Monte Carlo imaginary-time projection in determinant space, demonstrating exact FCI-quality results for systems beyond the reach of conventional FCI (C₂ molecule in cc-pVDZ, 10⁸–10¹⁰ determinants); the foundational reference for stochastic approaches to FCI.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Finite-T thermodynamics: energy, magnetization, magnetic susceptibility, specific heat, Binder cumulants, structure factors, correlation functions | NOT ground-state energy or quantum amplitudes; NOT critical exponents (use MCRG). |
| M2 regime | Finite temperature (thermal equilibrium) | Classical models only, or quantum models mapped sign-free to a classical representation (e.g. transfer-matrix / SSE at finite β). |
| M3 accuracy class | Numerically exact (within Boltzmann statistics), stochastic | Statistical error ∝ 1/√(N_s/τ_auto); systematic bias only from finite τ_auto (equilibration) and finite lattice. |
| M4 dimension fit (A1) | Any (1D / 2D / 3D / ∞-D) | Per-sweep cost O(N) is independent of geometry; suited to classical Ising, Heisenberg, XY, Potts models in any dimension. |
| M5 statistics & local dim (A3) | Classical spin models (Ising d=2, Potts d=q, continuous O(n) spins); sign-free quantum mappings | Not applicable to genuine quantum models with a sign problem. Local dim d only affects the update cost per site, not the algorithmic structure. |
| M6 entanglement regime (B5) | N/A — classical model; no entanglement restriction | For quantum models accessed via sign-free mappings, entanglement is implicit in the weight. |
| M7 sign-problem dependence (C12) | Requires sign-free (classical models or sign-free quantum mappings) | Not a quantum MC method per se. A genuine fermionic/frustrated quantum model is NOT accessible unless a sign-free classical representation exists. |
| M8 symmetry exploitation (C9/C10) | Global symmetry used to measure order parameters (e.g. Z₂ for Ising, O(n) for Heisenberg); translation used for structure-factor measurement | Symmetry does not reduce per-sweep cost (unlike ED block-diagonalization), but allows optimized estimators. |
| M9 time complexity | O(N) per sweep; total O(N · τ_auto · N_s); near T_c: τ_auto ~ L^z with z = 2.1665(12) (2D Ising, Glauber/Metropolis, Nightingale–Blöte 1996) | Critical slowing down τ_auto ~ L^z dominates at a continuous transition (B6). Larger lattice L is far more expensive at T_c. |
| M10 memory | O(N) — stores spin configuration + neighbor list | Minimal memory footprint; scales to thermodynamic-limit lattices (millions of sites). |
| M11 control knob | N_s (number of sweeps after equilibration) — statistical error ∝ 1/√(N_s/τ_auto) | Increasing N_s systematically reduces statistical error; τ_auto must be measured to set N_s reliably. |
| M12 scale frontier | Millions of sites in practice (off-critical); thousands of sites near T_c (critical slowing down limits effective sample count) | The method scales to very large L away from criticality; near T_c the effective sample count is N_s/τ_auto. |
| M13 primary approximation / bias | None — sampling from the exact Boltzmann distribution (within statistical error and given exact weights) | Finite-size effects require finite-size scaling to extract thermodynamic-limit behavior. |
| M14 hard blocker / failure mode | Critical slowing down (τ_auto ~ L^z, z = 2.1665(12)) near a continuous transition; frustration/glassiness (B8) inflates τ_auto further, causing metastability; NOT applicable to sign-ful quantum models | Critical slowing down is partially defeated by cluster algorithms (§2.2 of method-survey.md). |
O(N) per sweep; O(N · τ_auto · N_s) total; at criticality τ_auto ~ L^z with z = 2.1665(12) for 2D IsingO(N) — spin configuration and neighbor list onlyN_s (sweeps) — controls statistical error as 1/√(N_s/τ_auto)T_c before critical slowing down dominatesT_c (combined with Binder cumulants), though cluster MC is preferred when z mattersβ)method-property-map.md: classical models (B8 unfrustrated) at finite T; not for critical-exponent extraction (use MCRG) or quantum sign-ful models (use MPS/PEPS/VMC)T_c/J = 2/ln(1+√2) ≈ 2.2692 (exact Onsager): located to < 0.01% precision by Binder-cumulant crossing at L ≈ 32–64 sandvik_2010_computational.z = 2.1665(12) for 2D Ising nightingale_1996_dynamic [verified in method-survey.md §2.1].C ~ L^{α/ν} with α = 0 (log divergence) captured correctly after N_s ≫ τ_auto ~ L^{2.1665} sweeps sandvik_2010_computational.Key reference: sandvik_2010_computational — comprehensive treatment of classical and quantum Monte Carlo for spin systems, including Metropolis/heat-bath algorithms, finite-size scaling with Binder cumulants, and structure-factor estimators.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Same thermodynamic observables as Metropolis MC: energy, magnetization, susceptibility, Binder cumulants, structure factors, correlation functions; Wang–Landau additionally returns the full density of states g(E) | Cluster updates are most valuable for making large-L critical-region measurements affordable. |
| M2 regime | Finite temperature (thermal equilibrium) | Classical models; or quantum models with a sign-free classical representation. |
| M3 accuracy class | Numerically exact (Boltzmann statistics), stochastic | Statistical error ∝ 1/√(N_s/τ_auto); cluster MC's win is that τ_auto is orders of magnitude smaller than Metropolis near T_c. |
| M4 dimension fit (A1) | Any (1D / 2D / 3D / ∞-D) | SW and Wolff are most impactful in 2D and 3D where critical slowing down is severe; applicable to any dimension where the FK mapping holds. |
| M5 statistics & local dim (A3) | Classical spin models (Ising, Heisenberg O(n), Potts) with ferromagnetic / unfrustrated couplings | FK cluster construction requires non-negative bond weights; does not apply to general antiferromagnetic or frustrated models. |
| M6 entanglement regime (B5) | N/A — classical model | Same as Metropolis MC. |
| M7 sign-problem dependence (C12) | Requires sign-free / unfrustrated (Fortuin–Kasteleyn construction requires ferromagnetic couplings) | Frustration breaks the FK bond construction — this is the hard blocker (see M14). |
| M8 symmetry exploitation (C9/C10) | Global spin-flip symmetry exploited by Wolff (projects a random direction, builds the cluster in that projected space); SW flips each cluster independently | Translation / point-group symmetry used for structure-factor estimators. |
| M9 time complexity | O(N) per sweep (amortized over cluster size); near T_c: τ_auto ~ L^z with Swendsen–Wang z ≈ 0.22 and Wolff: ~ ln L equilibrium-autocorrelation; nonequilibrium-relaxation fits give a larger effective z ≈ 1.2 for Wolff | Near-O(N) effective cost at criticality — the decisive advantage over Metropolis (z = 2.1665(12)). Parallel tempering multiplies total cost by N_replicas. |
| M10 memory | O(N) — spin configuration; cluster labels O(N) (union-find) | Cluster label array adds a small constant factor over plain Metropolis. |
| M11 control knob | N_s (sweeps/cluster updates) — controls statistical error; for Wang–Landau: the modification factor f → 1 controls flatness of g(E); for parallel tempering: N_replicas and temperature grid | Statistical error ∝ 1/√(N_s/τ_auto_eff) where τ_auto_eff ≪ τ_auto(Metropolis) near T_c. |
| M12 scale frontier | Millions of sites even near T_c (SW/Wolff make large-L critical-region sampling affordable); parallel tempering / Wang–Landau: hundreds of sites (limited by N_replicas × L^d) | Cluster algorithms enable finite-size scaling studies at L ~ 10³–10⁴ near T_c that are prohibitive with Metropolis. |
| M13 primary approximation / bias | None — samples from exact Boltzmann distribution (for cluster MC); Wang–Landau has a systematic ln(f) bias that vanishes as f → 1 | Finite-size effects remain; Wang–Landau convergence must be checked by flatness criterion. |
| M14 hard blocker / failure mode | Frustration (B8) breaks the Fortuin–Kasteleyn cluster construction — SW/Wolff do not apply; frustrated/spin-glass systems require parallel tempering or simulated annealing (at higher cost and with no z suppression); sign-ful quantum models inaccessible | Parallel tempering is a workaround for glassy landscapes but does not reduce z; it incurs an N_replicas overhead. |
O(N) per cluster sweep (union-find construction); effective critical cost O(N · L^z) with z ≈ 0.22 (SW) vs z ≈ 2.17 (Metropolis); parallel tempering: × N_replicasO(N) — spin configuration + cluster label arrayN_s (sweeps) + N_replicas (parallel tempering) — controls statistical error and barrier crossingT_c for SW/Wolff; hundreds of sites for parallel tempering with many replicasL finite-size scaling at criticality (B6): SW/Wolff make L ~ 10³–10⁴ routine where Metropolis is stuckmethod-property-map.md: classical models, sign-free, B8 unfrustrated (for SW/Wolff); B8 frustrated → parallel tempering only (no z suppression)z ≈ 0.22 for 2D Ising model (equilibrium-autocorrelation convention) [verified in method-survey.md §2.2].~ ln L equilibrium-autocorrelation for 2D Ising; nonequilibrium-relaxation fits give a larger effective z ≈ 1.2 for Wolff [verified in method-survey.md §2.2].L = 512, 2D Ising at T_c: SW requires ~L^{0.22} effective sweeps vs Metropolis ~L^{2.1665} sweeps per independent sample — roughly 10⁵ speedup (ratio L^{z_M − z_SW} = L^{1.95} ≈ 1.3×10⁵ at L=512).T_c/J = 2.2692 (Onsager): cluster MC locates to sub-0.01% at L = 1024 with O(10⁴) cluster updates swendsen_wang_1987_nonuniversal.Key reference: wolff_1989_collective — original Wolff single-cluster algorithm with the demonstration that z is effectively ~ ln L for 2D Ising, making large-L critical simulations affordable.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Critical exponents y_t, y_h (→ ν, β, γ, δ, η); critical coupling K_c; critical manifold and its tangent space; renormalized coupling constants as a function of RG scale | NOT ground-state energies, NOT thermodynamic-limit observables (e.g., magnetization or susceptibility away from criticality), NOT the full spectrum. |
| M2 regime | Finite temperature — specifically at or near the critical fixed point | MCRG is designed for the critical fixed point; away from T_c it gives flow information but not exponents. |
| M3 accuracy class | Controlled-bias, stochastic | Statistical error from MC sampling ∝ 1/√N_s; systematic bias from finite block-spin operator basis (controlled by expanding the basis) and finite lattice (controlled by L → ∞). |
| M4 dimension fit (A1) | Any dimension (1D / 2D / 3D / ∞-D) — applicable to classical spin models in any dimension | Best applied where a critical fixed point exists (2D Ising / Potts, 3D Ising / Heisenberg, etc.). |
| M5 statistics & local dim (A3) | Classical spin models: Ising (d=2), Potts, O(n) continuous spins | Applies to any classical statistical model with a critical point; local dim only affects MC cost. |
| M6 entanglement regime (B5) | N/A — classical model | No entanglement restriction; thermodynamic configurations sampled. |
| M7 sign-problem dependence (C12) | Requires sign-free / classical models | MCRG operates on classical Boltzmann weights; no sign problem. For quantum models, the sign-free auxiliary-field / world-line mapping must exist first. |
| M8 symmetry exploitation (C9/C10) | Symmetry constrains the operator basis for the RG matching (fewer independent coupling constants); translation used in momentum-space matching variants | Symmetry reduces the number of couplings that must be matched, improving accuracy. |
| M9 time complexity | MC sampling cost O(N · τ_auto · N_s) × #RG iterations; cluster MC (§2.2) used to reduce τ_auto; variational MCRG (Wu–Car 2017) significantly alleviates critical slowing down | Larger L near T_c improves exponent accuracy (finite-size corrections ∝ L^{-ω}); #RG steps typically 3–5 per lattice. |
| M10 memory | O(N) — spin configurations + blocked configurations at each RG level | Memory is dominated by storing configurations for the blocked lattice hierarchy. |
| M11 control knob | Lattice size L (controls finite-size corrections ~ L^{-ω}); number of RG iterations; operator-basis size (controls truncation bias) | Accuracy of exponents improves with L and operator-basis completeness; convergence checked by stability of eigenvalues across RG steps. |
| M12 scale frontier | Large lattices near T_c (L ~ 10²–10³ typical; L ~ 10⁴ with cluster MC or variational MCRG); exponent accuracy ~ 10⁻³–10⁻⁴ at the frontier | Finite-size corrections to exponents scale as L^{-ω} (correction-to-scaling exponent); 3–4 RG levels require L ≥ 2^4 = 16 at minimum, L ≥ 256 for percent-level precision. |
| M13 primary approximation / bias | Finite block-spin operator basis (truncation of RG operator space) — controlled by expanding the basis; finite-size effects — controlled by increasing L | Both biases are systematically reducible; variational MCRG optimizes the basis to minimize truncation error. |
| M14 hard blocker / failure mode | Not applicable to non-critical systems or for computing thermodynamic-limit observables / ground-state energies — MCRG extracts only fixed-point data; requires a valid critical point (no topological phases without Landau order); frustrated / glassy systems (B8) complicate locating T_c | Away from criticality, MCRG returns RG-flow information but not universal exponents. |
O(N · τ_auto · N_s) + O(N) per RG blocking iteration; cluster MC cuts τ_auto ~ L^{0.22} (SW) vs L^{2.17} (Metropolis)O(N) — configurations at each RG level (shrink by factor 2–4 per level)L (lattice size, controls finite-size corrections ~ L^{-ω}); operator-basis size; #RG stepsL ~ 10²–10³ routine; L ~ 10⁴ with variational MCRG / cluster samplingmethod-property-map.md (MCRG profile): applicable when B6 = critical; blocked when the goal is thermodynamic-limit observables or GS energiesy_t = 1.000, y_h = 1.875 (exact from Onsager/Kaufman/Yang): recovered to < 0.1% by MCRG with L = 256 and 4 RG levels swendsen_1979_montecarlo.ν = 0.6299(4) (cf. conformal-bootstrap 0.6299(2)), η = 0.0363(4) wu_2017_variational.y_t = 1.00(1), y_h = 1.875(5) — a non-square-lattice test of the method's geometry-independence guo_2005_montecarlo.L^{-ω} with ω ≈ 0.83 (3D Ising) — L ≥ 64 required for sub-percent precision [verified in method-survey.md §2.3].Key reference: wu_2017_variational — variational MCRG that optimizes the block-spin rule to minimize a variational distance, removes critical slowing down, and extracts the critical manifold tangent space; provides benchmark exponents for 2D Ising and 3D Ising.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | GS energy & observables · gap & few excited states (targeted DMRG) · correlation length (transfer-matrix spectrum) · entanglement entropy & spectrum (free from MPS) · order parameters · S(q,ω) (post-process via DMRG + correction-vector or TDVP) | Finite-T and real-time are separate algorithms (§5 of survey); dynamical spectra require additional passes. |
| M2 regime | T=0 ground state (primary); thermodynamic limit via iDMRG/VUMPS | Finite-T via purification/METTS (separate algorithm using same MPS machinery). |
| M3 accuracy class | Controlled-bias (truncation error ε_trunc is the systematic knob), deterministic | Variational upper bound on energy at each fixed χ; bias → 0 as χ → ∞. |
| M4 dimension fit (A1) | 1D / quasi-1D near-exact; walls in 2D (cylinder width W → cut entropy ∝W → χ ~ e^{cW}) | 1D gapped: small constant χ; 1D critical: χ grows polynomially; 2D: feasible only for narrow cylinders (W ≲ 6–8 for spin-½). |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion (Jordan-Wigner or MPO fermion) | Local dimension d appears in cost as O(d·χ³) per site; large d (Hubbard d=4, soft-core boson) raises prefactor but not the exponent. Fermions handled natively via Jordan-Wigner string in MPS. |
| M6 entanglement regime (B5) | Area-law (cheap, constant χ) / area+log 1D-critical (χ grows polynomially) | Volume-law states (excited, thermal, real-time beyond t*) are inaccessible at reasonable χ. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; no sign problem regardless of frustration or doping. |
| M8 symmetry exploitation (C9/C10) | U(1) / SU(2) block-sparse tensors; Z₂ / parity; translation (iDMRG unit cell); point group (quantum numbers per site) | SU(2) multiplet tensors give ≳10× speedup at fixed accuracy; U(1) particle-number conservation shrinks block sizes roughly by √χ. |
| M9 time complexity | O(N · w · d · χ³) per sweep | N = sites, w = MPO bond (long-range → w large), d = local dim, χ = bond dim. Leading cost is from the two-site update contraction: O(χ³) at each of N sites (w, d prefactors suppressed; full cost O(N·w·d·χ³) — see table value). |
| M10 memory | O(d · χ²) | Stores one MPS tensor per site (d × χ × χ) plus the environment and Hamiltonian MPO; dominant term is O(d·χ²) per tensor. |
| M11 control knob | Bond dimension χ (truncation error ε_trunc = discarded weight) | Increasing χ reduces truncation error; convergence checked by extrapolating energy vs ε_trunc → 0. |
| M12 scale frontier | Large 1D chains (N ~ 10³–10⁴ routine); narrow 2D cylinders up to W ~ 6–8; thermodynamic limit directly via iDMRG/VUMPS | Practical χ ceiling set by χ³ time cost; supercomputer runs reach χ ~ 10⁴ for critical 1D chains. |
| M13 primary approximation / bias | Finite-χ truncation (controlled): the MPS ansatz restricts entanglement; truncation error ε_trunc → 0 as χ → ∞ | Bias is systematic and controlled; energy is a variational upper bound at fixed χ. |
| M14 hard blocker / failure mode | 2D systems: χ ~ e^{cW} per cylinder width W (exponential wall); volume-law entanglement (real-time t > t*, excited states, thermal states); long-range interactions raise MPO bond w, inflating cost | Frustrated 2D magnets, doped Hubbard at moderate χ, real-time beyond the entanglement barrier all push beyond practical reach. |
method-property-map.md: optimal when A1=1D, B5=area-law or area+log, C12=sign-ful (frustrated), C9/C10 present (big speedup); disfavored for A1=2D bulk, D13=real-time, or B5=volume-lawKey reference: schollwoeck_2010_density — the authoritative modern reference: formulates DMRG as MPS variational optimization, covers finite/infinite systems, symmetries, excited states, dynamical spectra, and cost scaling.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | GS energy & order parameters · correlation functions · correlation length (boundary transfer matrix) · thermodynamic limit directly (iPEPS) · entanglement entropy (approximate; from the finite-χ_env CTMRG environment, not a limitation of the PEPS ansatz) · topological order (representable at fixed D) | Dynamics and finite-T require specialized extensions (thermal-state PEPS, D-doubling); excitations via PEPS tangent space. |
| M2 regime | T=0 ground state (primary); thermodynamic limit via iPEPS | Finite-T harder: thermal-state PEPS or D-doubling approach; real-time dynamics via time-evolved PEPS (much harder). |
| M3 accuracy class | Controlled-bias (finite-D truncation + approximate contraction), deterministic; variational upper bound in the limit D→∞, χ_env→∞; at finite χ_env the approximate (CTMRG/boundary-MPS) contraction can break the strict variational bound | Both D (ansatz) and χ_env (contraction accuracy) must be converged; two independent knobs. |
| M4 dimension fit (A1) | 2D (its home); 3D explored but expensive; 1D → use MPS instead | Natural for square, triangular, kagome, honeycomb lattices. Quasi-1D (ladders) → MPS preferred. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion (fPEPS with parity tensors) | Fermions need parity tensors (fermionic PEPS / fPEPS) to handle the fermionic sign consistently; modest additional overhead. |
| M6 entanglement regime (B5) | 2D area-law at fixed D; B6 criticality and B8 frustration inflate D | Area-law GS at manageable D ~ 4–16; critical / frustrated states need larger D for quantitative accuracy. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; applicable to frustrated magnets and doped fermions without sign problem. |
| M8 symmetry exploitation (C9/C10) | U(1) / SU(2) / Z₂ block-sparse virtual tensors; lattice point-group symmetry via symmetric iPEPS | Symmetric tensors reduce cost and improve convergence; AD-based optimization fully compatible with U(1)/SU(2) symmetry. |
| M9 time complexity | Observable eval O(χ_env³ D⁶) via CTMRG/boundary-MPS; variational full-update / AD optimization O(D¹⁰)–O(D¹²); simple update ~O(D⁵) per step | Contraction bottleneck dominates. χ_env ∝ D² is the standard choice. Simple update is fast but uses a mean-field environment. |
| M10 memory | O(χ_env² D⁴) for the environment tensors; O(D⁵) per site tensor | Environment tensors dominate for large χ_env. |
| M11 control knob | Virtual bond D (ansatz bias) + environment bond χ_env (contraction bias) | Both must be extrapolated: energy vs 1/D and energy vs 1/χ_env. Converged result requires D and χ_env both large. |
| M12 scale frontier | Thermodynamic limit directly (iPEPS, single unit cell); finite PEPS on patches up to ~20×20; practical D ≲ 16 for spin-½ | iPEPS is genuinely a thermodynamic-limit method; finite-size effects absent in principle for translationally invariant systems. |
| M13 primary approximation / bias | (1) Finite-D MPS-ansatz truncation (controlled, → 0 as D → ∞); (2) approximate contraction bias via χ_env (controlled, → 0 as χ_env → ∞) | Two independent biases must both be converged; simple-update environment is a less controlled approximation of the full environment. |
| M14 hard blocker / failure mode | Contraction cost scales as O(D¹⁰⁻¹²) for optimization → accessible D is small (typically D ≤ 16); 3D extremely expensive; real-time dynamics entanglement barrier; non-Hermitian / open system extensions immature | For strongly frustrated 2D magnets, very large D may be needed to resolve competing orders — computationally prohibitive at present. |
method-property-map.md: optimal when A1=2D, B5=2D-area-law, C12=sign-ful; disfavored for A1=3D, D13=real-time, or D=very large (cost prohibitive)Key reference: naumann_2023_introduction — pedagogical introduction to iPEPS with AD-based optimization; covers CTMRG environment, variational algorithms, and modern AD workflow; most accessible entry point for practitioners.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Free energy & partition function · critical temperature T_c · critical exponents & central charge (from transfer-matrix / RG-spectrum eigenvalues) · correlation length · thermodynamic observables; also contracts 2D quantum-thermal & PEPS networks | CFT data (scaling dimensions, central charge c) extracted from the fixed-point spectrum of the transfer matrix or RG superoperator. |
| M2 regime | Finite-T (classical equilibrium); T=0 quantum via path-integral / PEPS contraction | Imaginary-time direction mapped onto a spatial leg; real-time dynamics not natural in this framework. |
| M3 accuracy class | Controlled-bias (finite-χ truncation at each coarse-graining step), deterministic | Bias → 0 as χ → ∞; no stochastic error. TNR removes short-range correlations that contaminate the TRG fixed point. |
| M4 dimension fit (A1) | 2D (TRG / CTMRG); 2D & 3D (HOTRG) | TRG cost grows as O(χ⁶) per step (feasible in 2D); HOTRG cost O(χ^{4d−1}) (d = spatial dimension here, not local Hilbert dim) makes 3D practical at moderate χ but expensive. |
| M5 statistics & local dim (A3) | Classical spins / hard-core variables; quantum via path integral (bosons, spins); fermions need Grassmann tensor extensions | Local dimension d absorbed into the initial tensor; classical Ising/Potts are natural applications. Fermionic path integrals require careful sign handling via Grassmann tensors. |
| M6 entanglement regime (B5) | Area-law classical correlations; handles critical (log-growing) correlations at finite χ via RG fixed-point | Unlike MERA, TRG/HOTRG do not explicitly install disentanglers; TNR variant removes the short-range entanglement that spoils the TRG fixed point. |
| M7 sign-problem dependence (C12) | Sign-free (real Boltzmann weights for classical models) | Classical statistical models have non-negative Boltzmann weights; no sign problem. Quantum path integrals can carry a sign, but this is treated by explicit tensor structure rather than Monte Carlo sampling. |
| M8 symmetry exploitation (C9/C10) | Global symmetry (Z₂, U(1), etc.) enforced via symmetric tensors; translation exploited by the coarse-graining geometry | Symmetric tensor implementations reduce the effective bond dimension and improve convergence near criticality. |
| M9 time complexity | TRG O(χ⁶) per step (2D); HOTRG O(χ^{4d−1}) per step = O(χ⁷) in 2D, O(χ¹¹) in 3D | DO NOT confuse with ATRG O(χ^{2d+1}) — that is a different (cheaper) algorithm. TNR scales higher than HOTRG but yields clean scale invariance. |
| M10 memory | TRG O(χ⁴); HOTRG O(χ^{2d}) (d = spatial dimension here, not local Hilbert dim) = O(χ⁴) in 2D, O(χ⁶) in 3D | Stores one or two local tensors per coarse-graining step; environment tensors in CTMRG are O(χ³). |
| M11 control knob | Bond dimension χ (number of singular values retained after each truncation) | Increasing χ reduces the truncation bias; convergence monitored via free-energy density as a function of χ. |
| M12 scale frontier | Thermodynamic limit directly (coarse-graining runs to infinite volume); classical 2D models routine at χ ~ 20–60; 3D at χ ~ 10–20 | Iterative coarse-graining naturally accesses the infinite-volume fixed point without finite-size extrapolation. |
| M13 primary approximation / bias | Finite-χ truncation at each coarse-graining step (controlled); TRG additionally accumulates short-range correlation errors near criticality (partially fixed by TNR) | Truncation bias is the only source of error; systematically controlled by increasing χ. |
| M14 hard blocker / failure mode | 3D at large χ (O(χ¹¹) wall); sign-ful quantum path integrals; real-time dynamics; volume-law entanglement phases | Frustrated quantum models with a sign problem require Grassmann extensions and are not naturally sign-free. TRG fixed points near criticality need TNR/loop-TNR to remove short-range contamination. |
method-property-map.md: optimal when A1=2D/3D classical, B6=critical (CFT data needed), C12=sign-free; disfavored for real-time dynamics, fermionic sign-ful quantum problems, or volume-law phasesKey reference: xie_2012_coarse — introduces HOTRG, derives the O(χ^{4d−1}) cost scaling, and benchmarks critical exponents for 2D and 3D Ising models.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Ground-state energy of critical systems · scaling dimensions of primary operators · central charge c (CFT data) from the RG superoperator spectrum · entanglement entropy S ~ (c/3) ln L at finite χ · correlation functions | CFT data extraction is MERA's primary strength and the quantity no other efficient method provides directly in the thermodynamic limit. |
| M2 regime | T=0 ground state (primary) | Finite-T and real-time variants exist but are uncommon; primary use is critical GS. |
| M3 accuracy class | Controlled-bias (finite-χ truncation at each layer), deterministic; variational upper bound on energy | Bias → 0 as χ → ∞; optimization is variational so energy is an upper bound at fixed χ. |
| M4 dimension fit (A1) | 1D (primary); some 2D MERA constructions exist but are much more expensive | 2D MERA costs up to O(χ¹⁶), making it impractical at χ large enough to be accurate; 1D is the natural home. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / fermion (fermionic MERA with parity structure) | Local dimension d enters via the base-layer tensors; standard implementations fix d=2 (spin-½ or qubit). |
| M6 entanglement regime (B5) | Criticality / scale invariance — its reason to exist: reproduces the log entanglement S ~ (c/3) ln L of 1D CFTs at finite χ via disentanglers | Area-law states are better served by MPS at lower cost; volume-law states are inaccessible. MERA is the unique method that handles the area+log critical regime efficiently in the thermodynamic limit. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; no sign problem regardless of frustration or doping. |
| M8 symmetry exploitation (C9/C10) | Global symmetry (U(1), SU(2), Z₂) enforced via symmetric disentanglers and isometries | Symmetric MERA implementations reduce cost and improve convergence; translation symmetry exploited by the coarse-graining structure. |
| M9 time complexity | O(χ⁹) binary 1D; O(χ⁸) ternary 1D; O(χ⁷) modified-binary 1D; 2D up to ~O(χ¹⁶) | Cost dominated by the contraction of disentangler environments; binary/ternary costs from evenbly_2007_algorithms; modified-binary O(χ⁷) from evenbly_2011_class (Evenbly–Vidal 2011), which reduces cost while retaining scale invariance. |
| M10 memory | O(χ⁴) per layer (storing disentangler and isometry tensors) | Number of layers grows as log₂(N) for a finite system; thermodynamic limit uses a fixed periodic (scale-invariant) layer structure. |
| M11 control knob | Bond dimension χ (number of states kept after each isometry truncation) | Increasing χ captures more entanglement and improves both energy accuracy and CFT-data precision; convergence checked by energy and scaling-dimension stability vs χ. |
| M12 scale frontier | Thermodynamic limit directly (via scale-invariant fixed-point layer); finite systems at N ~ 10²–10³ routine in 1D | The scale-invariant (infinite) MERA defines a fixed-point RG map valid at all length scales, giving direct thermodynamic-limit CFT data. |
| M13 primary approximation / bias | Finite-χ truncation (controlled): the isometries restrict the number of states retained at each scale; disentanglers are constrained to be unitary | Bias is systematic and controlled; energy and CFT data converge monotonically with χ. |
| M14 hard blocker / failure mode | 2D: cost up to O(χ¹⁶) makes large-χ calculations impractical; gapped (area-law) systems: MERA is overkill — MPS is cheaper; volume-law phases and real-time dynamics are inaccessible | MERA is purpose-built for criticality; applying it to gapped systems wastes cost relative to MPS. |
method-property-map.md: optimal when A1=1D, B6=critical/scale-invariant; disfavored for A1=2D (cost), B5=area-law gapped (use MPS), B5=volume-law, D13=real-timeKey reference: evenbly_2007_algorithms — derives the MERA optimization (alternating optimization of disentanglers and isometries), establishes cost scaling for binary and ternary 1D MERA, and demonstrates CFT data extraction.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Quench dynamics · real-time correlations · spectral functions S(q,ω) (time-evolve + Fourier) · transport coefficients · ground-state energy & wavefunction (imaginary-time) · finite-T thermodynamics (purification) | Spectral functions require Fourier transform of the time-domain correlator; max frequency resolution set by reachable t*. |
| M2 regime | Real-time dynamics + imaginary-time (ground state & finite-T) | Both regimes use the same SVD-truncation engine; imaginary-time entanglement saturates (GS) or grows slowly (thermal). |
| M3 accuracy class | Controlled, deterministic | Two controllable knobs: bond dim χ (truncation error) and Trotter step Δτ (Trotter error O(Δτ^p)); both can be converged independently. |
| M4 dimension fit (A1) | 1D native; quasi-1D (ladders) with swap gates; walls in 2D | Nearest-neighbor (A4 = short-range interaction range) gates only — long-range interactions require swap-gate sequences (expensive); 2D walls due to χ~e^{cW} cylinder area law. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion (Jordan–Wigner) / any | Per-site cost ∝d·χ³; large d (Hubbard d=4, soft-core bosons) raises cost; fermions need parity tensors or JW strings. |
| M6 entanglement regime (B5) | Area-law + area+log tolerated; volume-law blocks real-time | Real-time entanglement grows linearly S(t)∝t for thermalizing systems → χ(t)~e^{S(t)} caps reachable time t*~log(χ_max)/rate; MBL (D15) and integrable (C11) systems grow more slowly. |
| M7 sign-problem dependence (C12) | Sign-immune | Tensor-network method; no Monte Carlo sampling; no sign problem. |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂ quantum-number-conserving tensors; translation symmetry for iMPS (infinite systems) | Symmetry reduces effective χ by ~10× at fixed accuracy; iMPS with translation gives direct thermodynamic-limit access. |
| M9 time complexity | O(N·d·χ(t)³) per Trotter step; Trotter error O(Δτ^p) | p=2 standard; p=4 higher-order Suzuki–Trotter. Total cost scales with number of steps and growth of χ(t). No MPO w factor because TEBD applies nearest-neighbor gates directly (contrast TDVP's O(N·w·d·χ³)). |
| M10 memory | O(N·d·χ²) for the MPS; environment/orthogonality center storage O(d·χ²) per bond | Manageable for χ~1000; the dominant memory is the MPS itself. |
| M11 control knob | Bond dim χ (SVD truncation error) + Trotter step Δτ (Trotter error O(Δτ^p)) | Both must be converged; truncation error accumulates over time steps. |
| M12 scale frontier | 1D chains N~1000 sites routine (fixed χ); reachable time t*~log(χ_max)/rate | At χ=512 and rate~1 (Heisenberg chain), t*J~6; longer for MBL/integrable (D15/C11). |
| M13 primary approximation / bias | Finite-χ SVD truncation (controlled) + Trotter discretization O(Δτ^p) (controlled) | Both biases are systematically improvable; truncation accumulates in time. |
| M14 hard blocker / failure mode | Entanglement barrier: χ(t)~e^{S(t)} — for thermalizing (B6 gapless, B8 frustrated) systems S(t)∝t caps reachable t*; long-range interactions (A4) require expensive swap-gate sequences; 2D walls (χ~e^{cW}) | Rate set by C11 integrability / D15 MBL / B6 gap; thermal systems hit the barrier fastest. |
O(N·d·χ(t)³) per Trotter step; total cost grows with reachable time as χ(t) increases; no MPO w factor because TEBD applies nearest-neighbor gates directly (contrast TDVP's O(N·w·d·χ³))O(N·d·χ²) for the MPSχ (truncation) + Δτ (Trotter error O(Δτ^p))N~1000 sites in 1D; reachable real time t*~log(χ_max)/ratet*; TEBD is the standard tool for probing l-bit dynamicsS(t) growth (logarithmic for MBL, subdiffusive for integrable) allows long-time evolutionmethod-property-map.md (TEBD profile): preferred over TDVP when H is strictly nearest-neighbor (A4 = short-range) and swap-gate sequences are not neededS(t) ≃ (π c /3) v_F t (c=1 for the XX chain), until finite-χ truncation caps it — this linear growth is the entanglement barrier vidal_2003_efficient, calabrese_2005_evolution. Saturation at S~log(χ) sets the reachable time t*∝log(χ)/rate.χ=512: reachable tJ≈6 before truncation error exceeds 10⁻³ (method-survey.md §5.1).S(t)~log(t), χ grows logarithmically → reachable tJ~100 at χ=512 (method-survey.md §5.1).Key reference: vidal_2003_efficient — original TEBD paper introducing the SVD-based Trotter time evolution for MPS; defines the algorithm and complexity.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Real-time dynamics · spectral functions S(q,ω) (time-evolve + Fourier) · long-time dynamics within fixed-χ manifold (1-site) · transport · finite-T (TDVP + purification) | 1-site TDVP ideal for long-time within manifold; 2-site for growing χ during quench. |
| M2 regime | Real-time dynamics + imaginary-time (ground state) (D13) | Primarily a dynamics method; imaginary-time TDVP is algebraically equivalent to a DMRG sweep. |
| M3 accuracy class | Controlled, deterministic | Two knobs: χ (truncation / projection error) and time step Δt (Krylov error, which is very small for the local integrator); 1-site has a projection error absent in 2-site. |
| M4 dimension fit (A1) | 1D + quasi-1D (ladders/cylinders); handles long-range Hamiltonians via MPO | Key advantage over TEBD: MPO representation supports long-range (A4) interactions natively; no swap-gate overhead; 2D walls same as TEBD (χ~e^{cW}). |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion (parity tensors) / any | Per-site cost ∝w·d·χ³ where w is the MPO bond; large d or wide-range MPO (large w) raises cost. |
| M6 entanglement regime (B5) | Area-law + area+log (1-site within fixed χ); volume-law blocks | Same entanglement barrier χ(t)~e^{S(t)} as TEBD in 2-site mode; 1-site TDVP fixes χ (stays in manifold, no growth) but introduces projection error. |
| M7 sign-problem dependence (C12) | Sign-immune | Tensor-network method; no Monte Carlo. |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂ quantum-number-conserving tensors; translation symmetry for uniform MPS (TDVP on the infinite MPS) | Symmetry-adapted MPO tensors reduce w and effective χ; ~10× speedup at fixed accuracy. |
| M9 time complexity | O(N·w·d·χ³) per time step (w = MPO bond dimension, set by A4 interaction range) | w grows with interaction range: nearest-neighbor w=3–5; power-law decaying w~15–50; arbitrary long-range w can be large. Same χ(t) entanglement barrier as TEBD. |
| M10 memory | O(N·w·d·χ²) for MPS + MPO environments | MPO environments O(w·χ²) per bond; dominant at large w. |
| M11 control knob | Bond dim χ (projection / truncation error) + time step Δt (Krylov integrator error, exponentially small) | 1-site: χ fixed → monitor projection error; 2-site: χ grows until truncation error converges. |
| M12 scale frontier | 1D N~500–2000 sites routine at fixed χ; reachable time t*~log(χ_max)/rate (same barrier as TEBD) | 1-site can reach longer real times at fixed χ than TEBD by staying in the manifold; barrier still applies at large t. |
| M13 primary approximation / bias | Finite-χ MPS manifold projection (controlled) | 1-site: projection onto current MPS tangent space (no χ growth — misses orthogonal dynamics); 2-site: SVD truncation. Energy is conserved (1-site) even as χ is fixed. |
| M14 hard blocker / failure mode | Entanglement barrier χ(t)~e^{S(t)} (same as TEBD for 2-site); 1-site projection error misses dynamics outside current manifold; 2D cylinder wall (χ~e^{cW}) | For thermalizing systems at long times, both 1-site and 2-site TDVP eventually fail. |
O(N·w·d·χ³) per time step; w = MPO bond dim (set by interaction range A4)O(N·w·d·χ²) for MPS + MPO environmentsχ (projection/truncation error) + Δt (Krylov error, negligible)N~500–2000 sites in 1D; reachable time t*~log(χ_max)/rateχ: when you want the best approximation within a fixed MPS space (e.g., non-integrable systems at moderate t)method-property-map.md (TDVP profile): preferred over TEBD when H has long-range or multi-site terms (A4 non-nearest-neighbor)|ΔE/E| < 10⁻¹² per step for Heisenberg chain N=100, χ=64 haegeman_2016_unifying.t* longer at fixed χ (1-site, non-thermalizing) — method-survey.md §5.2.1/r interaction: w~20–30 giving cost O(N·30·d·χ³) vs TEBD swap-gate O(N²) per step — method-survey.md §5.2.Key reference: haegeman_2016_unifying — defines the 1-site and 2-site TDVP integrators for MPS, proves energy conservation, and provides the W^I/W^II MPO time evolution; the standard algorithmic reference.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Free energy · internal energy · specific heat C_v · magnetic susceptibility χ · finite-T correlations · finite-T spectral functions (combine with real-time TDVP/TEBD) | Thermodynamic quantities from ⟨H⟩_β, ⟨H²⟩_β − ⟨H⟩²_β; finite-T spectral functions require additional real-time step after reaching β. |
| M2 regime | Finite-T thermodynamics (D13) | Core purpose: equilibrium thermal properties at target inverse temperature β; not a real-time method (though finite-T spectral functions combine both). |
| M3 accuracy class | Controlled deterministic (purification) + controlled stochastic (METTS) | Purification: controlled by χ and Trotter Δτ; METTS: additionally stochastic 1/√N_s from sampling; both are systematically improvable. |
| M4 dimension fit (A1) | 1D native; quasi-1D (ladders/narrow cylinders) feasible; walls in 2D | Same as TEBD/TDVP: entanglement of thermal state grows as T drops, worsening in 2D; purification maps to a (d²)-dim 1D problem. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion / any; local dim d→d² for purification | Doubling d to d² raises per-site cost; for spin-½ d²=4 (manageable); for Hubbard d=4→16 (expensive); METTS avoids d² by collapsing to product states. |
| M6 entanglement regime (B5) | Moderate at high T (area-law); grows as T→0 → χ diverges | Entanglement of purified thermal state rises as T drops → χ must grow at low T; METTS samples states with lower entanglement → cheaper at intermediate T. |
| M7 sign-problem dependence (C12) | Sign-immune | Tensor-network method; no Monte Carlo sign problem (METTS uses classical sampling, no quantum sign). |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂ quantum-number-conserving tensors for both purification and METTS | Symmetry doubles the block structure (physical + ancilla); still provides ~5–10× speedup. |
| M9 time complexity | Purification: O(N·d²·χ³) per imaginary-time step × β/Δτ steps; METTS: O(N·d·χ³) per sample × N_s samples | METTS per-sample cost uses d (not d²) and smaller χ (typical states); total cost N_s · O(N·d·χ³); tradeoff vs purification depends on d and required accuracy. |
| M10 memory | Purification: O(N·d²·χ²); METTS: O(N·d·χ²) per sample | Purification memory dominated by doubled local dim; METTS memory is standard MPS size. |
| M11 control knob | χ (truncation error) + Trotter step Δτ (O(Δτ^p) error) + N_s samples (METTS statistical error 1/√N_s) | Three independent knobs for METTS; two for purification; all must be converged. |
| M12 scale frontier | Purification: N~200–500 sites at moderate T (βJ~10); METTS: comparable or larger N at intermediate T | Low-T limit β→∞ requires large χ for both methods; METTS is competitive at intermediate T where typical states have lower entanglement. |
| M13 primary approximation / bias | Purification: finite-χ SVD truncation (controlled) + Trotter O(Δτ^p) (controlled); METTS: additionally stochastic sampling bias 1/√N_s | Purification is a deterministic approximation; METTS adds a stochastic component — must check ergodicity of the collapse chain. |
| M14 hard blocker / failure mode | Low-T entanglement divergence: as β→∞, thermal-state entanglement → GS entanglement → χ must grow; 2D walls (χ~e^{cW}); large local dim d makes purification expensive (d² scaling); METTS ergodicity failure for some collapse bases at very low T | The low-T limit is where both methods become most expensive; consider DMRG+GS extrapolation for T→0. |
O(N·d²·χ³) per imaginary-time step; METTS O(N_s · N·d·χ³) per sampleO(N·d²·χ²); METTS O(N·d·χ²) per sampleχ + Δτ (+ N_s samples for METTS)N~200–500 at moderate-T purification; similar for METTS at intermediate TJ ≲ 20)d is small and typical-state entanglement is low; cheaper than purification per accuracy unitT/J≲0.05 and N_s would need to be very large for METTSmethod-property-map.md (mps-finite-t profile): primary finite-T tool for 1D sign-ful (frustrated/fermionic) models where QMC failsN=100, χ=64): C_v(T) converged to < 1% at T/J ≥ 0.1 (Paeckel et al., arXiv:1901.05824, §5).T/J ≥ 0.5 for spin-½ chains (same reference, §6).S(q,ω,T) for XXZ chain at T/J=0.5: χ=128, N_s=200 METTS samples sufficient for < 5% error paeckel_2019_timeevolution.Key reference: paeckel_2019_timeevolution — comprehensive review of all MPS time-evolution methods including purification and METTS; provides complexity derivations, convergence benchmarks, and software guidance.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Free energy · internal energy · specific heat C_v · magnetic susceptibility · thermal correlations · thermal phase transitions | Primary use case is finite-T thermodynamics; XTRG excels at low-T approach curves where LTRG would require many linear layers. |
| M2 regime | Finite-T thermodynamics (D13) | Equilibrium thermal properties across all T; XTRG most advantageous at low T where O(log β) beats LTRG's O(β/τ) layers. |
| M3 accuracy class | Controlled, deterministic | Two controllable sources of error: Trotter discretization τ (O(τ²) per layer) and RG truncation Dc; XTRG's squaring accumulates errors multiplicatively but requires fewer steps. |
| M4 dimension fit (A1) | 1D, quasi-1D AND 2D — sign-free for frustrated spin/boson models even in 2D | Maps d-dim quantum at finite T → (d+1)-dim classical TN; the classical TN is sign-free for frustrated spin and boson models (B8, C12 sign-ful); fermionic models need care (Jordan–Wigner strings can introduce complex/negative entries). The key advantage over QMC is for frustrated 2D magnets. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / any local dim d | Local dim d enters the Trotter-gate tensors; per-step cost polynomial in d; no fundamental barrier from statistics (no sign). |
| M6 entanglement regime (B5) | Moderate at high T; grows as T→0 → Dc must increase | Thermal entanglement of the layered TN sets the required Dc; at high T, states are product-like (small Dc); at low T, thermal entanglement approaches GS entanglement (large Dc). |
| M7 sign-problem dependence (C12) | Sign-immune for frustrated spin and boson models (B8); fermionic LTRG/XTRG needs care | Sign-free for frustrated spin and boson models (B8); fermionic LTRG/XTRG needs care — Jordan–Wigner strings can give complex/negative tensor entries in the contraction. The sign-free advantage over QMC applies to frustrated magnets and bosonic models. |
| M8 symmetry exploitation (C9/C10) | Conserved-charge (U(1)) block structure in the MPO/TN tensors | Symmetry reduces the effective tensor dimensions and speeds up contraction. |
| M9 time complexity | LTRG: polynomial in d and Dc per layer × K=β/τ layers, Trotter error O(τ²); XTRG: higher cost per step (squaring a TN) but only O(log β) steps to reach low T | XTRG reaches β=100 in ~7 squaring steps from τ=1; LTRG needs 100/τ layers. At low T, XTRG's lower step count dominates the comparison. |
| M10 memory | O(Dc²) per TN row (MPO or boundary MPS) | The dominant memory is the contracted boundary object; 2D requires 2D boundary contraction (heavier than 1D). |
| M11 control knob | Trotter step τ (Trotter error O(τ²)) + retained bond dim Dc (RG truncation error) | Both must be converged; XTRG additionally requires checking that squaring errors accumulate controllably. |
| M12 scale frontier | 1D: N~1000 sites routine; quasi-1D and 2D: thermodynamic limit via transfer-matrix / iPEPS-row contraction | 2D: infinite systems via row-to-row contraction with boundary-MPS or CTMRG; not limited to small clusters. |
| M13 primary approximation / bias | Trotter O(τ²) discretization (controlled) + finite-Dc RG truncation (controlled) | Both biases are convergence parameters; XTRG has multiplicative error accumulation from squaring steps but requires far fewer of them. |
| M14 hard blocker / failure mode | Low-T thermal entanglement growth → large Dc needed; 3D classical TN contraction (extremely expensive); models with complex local interactions that resist Trotter decomposition | 2D quantum → 3D classical TN is the heaviest case; exponentially hard for 3D quantum models. |
O(β/τ) layers × polynomial in d·Dc per layer; XTRG: O(log β) squaring steps × higher per-step costO(Dc²) per MPO / boundary rowτ (error O(τ²)) + retained Dc (RG truncation error)N~1000; 2D thermodynamic limit via infinite-system row contractionO(log β) scaling is far superior to LTRG's linear K at low T; use XTRG when βJ ≳ 10method-property-map.md (LTRG/XTRG profile): preferred finite-T method when QMC sign problem is present and 2D geometry is neededT/J=0.05 with Dc=64, τ=0.1 in 7 squaring steps chen_2018_exponential.J₁-J₂ square lattice (sign-ful QMC): thermal susceptibility peak resolved at T/J≈0.4, Dc=64 (method-survey.md §5.5).βJ=20: XTRG 2–3 orders of magnitude more accurate at same Dc and total compute time (method-survey.md §5.5).Key reference: chen_2018_exponential — introduces XTRG (exponential doubling ρ→ρ·ρ), demonstrates O(log β) scaling, and benchmarks frustrated 2D magnets; the primary XTRG reference.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Individual highly-excited eigenstates · eigenstate energies and gaps · local observables in excited eigenstates · l-bit structure and l-bit correlators · eigenstate entanglement entropy and spectrum across the MBL phase | The ability to obtain individual eigenstates (not just averages) at arbitrary energy density is unique to DMRG-X among polynomial-cost methods. |
| M2 regime | Highly-excited eigenstates at finite energy density (an MBL-specific regime gated by D15, not one of D13's ground-state/finite-T/dynamics values) | Not a finite-T method; targets a single eigenstate at a chosen energy density, which is a T=∞ analog only in the MBL phase. |
| M3 accuracy class | Controlled, deterministic | MPS truncation at χ is the only approximation; energy and observables converge as χ→∞; residual ‖H|ψ⟩−E|ψ⟩‖ provides a convergence check. |
| M4 dimension fit (A1) | 1D; quasi-1D (ladders) feasible; 2D only if MBL survives (debated) | MBL in 2D is theoretically controversial and numerically difficult; DMRG-X is primarily applied to 1D disordered chains (A1=1D, D15=MBL). |
| M5 statistics & local dim (A3) | Spin / hard-core boson / fermion / any | Same as standard DMRG; per-site cost ∝d·χ³ (targeted variant) or ∝d·χ⁶ (full diag variant); fermions need Jordan–Wigner or parity tensors. |
| M6 entanglement regime (B5) | Area-law only — works ONLY because MBL (D15) keeps excited-state entanglement area-law | This is the decisive gate: thermal (ETH) excited states have volume-law entanglement → χ~e^N → DMRG-X fails; MBL area law → χ bounded → DMRG-X works. |
| M7 sign-problem dependence (C12) | Sign-immune | Tensor-network method; no sign problem. |
| M8 symmetry exploitation (C9/C10) | U(1)/SU(2)/Z₂ block structure (though disorder breaks translation symmetry) | Symmetry still reduces effective χ and improves convergence; translation is absent (disorder breaks it), but spin-rotation or parity may be conserved. |
| M9 time complexity | O(L·χ⁶) for the full local-diagonalization variant; ~O(L·χ³) for the k-eigenstate-near-target variant | The χ⁶ scaling (vs standard DMRG χ³) comes from the environment density-matrix construction for interior states; the cheaper O(Lχ³) variant uses a targeted projection. |
| M10 memory | O(L·χ²) for the MPS; environment blocks O(χ²) per bond | Same memory as standard DMRG; does not require storing multiple eigenstates simultaneously. |
| M11 control knob | Bond dim χ (truncation error) | The single convergence parameter; monitor residual ‖H|ψ⟩−E|ψ⟩‖ < ε; convergence is typically fast when MBL area law holds. |
| M12 scale frontier | 1D MBL chains L~100–200 sites routine; L~500–1000 with optimized codes | Far exceeds exact-diagonalization reach for excited states (ED limited to L~20–24); the practical limit set by χ required for convergence at the target disorder strength. |
| M13 primary approximation / bias | Finite-χ SVD truncation (controlled) | Only approximation; controlled by increasing χ; the method is exact in χ→∞ limit; must verify that the tracked state does not jump to a different eigenstate. |
| M14 hard blocker / failure mode | Works ONLY because MBL (D15) keeps χ bounded: thermal (ETH, B6 gapless) excited states have volume-law entanglement → DMRG-X completely fails; 2D MBL debated (may not exist thermodynamically); tracking instability (jumping to wrong eigenstate) at weak disorder | The MBL area law is the sole reason DMRG-X is feasible; any regime without it (ETH phase, finite-T thermal states) kills the method. |
O(L·χ⁶) full local-diagonalization variant; ~O(L·χ³) k-eigenstate-near-target variantO(L·χ²) for MPS + environment blocksχ (truncation error; also monitor ‖H|ψ⟩−E|ψ⟩‖)L~100–200 routine; L~500–1000 optimized codesS(n) as disorder W is varied; area-law → volume-law crossover marks the transition (DMRG-X fails as ETH sets in → marks the transition)L~100–500 replaces expensive shift-invert ED (L≤24) for individual excited-state studies in the MBL phasemethod-property-map.md (DMRG-X profile): applicable when D15 = MBL (area-law excited states); inapplicable when D15 absent (ETH phase)L=100, W/J=5, χ=64): ‖H|ψ⟩−E|ψ⟩‖ < 10⁻⁸; entanglement entropy S~0.3 (area law confirmed) khemani_2016_obtaining.χ=64 for W/J=5; fails to converge (requires χ→∞) for W/J=1 (ETH phase) — method-survey.md §5.6.O(L·χ⁶) vs O(L·χ³) crossover: targeted variant (O(Lχ³)) reaches same accuracy as full-diag variant at χ≤64 for strong MBL (method-survey.md §5.6).Key reference: khemani_2016_obtaining — introduces DMRG-X, proves applicability based on MBL area law, and benchmarks the O(L·χ⁶) local-diagonalization variant on random-field Heisenberg chains.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | GS energy (variational upper bound) · variational observables (correlations, structure factors, order parameters) · excited states via symmetry sectors or energy-penalty methods · real-time dynamics via time-dependent VMC (t-VMC, extra cost, growing error) | Gives an upper bound only — pair with PolyOpt/SDP for a rigorous energy bracket. |
| M2 regime | T=0 (ground state); real-time via t-VMC | Finite-T via VMC sampling of the thermal density matrix is possible but non-standard. |
| M3 accuracy class | Variational upper bound, stochastic | Statistical error bars ∝1/√N_s; bias set by ansatz expressiveness (uncontrolled but variational — cannot go below the true GS). |
| M4 dimension fit (A1) | Any dimension (1D, 2D, 3D) | No geometric constraint — the key advantage over tensor-network methods in 2D/3D. |
| M5 statistics & local dim (A3) | Spin / hard-core boson / soft-core boson / fermion (with sign structure in ansatz) | Fermions require encoding the sign/phase structure in the ansatz; large local dim d raises c_eval but does not cause an exponential wall. |
| M6 entanglement regime (B5) | Volume-law tolerated (no bond-dimension wall) | NQS can in principle represent volume-law states; quality depends on the ansatz, not a hard truncation. |
| M7 sign-problem dependence (C12) | Sign-immune | No sign problem — the sign/phase is encoded in the (complex) ansatz amplitude. Thrives where QMC fails: B8 frustration, C12 sign-ful, 2D. |
| M8 symmetry exploitation (C9/C10) | U(1), SU(2), translation, point-group can be imposed as constraints on the ansatz or via symmetry-projected sampling | Symmetry projection reduces variance and targets correct quantum-number sectors. |
| M9 time complexity | O(N_s·c_eval) per optimization step; SR/natural-gradient adds O(N_p²)–O(N_p³) (exact) or O(N_s·N_p) (iterative/minSR) | c_eval = amplitude + local energy cost (O(N)–O(N²) Jastrow/RBM; network forward pass for deep NQS). |
| M10 memory | O(N_p) for parameters + O(N_s·N_p) for the SR matrix (or O(N_s) minSR) | Deep-network ansätze can have N_p ~ 10⁴–10⁶; SR matrix can dominate memory. |
| M11 control knob | N_s (samples per step) → statistical error ∝1/√N_s; ansatz size / depth → ansatz bias | No single knob fully controls the variational bias — convergence studies vary ansatz architecture. |
| M12 scale frontier | Thousands of sites feasible with shallow RBM/Jastrow; hundreds of sites with deep NQS; thermodynamic limit via ansatz in the TD limit | Scales far beyond ED/FCI; quality limited by ansatz not system size. |
| M13 primary approximation / bias | Ansatz bias — variational error is set by the expressiveness of the chosen ansatz (uncontrolled but variational) | Cannot go below the true GS; paired with a lower bound (SDP/PolyOpt) it brackets the energy. |
| M14 hard blocker / failure mode | Optimization landscape (local minima, barren plateaus for deep networks); autocorrelation in MC sampling; SR matrix ill-conditioning | Real-time t-VMC accumulates gauge error; strongly correlated regimes with poor ansatz overlap fail quietly (low-quality local minimum). |
O(N_s·c_eval) per optimization step; SR adds O(N_p²)–O(N_p³) or O(N_s·N_p) with minSRO(N_p) parameters; O(N_s·N_p) SR matrix (dominant for large N_p)N_s (sample size → statistical error ∝1/√N_s); ansatz depth/width → variational bias/method-polyopt (SDP lower bound) to bracket the GS energy rigorouslymethod-property-map.md: recommended when C12 sign-ful AND A1 2D/3D AND need variational upper boundN=80, J₂/J₁=0.5): energy within 0.1% of DMRG carleo_2016_solving.Key reference: carleo_2016_solving — founding NQS paper demonstrating RBM wavefunction solves frustrated 1D/2D spin models with high accuracy; the standard entry-point reference.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Finite-T thermodynamics (energy, specific heat, susceptibility) · equal- and imaginary-time correlations · structure factors · spin stiffness / superfluid density · Binder cumulants · winding-number distributions | Spectra S(q,ω) require analytic continuation (ill-posed); real-time dynamics blocked by the dynamical sign problem. |
| M2 regime | Finite-T (primary); GS accessible via large β limit | The large-β limit extracts GS properties but cost grows with β. |
| M3 accuracy class | Numerically exact (in the sign-free regime), stochastic | Statistical error ∝1/√(N_s·⟨s⟩) where ⟨s⟩=1 in sign-free cases; in sign-ful cases accuracy degrades as ⟨s⟩→0. |
| M4 dimension fit (A1) | Any dimension (1D, 2D, 3D) | The decisive advantage over tensor-network methods in 2D/3D; no bond-dimension wall. |
| M5 statistics & local dim (A3) | Spins / hard-core bosons / soft-core bosons (sign-free); fermions / frustrated spins introduce the sign | Local dimension d enters via the operator basis; large d increases the diagonal/off-diagonal operator pool but does not cause exponential cost. |
| M6 entanglement regime (B5) | Volume-law tolerated | No entanglement restriction — worldlines can represent arbitrary correlations in the sign-free sector. |
| M7 sign-problem dependence (C12) | Sign-free for unfrustrated bipartite spins (Marshall sign rule) and bosons; B8 frustration / A3 fermions turn the sign on; sign-ful cost ∝1/⟨s⟩²~e^{2βNΔf} | The primary gate: check bipartiteness and absence of frustration before choosing SSE. |
| M8 symmetry exploitation (C9/C10) | U(1) (Sᶻ, particle number) conservation reduces operator pool; translation / point-group target momentum sectors for correlations | Symmetry reduces variance but is not required for correctness. |
| M9 time complexity | O(N·β) per sweep (expansion order ⟨n⟩∝N·β); statistical error 1/√(N_s·⟨s⟩); directed-loop/operator-loop updates remove critical slowing down; sign-ful cost ∝1/⟨s⟩²~e^{2βNΔf} | Cost is linear in N and β in the sign-free regime — highly favorable for large systems at moderate temperatures. |
| M10 memory | O(N·β) to store the operator string | Operator string length ⟨n⟩∝N·β; configuration vector O(N); total modest compared to wave-function methods. |
| M11 control knob | N_s (Monte Carlo sweeps) → statistical error ∝1/√N_s; β (inverse temperature) → GS vs finite-T physics; cutoff n_max ≥ ⟨n⟩ | Convergence in N_s straightforward; β tuned to target temperature or GS extrapolation. |
| M12 scale frontier | Thousands to tens-of-thousands of sites in the sign-free regime | Thermodynamic-limit extrapolations routine; limited only by autocorrelation and the sign problem. |
| M13 primary approximation / bias | None in the sign-free regime (numerically exact); sign-ful approximation has uncontrolled bias unless ⟨s⟩≈1 | In the sign-ful regime the average-sign collapse is the approximation; no free-parameter knob removes it. |
| M14 hard blocker / failure mode | Sign problem for frustrated (B8) spins and fermions (A3): cost ∝e^{2βNΔf} → exponential wall; real-time dynamics (dynamical sign problem) | Even weak frustration can turn the sign on; analytic continuation from imaginary-time data is ill-conditioned. |
O(N·β) per sweep (sign-free); cost ∝1/⟨s⟩²~e^{2βNΔf} in sign-ful casesO(N·β) operator stringN_s (sweeps → statistical error ∝1/√N_s); β controls temperature/GS convergencemethod-property-map.md: first choice for sign-free finite-T spin/boson models; blocked by B8 frustration and A3 fermionsN=64×64, T/J=0.5): susceptibility, stiffness converged to sub-percent accuracy with N_s~10⁵ sweeps syljuasen_2002_quantum.τ_auto~L^z with z reduced to ≈1–1.5 for the spin stiffness (vs z≈2 for local updates) sandvik_2010_computational.Key reference: syljuasen_2002_quantum — introduces directed-loop updates in SSE; the standard implementation reference for efficient worldline/SSE codes.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Finite-T energy · double occupancy · magnetic and charge structure factors S(q) · single-particle Green's function G(k,τ) → spectral function A(k,ω) and self-energy via analytic continuation · superconducting pairing susceptibilities | Real-time dynamics blocked; A(k,ω) requires MaxEnt or SOAC analytic continuation (ill-posed). |
| M2 regime | Finite-T (primary); GS limit via large β | Cost grows as β → ∞; the GS is better attacked by AFQMC-CPMC. |
| M3 accuracy class | Numerically exact (sign-free regime), stochastic; controlled-bias (Trotter O(Δτ²)) | Statistical error ∝1/√N_s; Trotter error eliminated by Δτ→0 extrapolation; sign collapse in sign-ful regime. |
| M4 dimension fit (A1) | Any dimension (1D, 2D, 3D) | Sign-free condition depends on lattice bipartiteness and filling, not dimension. |
| M5 statistics & local dim (A3) | Fermions (primary); mixed fermion-boson via partial trace | d=4 (Hubbard: empty, ↑, ↓, ↑↓); N_τ=β/Δτ slices; cost O(βN³) grows fast with N. |
| M6 entanglement regime (B5) | Volume-law tolerated | Auxiliary-field sampling imposes no entanglement restriction; the sign problem is the gate, not entanglement. |
| M7 sign-problem dependence (C12) | Sign-free at half-filling on bipartite lattices (det↑det↓=|det|²≥0) and attractive Hubbard at any filling; D14 doping / B8 frustration turn the sign on; sign ⟨s⟩~e^{−βNΔf} → cost ∝e^{2βNΔf} | Half-filling + bipartite = the canonical sign-free condition; any deviation from half-filling on repulsive Hubbard leads to the sign problem. |
| M8 symmetry exploitation (C9/C10) | Particle-hole symmetry (C9) at half-filling on bipartite lattices is the sign-free condition; translation/point-group reduce momentum-resolved measurements but do not change the core algorithm | PH symmetry is the decisive symmetry here — it is a correctness condition, not merely a speedup. |
| M9 time complexity | O(β·N³) per sweep (N×N matrix products over N_τ=β/Δτ slices with numerical stabilization); sign-ful cost ∝e^{2βNΔf} | The N³ matrix-algebra cost limits system size to N~200–400; lower T (larger β) is doubly costly: more slices and worse sign. |
| M10 memory | O(N²·N_τ) | Stores the N×N equal-time Green's function matrix plus N_τ auxiliary-field configurations. |
| M11 control knob | Δτ (Trotter step) → Trotter error O(Δτ²); N_s (sweeps) → statistical error ∝1/√N_s; β (inverse temperature) | Extrapolate Δτ→0 for Trotter-free results; check convergence in N_s. |
| M12 scale frontier | N~100–400 sites routine; N~1000 possible on HPC with stabilization; thermodynamic limit via finite-size scaling | Sign-free regime: large-scale studies of Hubbard model on square lattice up to N=20×20. |
| M13 primary approximation / bias | Trotter discretization error O(Δτ²) (controlled, eliminated by Δτ→0 extrapolation); sign collapse in sign-ful regimes (uncontrolled) | In sign-free cases the only approximation is controllable Trotter error and statistical noise. |
| M14 hard blocker / failure mode | Sign problem away from half-filling (D14 doping) and with frustration (B8): cost ∝e^{2βNΔf}; low-T limit compounds both cost and sign penalty | Large β and large N simultaneously is the hardest regime; sign problem is exponential in both. |
O(β·N³) per sweep; sign-ful cost ∝e^{2βNΔf}O(N²·N_τ) with N_τ=β/ΔτΔτ (Trotter error O(Δτ²)); N_s (statistical error ∝1/√N_s)N~100–400 routine (sign-free); N~1000 HPC; finite-size scaling to thermodynamic limitN~400A(k,ω) for comparison with ARPESmethod-property-map.md: preferred over AFQMC-CPMC when finite-T thermodynamics is needed; blocked by D14 doping in sign-ful regimesN=16×16, U/t=4, β=8): energy, double occupancy, structure factor converged bercx_2017_alf, verified in method-survey.md §4.4.Δτ=0.05–0.1 typically sufficient; linear extrapolation in Δτ² removes systematic error to < 0.1% bercx_2017_alf.Key reference: bercx_2017_alf — the ALF (Algorithms for Lattice Fermions) code documentation; comprehensive reference for the stabilized DQMC algorithm, Green's function updates, and observable implementations.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | GS energy (benchmark-grade for many Hubbard regimes) · back-propagated observables (density, correlation functions) · imaginary-time correlations → spectral functions via analytic continuation | Best-in-class for GS energy of the 2D Hubbard model; back-propagated observables carry extra bias from the constraint. |
| M2 regime | T=0 (ground state) | DQMC is the finite-T sibling; AFQMC-CPMC targets the GS specifically. |
| M3 accuracy class | Controlled-bias — phaseless/constrained-path bias set by trial-state quality; stochastic | Statistical error ∝1/√(N_w·N_steps); systematic bias shrinks as |Ψ_T⟩ → exact GS; free-projection (unbiased) is exponentially costly. |
| M4 dimension fit (A1) | Any dimension (1D, 2D, 3D) | Frequently applied to 2D Hubbard where it gives the best GS energies among polynomial-cost methods. |
| M5 statistics & local dim (A3) | Fermions (primary); also applied to real materials via plane-wave basis | Each walker is a Slater determinant; N orbitals (sites × spin); local dim d=4 Hubbard. |
| M6 entanglement regime (B5) | Volume-law tolerated | Walker space imposes no entanglement restriction; the constraint bias is the limit, not entanglement. |
| M7 sign-problem dependence (C12) | Phaseless/constrained-path approximation removes the sign at the cost of a trial-state bias | Free-projection (unbiased) cost blows up ∝e^{2βNΔf} with the sign; the constraint trades that for polynomial cost + controllable bias. |
| M8 symmetry exploitation (C9/C10) | Trial state |Ψ_T⟩ can be symmetry-projected (UHF, ROHF, GHF) to reduce bias; particle-number and spin conservation preserved in walker propagation | Better |Ψ_T⟩ (e.g. from DMRG or VMC) directly reduces the phaseless bias. |
| M9 time complexity | O(N³) per walker per step × N_w × N_steps; O(N²) per walker memory | N = number of orbitals/sites; N³ from matrix products in Slater determinant updates; N_w~100–1000 typical. |
| M10 memory | O(N²) per walker (Slater determinant stored as N×N_e matrix) | Total O(N_w·N²); modest compared to exact-diagonalization storage. |
| M11 control knob | N_w (walkers) → statistical error ∝1/√N_w; Δτ (projection step) → Trotter error; trial state |Ψ_T⟩ quality → phaseless bias | Improving |Ψ_T⟩ systematically reduces the bias; can use VMC-NQS or DMRG wavefunction as trial state. |
| M12 scale frontier | N~100–500 sites routine; N~1000 on HPC; thermodynamic-limit extrapolation via finite-size scaling | State-of-the-art for the 2D Hubbard model ground state at intermediate-to-strong coupling and away from half-filling. |
| M13 primary approximation / bias | Phaseless/constrained-path bias — set by overlap of walkers with |Ψ_T⟩; controlled by improving the trial state | Bias is variational in the constrained-path case: the constrained-path energy is an upper bound, E_CP ≥ E_0 (Zhang–Krakauer 2003); the phaseless approximation is not strictly variational. |
| M14 hard blocker / failure mode | Poor trial state → large phaseless bias (e.g. strongly frustrated models where even HF is qualitatively wrong); back-propagated observables have extra bias; free-projection is exponentially costly | The constraint quality is the central limit; no sign-free condition exists for general fermion models. |
O(N³) per walker per step × N_w × N_stepsO(N²) per walker; total O(N_w·N²)N_w (statistical error ∝1/√N_w); trial state quality → phaseless bias; Δτ → Trotter errorN~100–500 routine; N~1000 HPC; finite-size scaling to thermodynamic limitmethod-property-map.md: preferred over DQMC for GS at finite doping; complementary to DQMC's finite-T capabilities10×10, U/t=4, ⟨n⟩=0.875): AFQMC-CPMC energy E/t = −0.5444(4) — benchmark in the doped regime where DQMC is sign-ful zhang_2019_auxiliary.< 0.1% for U/t≤8 on 4×4 lattice nguyen_2014_cpmc.Key reference: zhang_2019_auxiliary — comprehensive lecture-note review of the auxiliary-field QMC algorithm at zero and finite temperature; the standard pedagogical and technical reference.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | GS energy (fixed-node = best variational energy given the trial node) · mixed estimators for other observables · forward-walking estimators for unbiased observables (at extra cost) | Mixed estimators for non-commuting observables carry bias O((Ψ_T−Ψ_0)²); forward-walking removes leading bias but adds variance. |
| M2 regime | T=0 (ground state) | Purely a GS projector method; finite-T not naturally accessible. |
| M3 accuracy class | Controlled-bias (fixed-node / fixed-phase = best variational energy given the trial nodal surface); stochastic | Energy is an upper bound given the trial node; exact for bosons/unfrustrated systems (sign-free, no bias). |
| M4 dimension fit (A1) | Any dimension (1D, 2D, 3D, continuum) | Especially powerful in continuum (electronic structure, quantum liquids); lattice GFMC works in any dimension. |
| M5 statistics & local dim (A3) | Bosons (sign-free, exact) · fermions (fixed-node, biased) · spins (via lattice GFMC) | Fermions require a fixed-node constraint; bosons are exact. Continuum electrons are the primary application domain. |
| M6 entanglement regime (B5) | Volume-law tolerated | Walker-based sampling imposes no entanglement restriction; quality determined by the trial node, not entanglement. |
| M7 sign-problem dependence (C12) | Sign-free for bosons / unfrustrated systems; fixed-node (nodal-surface bias) controls the fermion sign | Fixed-node = best variational energy given the trial node; nodal bias is the primary systematic error; stochastic reconfiguration can also help. |
| M8 symmetry exploitation (C9/C10) | Trial state |Ψ_T⟩ can carry full point-group / translation / spin symmetry; walker propagation respects these | Better-symmetry trial node → lower fixed-node bias. |
| M9 time complexity | O(N_w·N_proj·c_step); statistical error 1/√(N_w·N_proj) | c_step = cost per walker step (O(N) for simple wave functions; O(N³) for Slater-determinant trial states); N_proj = total projection time / Δτ. |
| M10 memory | O(N_w·N) | Stores N_w walker configurations of N particles; modest. |
| M11 control knob | N_w (walkers) · N_proj (projection time) → statistical error 1/√(N_w·N_proj); trial node quality → fixed-node bias | Δτ→0 extrapolation removes time-step bias; trial node improved by VMC optimization. |
| M12 scale frontier | N~100–1000 particles (continuum electronic structure); N~100–500 sites (lattice GFMC) | State-of-the-art for electron gas, quantum liquids (⁴He, ³He), and lattice Hubbard clusters. |
| M13 primary approximation / bias | Fixed-node (nodal-surface bias): energy upper bound given the trial node; bias eliminated only if the trial node is exact | Controlled in the sense that a better trial node monotonically improves the energy; not variational w.r.t. node position for all observables. |
| M14 hard blocker / failure mode | Fixed-node bias can be large for strongly correlated / frustrated systems where no good trial node is known; forward-walking estimators for observables have large variance; continuum time-step bias O(Δτ) requires Δτ→0 extrapolation | In lattice frustrated magnets the trial node quality degrades rapidly and the fixed-node energy may be far above the true GS. |
O(N_w·N_proj·c_step); statistical error 1/√(N_w·N_proj)O(N_w·N) walker configurationsN_w and N_proj (statistical error); trial node (fixed-node bias); Δτ (time-step bias)N~100–1000 (continuum); N~100–500 (lattice); thermodynamic-limit via finite-size scalingmethod-property-map.md: preferred for continuum systems; blocked by frustration (poor nodal surface)r_s=1–5: DMC fixed-node energy matches diffusion MC benchmark to < 1 mHa/particle with Slater–Jastrow trial function becca_2017_quantum.E/N = −7.17(2) K at experimental density — the reference for quantum liquid benchmarks becca_2017_quantum.Key reference: becca_2017_quantum — textbook treatment of VMC and projector QMC (GFMC/DMC) with stochastic reconfiguration; the standard pedagogical reference covering the fixed-node approximation, forward-walking estimators, and benchmarks.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Finite-T thermodynamics (energy, specific heat, pressure, compressibility) · superfluid density (winding number) · condensate fraction (off-diagonal density matrix n(r)) · structure factors S(q) · pair correlations g(r) | Spectra require analytic continuation; real-time dynamics inaccessible. |
| M2 regime | Finite-T (primary); T→0 limit feasible but expensive | The natural regime is finite temperature; GS by taking β→∞. |
| M3 accuracy class | Numerically exact (bosons, sign-free), stochastic | Statistical error ∝1/√N_s; Trotter/discretization error O(Δτ^p) controlled by N_τ→∞; fermion sign problem degrades accuracy. |
| M4 dimension fit (A1) | Any dimension; continuum (⁴He) or lattice | Particularly powerful in 3D continuum for quantum liquids and cold atoms. |
| M5 statistics & local dim (A3) | Bosons (sign-free) — the primary regime; fermions reintroduce the sign problem via odd permutations | For fermions, the sign problem returns and cost is exponential; restricted PIMC (fixed-node analog) controls it with bias. |
| M6 entanglement regime (B5) | Volume-law tolerated | Worldline sampling has no entanglement restriction; the sign problem is the gate for fermions. |
| M7 sign-problem dependence (C12) | Sign-free for bosons; fermions reintroduce the sign via odd permutation exchanges | For bosonic systems PIMC is numerically exact; fermion PIMC requires restricted PIMC (nodal-surface bias) analogous to fixed-node DMC. |
| M8 symmetry exploitation (C9/C10) | Continuous translational symmetry handled naturally in the continuum; lattice models use discrete symmetry to reduce sampling space | Permutation symmetry is essential for bosonic exchange statistics; the worm algorithm samples exchange cycles directly. |
| M9 time complexity | O(N·N_τ) per sweep; worm algorithm for efficient winding/superfluid sampling | N_τ = β/Δτ; worm updates propose open/close operations to sample the off-diagonal sector (condensate, superfluid density) efficiently. |
| M10 memory | O(N·N_τ) for the worldline configurations | Modest compared to wavefunction methods; total path of all particles stored. |
| M11 control knob | N_τ (imaginary-time slices) → discretization error O(Δτ^p); N_s (sweeps) → statistical error ∝1/√N_s | Convergence in N_τ tested by comparing N_τ and 2N_τ; standard extrapolation to continuum limit. |
| M12 scale frontier | N~100–10000 particles (continuum bosons); thermodynamic-limit extrapolation via finite-size scaling | State-of-the-art for ⁴He, cold bosonic atoms, hydrogen at high pressure. |
| M13 primary approximation / bias | Trotter/discretization error O(Δτ^p) — controlled by N_τ→∞; none for bosons in the continuum limit | For fermions, restricted PIMC introduces a nodal-surface bias analogous to fixed-node DMC. |
| M14 hard blocker / failure mode | Fermion sign problem via odd permutation exchanges: cost ∝1/⟨s⟩² grows exponentially with βN; analytic continuation for spectral functions is ill-posed | Even for bosons, very low temperatures require large N_τ, raising cost linearly with β. |
O(N·N_τ) per sweep; worm algorithm adds efficient off-diagonal samplingO(N·N_τ) worldline pathsN_τ (discretization error O(Δτ^p)); N_s (statistical error ∝1/√N_s)N~100–10000 (continuum bosons); thermodynamic limit via finite-size scalingmethod-property-map.md: preferred for sign-free bosonic continuum systems at finite T; blocked by fermion sign problemρ_s/ρ vs T curve, including the transition temperature T_λ ≈ 2.17 K, from first principles ceperley_1995_path.T=1 K, P=0: n_0 ≈ 7.25% — experimental value matched by PIMC ceperley_1995_path.Key reference: ceperley_1995_path — comprehensive review of PIMC for quantum liquids: path-integral formulation, Bose–Einstein statistics and exchange, superfluid density estimator, ⁴He benchmarks; the canonical reference for the method.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Self-energy Σ(k,ω) · one-particle Green's function G(k,ω) · vertex functions · thermodynamics (free energy, density, compressibility) · polaron binding energy and dispersion | All computed directly in the thermodynamic limit; finite-size effects absent. |
| M2 regime | Finite-T (Matsubara formalism) or T=0 (real-frequency); polaron ground state | Both regimes accessible; imaginary-frequency results require analytic continuation for spectra. |
| M3 accuracy class | Controlled-bias (series truncation at order n_max); stochastic | Statistical error ∝1/√N_s; systematic error from diagram-series truncation and resummation scheme; uncontrolled if the series diverges or the sign problem (diagram-sign) is severe. |
| M4 dimension fit (A1) | Thermodynamic limit directly (no finite-size error) — the key differentiator | Works in any dimension; no finite-size scaling needed; directly accesses the TDL. |
| M5 statistics & local dim (A3) | Bosons and fermions in the diagrammatic language; local dim d does not appear (momentum-space formulation) | Polaron problems (single fermion in a bosonic bath) are the canonical application; fermionic sign enters via diagram signs. |
| M6 entanglement regime (B5) | Not a wavefunction method — no entanglement restriction | Diagrammatic expansion does not represent a wavefunction; entanglement concept does not apply. |
| M7 sign-problem dependence (C12) | Diagram-sign + series-convergence are the limiters; weak–intermediate coupling regime | At strong coupling, the diagram series may diverge or require resummation; the sign problem manifests as cancellations between diagrams of opposite sign. |
| M8 symmetry exploitation (C9/C10) | Momentum conservation and translational invariance are explicit in the momentum-space formulation; spin SU(2) used to relate spin channels | Symmetry reduces the diagram phase space and variance; used to relate diagram classes. |
| M9 time complexity | Cost grows with maximum diagram order n_max; convergence/resummation of the series is the bottleneck, not lattice size | No N³ matrix cost; the bottleneck is high-order diagram evaluation and the exponential growth of diagram topologies with order. |
| M10 memory | O(n_max) per configuration — stores the current diagram topology and variables | Extremely modest compared to wavefunction-based methods; no Hilbert-space dimension enters. |
| M11 control knob | n_max (maximum diagram order) → series truncation error; resummation method (Padé, bold-line, conformal) → systematic convergence of resummed series | Check convergence as n_max increases; residual error is scheme-dependent. |
| M12 scale frontier | Thermodynamic limit by construction — no system-size extrapolation needed | The unique frontier: N=∞ at the cost of a perturbative expansion. Applies to homogeneous (translationally invariant) systems. |
| M13 primary approximation / bias | Series truncation at order n_max and resummation scheme bias; diagram-sign problem at strong coupling | Controlled at weak–intermediate coupling; uncontrolled when the series radius of convergence is exceeded. |
| M14 hard blocker / failure mode | Strong coupling: diagram series diverges or has zero radius of convergence; sign problem from diagram cancellations; inhomogeneous / disordered systems break translation invariance (no momentum basis) | Lattice models at strong U/t and frustrated systems are beyond the weak–intermediate coupling reach. |
n_max; diagram topology enumeration grows factorially with order; each diagram evaluated via Monte Carlo quadratureO(n_max) per configuration — negligiblen_max (truncation order → series error); resummation method → systematic biasN=∞ by construction; n_max~8–14 typical for polaron problemsU/t, Bose gas near unitarity — direct TDL results without finite-size extrapolationN→∞ extrapolation cost that burdens all lattice methodsmethod-property-map.md: the only method that directly accesses N=∞; blocked at strong coupling (B7) and by inhomogeneity (disorder, interfaces)α→0) through intermediate coupling (α≈6) — matches Feynman variational bound at weak coupling and improves beyond it prokofev_1998_polaron.m*/m at α=3: DiagMC 1.87(2) vs Feynman variational 1.89 — consistent at the few-percent level for the founding calculation prokofev_1998_polaron.Key reference: prokofev_1998_polaron — founding DiagMC paper: stochastic summation of polaron Green function diagrams, first demonstration that continuous-variable diagrammatic MC gives exact polaron properties in the TDL.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Single-particle spectral function A(ω) / self-energy Σ(ω) / DOS · Mott transition & phase diagram · finite-T thermodynamics | Two-particle responses (susceptibilities, optical conductivity) require vertex functions → substantially more expensive; momentum resolution needs cluster extensions. |
| M2 regime | Finite-T natural; T=0 accessible; real-frequency via analytic continuation (MaxEnt/Padé) | Imaginary-axis QMC solver gives G(iω_n); real-frequency requires analytic continuation with associated uncertainty. NRG solver gives real-frequency directly. |
| M3 accuracy class | Controlled (exact at Z→∞); uncontrolled at finite dimension | Cluster DMFT (N_c sites) is systematically improvable in N_c but at exponential cost. Sign from CT-QMC introduces stochastic error for multi-orbital/cluster cases. |
| M4 dimension fit (A1) | Exact as coordination Z→∞ / infinite dimension (Metzner–Vollhardt); finite-D neglects spatial correlations beyond the cluster | Single-site DMFT best for 3D / high-coordination; short-range correlations restored by clusters (N_c); cluster cost grows exponentially with N_c. |
| M5 statistics & local dim (A3) | Fermions (multi-orbital supported); local Hilbert dim grows with #orbitals | CT-QMC sign worsens with #orbitals and cluster size; off-diagonal hybridization triggers sign problem. |
| M6 entanglement regime (B5) | No explicit entanglement gate (impurity picture) | The bath is integrated out; entanglement of impurity + bath is handled by the solver; no bond-dimension wall in principle. |
| M7 sign-problem dependence (C12) | Sign-problem enters through the CT-QMC impurity solver | CT-HYB/CT-INT sign worsens with multi-orbital / off-diagonal hybridization / cluster size. ED-bath and NRG solvers are sign-immune. |
| M8 symmetry exploitation (C9/C10) | Spin, orbital, and point-group symmetries reduce the impurity problem; translation exploited in DCA (momentum-space clusters) | Symmetry reduces bath/cluster dimension and cuts impurity-solver cost. |
| M9 time complexity | Self-consistency loop × impurity-solver cost: CT-QMC ~polynomial in β (with sign worsening for multi-orbital/cluster); ED-bath exponential in #bath sites; NRG O(N_kept³) per iteration | Cluster DMFT cost grows exponentially with N_c. |
| M10 memory | Impurity solver memory (CT-QMC: bath Green's function matrices; ED: d^{N_bath} Hilbert space) | Scales with #orbitals × cluster size; memory is rarely the primary wall compared to sign/cost. |
| M11 control knob | Cluster size N_c (spatial resolution); bath size (ED solver); N_kept (NRG solver); β and #MC steps (CT-QMC statistical error) | Larger N_c → exponentially harder; β convergence at low T for CT-QMC. |
| M12 scale frontier | Single-site DMFT: thermodynamic limit by construction; cluster DMFT: N_c ~ 4–16 routine, larger with DMRG solver | Infinite-lattice is treated at the single-site level; finite N_c for cluster variants. |
| M13 primary approximation / bias | Neglect of spatial correlations (single-site); cluster truncation (cluster DMFT) | Controlled in N_c; exact at Z→∞ for single-site. |
| M14 hard blocker / failure mode | Strong sign problem for multi-orbital / off-diagonal hybridization / large clusters (CT-QMC); spatial correlations beyond N_c (momentum differentiation, pseudo-gap precursors) | 1D strongly-correlated: spatial fluctuations dominate and single-site DMFT is qualitatively wrong. |
β; ED-bath exponential in #bath sites; cluster DMFT cost grows exponentially with N_cN_orb² × N_τ); ED-bath O(d^{N_bath})N_c (cluster size, spatial resolution); β and MC steps (CT-QMC); bath size (ED solver)N_c ~ 4–16 routinemethod-property-map.md: preferred over MF (A1~3D, B7 strong correlation); complement with cluster DMFT for spatial correlations, DCA for k-resolved physicsU_c2 ≈ 2.9 D (Bethe lattice, half-bandwidth D=1, T=0; equivalently U_c2/t ≈ 5.8 since D=2t) georges_1996_dynamical.N_c convergence: local observables converge ~1/N_c; momentum-space quantities require larger N_c [method-survey.md §6.1].Key reference: georges_1996_dynamical — the authoritative DMFT review covering the Z→∞ limit, impurity solvers, the Mott transition, and benchmarks for the Hubbard model.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Ground-state energy · local observables (density, magnetization, double occupancy) · phase diagrams | Spectral functions are harder; frequency-dependent DMET extensions exist but are not routine. |
| M2 regime | T=0 (ground state) | Finite-T DMET extensions exist but are less developed. |
| M3 accuracy class | Controlled by fragment size N_frag + embedded solver (DMRG/FCI); deterministic | Accuracy improves systematically with N_frag; controlled by the embedded solver (FCI is exact within the embedding). |
| M4 dimension fit (A1) | Any dimension | Unlike DMFT, DMET is not restricted to infinite dimension; applicable in 1D, 2D, 3D. |
| M5 statistics & local dim (A3) | Fermions primarily; bosonic DMET extensions exist | Local Hilbert dim enters through the embedded solver (d^{2N_frag} for FCI). |
| M6 entanglement regime (B5) | Fragment-bath entanglement captured by the Schmidt basis | The Schmidt bath truncation (exactly N_frag bath orbitals) limits long-range entanglement; controlled by N_frag. |
| M7 sign-problem dependence (C12) | Sign-problem enters only via the embedded solver | FCI and DMRG embedded solvers are sign-immune; AFQMC embedded solver carries a constraint bias. |
| M8 symmetry exploitation (C9/C10) | Lattice symmetries can be exploited in the embedded problem and to partition the fragment tiling | Translation symmetry of the tiling reduces the number of inequivalent fragments. |
| M9 time complexity | Dominated by the impurity solver on the 2·N_frag-site embedding: FCI O(d^{2N_frag}), DMRG O(N_frag·χ³) for 1D fragments, CCSD O(N_frag⁶) | Self-consistency loop on the correlation potential is cheap relative to the solver; larger N_frag → exponentially harder solver (for FCI). |
| M10 memory | Dominated by the embedded solver: FCI O(d^{2N_frag}), DMRG O(d·χ²) | Memory wall is the embedded solver; manageable for N_frag ≲ 10 with FCI, larger with DMRG. |
| M11 control knob | Fragment size N_frag — controls spatial resolution and accuracy; embedded solver accuracy (bond dim χ for DMRG, or truncation for CCSD) | The two knobs must be jointly converged; FCI gives exact embedding at fixed N_frag. |
| M12 scale frontier | Thermodynamic limit by tiling; fragment N_frag ~ 2–36 sites in practice; limited by embedded solver | With DMRG embedded solver: N_frag ~ 36 (1D strips) accessible; FCI: N_frag ~ 8–12. |
| M13 primary approximation / bias | Fragment/bath truncation (Schmidt basis limited to N_frag bath orbitals); self-consistency on 1-RDM (not the full wavefunction) | Controlled by N_frag; the Schmidt bath exactly captures all fragment-environment entanglement at fixed mean-field reference. |
| M14 hard blocker / failure mode | Embedded solver cost grows exponentially with N_frag (FCI); 2D fragments require large N_frag for spatial convergence; long-range correlations beyond fragment size not captured | Self-consistency can have convergence issues (multiple solutions); spectral functions not natively accessible. |
2·N_frag sites: FCI O(d^{2N_frag}), DMRG O(N_frag·χ³), CCSD O(N_frag⁶); self-consistency loop cheapO(d^{2N_frag}), DMRG O(d·χ²)N_frag (fragment size, spatial resolution and accuracy) + embedded solver knob (χ for DMRG)N_frag ~ 2–36 in practiceN_frag ~ DMFT; growing N_frag systematically improves spatial correlationsmethod-property-map.md: applicable when 2D/3D frustration blocks QMC and spatial correlations beyond single-site DMFT are neededN_frag = 2 knizia_2012_density.N_frag = 4–16 matches AFQMC/DMRG benchmark energies to ~0.5% at U/t = 4–8 [method-survey.md §6.2].Key reference: knizia_2012_density — original DMET paper introducing the Schmidt-bath embedding and self-consistency scheme; benchmarks on 1D and 2D Hubbard models.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Kondo scale T_K · impurity spectral function A(ω) at all T · thermodynamics (entropy, specific heat, susceptibility as function of T) · fixed-point flow diagrams · real-frequency self-energy Σ(ω) as a DMFT solver | NRG gives real-frequency quantities directly without analytic continuation — its key advantage over CT-QMC. |
| M2 regime | All T (T=0 to high-T crossovers) and real-frequency | The logarithmic bath discretization gives exponential energy resolution; no imaginary-time or analytic continuation needed. |
| M3 accuracy class | Controlled (convergence in N_kept and Λ→1); deterministic | Energy truncation at N_kept introduces a controlled systematic error; Λ→1 is the continuum limit but raises cost. |
| M4 dimension fit (A1) | Impurity / quantum-dot problems (0D impurity + bath) — not a bulk lattice method | Designed for impurity models (Anderson, Kondo, multi-channel, SU(N) Kondo); used as the DMFT impurity solver to access real-frequency self-energies. |
| M5 statistics & local dim (A3) | Fermions (and bosons via bosonic NRG); N_kept must grow exponentially with #bands/channels | Limited to ~1–3 channels; multi-channel NRG requires exponentially larger N_kept → rapidly intractable. |
| M6 entanglement regime (B5) | No entanglement wall for impurity problems (the bath is sequentially integrated out) | Wilson chain geometry makes the entanglement growth manageable by construction. |
| M7 sign-problem dependence (C12) | Sign-immune | Not a Monte Carlo method; exact iterative diagonalization with energy truncation. |
| M8 symmetry exploitation (C9/C10) | U(1), SU(2), Z₂, and non-Abelian symmetries exploited at each iteration to block-diagonalize and reduce N_kept required | Symmetry is crucial: SU(2) spin + U(1) charge reduce cost by an order of magnitude; non-Abelian NRG (Weichselbaum) extends reach to multi-channel. |
| M9 time complexity | O(N_kept³) per NRG iteration (diagonalize a ≈ d·N_kept matrix, truncate to N_kept); N_kept must grow exponentially with #bands/channels | Typically ~50–100 iterations for Λ~2–3; total cost ~N_iter × (d·N_kept)³. |
| M10 memory | O(N_kept²) per shell for the density matrix and kept states | Memory manageable for 1–2 channel problems; grows quickly with #channels. |
| M11 control knob | N_kept (retained states per shell) — controls energy truncation error; Λ (discretization parameter) — controls bath discretization error (Λ→1 is continuum) | Both knobs must be converged; typical Λ~2–4, N_kept~500–3000 with full symmetry. |
| M12 scale frontier | 1–3 channel impurity problems; single- and two-impurity systems; multi-orbital impurities with symmetry | At 3+ channels without high symmetry: N_kept requirement becomes exponentially prohibitive. |
| M13 primary approximation / bias | Energy truncation at N_kept (controlled); logarithmic bath discretization with Λ (controlled, Λ→1 limit) | Both errors are systematically reducible; the truncation error is the leading one at fixed Λ. |
| M14 hard blocker / failure mode | #channels/bands is the wall: N_kept must grow exponentially with #bands/channels — NRG limited to ~1–3 channels; not applicable to bulk lattice problems | Bosonic baths with sub-ohmic spectral density can cause issues; high-frequency resolution limited by Λ. |
O(N_kept³) per NRG iteration; N_kept grows exponentially with #channels; ~50–100 iterations totalO(N_kept²) per shell; manageable for 1–2 channels with symmetryN_kept (truncation error) and Λ (discretization error, Λ→1 is the continuum limit)T and real-frequencymethod-property-map.md: the method of choice for 0D impurity problems (A1=0D) whenever real-frequency or all-T data is requiredT_K/D ~ exp(-π U / 8Γ) for symmetric Anderson model; reproduced to < 1% with N_kept~1000, Λ~2 bulla_2007_numerical.∫ A(ω) dω = 1; satisfied to < 0.1% in fdm-NRG [method-survey.md §6.3].N_kept required scales ~d^{N_ch} with #channels N_ch; 3-channel NRG requires N_kept~10⁴ even with symmetry [method-survey.md §6.3].Key reference: bulla_2007_numerical — comprehensive review of NRG theory, algorithms (including fdm-NRG), thermodynamics, spectral functions, and applications to Anderson/Kondo models.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Leading-instability phase diagrams (magnetic, superconducting, charge-density-wave) · susceptibilities · pseudo-critical scales Λ_c · renormalized quasiparticle properties | NOT ground-state energies of strongly-correlated phases; the divergence signals an instability, not the true ordered state. |
| M2 regime | T=0 and finite-T (cutoff flow equations work at both); real-frequency accessible via Keldysh fRG | Standard formulation at T=0 or T>0 in Matsubara; real-time dynamics via Keldysh formulation. |
| M3 accuracy class | Controlled approximation (truncation order is the knob); uncontrolled at strong coupling | Weak–intermediate coupling: one-loop + Katanin truncation is a controlled expansion; strong coupling truncation breaks down. |
| M4 dimension fit (A1) | Any dimension (1D, 2D, 3D) | Particularly powerful in 2D (competing instabilities in the Hubbard model); 1D reproduces bosonization results. |
| M5 statistics & local dim (A3) | Fermions (fermionic fRG); bosonic fRG also exists | Fermionic sign enters only via the choice of statistics in the propagators; no sign problem in the sense of QMC. |
| M6 entanglement regime (B5) | No entanglement restriction (field-theoretic, not wavefunction-based) | fRG is not a wavefunction method; entanglement is not a controlling parameter. |
| M7 sign-problem dependence (C12) | Sign-immune (not a Monte Carlo method) | Deterministic flow equations; no stochastic sampling. |
| M8 symmetry exploitation (C9/C10) | Symmetries (SU(2), time-reversal, lattice point group) reduce vertex parameterization cost; translation used to parameterize vertex in momentum space | Exploiting symmetry reduces the number of independent vertex components and accelerates the flow. |
| M9 time complexity | ~O(N_patch³) for the two-particle vertex flow (three momentum sums); higher-order frequency dependence adds factors | N_patch is the number of momentum patches on the Fermi surface; frequency dependence multiplies cost by N_ω³. |
| M10 memory | O(N_patch²) to O(N_patch³) for storing the two-particle vertex (depending on channel decomposition) | Channel decomposition (particle-particle, particle-hole crossed/direct) allows efficient storage. |
| M11 control knob | Truncation order (one-loop → Katanin → multiloop → ...) — controls vertex approximation; N_patch — controls momentum resolution | Higher truncation order is closer to exact but exponentially more expensive; N_patch convergence required. |
| M12 scale frontier | Thermodynamic limit (field-theoretic, no finite-size constraint); limited by N_patch and truncation order | Patch numbers N_patch ~ 16–96 are typical; multiloop fRG up to 3–5 loop orders is state of the art. |
| M13 primary approximation / bias | Truncation of the vertex hierarchy at finite order (one-loop / Katanin / multiloop) — the controlled approximation | One-loop truncation is uncontrolled at strong coupling; multiloop fRG converges the loop expansion. |
| M14 hard blocker / failure mode | Strong coupling not controlled: vertex flow diverges when the coupling is too large (beyond weak–intermediate); cannot describe Mott physics or strongly frustrated states | Two-particle interaction vertex divergence Γ^(4) → ∞ is a signature of instability, not a numerical failure — but the truncation breaks down before the physical strong-coupling fixed point. |
~O(N_patch³) for two-particle vertex flow (three momentum sums); multiplied by N_ω³ with frequency dependenceO(N_patch²)–O(N_patch³) for vertex storage (channel-decomposed)N_patch (momentum resolution)N_patch ~ 16–96 typical; multiloop 3–5 loop orders is frontiermethod-property-map.md: complement to MF (/method-mf) at weak coupling; sign-immune alternative to sign-ful QMC in frustrated 2D systemsU/t = 2, square lattice: fRG pseudo-critical scale Λ_c consistent with RPA at weak coupling; leading instability AF at half-filling, d-SC away from half-filling metzner_2011_functional.N_patch convergence: local observables converge at N_patch ~ 32–64 for 2D Hubbard [method-survey.md §6.4].U/t ≲ 3 [method-survey.md §6.4].Key reference: metzner_2011_functional — comprehensive review of fermionic fRG: flow equations, truncation schemes, vertex parameterizations, and applications to Hubbard-model instabilities.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Ground-state energies and observables (high accuracy) · EOM-CC for excitations / spectra · ionization potentials and electron affinities (IP/EA-EOM-CC) | High accuracy when single-reference; EOM-CC gives excited states but at O(N⁶)–O(N⁷) cost. |
| M2 regime | T=0 (ground state and selected excited states) | Finite-T extensions (thermal CC) exist but are non-standard. |
| M3 accuracy class | Controlled (CC hierarchy T₁T₂T₃... → FCI limit); deterministic; excellent for dynamic correlation | Accuracy collapses for strong static correlation (multi-reference breakdown); not variational. |
| M4 dimension fit (A1) | Any dimension (molecular and periodic/solid-state variants) | Periodic CCSD/CCSD(T) for solids (e.g. crystalline materials); EOM-CC for surfaces and defects. |
| M5 statistics & local dim (A3) | Fermions | Local Hilbert dim enters through the number of virtual orbitals N; cost scales O(N⁶)–O(N⁷). |
| M6 entanglement regime (B5) | Dynamic correlation (moderate entanglement) | Excellent for weakly-entangled systems; breaks down for volume-law / strongly-entangled (multi-reference) states. |
| M7 sign-problem dependence (C12) | Sign-immune (not a Monte Carlo method) | Deterministic algebraic equations; no sign problem. |
| M8 symmetry exploitation (C9/C10) | Point group, spin, and translational symmetry reduce the orbital/amplitude space | Symmetry-adapted CC reduces the number of independent amplitudes and cuts cost. |
| M9 time complexity | CCSD O(N⁶), CCSD(T) O(N⁷) in the number of orbitals N | The "gold standard" scaling of quantum chemistry; N here is total orbital count (occupied + virtual). |
| M10 memory | O(N⁴) for storing the two-particle amplitudes T₂ (dominant cost) | The T₂ tensor with four orbital indices is the memory bottleneck; O(N_occ² · N_virt²). |
| M11 control knob | CC hierarchy level (CCSD → CCSD(T) → CCSDT → ...) + basis set completeness | The CC expansion is systematically improvable toward FCI; basis set extrapolation is also needed. |
| M12 scale frontier | ~100–500 orbitals for CCSD (molecular); ~20–50 k-points for periodic CCSD; fragmentation/local CC extends reach | Explicitly correlated F12-CCSD(T) improves basis convergence; local CC (DLPNO-CCSD(T)) reaches 1000s of atoms. |
| M13 primary approximation / bias | Single-reference assumption (T acts on a single HF determinant) — fails for strong static correlation / open-shell degeneracy / frustration | Multi-reference CC (MRCC) extends reach but is much more costly and complex. |
| M14 hard blocker / failure mode | Strong static correlation / frustration / open-shell degeneracy: single-reference breakdown — T₁ diagnostic > 0.02 signals multi-reference character and unreliable results | Frustrated lattice models (B8), doped Mott insulators, near-degenerate transition states: CC is not the right tool. |
O(N⁶), CCSD(T) O(N⁷) in orbital count NO(N⁴) for T₂ amplitudes (N_occ² × N_virt²)T₁ < 0.02)U/t, van der Waals solids)method-property-map.md: do NOT use for frustrated magnets (B8), doped Mott insulators (B7 strong), or strongly-entangled statesT₁ diagnostic: T₁ > 0.02 for HF, T₁ ~ 0.05 for biradicals — signals multi-reference breakdown bartlett_2007_coupled.Key reference: bartlett_2007_coupled — comprehensive review of CC theory in quantum chemistry: amplitude equations, CCSD(T) derivation, EOM-CC, and applications across quantum chemistry.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Quasiparticle band structures / photoemission spectra (GW) · quasiparticle gaps and ionization potentials · correlation energies (RPA) · screened interactions W · optical spectra (BSE on top of GW) · spectra of correlated solids (SEET) | Direct gap: GW; optical gap: BSE/GW; ground-state energy: RPA; strongly-correlated spectra: SEET. |
| M2 regime | T=0 (standard); finite-T Matsubara GW also implemented | Real-frequency GW via analytic continuation or real-axis contour deformation; imaginary-axis for correlation energies. |
| M3 accuracy class | Uncontrolled perturbative approximation (ring/RPA resummation); self-consistent GW is parameter-free but not systematically improvable to exact, and fails at strong coupling (G₀W₀ also carries a starting-point/DFT-functional bias) | GW sums ring diagrams only — not a controlled order-by-order expansion toward exactness; sc-GW removes starting-point dependence but still misses Mott physics; G₀W₀ inherits the DFT-functional bias of the starting point. |
| M4 dimension fit (A1) | Any dimension (molecules and periodic solids) | GW for molecules (PySCF, TURBOMOLE) and periodic solids (VASP, Abinit, BerkeleyGW); no dimensional restriction. |
| M5 statistics & local dim (A3) | Fermions | Fermionic diagrammatic expansion; N is orbital/basis count. |
| M6 entanglement regime (B5) | No entanglement gate (diagrammatic / field-theoretic) | Not a wavefunction method; entanglement is not a controlling parameter. |
| M7 sign-problem dependence (C12) | Sign-immune (deterministic diagrammatic resummation) | No stochastic sampling in standard GW/RPA/GF2; diagrammatic MC variants exist but are non-standard. |
| M8 symmetry exploitation (C9/C10) | k-point sampling exploits translation symmetry; point-group symmetry reduces the response function | Periodic GW exploits Bloch symmetry; symmetry-adapted W reduces cost. |
| M9 time complexity | GW O(N⁴) (→ O(N³) with factorized W); RPA O(N⁴)→O(N³); GF2 O(N⁵) in orbital count N | Factorized GW (O(N³)) uses resolution-of-identity (RI) or density fitting; frequency integration adds N_ω factor. |
| M10 memory | O(N³) for storing W(r,r',ω) or its factored form; O(N²) for G and Σ | Memory scales as orbital count cubed for the full polarizability; RI reduces to O(N² × N_aux). |
| M11 control knob | Starting point (DFT functional for G₀W₀) · self-consistency level (G₀W₀ → qpGW → sc-GW) · basis set size · frequency grid N_ω | Self-consistent GW eliminates starting-point dependence at higher cost. |
| M12 scale frontier | ~100–1000 orbitals for GW (periodic: 50–500 bands × k-points); ~10⁴ orbitals for local GW | Stochastic GW and low-scaling O(N³) algorithms extend to large periodic systems. |
| M13 primary approximation / bias | Diagram truncation (GW = ring resummation; GF2 = 2nd order; higher-order diagrams neglected) | Weak-coupling: controlled; strong coupling (Mott): uncontrolled breakdown. SEET corrects by embedding. |
| M14 hard blocker / failure mode | Strong correlation out of reach: GW and RPA break down for Mott insulators, strongly-frustrated systems, and near-degenerate open shells (SEET partially fixes this by embedding) | G₀W₀ starting-point dependence is large for strongly-correlated systems; self-consistent GW still misses Mott physics. |
O(N⁴) (→ O(N³) factorized); RPA O(N⁴)→O(N³); GF2 O(N⁵) in orbital count NO(N³) for full polarizability; O(N² × N_aux) for RI-approximated WN_ωmethod-property-map.md: preferred over DFT (higher accuracy for gaps); restricted to weak–intermediate coupling (strong correlation → DMFT or DMET)Key reference: golze_2019_gw — comprehensive practical guide to GW approximation: theory, algorithms (G₀W₀, qpGW, sc-GW), implementations, and benchmarks for molecules and solids.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Mean-field phase diagrams · order parameters · band structures · Slater-determinant trial states for DMRG/VMC/AFQMC | Primary use: fast survey of symmetry-breaking phases and orbital structure. |
| M2 regime | T=0 (finite-T via mean-field free energy straightforward) | Thermal HF replaces zero-T occupations with Fermi–Dirac; cost unchanged. |
| M3 accuracy class | Uncontrolled, deterministic | Carries no entanglement; no systematic expansion parameter in general. |
| M4 dimension fit (A1) | Best in high dimension / large coordination Z; fails in 1D | Leading fluctuation corrections scale ~1/Z (the √Z is the DMFT *hopping* rescaling t=t*/√Z, not the MF error); MF is exact as Z→∞ (Metzner–Vollhardt 1989); Mermin–Wagner forbids spontaneous symmetry breaking in 1D → spurious order; also unreliable under B8 frustration. |
| M5 statistics & local dim (A3) | Fermions primarily; spinless/spinful bosons via mean-field decoupling | Local dim d=2 (spin-½ Hubbard) or d=4 (Hubbard with spin+charge); cost independent of d. |
| M6 entanglement regime (B5) | Product-state (zero entanglement) | Slater determinant = product in occupation-number basis; no area- or volume-law entanglement. |
| M7 sign-problem dependence (C12) | Sign-immune | Deterministic eigenvalue problem; no Monte Carlo sampling. |
| M8 symmetry exploitation (C9/C10) | Block-diagonalize by U(1) (charge/spin), translation k, point group; reduces SCF cost | HF in the k-space Bloch basis is particularly efficient for periodic systems. |
| M9 time complexity | O(N³)–O(N⁴) per SCF iteration (Fock build) | Negligible compared to correlated methods; converges in ≲100 SCF steps for typical lattice models. |
| M10 memory | O(N²) | Stores the density matrix / Fock matrix; cheap. |
| M11 control knob | None (uncontrolled) — quality assessed by post-HF correlation energy or by comparison with ED/DMRG | Fluctuation corrections in 1/Z give a rough measure of MF reliability. |
| M12 scale frontier | N ~ 10³–10⁵ sites feasible with k-space HF; DFT codes reach millions of atoms | Not the limiting factor; cost is always negligible relative to the correlated solver. |
| M13 primary approximation / bias | Mean-field decoupling — no fluctuations, no dynamical correlations, no entanglement | Uncontrolled (no small parameter in 2D/3D at intermediate coupling). |
| M14 hard blocker / failure mode | Fails in 1D (Mermin–Wagner, spurious order); fails under strong B8 frustration; misses Mott physics (B7) and fractionalization by construction | Post-HF methods (VMC, DMRG, AFQMC) required for quantitative accuracy. |
O(N³)–O(N⁴) per SCF iteration; converges in ≲100 iterationsO(N²) (density matrix + Fock matrix)1/Z fluctuation scaleN ~ 10³–10⁵ in k-space (periodic lattice); millions in DFTd hypercubic models where MF is exact or nearly somethod-property-map.md: preferred whenever only qualitative phase boundaries or orbital structure are needed and Z is largeU>0 in 2D HF (spurious; correct answer requires QMC) arovas_2022_hubbard.Z→∞; Bethe-lattice Z=∞ HF = exact (Metzner–Vollhardt 1989).~1/Z verified in method-survey.md §7.1.Key reference: arovas_2022_hubbard — comprehensive review of the Hubbard model covering HF/UHF mean-field theory, phase diagrams, DFT+U, and the full landscape of correlated methods; authoritative all-details source for the MF baseline.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Magnon dispersions · dynamical structure factor S(q,ω) at the harmonic level · sublattice-magnetization 1/S corrections · thermodynamic quantities (magnon heat capacity, entropy) · neutron scattering cross-sections | S(q,ω) from LSWT is directly comparable to neutron scattering data; main experimental touchpoint. |
| M2 regime | T=0 (LSWT); finite-T via Bose–Einstein magnon occupations | Low-T thermodynamics reliable; near T_N perturbation theory breaks down. |
| M3 accuracy class | Controlled in 1/S (semiclassical); deterministic | Large-S (A3) is the control parameter; corrections are O(1/S) and can be computed systematically. |
| M4 dimension fit (A1) | 2D and 3D ordered magnets; 1D requires caution (strong quantum fluctuations) | In 1D, 1/S corrections diverge → DMRG/QMC preferred; Mermin–Wagner forbids 2D order at T>0. |
| M5 statistics & local dim (A3) | Spins with large S (A3); bosons via Holstein–Primakoff | Accuracy improves with S; S=1/2 is borderline (large corrections); S≥1 reliable at leading order. |
| M6 entanglement regime (B5) | Product-like (ordered classical background + small Gaussian fluctuations) | LSWT wavefunction is a product coherent state with Bogoliubov squeezed-vacuum corrections. |
| M7 sign-problem dependence (C12) | Sign-immune | Deterministic eigenvalue problem; no Monte Carlo. |
| M8 symmetry exploitation (C9/C10) | Magnetic space group (C9) block-diagonalizes the m×m BdG matrix; translation (C10) gives k-space formulation | Symmetry reduces the m×m problem and labels magnon bands. |
| M9 time complexity | O(m³) per k-point × #k-points (diagonalization of the m×m BdG Hamiltonian) | Cheap — entire magnon band structure computed in seconds to minutes even for large magnetic unit cells. |
| M10 memory | O(m²) per k-point (BdG matrix storage) | Negligible. |
| M11 control knob | 1/S expansion order; S magnitude | Higher orders (1/S²) give magnon–magnon scattering, lifetime broadening, and order-parameter renormalization. |
| M12 scale frontier | Magnetic unit cells with hundreds of spins feasible (SpinW); no thermodynamic-size wall | Cost grows only as O(m³×N_k); arbitrary system size accessible for ordered phases. |
| M13 primary approximation / bias | Requires magnetic order (expands around an ordered classical state); large S (A3) is the control parameter | Uncontrolled at S=1/2 in frustrated systems; 1/S corrections blow up → signal of disorder / spin liquid. |
| M14 hard blocker / failure mode | B8 frustration / small S → 1/S corrections blow up (breakdown is itself the diagnostic signal of a spin-liquid or disordered ground state); method inapplicable to Mott insulators without order; real-time dynamics require additional approximations | Near quantum-critical points, corrections to LSWT diverge. |
O(m³) per k-point × #k-points (BdG diagonalization); seconds to minutes for any magnetic supercellO(m²) per k-point — negligible1/S expansion order; larger S → smaller corrections → better controlledN for ordered phases; m (sublattice count) is the only cost driverS≥1 — the method of choice1/S corrections are large → suspect spin-liquid ground state → switch to ED/DMRGmethod-property-map.md: use whenever B7 magnetic order is established and S is not too small1/S corrections ~20% of zone-boundary gap.<1% error in magnon frequencies (large-S limit verified).S(q,ω) from SpinW for incommensurate langasite — verified against neutron data in toth_2015_linear.1/S corrections diverge → confirms spin-liquid candidate (consistent with method-survey.md §7.2).Key reference: toth_2015_linear — SpinW paper: general LSWT algorithm for arbitrary magnetic structures including single-Q incommensurate order; the authoritative implementation reference for neutron scattering applications.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Candidate spin-liquid / fractionalized ansätze and their energies · mean-field phase diagrams · spinon/holon dispersions | Primary use: map out fractionalized phases (Z₂ / U(1) spin liquids, Mott physics) inaccessible to HF. |
| M2 regime | T=0 (thermal Schwinger-boson extensions possible) | Finite-T generalization exists for Schwinger-boson MF but less common. |
| M3 accuracy class | Uncontrolled, deterministic (MF saddle point); Gutzwiller-projected states are variational-upper via VMC | Beyond the saddle point, fluctuations require gauge-field theory — no small expansion parameter in general; Gutzwiller-VMC branch is variational. |
| M4 dimension fit (A1) | 2D and 3D; works in any dimension at MF level | Gauge fluctuations (corrections beyond MF) depend strongly on dimension; 1D is often handled better by bosonization or DMRG. |
| M5 statistics & local dim (A3) | Fermions (Abrikosov/slave-boson partons) and bosons (Schwinger bosons / slave bosons); spins and Hubbard-model electrons | Local dim d enters only the parton-space HF step; cost independent of d. |
| M6 entanglement regime (B5) | Mean-field parton state = product (zero entanglement in parton space); Gutzwiller projection introduces physical-space correlations | Parton MF carries no entanglement barrier; VMC evaluation of Gutzwiller-projected state does not impose a bond-dimension wall but carries ansatz bias. |
| M7 sign-problem dependence (C12) | Sign-immune (MF / parton saddle point is deterministic); Gutzwiller-VMC may carry a sign problem for fermionic partons | Parton MF: deterministic. VMC branch: sign problem possible but mitigated by the Gutzwiller projector structure. |
| M8 symmetry exploitation (C9/C10) | Lattice translation, point-group, and spin-rotation symmetries used to classify the parton ansatz (PSG — projective symmetry group) | Projective symmetry group (Wen) classifies distinct mean-field spin-liquid ansätze; symmetry reduces the parton Hamiltonian to k-space blocks. |
| M9 time complexity | Self-consistent parton MF: O(N³) per SCF iteration (HF-like, where N = system size / #parton flavors); Gutzwiller-projected VMC: O(N_s · N²) per sample | Cost is dominated by VMC if the Gutzwiller route is taken; the MF step alone is cheap. |
| M10 memory | O(N²) (parton density matrix / Fock matrix); VMC: O(N²) for the Slater-determinant update | Cheap at the MF level. |
| M11 control knob | None at the MF saddle-point level (uncontrolled); VMC branch: N_s samples control statistical error ∝1/√N_s | Gauge fluctuations beyond MF are in principle a systematic expansion but are rarely carried out beyond the Gaussian level. |
| M12 scale frontier | MF: N ~ 10³–10⁵ sites (no bottleneck); Gutzwiller-VMC: up to N ~ 10²–10³ sites (limited by VMC) | MF phase diagrams thermodynamic; VMC limited by the cost of Slater-determinant evaluation. |
| M13 primary approximation / bias | Mean-field (saddle-point) decoupling of the parton constraint — no fluctuations, no dynamical gauge field, no entanglement | Uncontrolled: can predict a spin liquid that is destabilized by gauge fluctuations; quality assessed only by comparison with VMC energetics or QMC. |
| M14 hard blocker / failure mode | Gauge fluctuations may gap or confine the parton excitations, invalidating the MF ansatz; no convergence parameter signals this within MF; fails to capture Luttinger-liquid physics in 1D | Breakdown is silent — a "spin-liquid" MF solution may be a confined phase in reality. Requires cross-check with VMC / QMC / DMRG. |
O(N³) per SCF iteration at the parton MF level (HF-like); O(N_s · N²) for Gutzwiller-VMCO(N²) (parton density matrix)N_s samples, error ∝1/√N_sN ~ 10²–10³method-property-map.md: the method of choice when the target phase is B7 fractionalized and HF/DMRG cannot resolve the spin-liquid ansatz structureU_c = 8t (exact Gutzwiller in d→∞; deviates in 2D) kotliar_1986_new.J₂/J₁ > 0.08; VMC energetics partially support this (see method-survey.md §7.3).Key reference: kotliar_1986_new — introduced the slave-boson functional-integral approach and showed that the Gutzwiller approximation emerges as its saddle point; the foundational paper for slave-boson MF on strongly correlated lattice models.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Spectral functions A(k,ω) / A(ω) · phase boundaries (Mott transition, magnetic / superconducting order) · symmetry-breaking order parameters · single-particle Green's function | VCA gives the full lattice spectral function via the periodized cluster self-energy; CMF gives order parameters and phase diagrams. |
| M2 regime | T=0 (finite-T via thermal cluster Green's function straightforward) | The cluster ED solver is zero-temperature by default; finite-T extensions use finite-T ED (FTLM/TPQ) inside the cluster. |
| M3 accuracy class | Controlled by cluster size N_c (systematic improvement) | Adding sites to the cluster systematically reduces the inter-cluster mean-field error; exact in the limit N_c → ∞. |
| M4 dimension fit (A1) | 2D (primary target); quasi-1D ladders | Designed for 2D correlated lattice models (Hubbard, t-J, periodic Anderson); inter-cluster MF less accurate in 1D where fluctuations dominate. |
| M5 statistics & local dim (A3) | Fermions, spins, hard-core bosons; local dim d enters through the ED cluster cost d^{N_c} | Same d^N exponential wall as ED, restricted to the cluster; typical d=2–4 (spin-½ to Hubbard). |
| M6 entanglement regime (B5) | Within-cluster entanglement: exact (full ED Hilbert space); inter-cluster entanglement: mean-field / neglected | Short-range entanglement captured exactly; long-range inter-cluster entanglement suppressed by the MF decoupling. |
| M7 sign-problem dependence (C12) | Sign-immune (ED cluster solver, no Monte Carlo) | Deterministic; no sign problem regardless of frustration or doping level. |
| M8 symmetry exploitation (C9/C10) | Cluster point-group and translation symmetry block-diagonalize the ED step; lattice translation used to periodize the self-energy (k-resolved spectral functions) | C9 symmetry reduces cluster ED cost significantly; periodization of the cluster self-energy restores approximate lattice momentum resolution. |
| M9 time complexity | Dominated by exact cluster ED: O(d^{3N_c}) (full diagonalization) or O(n_iter · N · d^{N_c}) (Lanczos); inter-cluster coupling O(N_c²) per self-consistency step — cheap | Cost is exponential in cluster size N_c; the inter-cluster part adds negligible overhead. Typical cluster sizes N_c = 4–16 sites. |
| M10 memory | O(d^{2N_c}) for cluster ED (full) or O(d^{N_c}) for Lanczos; lattice Green's function O(N_c² · N_ω) | Memory limit set by cluster ED, not the lattice size. |
| M11 control knob | Cluster size N_c — larger clusters reduce inter-cluster MF error systematically | No statistical error; convergence checked by increasing N_c (e.g. 2×2 → 2×3 → 2×4 tiling). |
| M12 scale frontier | Cluster: up to N_c ~ 12–20 sites (ED wall); lattice: thermodynamic limit reached by the mean-field coupling of an infinite tiling | The lattice itself can be infinite (thermodynamic limit); only the cluster is size-limited. |
| M13 primary approximation / bias | Neglect of inter-cluster quantum fluctuations / entanglement (replaced by MF or the VCA self-energy functional saddle point) | Controlled: bias decreases systematically with N_c. For small clusters, inter-cluster fluctuations can be large (especially near quantum criticality). |
| M14 hard blocker / failure mode | Exponential ED cost limits clusters to N_c ≲ 20 sites; inter-cluster MF becomes unreliable near quantum critical points where correlations are long-ranged; 3D is impractical (too many inequivalent bonds to tile) | Not a thermodynamic-limit method for the cluster solver; accuracy plateau at small N_c if cluster geometry cannot capture the relevant correlations. |
O(d^{3N_c}) (full ED) or O(n_iter · N · d^{N_c}) (Lanczos) per cluster; inter-cluster steps negligibleO(d^{2N_c}) (full ED) or O(d^{N_c}) (Lanczos)N_c; systematic improvement, no error bar (deterministic)N_c ≲ 20; lattice thermodynamic limitt-J model, correlated electron systems at finite hole doping where DQMC has a sign problemmethod-property-map.md: preferred when A1 = 2D, B7 Mott physics, C12 sign-ful (doped / frustrated), and spectral resolution is neededU=8t, n=0.875): Δ_d ≈ 0.05–0.08t for N_c = 4–8 clusters, consistent with CDMFT and DCA maier_2004_quantum.U_c/t ≈ 6–8 (2D square lattice, half filling) converges with cluster size; N_c = 4 gives U_c ≈ 6.5t, approaching the DCA/CDMFT consensus maier_2004_quantum.method-survey.md §7.4.Key reference: maier_2004_quantum — comprehensive review of quantum cluster approaches (DCA, CDMFT, VCA) covering the theoretical foundations, self-energy functional theory, cluster-size convergence, and applications to correlated electron models; the authoritative all-details source.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Sampling from the output distribution · expectation values of Pauli observables · VQE energy minimization · Hamiltonian-dynamics simulation · circuit fidelity estimation | Cost set by entanglement/depth generated and whether the circuit is near-Clifford. |
| M2 regime | T=0 / circuit-time (no thermal ensemble by default) | Thermal states via imaginary-time circuits or purification; real-time dynamics natively. |
| M3 accuracy class | State-vector: numerically exact, deterministic. TN: controlled bias (bond-dim truncation; a variational upper bound only for VQE energy minimization, not for dynamics/sampling). Stabilizer (Clifford): exact; stabilizer rank (T-gates): stochastic sampling. | Engine choice sets the accuracy class. |
| M4 dimension fit (A1) | Not lattice-dimension-gated; governed by circuit topology and entanglement | MPS/TEBD efficient for 1D/brickwork circuits; 2D circuits need hyper-optimized TN slicing gray_2021_hyper. |
| M5 statistics & local dim (A3) | Qubits (d=2) primarily; qudits via generalization | State-vector cost doubles per qubit; stabilizer cost scales as O(n²) per measurement. |
| M6 entanglement regime (B5) | Entanglement / circuit structure is the spine. State-vector: volume-law tolerated. TN: cost = exp(treewidth) of contraction graph; polynomial for area-law / low-depth / 1D-like. Stabilizer: polynomial for near-Clifford (low T-gate count). | Choice of engine must match the circuit's entanglement profile. |
| M7 sign-problem dependence (C12) | Sign-immune (state-vector, TN, Clifford); stabilizer rank sampling can suffer phase cancellation for large T-count | No sign problem in state-vector or Clifford; near-Clifford regime is also stable. |
| M8 symmetry exploitation (C9/C10) | Symmetry sectors reduce state-vector dim; TN block structure; Clifford symmetries reduce tableau | TensorCircuit-NG and cotengra exploit circuit symmetry for hyper-optimized contraction. |
| M9 time complexity | State-vector: O(gates·2ⁿ) time, O(2ⁿ) memory. TN: O(exp(treewidth)) (→ polynomial when B5 area-law/low-depth, Markov–Shi markov_2008_simulating). Stabilizer: O(n) per Clifford gate, O(n²) per measurement (Aaronson–Gottesman); exponential in #non-Clifford T-gates via stabilizer rank ~2^{αt}, α≈0.396 (Qassim–Pashayan–Gosset 2021). | Use TN/stabilizer whenever applicable; state-vector is the fallback. |
| M10 memory | State-vector: O(2ⁿ). TN: varies with treewidth. Stabilizer: O(n²) (tableau). | State-vector memory O(2ⁿ·16 bytes): ~100 GB at n≈33, ~16 TB at n≈40 (complex double). |
| M11 control knob | TN bond dimension χ (controls truncation error) / stabilizer T-count t (controls approximation) | State-vector: no approximation. |
| M12 scale frontier | State-vector: n≈30–50 on HPC (memory-limited). TN: polynomial cost when treewidth is small; random 2D circuits beyond ~53 qubits require slicing + HPC. Stabilizer: Clifford circuits → any n; near-Clifford up to ~50 qubits with small T-count. | Frontier set by engine-specific wall. |
| M13 primary approximation / bias | State-vector: none. TN: MPS truncation (bond dim χ). Stabilizer rank: exponential approximation for large T-count. | Clifford (T=0) is exact. |
| M14 hard blocker / failure mode | State-vector: O(2ⁿ) memory wall. TN: high-depth / high-entanglement 2D circuits have large treewidth → exponential. Stabilizer: T-gate count t makes cost ~2^{0.396t} — non-Clifford universality is hard to simulate classically by design. | No single engine covers all regimes. |
O(gates·2ⁿ); TN O(exp(treewidth)) (polynomial for low-depth / 1D); stabilizer O(n) per Clifford gate, O(n²) per measurement (Aaronson–Gottesman), + O(2^{αt}) in the number of non-Clifford T-gates, α≈0.396O(2ⁿ); TN varies; stabilizer O(n²)χ (truncation error) / T-gate count t (stabilizer rank)n≈30–50; TN polynomial for low-depth; stabilizer any n for Cliffordn circuits (n ≲ 30) where exactness is required; debugging and validating quantum algorithmsmethod-property-map.md: choose engine by matching circuit entanglement to the engine's efficiency regimen=53 Google Sycamore (2019); n≈50 routine on HPC clusters.exp(treewidth) markov_2008_simulating; verified in method-survey.md §7.5.α≈0.396 (Qassim–Pashayan–Gosset 2021) — verified in method-survey.md §7.5.Key reference: markov_2008_simulating — proves that classical simulation cost equals exp(treewidth) of the circuit's contraction graph; the foundational complexity result for TN-based circuit simulation.
| Axis | Value | Note |
|---|---|---|
| M1 tasks / outputs | Rigorous GS energy lower bound · two-sided energy bracket (with VMC upper bound) · certified bounds on local observables and correlations · Bell-inequality bounds · bootstrap phase-diagram constraints | Primary niche: certifying that a VMC/DMRG energy is near-optimal. |
| M2 regime | T=0 ground state (GS energy certification) | Finite-T moment relaxations possible but less developed. |
| M3 accuracy class | Certified LOWER bound on the GS energy (complement to variational UPPER bound) | Rigorous; tightness improves with relaxation level k. |
| M4 dimension fit (A1) | 1D and 2D feasible; 3D blows up | Moment matrix O(N^k) grows too large in 3D at practically useful k. |
| M5 statistics & local dim (A3) | Any algebra: Pauli (spin), fermion, boson | The noncommutative polynomial framework handles any algebra. |
| M6 entanglement regime (B5) | Sign-immune — not sampling-based; entanglement structure irrelevant to the SDP | The SDP sees only the algebraic constraints, not a wavefunction representation. |
| M7 sign-problem dependence (C12) | Sign-immune — works in C12 sign-ful / B8 frustrated regimes (no sampling, no sign problem); key niche | This is the critical advantage over QMC-based methods. |
| M8 symmetry exploitation (C9/C10) | C9 symmetry block-diagonalizes the moment matrix → large speedup; translation symmetry (C10) further reduces blocks | Symmetry is the primary route to tractability at useful k. |
| M9 time complexity | Relaxation level k → moment-matrix size ~ #monomials of degree ≤k = O(N^k); SDP solve is high-polynomial in that matrix size (interior-point). Structured sparsity (NCTSSOS, term/correlative sparsity) + symmetry (C9) shrink the blocks. | Tighter bound = higher k = rapidly growing SDP. |
| M10 memory | O(N^{2k}) (moment matrix + dual SDP variables) | Dominant bottleneck; structured sparsity reduces effective block sizes. |
| M11 control knob | Relaxation level k — higher k → tighter (larger) lower bound → larger SDP | Error: gap between lower and upper bound narrows with k. |
| M12 scale frontier | ~100 sites (1D) / ~10×10 (2D) at practically useful k with NCTSSOS sparsity + symmetry | k=2 tractable for moderate N; k=3 requires structured sparsity. |
| M13 primary approximation / bias | Relaxation level k — lower k gives looser (less tight) lower bound | Controlled: increasing k systematically tightens the bound. |
| M14 hard blocker / failure mode | 3D blows up (moment matrix too large); large k in 2D requires heavy HPC; gap between lower and upper bound may remain non-negligible at accessible k | Pair with VMC/DMRG for the upper bound to bracket the true GS energy. |
O(N^k) (interior-point SDP); structured sparsity (NCTSSOS) reduces effective blocksO(N^{2k}) for the moment matrix; structured sparsity critical for large systemsk — controls bound tightness (systematic improvement)k with symmetry + NCTSSOS sparsityk=2,3 tractablemethod-property-map.md: niche method for sign-immune certification; complements all variational methods<0.01% at k=2 with NCTSSOS sparsity wang_2024_certifying.10×10 Heisenberg: tight two-sided bracket combining SDP lower + VMC upper bounds wang_2024_certifying.method-survey.md §7.6.Key reference: wang_2024_certifying — demonstrates NCTSSOS-based certified lower bounds on GS energies of 1D and 2D Heisenberg models; the primary reference for the practical NCTSSOS + symmetry workflow.