The harness's verification-oracle layer: 63 models whose exact results — spectra, ground energies, gaps, degeneracies — serve as ground truth for checking ED, DMRG, QMC, and VMC runs. Open a row for the Hamiltonian, the solvability statement, and the benchmark values.
quadratic Hamiltonians — Jordan–Wigner, BdG, Bloch: one single-particle matrix yields the entire spectrum
T1: the chain maps to free fermions via a Jordan–Wigner transformation, giving a quadratic (Bogoliubov-diagonalizable) fermion Hamiltonian. Everything reported here — the full spectrum (via the single-particle dispersion ε(k)), the ground-state energy at finite L and in the thermodynamic limit, the gap, and the transverse magnetization — is exact for any L, J, h. The model is exactly solvable in its entirety — there is no approximation anywhere. Not exact: nothing; everything about this model is exact. Some exact quantities are simply not implemented in oracle.py (out of this card's scope, ground-state statics only): longitudinal spin-spin correlators ⟨σ^z_i σ^z_j⟩ (exactly known as Toeplitz determinants of the JW correlation matrix, requiring determinant/Pfaffian machinery), and finite-temperature and non-equilibrium quench quantities (also exactly computable from the free-fermion solution).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | h = J = 1 (critical) | -4/π ≈ -1.27324 | Pfeuty 1970 |
gap_single_fermion | h = 1.5, J = 1 | 1 | Pfeuty 1970 |
m (elliptic-integral parameter, m = k²) | definition | m = 4Jh/(J+h)² | Pfeuty 1970 |
T1: the Jordan–Wigner transformation c_i = (∏_{j<i} σ^z_j)\,σ^-_i maps the chain to free fermions with pairing, i.e. a quadratic (Bogoliubov-diagonalizable) Hamiltonian. Everything reported here — the full single-particle spectrum via ε(k), the ground-state energy per site at finite L and in the thermodynamic limit, and the gap — is exact for any L, γ, h. The model is exactly solvable in its entirety; there is no approximation. Not exact: nothing about this model is approximate. Exact quantities not implemented here (out of this card's ground-state-statics scope): the Barouch–McCoy spin–spin correlators ⟨S^x_i S^x_j⟩, ⟨S^y_i S^y_j⟩, ⟨S^z_i S^z_j⟩ (Toeplitz-determinant / Pfaffian expressions of the free-fermion correlation matrix) and the transverse magnetization, plus all finite-T and quench quantities (also exactly computable from the same free-fermion solution).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site (thermo) | γ=0, h=0 (XX) | -1/π ≈ -0.318310 | Lieb–Schultz–Mattis 1961 |
e0_per_site (L=8, ED) | γ=0, h=0 | -0.3266407412 (sector ABC) | self-test |
e0_per_site (L=8, ED) | γ=0.7, h=0.3 | -0.4405791876 (sector PBC) | self-test |
e0_per_site (L=8, ED) | γ=1.0, h=0.5 | -0.5318176397 (sector ABC) | self-test |
e0_per_site (thermo) | γ=1, J=h=1 (Ising critical) | -2/π ≈ -0.636620 | TFIM map Lieb–Schultz–Mattis 1961 |
T1: the Hamiltonian is quadratic in the fermions and is diagonalized exactly by a Bogoliubov–de Gennes (BdG) transformation. Everything reported here — the full single-particle spectrum (via ε(k)), the ground-state energy per site, the bulk gap, and the topological flag — is exact for any L, μ, t, Δ. The model is exactly solvable in its entirety; there is no approximation. Not exact: nothing about this model is approximate. Some exact quantities are simply out of this card's scope (bulk-statics only): the OBC Majorana end-mode wavefunctions and the exponentially small even/odd parity splitting ∝ e^{−L/ξ} (exactly computable from the real-space Nambu matrix), the bulk Z₂ invariant, and quench/braiding dynamics (all exactly tractable because the model stays quadratic).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site | μ=0, t=Δ=1 (sweet spot) | -1 | Kitaev 2001 |
gap (flat band) | μ=0, t=Δ=1 | 2 (= 2t) | Kitaev 2001 |
gap | μ=2t (transition) | 0 | Kitaev 2001 |
topological | |μ| < 2t / |μ| > 2t | True / False | Kitaev 2001 |
T1: the Hamiltonian remains quadratic in the fermions for any pairing range, so it is diagonalized exactly by a Bogoliubov–de Gennes transformation. The ground-state energy per site and the BdG single-particle spectrum (hence the gap) reported here are exact for any L, μ, t, Δ, α. Not exact: nothing about this model is approximate. What is out of this card's scope (bulk spectrum/energy only): the edge physics that makes the long-range model interesting — for α < 1 the model hosts massive Dirac-like edge modes and violates the entanglement area law (logarithmic / super-area-law growth), and the pairing-induced correlations decay algebraically rather than exponentially; these are exactly studied from the same BdG solution but are not computed by the script Vodola 2014.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site | α=30, μ=0.5, t=Δ=1, L=64 | -1.0156870128 (= NN Kitaev) | Kitaev 2001 |
e0_per_site (L=8, ED) | α=1.5, μ=0.5, t=Δ=1 | -1.0128951194 | self-test |
e0_per_site | α=1.5, μ=0, t=Δ=1, L=64 | -1.0301797662 | this card |
T1: the Hamiltonian is quadratic (bilinear, number-conserving) in the fermions, diagonalized exactly by Fourier transform (PBC, bulk bands + winding number) or by direct real-space diagonalization of the 2L×2L OBC hopping matrix (edge modes). Everything reported here — the bulk dispersion, the winding invariant, the bulk gap, the OBC edge-mode count, the edge-mode decay length, and the filled-band (half-filling) ground-state energy — is exact for any L, v, w. The model is exactly solvable in its entirety; there is no approximation anywhere. Not exact: nothing about this model is approximate. Some exact quantities are simply out of this card's scope (ground-state statics only): the exponentially small even/odd finite-size splitting of the two near-zero OBC edge modes, the Zak/Berry phase as a continuous quantity (only the quantized winding number is reported), single-particle correlation functions, and quench/dynamics (all exactly tractable because the model stays quadratic).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
winding | v=0.5, w=1.0 | 1 (topological) | SSH 1979 |
winding | v=1.0, w=0.5 | 0 (trivial) | SSH 1979 |
gap | v=0.5, w=1.0 | 1 (= 2|v-w|) | SSH 1979 |
n_edge_modes_obc | v=0.5, w=1.0, OBC | 2 | SSH 1979 |
n_edge_modes_obc | v=1.0, w=0.5, OBC | 0 | SSH 1979 |
edge_decay_length | v=0.5, w=1.0 | 1/\ln 2 \approx 1.4427 | SSH 1979 |
T1: Kitaev's four-Majorana representation σ^a_j = i\,b^a_j c_j (with the on-site gauge constraint b^x b^y b^z c = 1) turns each bond operator \hat u_{jk} = i\,b^a_j b^a_k into a conserved Z₂ gauge field, and every plaquette flux W_p = \prod \hat u is a constant of motion. In each fixed flux sector the Hamiltonian is a quadratic (free) Majorana hopping of the single c-Majorana per site — σ^a_j σ^a_k = -i\,\hat u_{jk} c_j c_k, so H = i\sum_{\langle jk\rangle,\,j\in A} J_a \hat u_{jk} c_j c_k — diagonalized exactly by Fourier transform, giving ε(k) = 2|f(k)|. By Lieb's theorem the ground state lies in the flux-free sector (W_p = +1 on every plaquette), realized by the gauge \hat u_{jk} = +1; the quantities this card reports (e0_per_site, gap, phase) are the exact flux-free-sector values.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site | isotropic J_x=J_y=J_z=1 (Pauli convention) | −0.787299 (= −½⟨|f|⟩_BZ, ⟨|f|⟩ ≈ 1.574597) | Kitaev 2006 |
gap | isotropic (B phase, Dirac point) | 0 | Kitaev 2006 |
e0_per_site | decoupled dimers J_x=J_y=0, J_z=1 | −1/2 | derived above |
gap | A phase J_z=2.5, J_x=J_y=1 | 1 (= 2(J_z−J_x−J_y)) | derived above, Kitaev 2006 |
T1: the Hamiltonian is quadratic in the fermions and translationally invariant, block-diagonalized exactly by Fourier transform into the 2×2 BdG matrix above and diagonalized exactly for every k. Everything reported here — the first Chern number of the filled BdG band (Fukui–Hatsugai–Suzuki lattice-gauge invariant, _lib.topology.chern) and the minimum band gap over the BZ — follows from the exact Bloch Hamiltonian with no approximation. Not exact: nothing about this model is approximate. The gap is obtained by a BZ grid minimum that converges to (and, because the closing momenta (0,0),(0,π),(π,0),(π,π) lie exactly on an even grid, reaches) the exact value rather than a closed-form substitution. Out of this card's scope entirely (still exact from the same quadratic solution, just not implemented): the real-space chiral Majorana edge mode and its dispersion under OBC, the BdG quasiparticle wavefunctions and the half-quantum-vortex Majorana zero mode, and finite-T/dynamical quantities.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
chern | μ=−2, t=1, Δ=0.5 (weak pairing) | ±1 (+1 this convention) | Read–Green 2000 |
chern | μ=+2, t=1, Δ=0.5 (weak pairing) | ∓1 (−1 this convention) — opposite sign to μ=−2 | Read–Green 2000 |
chern | μ=−5, t=1, Δ=0.5 (strong pairing) | 0 | Read–Green 2000 |
gap | μ ∈ {−4, 0, +4}, t=1, Δ=0.5 | 0 (critical) | derived above |
T1: each lattice's Hamiltonian is quadratic and translationally invariant, block-diagonalized exactly by Fourier transform into an n×n Bloch matrix per crystal momentum (n = sites per unit cell: 1, 2, 3, 3). Everything reported here — the full band structure (via numerical diagonalization of the Bloch matrix on a fine u,v grid), the bandwidth, the flat-band energy where one exists, the honeycomb Dirac-point gap (evaluated exactly at the analytic Dirac point, not by grid search), and a grid-histogram estimate of the DOS peak — is obtained from the exact Bloch Hamiltonian; the only numerical ingredients are the (exact, textbook) matrix diagonalization and, for dos_van_hove only, a finite-resolution energy histogram (see below). The model is exactly solvable in its entirety; there is no approximation in the Hamiltonian or its diagonalization. Not exact: nothing about the band structure itself is approximate. dos_van_hove specifically is computed as the bin-center of the highest-count bin of a 400-bin histogram over the sampled grid's eigenvalues — a numerical estimate whose value depends on grid/bin resolution (stated explicitly here, not a hidden imprecision: the underlying spectrum is exact, only the *peak-finding* is a numerical convenience). For the square lattice this estimate converges to the exactly-derivable value E=0 (see below); for the flat-band lattices (kagome, Lieb) the histogram peak is dominated by the flat band itself, which is correct — a flat band is an even stronger DOS singularity than a van Hove point — but is a different phenomenon than a van Hove saddle-point divergence. Out of this card's scope entirely (not attempted): real-space flat-band eigenstate wavefunctions (see below for the one-line description), correlation functions, and interacting extensions.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
bandwidth | square, t=1 | 8 | derived above |
bandwidth | honeycomb, t=1 | 6 (= 2 \times 3t) | derived above |
dirac_point_gap | honeycomb, at K | 0 | derived above |
flat_band_energy | kagome, t=1 | 2 | Bergman 2008 |
flat_band_energy | Lieb, t=1 | 0 | Bergman 2008 |
bandwidth | Lieb, t=1 | 4\sqrt2 \approx 5.657 | derived above |
T1: at rational flux α = p/q the Peierls-phase Hamiltonian is exactly block-diagonalized by Fourier transform into the q×q Harper matrix above, diagonalized exactly by numpy.linalg.eigvalsh (textbook, exact matrix diagonalization — the only "numerical" step, with no approximation in the physics). Everything reported here — the full q-subband spectrum (band_edges), the per-band Chern numbers (Fukui–Hatsugai–Suzuki, _lib.topology.chern, cumulative-occupied differencing C_n = \text{chern}(H, n+1) - \text{chern}(H, n)), and the joint Chern number of any exactly-touching subband group — is exact for any coprime (p,q). The model is exactly solvable in its entirety at every rational flux. Not exact: nothing about this model is approximate. Two quantities are obtained via numerical routines that converge to the exact value rather than a closed-form substitution: the Fukui–Hatsugai–Suzuki Chern number is a lattice-gauge discretization of the exact Berry-curvature integral that returns the exact integer once nk is fine enough to avoid landing exactly on a degeneracy (it is not a continuum-limit approximation — any sufficiently fine grid gives the exact topological invariant, away from an actual gap closing), and band_edges is a finite-grid min/max scan of the exact eigenvalues that converges to the true extremum as nk_bands → ∞ (no closed form is derived for it here, though one exists band-by-band). Out of this card's scope entirely (not attempted): irrational α (Cantor-set spectrum, not tractable by a finite Harper matrix), strip/ribbon edge-mode spectra, and the Hofstadter–Hubbard interacting extension (sign-problem-ful).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
chern_numbers | p=1, q=3 | [1, -2, 1] | Hofstadter 1976 (TKNN Diophantine labeling) |
n_bands | p=1, q=3 | 3 (=q) | Hofstadter 1976 |
band_touching_groups | p=1, q=4 | [{bands:[1,2], joint_chern:-2}] | derived above (numerical gap search + Diophantine cross-check) |
band_touching_groups | p=1, q=6 | [{bands:[2,3], joint_chern:-4}] (touching at (k_x,k_y)=(\pi/6,\pi)) | derived above (numerical gap search + Diophantine cross-check) |
sum(chern_numbers) | any coprime p,q | 0 (telescoping, C_q - C_0 = 0-0) | derived above |
T1: the Hamiltonian is quadratic and translationally invariant, block-diagonalized exactly by Fourier transform into the 2×2 Bloch matrix above, diagonalized exactly by numpy.linalg.eigvalsh for any k. Everything reported here — the Chern number of the filled lower band (Fukui–Hatsugai–Suzuki lattice-gauge invariant, _lib.topology.chern) and the closed-form phase boundary M = \pm3\sqrt3\,t_2\sin\phi (derived below from the exact Dirac-point locations) — follows from the exact Bloch Hamiltonian with no uncontrolled approximation. Not exact: nothing about this model is approximate. One quantity is obtained via a numerical routine that converges to the exact value rather than a closed-form substitution: gap (minimum direct gap over the BZ) is found by a finite grid scan (nk_gap=200) followed by a local Nelder–Mead refinement, converging to the true minimum but not evaluated from a closed form for generic M — although the *location* of that minimum (the Dirac point K or K') is known in closed form and is what the phase_boundary_M closed-form result below is derived from directly (so the phase boundary itself is exact even though the generic-M gap value is found numerically). Out of this card's scope entirely (not attempted): the real-space ribbon/edge-mode spectrum, the Hall conductance as a transport (Kubo) calculation (equal to chern by TKNN, not separately computed), and the interacting Haldane–Hubbard extension (sign-problem-ful).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
phase_boundary_M | t_2=1/3, φ=π/2 | \sqrt3 ≈ 1.7321 (=3\sqrt3\cdot\frac13\cdot1) | derived above, Haldane 1988 |
chern | M=0, t_2=1/3, φ=π/2 | -1 | derived above |
chern | M=2.0, t_2=1/3, φ=π/2 | 0 | derived above (2.0 > 1.7321) |
gap | M=phase_boundary_M, t_2=1/3, φ=π/2 | 0 (closes at K) | derived above |
T1: in 1D, the tight-binding eigenvalue equation at energy E is equivalent to an SL(2,ℝ) transfer-matrix recursion (ψ_{i+1}, ψ_i)^T = T_i (ψ_i, ψ_{i-1})^T with T_i = [[E - ε_i, -1], [1, 0]], det T_i = 1. By the Furstenberg theorem, the ordered product T_N ⋯ T_1 grows almost surely as e^{γN} for a disorder-realization-independent (self-averaging) Lyapunov exponent γ(E, W) > 0 whenever the disorder has a continuous, non-degenerate distribution — which the box distribution is. γ is the inverse localization length: ξ_loc = 1/γ. This script estimates γ by the standard QR (Benettin) algorithm on n_steps = 10^6 transfer-matrix multiplications, re-orthogonalizing every 10 steps to avoid over/underflow; this is a numerically exact, non-perturbative estimator of a quantity that is *itself* exact in the N → ∞ limit (finite n_steps gives a statistical estimate with O(1/√n_steps) sampling error, not a systematic bias, since disorder is self-averaging along the chain — no ensemble averaging over independent realizations is needed for a single long chain). Not exact: the *closed-form* thouless_perturbative = W²/(96(1-(E/2)²)) reported alongside lyapunov is a second-order (Born) weak-disorder approximation Thouless 1972 — valid only for W ≪ 1 (in units of the 4t = 4 bandwidth) and *not* valid near the band edges E → ±2 (where it diverges, 1-(E/2)² → 0) or at the exact band center E = 0 (band-center anomaly, below). lyapunov (the transfer-matrix output) is the exact quantity; thouless_perturbative is a cross-check formula, not a substitute.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
lyapunov | E=0.5, W=1.0, n_steps=10^6, seed=1 | ≈ 0.0111 (within 0.7–1.4× of thouless_perturbative) | self-test |
thouless_perturbative | E=0.5, W=1.0 | 1/90 ≈ 0.01111 | Thouless 1972 |
γ(E=0, W) (band-center anomaly) | W → 0 | ≈ W²/105.2, not W²/96 | Kappus–Wegner 1981 |
Localization at any W>0, 1D | all E, W | every eigenstate exponentially localized | Ashcroft 1976 |
T1: H is already quadratic in (x_i, p_i) — a normal-mode (Fourier) transformation diagonalizes it exactly for any L, κ, m, giving a set of L independent harmonic oscillators of frequency ω(k_n) = 2\sqrt{κ/m}\,|\sin(k_n/2)|, k_n = 2πn/L. Everything reported here — the finite-L zero-point energy, its thermodynamic limit, and the sound speed — is exact. The model is exactly solvable in its entirety; there is no approximation anywhere. Not exact: nothing about this model is approximate. Exact quantities not implemented here (out of this card's ground-state-statics scope): finite-temperature phonon occupation ⟨n_k⟩ = 1/(e^{ω(k)/T}-1) and the associated thermal energy/specific heat, real-space displacement–displacement correlators ⟨x_i x_j⟩ (a closed-form Bessel-function-type sum over k), and the classical-statistical-mechanics partition function (also exactly Gaussian) — all straightforwardly exact from the same normal-mode solution.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | κ=m=1 | 2/π ≈ 0.636620 | normal-mode sum |
sound_speed | κ=m=1 | 1 | dispersion slope |
| Dynamical-matrix spectrum | L=6, κ=m=1 | {0, 1, 1, 3, 3, 4} (= ω²) | eigenvalue check |
sound_speed | κ=2, m=0.5 | 2 | dispersion slope |
T1: the Bogoliubov-truncated Hamiltonian is quadratic in the bosonic fluctuation operators and is diagonalized exactly (for any g, n, m) by a k-space Bogoliubov transformation, giving the excitation spectrum ε(k) = \sqrt{ε_k(ε_k+2gn)}. Everything reported here — the dispersion, its k→0 phonon slope (sound speed), the healing length, and the momentum-integrated quantum depletion — is an exact closed-form (or exactly-convergent-integral) property of the Bogoliubov Hamiltonian itself: tier A *as a statement about that quadratic theory*. Not exact: the Bogoliubov Hamiltonian is not the true interacting-boson Hamiltonian — it is the leading-order term of an expansion in the diluteness parameter na³ ≪ 1 (dropping cubic-and-higher fluctuation terms, i.e. exchange/beyond-mean-field corrections to the depletion and the ground-state energy, captured at the next order by the Lee–Huang–Yang correction). As a statement about a real interacting Bose gas (e.g. liquid ⁴He, where na³ ~ O(1), or even a dilute ultracold gas at the percent-level depletion corrections), Bogoliubov theory is only tier D — exact in the dilute/weak-coupling limit na³ → 0, not for the true many-body ground state at finite coupling. The quantity depletion_3d itself is a direct diagnostic of how good the approximation is: it *is* (to leading order) the fraction of atoms depleted from the condensate by interactions, and the theory is self-consistent only where this fraction is small.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
sound_speed_closed | g=0.1, n=1, m=1 | √0.1 ≈ 0.316228 | Bogoliubov 1947 |
healing_length | g=0.1, n=1, m=1 | 1/√0.2 ≈ 2.236068 | Bogoliubov 1947 |
depletion_3d_closed | g=0.1, n=1, m=1 (a ≈ 7.96×10⁻³, na³ ≈ 5.04×10⁻⁷) | ≈ 1.068×10⁻³ | Bogoliubov 1947 |
| Validity regime | — | closed forms trustworthy for na³ ≲ 10⁻³; He-4-like na³ ~ O(1) is outside the theory's regime | tier statement above |
classical partition functions in 2D — Onsager transfer matrices, Kasteleyn Pfaffians, vertex weights
T2 (2D classical / transfer matrix): the model is solved by diagonalizing the row-to-row transfer matrix. Onsager (1944) obtained the thermodynamic-limit free energy; Kaufman (1949) recast the transfer matrix as a product of anticommuting spinor rotations, giving the exact partition function of any finite $m\times n$ torus as a sum of four products over Bogoliubov angles $\gamma_r$. Exactly known and implemented here: the critical temperature, the thermodynamic-limit free energy (double integral), internal energy (elliptic-$K$ closed form), the spontaneous magnetization for $T<T_c$ (Yang 1952), and the finite-torus $\ln Z$ (Kaufman). Not exact: nothing — everything about this model is exact. Some exact quantities are simply not implemented in oracle.py (out of this card's scope): the two-spin correlation functions $\langle s_0 s_r\rangle$ (exactly known as Toeplitz determinants — the diagonal correlator is a closed Toeplitz form) and the finite-torus magnetization. The magnetization formula below is a genuine closed form valid only in the ordered phase $T<T_c$; it does not describe $T\ge T_c$ (where $m=0$ identically) and is not an approximation to anything above $T_c$.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
tc | $J=1$ | $2/\ln(1+\sqrt2) \approx 2.2691853$ | Onsager 1944 |
internal_energy | $T=T_c$ | $-\sqrt2 \approx -1.4142136$ (per site, $J=1$) | Onsager 1944 |
magnetization | $T=2$, $J=1$ | $(1-\sinh^{-4}1)^{1/8} = 0.9113194$ | Onsager 1944 |
T2 (2D classical / transfer matrix): the model is solved by diagonalizing the row-to-row transfer matrix. Exactly known and implemented here: the FM critical temperature (closed form), the AFM $T=0$ residual entropy (Wannier's exact integral, in closed form via the Clausen function), and the exact finite $L\times W$-torus $\ln Z$ for both signs of $J$ from the transfer matrix, ground-truthed against brute-force enumeration. Not exact: nothing — the triangular Ising model is exactly solvable in full (Wannier 1950; the free energy, specific heat, and — for the FM — critical behaviour are all exactly known). Some exact quantities are simply not implemented in oracle.py (out of this card's scope): the full thermodynamic-limit free energy $f(T)$ (the finite-width transfer matrix gives only a width-limited strip value, documented below as such), the AFM spin-spin correlations (algebraically decaying, exactly known Wannier 1950), and the FM critical exponents. The AFM residual entropy is a genuine $T=0$ ground-state entropy, not an approximation to a finite-$T$ quantity.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
tc_ferro | $J=1$ | $4/\ln 3 \approx 3.6409569$ | Wannier 1950 |
residual_entropy_afm | $J<0$, $T\to 0$ | $\mathrm{Cl}_2(\pi/3)/\pi \approx 0.3230659$ | Wannier 1950 |
logZ_finite | $4\times4$ torus, $\beta=0.8$, $J=1$ | $39.0942362$ (= enumeration) | this work |
T2 (2D classical / transfer matrix), by the Kasteleyn Pfaffian method. Kasteleyn (1961) showed that the perfect matchings of a planar lattice are counted by the Pfaffian of a suitably oriented (signed) adjacency matrix: choosing edge orientations so that every elementary face is "clockwise-odd" makes every matching contribute $+1$ to the Pfaffian, so $Z = |\mathrm{Pf}(A)| = \sqrt{|\det A|}$. Diagonalizing $A$ for the open $m\times n$ grid gives the eigenvalues $\pm 2i\cos\frac{j\pi}{m+1} \pm 2\cos\frac{k\pi}{n+1}$ and hence the closed product form below. Everything reported — the exact integer count for any finite board and the thermodynamic-limit entropy — is exact. Not exact: nothing about the *open-boundary* count; it is exact and closed-form. What is out of scope (exactly solvable, but not implemented here): the torus (periodic) dimer count, which on a genus-1 surface is not a single Pfaffian but a signed combination of four Pfaffians (the four spin structures) — exactly known, but a different formula from the one shipped; and weighted/monomer-doped variants. The card is honest about this: the product formula here is for open boundaries only.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
| $n_\text{coverings}$ | $2\times2$ | $2$ | Kasteleyn 1961 |
| $n_\text{coverings}$ | $2\times3$ | $3$ | Kasteleyn 1961 |
| $n_\text{coverings}$ | $4\times4$ | $36$ | Kasteleyn 1961 |
| $n_\text{coverings}$ | $8\times8$ open | $12\,988\,816$ | Kasteleyn 1961 |
T2 (2D classical / transfer matrix): the model is solved by diagonalizing the row-to-row transfer matrix, whose eigenvectors are the Bethe-ansatz states of the associated XXZ chain (Lieb 1967; Sutherland; Lieb–Wu). Lieb obtained the thermodynamic-limit free energy in every regime; at the ice point it gives the residual entropy of square ice $\ln W$, $W=(4/3)^{3/2}$. Exactly known and implemented here: Lieb's square-ice constant (algebraic closed form), the finite $N\times M$ torus partition function via the transfer matrix (cross-checked against brute-force enumeration — the ground truth), and the disordered-regime free-energy integral. Not exact: nothing — the six-vertex model is fully integrable and every quantity here is exact. Some exact quantities are simply out of this card's scope (not implemented): the ordered-regime (ferroelectric $\Delta>1$, antiferroelectric $\Delta<-1$) free energies, the correlation-length exponents, and the arctic-curve / domain-wall boundary results. The convergence anchor $\Lambda_{\max}(N)^{1/N}\to W$ is genuinely slow (power-law in $1/N$); the exact-count anchor is the real gate, and the card says so.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
lieb_constant | ice point $a=b=c$ | $(4/3)^{3/2}=1.5396007$ | Lieb 1967 |
entropy_per_vertex | ice point | $\tfrac32\ln\tfrac43 = 0.4315231$ | Lieb 1967 |
logZ_torus | $4\times4$, $a=b=c=1$ | $\ln 2970 = 7.9963172$ | Lieb 1967 |
free_energy_disordered | ice point | $=\ln W = 0.4315231$ | Lieb 1967 |
T2 (2D classical / transfer matrix): Baxter (1971) solved the zero-field eight-vertex model by the star–triangle / Yang–Baxter equation, showing that transfer matrices with common $\Delta,\Gamma$ commute; the free energy follows from an elliptic-function parametrization and has a continuously varying critical exponent — the archetype of non-universal criticality. Not exact: nothing — the model is exactly solvable in its entirety; nothing about it is approximate.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
free_energy_free_fermion | $a=b=1,c=\sqrt2,d=0$ | $=$ six-vertex $\Delta{=}0$ value $0.5831218$ | Fan–Wu 1970 |
free_energy_free_fermion | $a=b=1,c=1.2,d=\sqrt{2-1.44}$ | $0.6803218$ | Fan–Wu 1970 |
Delta | free-fermion manifold | $0$ | Baxter 1971 |
logZ_torus | $4\times4$, $a{=}b{=}c{=}1,d{=}0$ | $\ln 2970$ (six-vertex ice point) | Baxter 1971 |
T2 (2D classical / transfer matrix): Baxter (1980) solved the hard-hexagon model exactly using the corner-transfer-matrix (CTM) method — the "miracle" being that the sublattice densities and the free energy come out as Rogers–Ramanujan functions, and the critical activity is the algebraic number $z_c=(11+5\sqrt5)/2=\varphi^5$. Not exact: nothing — the model is exactly solvable in its entirety; nothing about it is approximate.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
zc | — | $(11+5\sqrt5)/2=\varphi^5=11.0901699$ | Baxter 1980 |
logZ_torus | $4\times4$, $z=1$ | $\ln 201 = 5.3033049$ (independent-set count) | Baxter 1980 |
density_width6 | $z\to0$ | $\rho\to z-7z^2+\cdots$ | Baxter 1980 |
| exponents (tabulated) | at $z_c$ | $\alpha=1/3,\ \beta=1/9$ (3-state Potts) | Baxter 1980 |
interacting but integrable — Bethe rapidities and Yang–Baxter give exact energies, gaps, and thermodynamics
T3 (Bethe ansatz / Yang–Baxter): the chain is integrable by Bethe's 1931 coordinate ansatz Bethe 1931. Working in the S^z sectors above the ferromagnetic reference |↑↑…↑⟩ (energy N/4), an M-magnon eigenstate is a superposition of M down-spins whose amplitudes are plane waves with two-body scattering phases fixed by the Yang–Baxter/Bethe equations. Each eigenstate is labelled by a set of rapidities {λ_j} solving the Bethe equations; the antiferromagnetic ground state sits in the S^z=0 sector (M=N/2) and is the "1-string" solution — all rapidities are real (no bound states/strings, which would raise the energy). The exact finite-N ground energy (from the real-root solver), the thermodynamic-limit energy 1/4 − ln 2, and the spinon velocity π/2 are all reported here. Not exact in closed form (Tier B, not A): the full spectrum and dynamical/finite-T quantities require solving the Bethe equations state-by-state or the thermodynamic Bethe ansatz — integrable but not a single closed form; correlation functions are known only through much heavier (determinant / vertex-operator) machinery, out of this card's scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | J=1, N→∞ | 1/4 − ln 2 ≈ −0.4431472 | Hulthen 1938 |
e0_per_site_finite | J=1, N=12 | −0.4489492 (converges to 1/4 − ln 2) | Bethe 1931 |
spinon_velocity | J=1 | π/2 ≈ 1.5707963 | des–Cloizeaux–Pearson 1962 |
T3 (Bethe ansatz / Yang–Baxter): the XXZ chain is Bethe-ansatz integrable for every anisotropy Δ — Yang & Yang proved Bethe's hypothesis for the ground state and worked out the thermodynamic ground-state energy Yang–Yang 1966, Yang–Yang 1966b. The U(1) symmetry (S^z_{tot} conserved) organises the ansatz exactly as for the isotropic chain, but the two-magnon scattering phase now carries the anisotropy through Δ = \cos γ (|Δ|<1) or Δ = \cosh η (Δ>1). The ground state is again the packed all-real-root solution in the S^z=0 sector; in the thermodynamic limit the root density solves a linear integral equation whose kernel depends on γ/η, and the ground energy per site becomes a closed-form integral (|Δ|<1) or a rapidly convergent series (Δ>1). Reported here: the thermodynamic-limit e0(Δ) across all three regimes, and the spin gap for Δ>1. Not exact in closed form (Tier B): the full excitation spectrum, correlation functions, and finite-T thermodynamics need state-by-state Bethe roots / TBA — integrable but not single closed forms; out of this card's scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site | Δ=0 (XX) | −1/π ≈ −0.3183099 | Yang–Yang 1966b |
e0_per_site | Δ=1 (isotropic) | 1/4 − ln 2 ≈ −0.4431472 | Hulthen 1938 |
e0_per_site | Δ=2 | −0.6172220 | Yang–Yang 1966b |
gap | Δ=2 | 0.3898023 | des–Cloizeaux–Gaudin 1966 |
T3 (Yang–Baxter): the XYZ chain is the row-to-row transfer-matrix Hamiltonian of Baxter's eight-vertex model; the eight-vertex R-matrix satisfies the Yang–Baxter equation with an elliptic parametrization of the spectral parameter, and Baxter's solution gives the exact thermodynamic ground-state energy per site as a closed-form elliptic-function expression Baxter 1972. The model is Bethe-ansatz integrable for every (J_x, J_y, J_z). Not exact in closed form (Tier B): the full spectrum, correlation functions, and finite-T thermodynamics need the (elliptic) Bethe equations / TBA state-by-state — integrable but not single closed forms; out of scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site (exact, XXZ line) | (1,1,0.5) | -0.3750000 | Yang–Yang 1966b via xxz-chain |
e0_per_site (exact, XXZ line) | (1,1,2.0) | -0.6172220 | Yang–Yang 1966b via xxz-chain |
e0_per_site (exact, XY line) | (1,1,0) (XX) | -1/π ≈ -0.3183099 | Lieb–Schultz–Mattis 1961 via xy-chain |
e0_per_site (exact, XY line) | (1.4,0.6,0) | -0.3662651 | Lieb–Schultz–Mattis 1961 via xy-chain |
e0_per_site (ED-extrap, generic) | (1,0.7,0.4) | -0.3199262 (non-closed-form; L=10 ED -0.3253344) | this card; exact form Baxter 1972 |
e0_per_site (ED-extrap, generic) | (1,0.8,0.6) | -0.3580140 (non-closed-form; L=10 ED -0.3647016) | this card; exact form Baxter 1972 |
T3 (Bethe ansatz / Yang–Baxter): the spin-1 R-matrix obtained by fusion of two spin-½ (six-vertex) R-matrices satisfies the Yang–Baxter equation, making this chain integrable; the transfer matrix generates a commuting family and the spectrum follows from a nested/higher-spin Bethe ansatz Zamolodchikov–Fateev 1980, Takhtajan 1982, Babujian 1983. The model is gapless and critical: its low-energy theory is the SU(2)_2 Wess–Zumino–Witten conformal field theory, central charge c = 3S/(S+1) = 3/2 at S=1; elementary excitations are a gapless spin-½ spinon doublet. Not exact in closed form (Tier B): as with the other integrable chains in this catalog, the full spectrum and dynamical/finite-T quantities require solving the Bethe equations state-by-state or the TBA — integrable but not single closed forms; correlation functions need heavier machinery. Out of this card's scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
E0 (total, ED) | N=8, PBC | -32.6427397329 | finite-L ED reference, this card |
e0 = E0/N (ED) | N=8, PBC | -4.0803424666 | finite-L ED reference, this card |
e0 = E0/N (ED) | N=6, PBC | -4.1462349785 | finite-L ED reference, this card |
central charge c | thermodynamic | 3/2 (SU(2)_2 WZW) | Takhtajan 1982, Babujian 1983 |
e0 (thermodynamic) | N→∞ | *no web-verified numeric value quoted here* | Babujian 1983 |
T3 (Bethe ansatz / Yang–Baxter, Yangian): the 1/r² chord exchange is the point where the antiferromagnetic ground state is known exactly and in closed form. Haldane Haldane–HS 1988 and Shastry Shastry 1988 showed independently (1988) that the singlet ground state is a Jastrow–Gutzwiller wavefunction — the U→∞ (Gutzwiller-projected) filled Fermi sea, i.e. Anderson's resonating-valence-bond state — with an exactly evaluable energy. The finite-N ground energy is the closed form E0(N) = −J(π²/24)(N + 5/N); per site e0(N) = −(π²/24)(1 + 5/N²) → −π²/24 as N→∞. The full spectrum organises into "supermultiplet" degeneracies, observed in Haldane–HS 1988 as the signature of a hidden continuous symmetry and later identified as a Y(su(2)) Yangian present already at finite N (Haldane–Ha–Talstra–Bernard–Pasquier 1992, standard); the elementary excitations are spinons obeying semionic (half-)statistics and are non-interacting (free) in this model — the 1/r² chain is the archetypal "ideal spinon gas" (Haldane 1991, standard). Tier B, not A: although the ground state and static structure factor are closed-form, the complete finite-T/dynamical solution still goes through the Yangian/asymptotic-Bethe machinery, so we treat it as integrable rather than fully closed.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | J=1, N→∞ | −π²/24 ≈ −0.4112335 | Haldane–HS 1988 |
e0_per_site | J=1, N=12 | −0.4255125 (= −(π²/24)(1+5/144)) | Haldane–HS 1988 |
E0 (total, closed form = ED) | J=1, N=10 | −4.3179519255 (= −(π²/24)(10+1/2)) | Haldane–HS 1988, Shastry 1988 |
E0 (total, closed form = ED) | J=1, N=8 | −3.5468890816 | Haldane–HS 1988, Shastry 1988 |
T3 (Bethe ansatz / Yang–Baxter, elliptic): the Inozemtsev chain is integrable at every κ — it possesses a full tower of commuting conserved charges, and its spectrum is accessible through a (coordinate/asymptotic) Bethe ansatz with elliptic two-body data Inozemtsev 1990. It is the spin analogue of the elliptic Calogero–Moser–Sutherland system, and the two limits inherit the closed-form solvability of their endpoints (Bethe ansatz at heisenberg-xxx, Yangian/Jastrow at haldane-shastry). Tier B, not A: integrable but not a single closed form — at generic κ there is no elementary expression for the ground energy; the finite-N/thermodynamic energetics require solving the elliptic Bethe equations state-by-state, and correlation functions need heavier machinery. Out of this card's scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
E0 (total, ED) | N=8, κ=1, hyperbolic, PBC | −3.5083521420 | finite-L ED reference, this card |
e0 = E0/N (ED) | N=8, κ=1, hyperbolic, PBC | −0.4385440178 | finite-L ED reference, this card |
e0 (limit κ→∞) | N→∞ | 1/4 − ln 2 ≈ −0.4431472 (XXX) | Inozemtsev 1990 → heisenberg-xxx |
e0 (limit κ→0) | N→∞ | −π²/24 ≈ −0.4112335 (Haldane–Shastry) | Inozemtsev 1990 → haldane-shastry |
T3 (Bethe ansatz): the one-dimensional Hubbard model is integrable by the nested Bethe ansatz — Lieb & Wu (1968) reduced the eigenvalue problem to a coupled set of equations for charge momenta k_j and spin rapidities λ_α Lieb–Wu 1968. At half filling the ground-state charge distribution fills the whole Brillouin zone and the spin rapidities condense onto the real line; the coupled equations decouple by Fourier transform into a single closed-form integral for the energy per site,
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | U=0 (free) | -4/π ≈ -1.2732395 | Lieb–Wu 1968 |
e0_thermodynamic | U=4 | -0.5737294 | Lieb–Wu 1968 |
e0_thermodynamic | U=8 | -0.3275305 | Lieb–Wu 1968 |
mott_gap | U=4 | 1.2867270 | Ovchinnikov 1970 |
mott_gap | U=8 | 4.6795171 | Ovchinnikov 1970 |
e0_strong_coupling_asymptote | U→∞ | -4 ln2/U (ratio e0/asymptote → 1) | Lieb–Wu 1968 |
T3 (Bethe ansatz): at J = 2t the t-J chain is Bethe-ansatz integrable and carries a graded su(2|1) supersymmetry — the three local states {∅, ↑, ↓} (one boson, two fermions) form a fundamental of the superalgebra, and the Hamiltonian is (up to a constant and chemical potential) the sum of nearest-neighbour graded permutation operators. Sutherland's multicomponent Bethe ansatz Sutherland 1975 and Schlottmann's narrow-band solution Schlottmann 1987 give the exact spectrum and thermodynamics via nested Bethe / thermodynamic-Bethe-ansatz equations. Not exact in closed form (Tier B): the ground-state energy at generic filling, the full spectrum, and finite-T thermodynamics require solving the (nested) Bethe equations state-by-state — integrable but not single closed forms.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
lowest_supermultiplet_energy | L=6, N=6↔5, J=2t | -(2+√3) ≈ -3.7320508 (0 shared at J=1.9t) | this card; Sutherland 1975 |
n_susy_supermultiplets | L=6, N=6↔5, J=2t | 21 shared eigenvalues (1e-8) | this card |
e0_halffilling_thermo | n=1, N→∞ | -2 ln2 ≈ -1.3862944 (Heisenberg limit) | Sutherland 1975 |
e0_halffilling_per_site | n=1, L=6 OBC | -1.2478590 (finite L; → -2 ln2) | this card |
generic-filling e_0 | 0<n<1 | Sutherland/Schlottmann TBA (not coded) | Sutherland 1975, Schlottmann 1987 |
T3 (coordinate/nested Bethe ansatz): the δ-interacting Bose gas is integrable Lieb–Liniger 1963. Every eigenstate is a Bethe wavefunction — a sum over permutations of N plane waves whose two-body scattering phase θ(k) = 2\arctan(k/c) fixes all many-body amplitudes — labelled by quasi-momenta (rapidities) {k_j} solving the Bethe equations. The ground state fills a symmetric Fermi sea of rapidities [-q, q]; in the thermodynamic limit the rapidity density ρ(k) obeys the linear Lieb integral equation, from which e(γ) is computed here to machine precision. Tier B, not A: e(γ) (and the two elementary excitation branches, the type-I/Bogoliubov and type-II/hole modes) are exactly characterised, but as solutions of integral / transcendental equations, not a single closed form; correlation functions require much heavier machinery (Lenard determinants, quantum inverse scattering) and are out of scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_gamma | γ = 1 | ≈ 0.639 | Lieb–Liniger 1963 |
e_gamma | γ → ∞ | π²/3 ≈ 3.289868 | Lieb–Liniger 1963 |
e_gamma | γ = 1000 | ≈ 3.27675 ((π²/3)(1−4/γ+12/γ²)) | Lieb–Liniger 1963 |
e_gamma | γ = 0.01 | ≈ 0.009576 (γ−(4/3π)γ^{3/2}) | Lieb–Liniger 1963 |
Tier A: the Tonks–Girardeau gas is exactly solvable in closed form via Girardeau’s Bose–Fermi mapping Girardeau 1960. An impenetrable-boson wavefunction is the modulus of a free spinless-fermion Slater determinant, Ψ_B = |Ψ_F| = Ψ_F · ∏_{i<j}\mathrm{sgn}(x_i-x_j). Consequently every local, permutation-symmetric observable is identical to the free Fermi gas: the ground state is a filled Fermi sea to k_F = πn, and the ground-state energy, density profile, pair correlation, and density–density response are literally the free-fermion ones. The mapping is exact for all N — hence tier A, the strongest in this catalog — and needs no integral equation. What is exact but not a filled-Fermi-sea number: the momentum distribution n(k) (see below).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_ll_units | LL units | π²/3 ≈ 3.289868 | Girardeau 1960 |
E_over_N | n=1, m=1/2, ħ=1 | π²/3 ≈ 3.289868 | Girardeau 1960 |
k_F | n=1 | π ≈ 3.141593 | Girardeau 1960 |
g2_zero | — | 0 (exact) | Girardeau 1960 |
T3 (nested Bethe ansatz): the two-component δ-Fermi gas is integrable Yang 1967, Gaudin 1967. Because the internal (spin) degrees of freedom scatter non-trivially, a *single* Bethe ansatz is not enough — Yang’s diagonalization of the spin transfer matrix requires a second Bethe ansatz nested inside the first, introducing spin rapidities λ_α alongside the charge rapidities k_j. The balanced ground state fills a charge Fermi sea [-Q, Q] and a spin sea over the entire real axis (B = ∞, since there is no field). The densities obey two coupled linear integral equations; e(γ) is computed here to machine precision. Tier B, not A: e(γ), the spin/charge velocities, and the (gapless) excitation structure are exactly characterised as integral-equation solutions, not closed forms; correlators need heavier machinery and are out of scope.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_gamma | γ → 0 | π²/12 ≈ 0.822467 | Gaudin 1967 |
e_gamma | γ = 1 | ≈ 1.250 | Yang 1967 |
e_gamma | γ → ∞ | π²/3 ≈ 3.289868 | Yang 1967 |
e_gamma | γ = 0.05 | ≈ 0.8473 (π²/12 + γ/2) | Gaudin 1967 |
T3 (asymptotic Bethe ansatz): the trigonometric 1/sin² interaction is integrable and the entire spectrum is closed-form Sutherland 1971. Every eigenstate carries a Jastrow factor ∏_{i<j}|\sin(\pi(x_i-x_j)/L)|^\lambda times a symmetric polynomial that is a Jack polynomial in the z_i = e^{2\pi i x_i/L}, and is labelled by a non-decreasing integer sequence I_1 ≤ … ≤ I_N. The pseudo-momenta are k_i = (2\pi/L)[\,I_i + (\lambda/2)(2i-N-1)\,] and the energy is E = Σ_i k_i² — the (\lambda/2)(2i-N-1) term is the asymptotic-Bethe statistical shift. Tier B, script-P rationale: the spectrum, ground-state energy and gaps are genuinely closed-form (scripted here, and *exact* — this is unusually strong for a Tier-B card); the dynamical density-density and one-body Green's functions are *also* exactly known, at every rational λ, from the Jack-polynomial expansion Ha 1994, but that machinery is not scripted — it is tabulated below. Hence P, partial.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0 | N=5, λ=2, L=2π | 40 | Sutherland 1971 |
e0 | N=2, λ=2, L=2π | 2 (=2λ²(π/L)², two-body exact) | this card; Sutherland 1971 |
e0 | N=3, λ=1, L=2π | 2 (= free-fermion Σk²) | Sutherland 1971 |
first_gap | N=5, λ=2, L=2π | 5 (CM-boost branch, min(N,1+λ(N-1))) | this card |
first_gap | N=3, λ=1, L=2π | 3 (=N; branches degenerate at λ=1) | this card |
E_0/N (thermodynamic) | N→∞, density n | (π²n²/3)λ² | Sutherland 1971 |
T3 (Bethe ansatz): the single-channel spin-1/2 Kondo model is exactly solvable by the Bethe ansatz — after linearizing the conduction dispersion, the electron–impurity scattering is factorizable (Yang–Baxter) and the spectrum and full thermodynamics follow from Bethe/thermodynamic-Bethe-ansatz equations Andrei–Furuya–Lowenstein 1983, Tsvelick–Wiegmann 1983. This yields the impurity free energy, entropy, and magnetic susceptibility at all temperatures and fields as universal functions of T/T_K and H/T_K, with the crossover from a free local moment to a screened singlet controlled by the single scale T_K. This is Tier B (integrable): the exact results are universal scaling functions and closed-form limits (below), obtained from the Bethe/TBA equations rather than a single elementary closed form; finite-L/lattice energetics still require ED (the pinned rows below). This is a T-flag card — no oracle.py; the exact content is the integrability facts and universal numbers below plus finite-size ED reference rows computed once for this card.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
Wilson ratio R | spin-1/2, 1 channel | 2 (exact) | Andrei–Furuya–Lowenstein 1983 |
S_{\mathrm{imp}} (T\gg T_K) | zero field | \ln 2 | Tsvelick–Wiegmann 1983 |
S_{\mathrm{imp}} (T\ll T_K) | zero field | 0 | Tsvelick–Wiegmann 1983 |
E_0 (total, ED) | L=5, J=1, t=1, N_e=5 | -5.8397495469 | finite-L ED reference, this card |
E_0 (total, ED) | L=5, J=0, t=1, N_e=5 | -5.4641016151 | finite-L ED reference, this card |
| singlet–excitation gap | L=5, J=1, N_e=5 | 0.4167212187 | finite-L ED reference, this card |
T3 (Bethe ansatz): the symmetric single-impurity Anderson model is exactly solvable by the Bethe ansatz — with a linearized conduction band the model is integrable and its impurity thermodynamics (free energy, occupation, charge and spin susceptibilities, specific heat) follow as exact universal functions from the Bethe/thermodynamic-Bethe-ansatz equations Kawakami–Okiji 1981, Tsvelick–Wiegmann 1983. In the local-moment regime (U\gg\Gamma) the spin sector reduces to the Kondo problem with J=8V^2/U, reproducing the Kondo scale T_K and the universal spin thermodynamics; away from it the exact solution also captures the mixed-valence/charge-fluctuation crossover. This is Tier B (integrable): the exact content is universal scaling functions and closed-form limits (below) from the Bethe/TBA equations, not a single elementary closed form; finite-bath energetics require ED (the pinned rows). This is a T-flag card — no oracle.py; the exact content is the integrability facts plus finite-size ED reference rows computed once for this card.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
⟨n_d⟩ (symmetric) | ε_d=-U/2 | 1 (exact, PH symmetry) | Kawakami–Okiji 1981 |
Wilson ratio R_W | Kondo limit | 2 (exact) | Tsvelick–Wiegmann 1983 |
E_0 (total, ED) | L=5, U=2, V=1, t=1, N_e=6 | -7.5645006223 | finite-bath ED reference, this card |
E_0 (total, ED) | L=5, U=0, V=1, t=1, N_e=6 | -6.9879184149 | finite-bath ED reference, this card |
| first excitation gap | L=5, U=2, N_e=6 | 0.7216924611 | finite-bath ED reference, this card |
T3 (Bethe ansatz / Richardson–Gaudin): the reduced BCS Hamiltonian is exactly solvable by Richardson's 1963 exact eigenstate construction, a Bethe-ansatz-type solution Richardson 1963. In the seniority-zero sector the M-pair eigenstates are products of collective pair-creation operators whose M spectral parameters E_a (the Richardson roots) solve the coupled Richardson equations, and the eigen-energy is the plain sum of roots E=\sum_a E_a. Written consistently with the Hamiltonian above (attractive -g pairing, pair-level energy 2ε_j), the equations are 1 - g\sum_{j} 1/(2ε_j-E_a) - 2g\sum_{b\ne a} 1/(E_a-E_b)=0 for each a. As g\to0^+ the roots start at 2ε_a of the M lowest levels; as g grows a pair of real roots can collide at a level 2ε_j and split into a complex-conjugate pair, so the roots must be tracked in the complex plane. This is Tier B (integrable), not a single closed form: the roots are the solution of M coupled nonlinear (Bethe) equations, solved here by homotopy in g, not a closed-form expression.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0 | N=6, M=3, g=0.1 | 11.6798192419 | Richardson 1963 (roots = ED) |
e0 | N=6, M=3, g=0.5 | 9.8015279710 | Richardson 1963 (roots = ED) |
e0 | N=6, M=3, g=1.0 | 5.8108407474 | Richardson 1963 (roots = ED) |
e0 (g\to0) | N=6, M=3 | 2(ε_1{+}ε_2{+}ε_3)=12 | Richardson 1963 |
roots at g=1.0 | N=6, M=3 | one real \approx0.8319, one conjugate pair 2.4894\pm2.5843i | this card (solver) |
T3 (Bethe ansatz / Yang–Baxter, rational Gaudin): the central-spin model is a rational Gaudin magnet — an integrable system whose N_b+1 Gaudin Hamiltonians H_i=\sum_{k\ne i}(\mathbf{S}_i\cdot\mathbf{S}_k)/(u_i-u_k) mutually commute Gaudin 1976. This card ships the conserved-charge construction: with the inhomogeneities u=(0,-1,\dots,-N_b) the central-spin Hamiltonian is exactly one of these commuting charges, H=H_0 (because A_j=1/(u_0-u_j)=1/j), so its integrability is proven in-card by verifying that all pairwise commutators [H_i,H_j] vanish numerically, and its ground-state energy is supplied by exact diagonalization of H (the exact numbers), with [H,\mathbf{S}^2_{\mathrm{tot}}]=0 organising the spectrum into total-spin multiplets. This is Tier B (integrable): the exact structure (commuting charges + Bethe/Gaudin eigenstates) is scripted, but the ground energy at generic 1/j couplings is an ED number, not a closed form — only the uniform-coupling limit has the closed form below.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0 (A_j=1/j) | N_b=5 | -0.9148146832 | this card (ED = Gaudin charge H_0) |
e0 (A_j=1/j) | N_b=4 | -0.8853708830 | this card (ED = Gaudin charge H_0) |
E_0 (uniform A=1) | N_b=5 | -\tfrac14(N_b+2)=-1.75 | Gaudin 1976 (SU(2) closed form) |
E_0 (uniform A=1) | N_b=4 | -\tfrac14(N_b+2)=-1.5 | Gaudin 1976 (SU(2) closed form) |
| max commutator | [H_i,H_j], N_b\in\{4,5\} | \sim10^{-16} (integrable) | this card (measured) |
T3 (Yang–Baxter, but off the difference property): the chiral Potts Boltzmann weights satisfy the star–triangle (Yang–Baxter) relations with rapidities on a higher-genus curve Baxter–Perk–Au–Yang 1988; the model is integrable (commuting transfer matrices, infinite conserved charges) yet the rapidity curve's genus >1 blocks the elementary uniformisation that gives closed-form Bethe roots for the six/eight-vertex family. At the superintegrable point the quantum chain gains an Onsager-algebra symmetry von–Gehlen–Rittenberg 1985: writing H = H_X + H_Z (the field and bond k-sums above) with normalised generators A_0 \propto H_X, A_1 \propto H_Z, the pair satisfies the Dolan–Grady relations [A_0,[A_0,[A_0,A_1]]]=16\,[A_0,A_1] (and 0\leftrightarrow1), generating the Onsager algebra; eigenstates fall into Ising-like multiplets and the energy levels are E=a+bk'+\sum_j m_j\sqrt{1+2k'\cos\theta_j+k'^2} with occupation numbers m_j\in\{0,\dots,N-1\} and Ising-like \theta_j — a *parafermionic* free-particle-like spectrum. Tier B, not A: superintegrability makes the spectrum highly structured (and the order parameter exactly known, below), but the general model is not a single closed form, and finite-L energies/correlators still require the Onsager-sector machinery or ED. Out of this card's scope beyond the pinned numbers.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
E_0 (total, ED) | N=3, L=8, PBC | −20.0323794586 | finite-L ED reference, this card |
e0 = E_0/L (ED) | N=3, L=8, PBC | −2.5040474323 | finite-L ED reference, this card |
E_0 (total, ED) | N=3, L=6, PBC | −15.0702102767 | finite-L ED reference, this card |
e0 = E_0/L (ED) | N=3, L=6, PBC | −2.5117017128 | finite-L ED reference, this card |
| Dolan–Grady residual | N=3, L=4, \alpha_k=2/(1-\omega^{-k}) | 3.8×10^{-16} (holds) | this card (measured) |
| Dolan–Grady residual | N=3, L=4, \alpha_k=1/\sin(\pi k/3) (achiral) | 0.47 (fails — different model) | this card (measured) |
| order-parameter exponent | \beta_n, any N | n(N-n)/(2N^2) (N=3: 1/9) | Baxter 2005 |
e0 (thermodynamic) | N=3, L→∞ | *no web-verified numeric value quoted here* | von–Gehlen–Rittenberg 1985, Baxter–Perk–Au–Yang 1988 |
every term commutes — stabilizer algebra and fusion categories fix the exact spectrum and topological degeneracy
T4: the toric code is a commuting-projector stabilizer Hamiltonian. The ground space is the simultaneous +1 eigenspace of every A_v and B_p, and the entire spectrum is enumerated by counting violated stabilizers: E = -N_s + 2\,(\#\text{violated}), where N_s = 2L² is the number of stabilizer terms. Everything reported — the torus ground-state degeneracy gsd and the excitation gap gap_pair — is exact for every L; there is no approximation anywhere. Not exact: nothing about this model is approximate. Exact content deliberately out of this card's scope (all still exact from the same stabilizer structure, just not implemented in oracle.py): the anyon braiding data, the topological entanglement entropy γ = ln 2, the Wilson/'t Hooft loop logical operators and their algebra, and planar-boundary code distances. The gsd is obtained from an exact GF(2) rank (_lib.gf2.stabilizer_gsd_log2, which also asserts pairwise commutation), not from a numerical eigensolver.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
gsd | torus, any L | 4 (= 2^{2g}, g=1) | Kitaev 2003 |
gap_pair | torus, J = 1 | 4 (minimal anyon pair, ΔE = 4) | Kitaev 2003 |
n_qubits | L×L torus | 2L² | — |
T4: D(G) is a commuting-projector stabilizer(-like) lattice gauge theory, exactly solvable in its entirety. Its anyons are the irreducible representations of the Drinfeld double, labelled by pairs ([c], \pi) with [c] a conjugacy class of G and \pi an irrep of the centralizer Z(c) of a representative; hence the anyon count and the torus ground-state degeneracy are exact:
| Quantity | Params | Exact value | Source |
|---|---|---|---|
gsd_torus | G = Z2 | 4 (= toric code) | Kitaev 2003 |
gsd_torus | G = Z3 | 9 (= 3²) | Kitaev 2003 |
gsd_torus | G = Z4 | 16 (= 4²) | Kitaev 2003 |
gsd_torus | G = S3 | 8 (= 3+2+3, non-abelian) | Kitaev 2003 |
n_anyons | any listed G | = gsd_torus | Kitaev 2003 |
T4: the Levin–Wen model is a commuting-projector Hamiltonian, exactly solvable in its entirety. From the fusion tensor N^k_{ij} alone, the quantum dimensions, total quantum dimension, doubled anyon count, and torus GSD are exact:
| Quantity | Params | Exact value | Source |
|---|---|---|---|
d_max (= d_\tau) | fibonacci | \varphi = (1+\sqrt5)/2 ≈ 1.61803 | Levin–Wen 2005 |
total_quantum_dim_sq | fibonacci | 1 + \varphi^2 ≈ 3.61803 | Levin–Wen 2005 |
gsd_torus | fibonacci | 4 (doubled Fibonacci) | Levin–Wen 2005 |
d_max (= d_\sigma) | ising | \sqrt2 ≈ 1.41421 | Levin–Wen 2005 |
total_quantum_dim_sq | ising | 4 (= 1+2+1) | Levin–Wen 2005 |
gsd_torus | ising | 9 (doubled Ising) | Levin–Wen 2005 |
T4: the color code is a commuting-projector stabilizer Hamiltonian. The ground space is the simultaneous +1 eigenspace of every X_f and Z_f, and the full spectrum is E = -N_s + 2\,(\#\text{violated}) with N_s = 2\,cells². The reported gsd is exact for every valid torus size; there is no approximation. Not exact: nothing about this model is approximate. Exact content deliberately out of this card's scope (all still exact from the same stabilizer structure, just not implemented): the string/logical operators and code distance, the anyon/charge-flux labelling by colour, and the transversal Clifford gates that make the color code notable. The gsd comes from an exact GF(2) rank (_lib.gf2.stabilizer_gsd_log2, which asserts pairwise commutation internally — reaching the correct value is itself a proof the honeycomb-torus stabilizers all commute), not from an eigensolver.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
gsd | torus, cells ≡ 0 (mod 3) | 16 (= 2^{4g}, g=1 = 4²) | Bombin–Martin–Delgado 2006 |
n_qubits | cells×cells torus | 2\,cells² (= 18 at cells=3) | — |
n_hexagons | cells×cells torus | cells² (= 9 at cells=3) | — |
T4: the X-cube model is a commuting-projector stabilizer Hamiltonian. The ground space is the simultaneous +1 eigenspace of every cube A_c and every planar cross B^{μν}_v, and the entire spectrum is enumerated by counting violated stabilizers, E = -N_s + 2\,(\#\text{violated}). The reported subextensive degeneracy gsd_log2 = \log_2 GSD is exact for every L; there is no approximation. Not exact: nothing about this model is approximate. Exact content deliberately out of this card's scope (all still exact from the same stabilizer structure, just not implemented in oracle.py): the fracton/lineon excitation content and their restricted mobility, the fractal/planar logical-operator membranes, the entanglement structure, and the RG/coupled-layer construction relating X-cube to stacks of 2D toric codes. The gsd_log2 is obtained from an exact GF(2) rank (_lib.gf2.stabilizer_gsd_log2, which also asserts pairwise commutation), not from a numerical eigensolver.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
gsd_log2 | L=2 torus | 9 (= 6·2-3) | Vijay–Haah–Fu 2016 |
gsd_log2 | L=3 torus | 15 (= 6·3-3) | Vijay–Haah–Fu 2016 |
gsd_log2 | L×L×L torus | 6L - 3 (subextensive) | Vijay–Haah–Fu 2016 |
n_qubits | L×L×L torus | 3L³ | — |
T4: the Haah code is a commuting-projector stabilizer Hamiltonian. The ground space is the simultaneous +1 eigenspace of every A_s, B_s, and the full spectrum is E = -N_s + 2\,(\#\text{violated}) with N_s = 2L³. The reported degeneracy gsd_log2 = k = \log_2 GSD is an exact integer for every L, obtained from an exact GF(2) rank (_lib.gf2.stabilizer_gsd_log2, which also asserts pairwise commutation — reaching a value at all certifies the transcription commutes). Not exact: nothing is approximated — k(L) is computed exactly by integer GF(2) elimination. What is *absent* is a closed form: k(L) has a subextensive, number-theoretic dependence on L with no simple formula (only the bound 2 ≤ k ≤ 4L - 2 and the polynomial-algebra origin below are stated in closed form) Haah 2011. Exact content deliberately out of this card's scope (still exact from the same stabilizer structure, just not implemented): the fractal logical-operator supports, the code distance d ~ 2^{...} growth, the immobile-fracton excitation structure, and the polynomial/homological classification of k(L).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
gsd_log2 (k) | L=2 torus | 6 (= 4L-2, saturates bound) | computed here (GF(2) rank); bound Haah 2011 |
gsd_log2 (k) | L=3 torus | 2 (generic; 3 factor-free) | computed here; matches generic-k statement Haah 2011 |
gsd_log2 (k) | L=4 torus | 14 (= 4L-2, saturates bound) | computed here (GF(2) rank); bound Haah 2011 |
gsd_log2 (k) | general L | 2 ≤ k ≤ 4L-2, no closed form | Haah 2011 |
n_qubits | L×L×L torus | 2L³ | — |
T4: a commuting-projector stabilizer Hamiltonian. The ground space is the simultaneous +1 eigenspace of every K_i; the full spectrum is E = -N_s + 2\,(\#\text{violated}) with N_s the number of stabilizers. Everything reported — the PBC/OBC degeneracies, the gap, and the string-order parameter — is exact for every L; no approximation anywhere. Not exact: nothing about this model is approximate. Exact content deliberately out of this card's scope (all still exact, just not implemented): the explicit edge-mode logical operators of the OBC degeneracy, the full entanglement spectrum (doubly degenerate — the SPT fingerprint), and the MBQC gate-teleportation identities. gsd_pbc/gsd_obc come from an exact GF(2) rank; string_order is proved exactly by GF(2) stabilizer-membership and cross-checked in ED.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
gsd_pbc | L-site ring | 1 (unique, bulk-trivial) | Raussendorf–Briegel 2001 |
gsd_obc | open chain | 4 (= 2², two edge modes) | Raussendorf–Briegel 2001 |
gap | PBC, coupling 1 | 2 (one flipped K_i) | Raussendorf–Briegel 2001 |
string_order | ⟨\prod K_i⟩ | 1 (product of stabilizers) | derived above |
the ground state is exact even when the spectrum is not — valence bonds, dimer coverings, scar towers
T5 (frustration-free / exact eigenstates): on each bond the local term equals 2P₂ − 2/3, where P₂ projects the bond onto total spin 2 (for two spin-1's the bilinear-plus-⅓-biquadratic form takes the value 4/3 in the spin-2 channel and −2/3 in the spin-0 and spin-1 channels, i.e. 2P₂ − 2/3). So H = 2Σ_bond P₂ − 2N/3 with P₂ ≥ 0 and N = L bonds (PBC). The valence-bond-solid (VBS) state — the exact bond-dimension-2 matrix-product state with A^{+}=√(2/3)σ^+, A^{0}=−√(1/3)σ^z, A^{−}=−√(2/3)σ^- — carries zero weight in every bond's spin-2 channel, so it is annihilated by every P₂ and is the exact ground state with energy E₀ = −2N/3. It is unique on the periodic chain and 4-fold degenerate with open boundaries (two emergent spin-½ edge modes). What is exact is fully closed-form: the ground energy −2/3 per site, the two-point function ⟨S^z_iS^z_{i+r}⟩ = (4/3)(−1/3)^r, and the string order O^z = −4/9, all read off the 2×2 transfer matrix (eigenvalues 1, −1/3). Not exact: everything except the ground state. The generic excited spectrum is *not* solvable in closed form, and in particular the Haldane gap above the VBS is not a closed-form quantity — it is only known numerically (ED ≈ 0.70 on the L=8 PBC cluster reported here; the rigorous lower bound of AKLT/Knabe is a bound, not the gap). This is a genuine Tier-C card: only the ground state (and its exact correlators) is exact, not the spectrum.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site | AKLT point, any L | −2/3 | AKLT 1987 |
szsz_r1 = ⟨S^z_iS^z_{i+1}⟩ | infinite chain | −4/9 | AKLT 1987 |
szsz_r2 = ⟨S^z_iS^z_{i+2}⟩ | infinite chain | 4/27 | AKLT 1987 |
string_order O^z | infinite chain | −4/9 | AKLT 1987 |
| GS degeneracy | PBC / OBC | 1 / 4 | AKLT 1987 |
| Haldane gap | L=8 PBC (numerical) | ≈ 0.700 (not closed form) | ED (this card) |
T5 (frustration-free / exact ground state), Tier C: since every bond projector P^{(3)}\ge0, H=\sum P^{(3)}\ge0 has spectrum floor E\ge0 (a frustration-free parent Hamiltonian). The honeycomb valence-bond solid — put three virtual spin-\tfrac12's on each site (one per bond) and symmetrise to spin-\tfrac32, and place a singlet on every bond — caps the total spin of any bond at 2, so it carries zero total-spin-3 weight and is annihilated by every P^{(3)}: it is an exact E=0 ground state AKLT 1988. This card verifies that exactly on the 8-site torus: (i) sparse ED gives ground energy E_0\approx6\times10^{-16} (the frustration-free floor 0 to 1e-10) with spectrum floor E\ge0, a *unique* zero-energy ground state, and a positive gap \approx0.092 on this cluster (numerical, finite-size); and (ii) the explicit VBS/PEPS state is constructed (contracting the bond singlets through the on-site spin-3/2 symmetrisers) and shown to be annihilated at operator level, \|H|\psi\rangle\|\approx7\times10^{-16}<1e-10, and to coincide with the ED ground state (overlap 1). Not exact / P-scope: everything except the ground state at this point. The thermodynamic-limit correlation functions, the disordered (no Néel) nature of the state, and — crucially — the nonzero spectral gap are *not* reproduced here; the gap in particular is a hard theorem, proved only in 2020 Pomata–Wei 2020, Lemm–Sandvik–Wang 2020, and is tabulated with citations, not scripted. The small torus's 0.092 gap is a finite-cluster number, not the proven thermodynamic gap.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
spectrum_floor | honeycomb, any size | E\ge0 (frustration-free) | AKLT 1988 |
e0 | 2×2 torus (8 sites) | 0 | AKLT 1988 |
vbs_residual |Hψ| | explicit PEPS, 8 sites | 0 (<1e-10) | AKLT 1988 |
gsd_observed | 2×2 torus | 1 (unique) | ED (this card) |
gap_observed | 2×2 torus (numerical) | ≈0.092 (finite-size, not the theorem) | ED (this card) |
| spectral gap | thermodynamic limit | nonzero (proven) — tabulated | Pomata–Wei 2020, Lemm–Sandvik–Wang 2020 |
T5 (frustration-free / exact eigenstates): at J₂/J₁ = 1/2 the two nearest-neighbour dimer coverings — the singlet product |D_A⟩ on bonds (1,2)(3,4)… and |D_B⟩ on bonds (2,3)(4,5)…(L,1) — are exact ground states with energy E₀ = −3JL/8. Each singlet contributes ⟨S_i·S_j⟩ = −3/4; the frustrating next-nearest-neighbour term annihilates a dimer product because any spin sits on a triangle with the two members a,b of a neighbouring singlet, and (S_a+S_b) annihilates that singlet (total spin 0), so the extra bonds cost nothing. The ground space is exactly 2-fold (the two coverings, which become orthogonal and degenerate as L→∞), and E₀/L = −3/8 is exact for every L∈4ℤ. Not exact: everything except the ground state. Only these two dimer states are exact eigenstates — the rest of the spectrum, the excitations (deconfined spinons / a triplet continuum), and the spin gap are not closed-form; the gap is positive but known only numerically (ED ≈ 0.32 at L=12, this convention). This is a genuine Tier-C card: the exact content is the ground state, not the spectrum.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_per_site | MG point, L∈4ℤ | −3/8 | Majumdar–Ghosh 1969 |
e0_total | L=12 | −9/2 | Majumdar–Ghosh 1969 |
ground_degeneracy | PBC, L∈4ℤ | 2 | Majumdar–Ghosh 1969 |
gap_ed | L=12 (numerical) | ≈ 0.319 (not closed form) | ED (this card) |
T5 (frustration-free / exact eigenstates): the orthogonal-dimer geometry makes the dimer-singlet product |Ψ⟩ = ∏_{\rm dimers}|{\rm singlet}⟩ an exact eigenstate of the *full* H for all J/J', with energy E = −(3/4)\,J'\,N_{\rm dimer} (per spin −3J'/8). The proof is the orthogonality miracle: every square-lattice bond attaches an external spin to *both* members of a neighbouring singlet, so its two contributions combine into J\,S_c·(S_a+S_b) and vanish on the singlet (S_a+S_b=0) — the J bonds contribute exactly zero and only the J' dimer bonds count (−3/4 each). This exact eigenstate is the ground state only in the dimer phase (small J/J'); above a level crossing the ground state is a *different, non-exact* state (plaquette-singlet, then Néel). Not exact: everything except the dimer state. The generic spectrum, the competing plaquette/Néel ground states, the triplet dispersion, and the transition point are all numerical — the exact-dimer→plaquette boundary is J/J' = 0.675(2) in the thermodynamic limit Corboz–Mila 2013, quoted from the literature, not derived here. This is a genuine Tier-C card: only the dimer product state is exact, and only in part of the phase diagram.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_dimer_per_spin | any J/J', J'=1 | −3/8 | Shastry–Sutherland 1981 |
e_dimer_exact | 16-site cluster, J'=1 | −6 (= −¾·8) | Shastry–Sutherland 1981 |
dimer_is_ground_state | J/J' = 0.5 / 0.9 | True / False | ED (this card) |
| dimer→plaquette boundary | thermodynamic | J/J' = 0.675(2) | Corboz–Mila 2013 |
T5 (frustration-free / exact eigenstates): at the RK point each plaquette term factorises as h_□ = (|=⟩−|‖⟩)(⟨=|−⟨‖|), a rank-1 positive-semidefinite operator (expand: v(|=⟩⟨=|+|‖⟩⟨‖|) − t(|=⟩⟨‖|+|‖⟩⟨=|) at t=v). Hence H_RK = Σ_□ h_□ ≥ 0 is frustration-free with spectrum floor E ≥ 0 (verified numerically: min eigenvalue ≈ −3e-15 on the 4×4 torus). The equal-weight superposition of the coverings in any flip-connected set is an exact E=0 ground state: for every plaquette the local move pairs a =-covering with its ‖-partner at equal amplitude, so (⟨=|−⟨‖|)|\psi⟩ = 1−1 = 0 term by term. Winding numbers (W_x,W_y) — signed counts of dimers crossing a reference cut, cut-independent and preserved by every flip — block-diagonalise H, and the equal-weight state of each winding sector is a zero-energy state. Not exact: everything except this ground manifold at t=v. The gap, the excited spectrum, the columnar↔plaquette↔staggered transitions of the square-lattice QDM, and the triangular-lattice deconfined liquid are not closed-form — numerical / cited. This is a genuine Tier-C card: exact only at one point, and only the ground states there.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
n_coverings | 4×4 torus | 272 | this card (3 methods) |
n_winding_sectors | 4×4 torus | 13 | this card |
gsd_zero_modes | 4×4 torus, RK point | 17 (observed) | ED (this card) |
spectrum_floor | RK point | ≥ 0 (frustration-free) | Rokhsar–Kivelson 1988 |
| dimer correlators | 4×4 torus, |ψ_RK⟩ | = classical dimer average (identity) | Rokhsar–Kivelson 1988 |
T5 (frustration-free / exact eigenstates): H is a sum of projectors (Π_{j,j+1} ≥ 0 and boundary projectors ≥ 0), so H ≥ 0 and any state annihilated by every term has E=0. The bulk moves connect all height-≥0 walks with the same endpoints at equal amplitude, and the boundary terms select endpoints 0→0; the unique common zero of all terms is therefore the uniform superposition of all Motzkin walks of length n, |GS⟩ ∝ Σ_{\rm Motzkin}|walk⟩, with E_0 = 0. Uniqueness is genuine (verified: the second-lowest ED eigenvalue is > 0 at n∈{4,6,8}; the gap closes only polynomially in 1/n Bravyi–Et–Al 2012). The number of Motzkin walks is the Motzkin number M_n = 1,1,2,4,9,21,51,127,323,…. Not exact: everything except the ground state. The excited spectrum and the exact gap are not closed-form (only the 1/\mathrm{poly}(n) scaling is proven), and the half-chain entanglement — computed exactly here from path-counting Schmidt weights — grows like ~(1/2)\ln n + O(1) Bravyi–Et–Al 2012, reported observed-as-observed, not as a fitted exponent. A genuine Tier-C card: exact ground state (and its exact Schmidt spectrum), not the spectrum.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
motzkin_number | n=8 | 323 | Bravyi–Et–Al 2012 |
e0 | any n | 0 (frustration-free) | Bravyi–Et–Al 2012 |
| GS uniqueness | n∈{4,6,8} | unique (gap >0) | ED (this card) |
half_chain_entropy | n=8 (Motzkin) | ≈ 1.2206 | this card (exact Schmidt) |
| entropy law | n→∞ | ~(1/2)\ln n + O(1) (observed) | Bravyi–Et–Al 2012 |
| Fredkin Dyck count | n=8 | C_4 = 14 | Salberger–Korepin 2017 |
T5 (frustration-free / exact eigenstates), but the load-bearing statement is an operator identity, not a variational ground state. The staggered η-pairing operator obeys, on any bipartite lattice with L even, [H,\eta^\dagger]=U\eta^\dagger — verified as a matrix identity at L=4,6 to < 1e-12. The derivation splits cleanly: the kinetic term commutes with \eta^\dagger by itself (each bond (i,j) contributes (-1)^i+(-1)^j=0 because i,j sit on opposite sublattices), and the interaction gives [U\sum_i n_{i\uparrow}n_{i\downarrow},\eta^\dagger]=U\sum_i(-1)^i c^\dagger_{i\uparrow}c^\dagger_{i\downarrow}=U\eta^\dagger since [n_{i\uparrow}n_{i\downarrow},c^\dagger_{i\uparrow}c^\dagger_{i\downarrow}]=c^\dagger_{i\uparrow}c^\dagger_{i\downarrow}. Consequently \eta^{\dagger m}|{\rm vac}\rangle is an exact eigenstate with energy E_m = mU (induction: H\eta^{\dagger m}|{\rm vac}\rangle = mU\,\eta^{\dagger m}|{\rm vac}\rangle), sitting in the (N_\uparrow,N_\downarrow)=(m,m), S=0, S^z=0 sector (each \eta^\dagger deposits a pair at the antiferromagnetic wavevector, so the total momentum is m\pi). These are exact EXCITED states, not ground states — for U>0 they lie at positive energy mU, whereas the true ground state of the *same* (m,m) sector is bound (negative energy), strictly and by a wide margin below mU (identity-proof anchor: at U=4, L=6, m=2 the sector ground energy is ≈ −4.70 < 8 = mU). What is exact is only this η-pairing tower and its correlators; the generic Hubbard spectrum is *not* closed-form (it needs the Lieb–Wu Bethe ansatz — see hubbard-1d-lieb-wu). Convention caveat: with the particle-hole-symmetric U\sum(n_\uparrow-\tfrac12)(n_\downarrow-\tfrac12) the shift becomes 0 (η-states degenerate with Néel states, the SO(4) multiplet); with a -\mu N term it is (U-2\mu) since [N,\eta^\dagger]=2\eta^\dagger. This card uses the bare H → shift U.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
eta_shift [H,η†]/η† | bipartite, L even, bare H | U | Yang 1989 |
energy_m_pair | any L,U; m pairs | mU (exact excited) | Yang 1989 |
norm_sq_closed | L=6, m=2 | 60 = 2L(L−1) | Yang 1989 |
odlro_offdiag | L=6, m=2 | 4/15 ≈ 0.2667 | Yang 1989 |
pair_density_diag | L=6, m=2 | 1/3 | Yang 1989 |
sector_gs_energy | L=6, U=4, m=2 | ≈ −4.70 (≪ mU=8) | ED (this card) |
T5 (exact eigenstates), Tier C: PXP is non-integrable (Wigner–Dyson bulk, ETH) yet hosts a handful of exact eigenstates — quantum many-body scars — that violate ETH. The kinematics are exact combinatorics: the blockade restricts the Hilbert space to bitstrings with no two adjacent 1s, of dimension F_{N+2} (OBC Fibonacci) and L_N (PBC Lucas) — verified against a direct enumerator for N\le 14 (e.g. N=12: 377=F_{14} OBC, 322=L_{12} PBC). Lin–Motrunich Lin–Motrunich 2019 found exact scar eigenstates as low-bond-dimension matrix product states: block two physical sites (2b,2b{+}1) into one block-site with allowed states O=(00) and the two single-excitation states, and use the bond-dimension-2 MPS |\Gamma_{a,b}\rangle=\sum v_a^{\mathsf T}A^{s_1}\cdots A^{s_{N/2}}v_b|s\rangle with A^{O}=\bigl[\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr], the two excitation matrices \bigl[\begin{smallmatrix}\sqrt2&0\\0&0\end{smallmatrix}\bigr] (for block (1,0)) and \bigl[\begin{smallmatrix}0&0\\0&-\sqrt2\end{smallmatrix}\bigr] (for (0,1)) — the assignment fixed by the blockade (the product vanishes on forbidden adjacencies) — and boundary vectors v_1=(1,1)^{\mathsf T}, v_2=(1,-1)^{\mathsf T}. Then |\Gamma_{1,1}\rangle,|\Gamma_{2,2}\rangle are exact E=0 eigenstates and |\Gamma_{1,2}\rangle,|\Gamma_{2,1}\rangle are exact E=\pm\sqrt2 eigenstates in this X=\sigma^x convention — verified at N=8,12 OBC as operator-level eigenstates (\|H|\Gamma\rangle-E|\Gamma\rangle\|<1e-10). Not exact: everything else. These few MPS states are the only closed-form eigenstates; the rest of the spectrum is thermal. The scars are the mechanism of the |Z_2\rangle-quench revivals but the revival dynamics itself, the approximate su(2) tower, and the bulk spectrum are numerical, not closed-form.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
dim_constrained_obc | N=12 OBC | 377 = F_{14} | serbyn_2020_quantum |
dim_constrained_pbc | N=12 PBC | 322 = L_{12} | serbyn_2020_quantum |
E_scar_plus / E_scar_minus | N=8,12 OBC, X=σ^x | \pm\sqrt2 | Lin–Motrunich 2019 |
eth_scar_entropy | N=12, +\sqrt2 scar | \ln 2 ≈ 0.693 (vs mean ≈1.90) | Lin–Motrunich 2019 |
eth_n_zero_modes | N=12 OBC | 13 (observed) | Lin–Motrunich 2019 |
all-to-all couplings, single modes, random ensembles — mean field and large N become exact
T6 (collective / large-N): the Hamiltonian is built entirely from components of the total spin, so [S², H] = 0 and H is block-diagonal in the total-spin sectors j. The ground state lives in the maximal-spin sector j = N/2 (dimension N+1): both energy-lowering terms −(J/N)S_x² and −h S_z are minimized by the largest available spin length (ferromagnetic ordering), so the (N+1)-dimensional block carries the GS exactly at any finite N — an exponential 2^N → N+1 collapse. Quantizing along x makes S_x diagonal and S_z a ladder, so the block is tridiagonal and diagonalized to machine precision by eigh_tridiagonal for N up to millions. In the thermodynamic limit spin-coherent (mean-field) minimization becomes exact and gives closed-form energies in both phases, with a second-order QPT at h = J. Not exact: nothing is approximate at finite N (the block diagonalization is exact); the *closed-form* e₀(h/J) and the N^{−1/3} critical-gap scaling are thermodynamic-limit statements, approached as O(1/N) by the finite-N block. This is a Tier-A card: the GS sector is exactly solvable in full, and the whole j = N/2 spectrum is available from the tridiagonal block.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | h = J (critical) | −J/2 | Lipkin–Meshkov–Glick 1965 |
e0_thermodynamic | broken, h = 0.5, J = 1 | −0.3125 | Lipkin–Meshkov–Glick 1965 |
e0_thermodynamic | symmetric, h = 1.5, J = 1 | −0.75 | Lipkin–Meshkov–Glick 1965 |
sx_order_thermo | broken, h = 0.5, J = 1 | √3/4 ≈ 0.4330 | Lipkin–Meshkov–Glick 1965 |
| critical gap exponent | h = J, N ∈ {200..1600} | ≈ −1/3 (observed −0.32) | Dusuel–Vidal 2005 |
T6 (collective spin + single mode). Tavis–Cummings (RWA) conserves the excitation number C = a†a + J_z + N/2, so H_TC is exactly block-diagonal: sector C = c is spanned by |n = c−k, J_z = −j+k⟩, k = 0…min(c,N), giving blocks of dimension dim(c) = min(c,N)+1 = min(c+1, N+1). Each block is a small tridiagonal matrix in k, diagonalized directly — this finite-block structure is the exact solution (Tavis–Cummings is Richardson–Gaudin / Bethe-ansatz integrable, but at collective j the blocks are already finite). H_TC at N = 1 reduces exactly to the Jaynes–Cummings model. Not exact for H_TC: nothing — the block spectra are exact; the Fock-truncated full-space ED in the script is only a cross-check.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
| TC block dimension | sector C = c, N atoms | min(c+1, N+1) | Tavis–Cummings 1968 |
tc_e_ground | N = 4, resonant g = 0.5 | −N/2 = −2 (all atoms down, C=0) | Tavis–Cummings 1968 |
Dicke λ_c | resonant ω = ω₀ = 1 | 0.5 | Emary–Brandes 2003 |
Dicke λ_c | general | √(ω ω₀)/2 | Emary–Brandes 2003 |
N = 1 spectrum | any g, ω₀ | Jaynes–Cummings dressed states | Jaynes–Cummings 1963 |
T6 (single-mode / collective): the excitation number C = a†a + (σ^z+1)/2 (photons + atomic excitation) commutes with H, so the infinite-dimensional space decomposes into a one-dimensional C = 0 sector {|0,↓⟩} plus two-dimensional blocks {|n,↑⟩, |n+1,↓⟩} for each C = n+1 ≥ 1. Diagonalizing the 2×2 block gives the dressed-state energies E_±(n) = ω(n+½) ± √(g²(n+1) + δ²/4) in closed form for every level — Tier A, the entire spectrum is exact and algebraic. The C = 0 ground state |0,↓⟩ is uncoupled with energy exactly −ω₀/2 (it is the many-body ground state in the weak-coupling / near-resonant regime; at strong g a dressed level E_−(0) dips below it). Not exact: nothing — every eigenvalue is closed-form. The truncated-boson full-space ED in the script is only a cross-check; its low-lying levels reproduce the block spectra and are stable under doubling the Fock cutoff (a convergence check, not an approximation to the exact answer).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_ground (|0,↓⟩) | any g, resonant ω₀ = 1 | −0.5 | Jaynes–Cummings 1963 |
| vacuum Rabi splitting | resonant δ = 0, n = 0 | 2g | Jaynes–Cummings 1963 |
E_±(0) | resonant δ = 0, ω = 1 | ½ ± g | Jaynes–Cummings 1963 |
E_±(n) | detuned | ω(n+½) ± √(g²(n+1)+δ²/4) | Jaynes–Cummings 1963 |
T6 (single-mode spin-boson). The Rabi model has only a Z₂ parity symmetry Π = σ^z exp(iπ a†a) ([H_R, Π] = 0), yet Braak (2011) showed it is exactly solvable: the *regular* spectrum in each parity sector is given by the zeros of a transcendental G-function G_±(x), and the corresponding energies are E = x − g²/ω. This is Tier B — exact but transcendental: there is no closed algebraic form, only the roots of G_±. The rare *exceptional* (doubly parity-degenerate, "baseline") eigenvalues E = nω − g²/ω appear only for special (g, Δ) tuned so K_n(n) = 0; they are not scanned here. Not closed-form: the eigenvalues themselves (they are roots of G_±); everything else — the recurrence, the parity structure, the E = x − g²/ω map — is exact and explicit.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_ground | g=0.7, Δ=0.4, ω=1 (Braak Fig. 1) | ≈ −0.70781 (root of G_-) | Braak 2011 |
parity-+ root count | g=0.7, Δ=0.4, x ∈ [−1,5] | 6 | Braak 2011 |
parity-− root count | g=0.7, Δ=0.4, x ∈ [−1,5] | 5 (incl. ground state) | Braak 2011 |
| energy from root | any | E = x − g²/ω | Braak 2011 |
| exceptional eigenvalues | K_n(n)=0 | E = nω − g²/ω | Braak 2011 |
T6 / Tier D (exact in the N→∞ limit): after disorder averaging, the 1/N^3 variance makes only "melonic" diagrams survive, and the theory closes into the Schwinger–Dyson equations Maldacena–Stanford 2016 Eq. (2.6), 1/G(iω) = -iω - Σ(iω), Σ(τ) = J^2 G(τ)^{q-1}, which are *exact* at N=∞. Their strong-coupling (IR) solution is conformal, G_c(τ)∝\mathrm{sgn}(τ)/|τ|^{2Δ} with fermion dimension Δ=1/q=1/4, and the model is a non-Fermi liquid: maximal chaos (Lyapunov exponent saturating λ_L=2πk_BT/ℏ) and an extensive zero-temperature entropy S_0/N≈0.2324. Not exact at finite N: the finite-N model is fully chaotic (random-matrix level statistics) and requires ED. So this is a D-tier / P-script card — the exact content is the N→∞ limit (tabulated conformal data + the scripted SD entropy trend) and the finite-N ED level statistics are exact reference numbers for whatever N is diagonalised, *not* a thermodynamic-limit claim.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
⟨r̃⟩ (level class) | N=16, mod 8=0 | 0.531 → GOE 0.5307 | Garcia–Garcia–Verbaarschot 2016, Atas–Et–Al 2013 |
⟨r̃⟩ (level class) | N=12, mod 8=4 | 0.678 → GSE 0.6744 | You–Ludwig–Xu 2017, Atas–Et–Al 2013 |
| Kramers pair-gap | N=12 (GSE sector) | <1e-9 (doubling exact) | this card (identity-proof) |
Δ (fermion dimension) | q=4 | 1/4 (tabulated) | Maldacena–Stanford 2016 |
conformal amplitude b | βJ=40 | 0.524 → (4π)^{-1/4}=0.5311 | this card / Maldacena–Stanford 2016 |
S/N high-T | βJ→0 | \tfrac12\ln2=0.34657 | this card |
S/N at βJ=40 | SD, observed | ≈0.241 (decreasing → S_0) | this card (observed) |
S_0/N (zero-T entropy) | q=4, N→∞ | ≈0.2324 (tabulated) | Maldacena–Stanford 2016 |
E/N (ground energy) | SD βJ=40 | ≈-0.0405 | this card / Garcia–Garcia–Verbaarschot 2016 |
T6 (collective / large-N): the model is the infinite-range (mean-field) TFIM — every pair couples equally, so H is a function of the total spin alone, [S², H] = 0, and the ground state sits in the maximal-spin sector j = N/2 (dimension N+1). Quantizing along z makes S_z² diagonal and S_x a ladder, giving a tridiagonal (N+1)-block diagonalized to machine precision at any finite N. The Tier-D framing is deliberate: the collective diagonalization is *exact at every finite N*, but the *thermodynamic* results — the closed-form e₀(h), the h = J quantum critical point, and the mean-field exponents — are exact only in the N → ∞ limit, where spin-coherent (mean-field) minimization becomes exact for all-to-all coupling. The finite-N block approaches them as O(1/N). Not exact (at finite N): the closed-form energies and the sharp transition; both are limit statements. What *is* exact at finite N is the N+1-dimensional collective spectrum itself.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e0_thermodynamic | h = J (critical) | −J | Botet–Jullien 1983 |
e0_thermodynamic | ferromagnetic, h = 0.5, J = 1 | −0.625 | Botet–Jullien 1983 |
e0_thermodynamic | paramagnetic, h = 1.5, J = 1 | −1.5 | Botet–Jullien 1983 |
mz_order_thermo | ferromagnetic, h = 0.5, J = 1 | √3/2 ≈ 0.8660 | Botet–Jullien 1983 |
| critical field | any | h = J (mean-field, β = 1/2) | Botet–Jullien 1983 |
T6 (random): the classical Gaussian ensembles are exactly characterised at the level of their spectral correlation functions. Two closed-form ingredients are pinned here. (1) Wigner spacing surmise: the 2×2 result P_β(s) = a_β s^{β} e^{-b_β s^2} is, to <1% deviation, the large-N nearest-neighbour spacing distribution Wigner 1951; the two constants a_β, b_β are fixed uniquely by imposing unit normalisation ∫P ds = 1 and unit mean spacing ∫sP ds = 1 — the oracle re-derives them numerically to 1e-8. (2) Ratio surmise: Atas–Bogomolny–Giraud–Roux Atas–Et–Al 2013 computed the exact 3×3 distribution P(r) = Z_β^{-1}(r+r^2)^{β}/(1+r+r^2)^{1+3β/2} and the mean ⟨r̃⟩; the Poisson value ⟨r̃⟩ = 2\ln 2 - 1 is exact (integrable levels), while the three Gaussian ensembles give the numerical large-matrix values 0.5307/0.5996/0.6744. Tier A: the spacing/ratio distributions are known in closed form; the sampled anchors are exact realisations of these ensembles, and the GSE construction is a genuine quaternionic self-dual matrix whose Kramers doubling is proven numerically before deduplication.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
⟨r̃⟩ Poisson | integrable levels | 2\ln2-1 = 0.386294 (exact) | Atas–Et–Al 2013 |
⟨r̃⟩ GOE | β=1, large matrix | 0.5307 (surmise 4-2\sqrt3=0.53590) | Atas–Et–Al 2013 |
⟨r̃⟩ GUE | β=2, large matrix | 0.5996 (surmise 0.60266) | Atas–Et–Al 2013 |
⟨r̃⟩ GSE | β=4, large matrix | 0.6744 (surmise 0.67617) | Atas–Et–Al 2013 |
sampled ⟨r̃⟩ GSE | n=350, 20 real, seed 1 | 0.675 (in 0.6744±0.013) | this card (measured) |
| Kramers pair-gap | GSE n=350 | <1e-9 (doubling exact) | this card (identity-proof) |
| GUE pair-gap (control) | n=400 | >1e-3 (no doubling) | this card (negative control) |
| Wigner surmise norm/mean | β∈{1,2,4} | ∫P=∫sP=1 to 1e-8 | this card Wigner 1951 |
T6 / Tier D (exact in d=∞): because each n^f_i is conserved, tracing out the f-electrons leaves, for every frozen f-configuration, a free-fermion c-problem — the partition function is a *classical* sum over f-configurations of free-fermion determinants (manifestly positive, hence sign-free). In the limit Z→∞ the c-electron self-energy becomes purely local, and the lattice problem collapses onto a single-site impurity coupled to a self-consistent bath that closes exactly Brandt–Mielsch 1989: the DMFT equations are not an approximation but the exact d=∞ solution. This yields the exact c-spectral function, the metal–insulator transition (the DOS splits into lower/upper sub-bands separated by a gap ≈U at large U), and the checkerboard charge-density-wave transition temperature T_c(U) Freericks–Zlatic 2003. Not exact in finite d: finite-dimensional Falicov–Kimball is not integrable (though still sign-free); the DMFT solution and the closed-form results below are d=∞ statements. The finite-L numbers pinned here are exact ED references for a specific cluster, not thermodynamic or d=∞ values.
| Quantity | Params | Exact value | Source |
|---|---|---|---|
annealed GS E_0 | U=0 (free) | -4.0000000000 | finite-L ED reference, this card |
annealed GS E_0 | U=2 | -2.0644951022 | finite-L ED reference, this card |
annealed GS E_0 | U=4 | -1.3005630797 | finite-L ED reference, this card |
optimal f-config | U=2 | checkerboard 101010 | this card (identity-proof) |
T=0 order (bipartite, half-filling) | d=∞, any U>0 | checkerboard CDW (exact) | Brandt–Mielsch 1989, Freericks–Zlatic 2003 |
CDW T_c(U) | d=∞, half-filling | >0, non-monotonic in U | Freericks–Zlatic 2003 |
c-charge gap | d=∞, large U | ≈U (band splitting) | Freericks–Zlatic 2003 |
exact maps between models and exactly evolvable dynamics — Kramers–Wannier to dual-unitary circuits
T7: Kramers–Wannier duality maps the ordered (low-T) and disordered (high-T) Ising descriptions onto each other, and its self-dual fixed point locates the exact critical point. Two exact, scripted faces:
| Quantity | Params | Exact value | Source |
|---|---|---|---|
e_0(J,h) − e_0(h,J) (TFIM) | L=16, any (J,h) | 0 (< 10^{-12}) | Kramers–Wannier 1941 |
self-dual invariant residual g(K)−g(K^\ast) | K∈\{0.3,0.6\} | 0 (< 10^{-8}) | this card (derived) |
\sinh 2K\,\sinh 2K^\ast | any K | 1 | Kramers–Wannier 1941 |
self-dual coupling K_c | \sinh 2K_c=1 | \tfrac12\ln(1+\sqrt2)=0.4406867935 | Kramers–Wannier 1941 |
K_c vs Onsager J/T_c | — | equal (< 10^{-12}) | this card + ising-2d-onsager |
T7: exact model-to-model maps. A duality is not itself a solution — it transports an exact result from one card to another (free-fermionization, a critical-point location, a transfer-matrix/Hamiltonian limit, a strong-coupling projection). This card is the index of those maps; the arithmetic lives on the linked cards. It is a T-flag pointer card by design: nothing here is independently scripted.
T7: dual-unitarity is an exact algebraic property (a single reshuffled-gate unitarity condition), and it makes the light-cone dynamics exactly computable. Everything reported is exact for any L: (a) the dual-unitarity certificate ‖Ũ†Ũ − 1‖; (b) infinite-temperature two-point correlators ⟨σ^z_x(t) σ^z_0(0)⟩/2^L, which vanish exactly off the light cone and on the edge equal a single-site quantum channel iterated 2t times Bertini–Kos–Prosen 2019. Not exact / out of scope: correlators at finite temperature or of multi-site operators, entanglement growth (exactly linear for dual-unitary but not scripted here), and the spectral form factor of a *single* dual-unitary circuit — the SFF lives on the sibling kicked-ising-floquet card. Finite-size caveat (scripted): on an L-ring the two light cones wrap once 2\cdot 2t \ge L, so the exact off-cone *vanishing* is asserted only in the clean window t \le L/4; the on-edge value stays exact as long as the ±2t edges do not collide (checked to t=3 at L=10).
| Quantity | Params | Exact value | Source |
|---|---|---|---|
‖Ũ†Ũ − 1‖ (self-dual KIM gate) | J=b=π/4 | 0 (< 10^{-15}) | Bertini–Kos–Prosen 2019 |
‖Ũ†Ũ − 1‖ (control) | J=π/4, b=π/6 | 3/4 (fails) | this card (measured) |
off-cone ⟨σ^z_x(t) σ^z_0⟩/2^L | generic DU gate, x≠±2t, t≤L/4 | 0 (< 10^{-12}) | Bertini–Kos–Prosen 2019 |
on-edge ⟨σ^z_{-2t}(t) σ^z_0⟩/2^L | generic DU gate (J_z=0.7), t=1 | -0.170542 = (S^2)_{zz} | this card (channel) |
on-edge ⟨σ^z_{-2t}(t) σ^z_0⟩/2^L | same, t=2 | -0.457743 = (S^4)_{zz} | this card (channel) |
on-edge ⟨σ^z_{-2t}(t) σ^z_0⟩/2^L | same, t=3 (L=10) | +0.878273 = (S^6)_{zz} | this card (channel) |
T7: at the self-dual point the model is exactly solvable *for spectral statistics*. The exact statement is the disorder-averaged spectral form factor Bertini–Kos–Prosen 2018. Define, for integer Floquet time t > 0,
| Quantity | Params | Exact value | Source |
|---|---|---|---|
BKP odd-t SFF (thermodynamic limit) | t odd, t≤5 | \overline{K(t)} = 2t-1 | Bertini–Kos–Prosen 2018 |
BKP odd-t SFF (thermodynamic limit) | t odd, t≥7 | \overline{K(t)} = 2t | Bertini–Kos–Prosen 2018 |
| RMT comparison curve | COE, N=2^L | K_{COE}=2t-t\ln(1+2t/N) | Bertini–Kos–Prosen 2018 |
self-dual K(1),K(3),K(5) (measured) | L=8, 400 real, seed 7 | 1.01, 5.30, 8.06 (vs 1,5,9) | this card (measured) |
self-dual K(1),K(3),K(5) (measured) | L=10, 80 real, seed 7 | 0.94, 4.52, 8.50 (vs 1,5,9) | this card (measured) |
control K(1) (b=π/5) | L=8 / L=10, seed 7 | 8.7 / 13.9 (off band) | this card (measured) |