TFIM — Transverse-Field Ising Model

Overview

The one-dimensional transverse-field Ising model is the canonical exactly-solvable quantum many-body system with a quantum phase transition. QAtlas solves it via the Jordan-Wigner transformation followed by Bogoliubov-de Gennes (BdG) diagonalisation; PBC additionally requires parity-projected (NS+R) sectors.

Hamiltonian

\[H = -J \sum_{i} \sigma^z_i \sigma^z_{i+1} - h \sum_{i} \sigma^x_i\]

(σ-convention: eigenvalues ±1.) Parameters: Ising coupling $J$ (default 1.0), transverse field $h$.

Phase diagram:

\[h/J < 1\]
: ferromagnetic ordered phase ($\langle \sigma^z\rangle\neq0$)
\[h/J = 1\]
: quantum critical point (Ising CFT, $c=1/2$)
\[h/J > 1\]
: quantum paramagnetic phase ($\langle \sigma^z\rangle = 0$)

Universality: the critical point belongs to the 2D Ising universality class via the quantum-classical mapping (1+1D quantum ↔ 2D classical).

Coverage Matrix

The table below reflects TFIM_registry.jl @register entries. ✅ marks a native fetch method; "conversion" means routed through another granularity by core/registry.jl.

Quantity	OBC	PBC	Infinite
`Energy` `{:total}`	✅ BdG	conversion	—
`Energy` `{:per_site}`	conversion	✅ BdG (NS+R)	✅ closed-form
`FreeEnergy`	✅	✅ NS+R	✅
`ThermalEntropy`	✅	✅ NS+R	✅
`SpecificHeat`	✅	✅ NS+R	✅
`MagnetizationX`	✅	✅ NS+R	✅
`MagnetizationZ`	0 by Z₂	—	✅ Pfeuty $m_z = (1-(h/J)^2)^{1/8}$
`SusceptibilityXX`	✅ variance	✅ NS+R	✅ Kubo (Calabrese-Mussardo)
`SusceptibilityZZ`	✅ Wick	—	✅ `N_proxy=80`
`CorrelationLength`	—	—	✅ $\xi = 1/(2\lvert h-J\rvert)$
`MassGap`	✅	✅	✅ $\Delta = 2\lvert h-J\rvert$
`XXCorrelation` `{:static}`	✅ Pfaffian	—	✅ proxy
`XXCorrelation` `{:connected}`	✅	—	✅
`XXCorrelation` `{:dynamic}`	✅	—	—
`ZZCorrelation` `{:static}`	✅	—	—
`ZZCorrelation` `{:connected}`	✅ (= static, Z₂)	—	—
`ZZCorrelation` `{:dynamic}`	✅	—	—
`ZZCorrelation` `{:lightcone}`	✅	—	—
`ZZStructureFactor`	✅	—	✅ static + dynamic (proxy)
`VonNeumannEntropy`	✅ Peschel	—	✅ CC (T=0 crit/gapped + T>0 crit)
`RenyiEntropy(α)`	✅ Peschel	—	✅ CC
`EnergyLocal`	✅	—	—
`MagnetizationXLocal` `{:equilibrium}`	✅	—	—
`MagnetizationXLocal` `{:quench}`	✅	—	✅ closed-form k-integral (#145)
`MagnetizationZLocal`	✅	—	—
`SpontaneousMagnetization`	—	—	✅ alias of `MagnetizationZ`
`CentralCharge`	—	—	✅ 1/2 (critical) / 0

YY observables (YYCorrelation, SusceptibilityYY, MagnetizationY) are intentionally not implemented — the σʸ JW string makes OBC contractions expensive; tracked as Tier 3 in issue #110.

Boundary Conditions

QAtlas supports three boundary conditions for the TFIM, each with different physical content:

BC	`fetch` argument	BdG size	Physical setting
OBC	`OBC(N)`	$2N \times 2N$ (numerical)	Open chain, $N$ sites, $N-1$ bonds
PBC	`PBC(N)`	parity-projected NS+R	Ring of $N$ sites, $N$ bonds
Infinite	`Infinite()`	$k$-integral	Thermodynamic limit, PBC $N \to \infty$

OBC: the BdG matrix is diagonalised numerically. Boundary effects include the Z₂ tunneling splitting in the ordered phase and the $O(1/N)$ boundary correction at criticality. See gap analysis below.

PBC: the JW transformation produces a fermion parity factor that splits the partition function into Neveu-Schwarz (anti-periodic) and Ramond (periodic) sectors with both signs of the parity projector (LSM). QAtlas evaluates all four (NS±, R±). The Ramond k=0 zero mode at criticality is handled explicitly.

Infinite: the quasiparticle dispersion $\Lambda(k) = 2\sqrt{J^2 + h^2 - 2Jh\cos k}$ is integrated over the Brillouin zone using Gauss-Kronrod quadrature (QuadGK.jl).

v0.17 / v0.18 Highlights

Status: Unstable (v0.18.x)

The PBC thermodynamics, Z-axis Infinite surface, XX static / connected via Pfaffian, Calabrese-Cardy entanglement at Infinite, and dynamic structure-factor helpers are new in v0.17–v0.18. Method signatures, granularity conventions, and keyword-argument names (N_proxy, ω, ℓ, beta) may change in v0.19. Call sites should use the public QAtlas.fetch(model, quantity, bc; ...) interface and must not depend on internal helpers (the _tfim_* prefixed functions).

1. PBC free-fermion thermodynamics (v0.17)

Jordan-Wigner with a fermion parity factor splits $Z$ into Neveu-Schwarz and Ramond sectors with both parity-projector signs. QAtlas sums all four sectors (NS+, NS−, R+, R−); at the critical point the R-sector $k=0$ zero mode is handled explicitly.

m  = TFIM(; J=1.0, h=0.5)
β  = 1.0
QAtlas.fetch(m, FreeEnergy(),       PBC(8); beta=β)
QAtlas.fetch(m, MagnetizationX(),   PBC(8); beta=β)
QAtlas.fetch(m, SusceptibilityXX(), PBC(8); beta=β)
QAtlas.fetch(m, MassGap(),          PBC(8))

References: Lieb-Schultz-Mattis (1961); Sachdev §4.2. Source: TFIM_pbc_thermal.jl.

2. Z-axis Infinite — Pfeuty closed forms (v0.17)

Quantity	Formula
`MagnetizationZ` (= `SpontaneousMagnetization`)	$m_z = (1 - (h/J)^2)^{1/8}\;\;(h<J)$, else 0
`CorrelationLength`	$\xi = 1/(2\lvert h-J\rvert)$ (Inf at criticality)
`SusceptibilityZZ`	OBC large-$N$ proxy via `N_proxy` kwarg
`ZZStructureFactor`	static $S_{zz}(q)$ from Fourier of large-$N$ correlator

QAtlas.fetch(TFIM(; J=1.0, h=0.5), MagnetizationZ(),    Infinite())  # ≈ 0.985
QAtlas.fetch(TFIM(; J=1.0, h=0.7), CorrelationLength(), Infinite())  # 1/0.6 ≈ 1.667

Source: TFIM_zaxis.jl.

3. XX static / connected via Pfaffian Wick (v0.18)

OBC static $\langle\sigma^x_i\sigma^x_j\rangle$ is the $t=0$ limit of the existing dynamic Wick contraction, evaluated as a real Pfaffian over the Majorana covariance block. The connected variant subtracts $\langle\sigma^x_i\rangle\langle\sigma^x_j\rangle$. Infinite uses the OBC large-$N$ proxy (N_proxy kwarg).

m = TFIM(; J=1.0, h=0.7)
QAtlas.fetch(m, XXCorrelation{:static}(),    OBC(8); beta=Inf, i=3, j=5)
QAtlas.fetch(m, XXCorrelation{:connected}(), OBC(8); beta=Inf, i=3, j=5)
QAtlas.fetch(m, XXCorrelation{:static}(),    Infinite(); i=3, j=5, N_proxy=80)

YY OBC remains unimplemented (issue #110, Tier 3). Source: TFIM_xx_static.jl.

4. Calabrese-Cardy Infinite entanglement (v0.18)

The thermodynamic-limit von Neumann and Rényi entropies are evaluated in closed form via the Calabrese-Cardy formula. Coverage:

Region	$T = 0$	$T > 0$
Critical (h = J)	$S = (c/3)\,\log(2\ell)$	$S = (c/3)\,\log\!\left[(2\beta/\pi)\sinh(\pi\ell/\beta)\right]$
Gapped (h ≠ J)	$S = (c/6)\,\log(2\xi\,\sinh(\ell/\xi))$	error (deferred — see issue #110)

with $c = 1/2$ for Ising. The Rényi $\alpha\neq 1$ prefactor is $(c/12)(1 + 1/\alpha)$.

QAtlas.fetch(TFIM(; J=1.0, h=1.0), VonNeumannEntropy(), Infinite(); ℓ=50)
# ≈ (1/6) log(100) — critical T=0

QAtlas.fetch(TFIM(; J=1.0, h=0.5), RenyiEntropy(2.0),   Infinite(); ℓ=20)
# Rényi-2, gapped CC

QAtlas.fetch(TFIM(; J=1.0, h=1.0), VonNeumannEntropy(), Infinite();
             ℓ=20, beta=4.0)
# critical T>0

Source: TFIM_cft_entanglement.jl.

5. Dynamic structure factor at Infinite (v0.18, proxy)

ZZStructureFactor at Infinite() is router-dispatched on the optional ω keyword:

ω === nothing → existing static proxy (Fourier of static correlator)
ω::Real → dynamic proxy (time-evolution + Fourier of dynamic correlator)

Two helpers are exported for analytic post-processing:

tfim_quasiparticle_dispersion(model, k) -> Float64 — closed-form Bogoliubov dispersion $\Lambda(k)$.
tfim_two_spinon_dos(model, ω; q_total = 0.0) -> Float64 — two-spinon density of states at fixed total momentum, used to identify the continuum threshold.

m = TFIM(; J=1.0, h=1.0)
QAtlas.fetch(m, ZZStructureFactor(), Infinite(); q=π/2, ω=1.5)
tfim_quasiparticle_dispersion(m, π/2)
tfim_two_spinon_dos(m, 1.5; q_total=0.0)

Closed-form form-factor expansion (Calabrese-Mussardo) is not yet implemented — issue #110. Source: TFIM_infinite_dynamics.jl.

Ground-State Energy

Statement

The ground-state energy of the OBC TFIM with $N$ sites is

\[E_0 = -\sum_{n=1}^{N} \frac{\Lambda_n}{2}\]

where $\{\Lambda_n\}$ are the positive eigenvalues of the $2N \times 2N$ BdG matrix. At finite temperature $\beta = 1/(k_B T)$:

\[\langle H \rangle(\beta) = -\sum_{n=1}^{N} \frac{\Lambda_n}{2} \tanh\!\left(\frac{\beta \Lambda_n}{2}\right)\]

Derivation

The TFIM is solved exactly via the Jordan-Wigner transformation, which maps the spin chain to free fermions after a Kramers-Wannier duality step. The full derivation — including why the duality is needed for the $\sigma^z\sigma^z$ convention and the explicit construction of the BdG matrix — is given in the calculation note JW-TFIM-BdG.

The result is a $2N \times 2N$ real symmetric BdG matrix whose eigenvalues come in $\pm\Lambda_n$ pairs. The positive eigenvalues $\Lambda_n > 0$ are the quasiparticle energies, and the total energy at inverse temperature $\beta$ is:

\[\langle H \rangle = -\sum_n \frac{\Lambda_n}{2} \tanh\!\left(\frac{\beta \Lambda_n}{2}\right)\]

Thermodynamic limit

For PBC in the $N \to \infty$ limit, the quasiparticle dispersion is $\Lambda(k) = 2\sqrt{J^2 + h^2 - 2Jh\cos k}$, and the energy per site becomes a $k$-integral evaluated by Gauss-Kronrod quadrature (QuadGK.jl).

References

P. Pfeuty, "The one-dimensional Ising model with a transverse field", Ann. Phys. 57, 79 (1970) — exact solution of the 1D TFIM.
E. Lieb, T. Schultz, D. Mattis, "Two Soluble Models of an Antiferromagnetic Chain", Ann. Phys. 16, 407 (1961) — JW transformation for spin chains.
S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2011), Ch. 5 — pedagogical treatment.

QAtlas API

m = TFIM(; J=1.0, h=0.5)

# Ground-state energy (β → ∞), OBC, N=16 — total
E₀ = QAtlas.fetch(m, Energy{:total}(), OBC(16))

# Finite-temperature total energy
Eβ = QAtlas.fetch(m, Energy{:total}(), OBC(16); beta=2.0)

# Thermodynamic limit (PBC, N→∞) — per site
ε  = QAtlas.fetch(m, Energy{:per_site}(), Infinite(); beta=2.0)

Verification

Test file	Method	What is checked
`test_tfim_gap_closure.jl`	Dense ED via `build_tfim`	$E_0^{\text{ED}} = E_0^{\text{BdG}}$ for $N = 4, 6, 8$
`test_universality_cross_check.jl`	BdG at $N = 200$	$E_0/N \to -4J/\pi$ at $h = J$

Finite-Temperature Observables

Statement

At inverse temperature $\beta$ and for $N$ sites (OBC), the following quantities are computed from the BdG spectrum $\{\Lambda_n\}$:

Quantity	Formula	Type
Free energy	$F = -\frac{1}{\beta}\sum_n \ln\!\left[2\cosh(\beta\Lambda_n/2)\right]$	`FreeEnergy`
Entropy	$S = \beta(\langle H \rangle - F)$	`ThermalEntropy`
Specific heat	$C_v = -\beta^2\,\partial \langle H \rangle / \partial \beta$	`SpecificHeat`
Mag. (X)	$\langle\sigma^x\rangle$ from the Bogoliubov occupation	`MagnetizationX`
Susc. (XX)	Variance of $\sum_i \sigma^x_i$ (OBC); Kubo at Infinite	`SusceptibilityXX`

PBC ⇒ all of the above with parity-projected NS+R sums (v0.17).

Derivation

All quantities follow from the free-fermion partition function. For independent modes with energies $\Lambda_n$:

\[\mathcal{Z} = \prod_n 2\cosh\!\left(\frac{\beta\Lambda_n}{2}\right)\]

The free energy is $F = -\beta^{-1}\ln\mathcal{Z}$, and all other thermodynamic quantities follow from $\beta$-derivatives.

References

S. Sachdev, Quantum Phase Transitions (2011), Ch. 5.3.
QAtlas: src/models/quantum/TFIM/TFIM_thermal.jl, TFIM_pbc_thermal.jl — full implementation.

QAtlas API

m = TFIM(; J=1.0, h=0.5)
β = 2.0

F  = QAtlas.fetch(m, FreeEnergy(),       OBC(16); beta=β)
S  = QAtlas.fetch(m, ThermalEntropy(),   OBC(16); beta=β)
Cv = QAtlas.fetch(m, SpecificHeat(),     OBC(16); beta=β)
Mx = QAtlas.fetch(m, MagnetizationX(),   OBC(16); beta=β)
χ  = QAtlas.fetch(m, SusceptibilityXX(), OBC(16); beta=β)

# PBC variants (NS+R) — v0.17
F_pbc = QAtlas.fetch(m, FreeEnergy(),     PBC(16); beta=β)
Mx_pbc = QAtlas.fetch(m, MagnetizationX(), PBC(16); beta=β)

# Infinite — closed-form k-integrals
F_inf  = QAtlas.fetch(m, FreeEnergy(),     Infinite(); beta=β)
Mx_inf = QAtlas.fetch(m, MagnetizationX(), Infinite(); beta=β)

Verification

Test file	Method	What is checked
`test_TFIM_thermal.jl`	Dense ED ($N \leq 10$)	Exact match of $F$, $S$, $C_v$, $M_x$ vs. ED
`test_TFIM_pbc_thermal.jl`	NS+R vs. ED ($N\leq8$)	PBC parity-projected sums match exact ring partition

Energy Gap and Quantum Phase Transition

Statement

The many-body energy gap $\Delta = E_1 - E_0$ equals the smallest BdG quasiparticle energy $\Lambda_{\min}$. In the thermodynamic limit:

\[\Delta = 2|J - h|\]

At the critical point $h = J$, the gap closes as $\Delta \sim N^{-z}$ with dynamic exponent $z = 1$.

Physical Context

Ordered phase ($h < J$): for OBC with finite $N$, the "gap" seen by exact diagonalisation is actually the Z₂ tunneling splitting between $|\!\uparrow\cdots\uparrow\rangle$ and $|\!\downarrow\cdots\downarrow\rangle$, which is exponentially small in $N$. This is distinct from the physical excitation gap $\Delta \approx 2(J - h)$.
Critical point ($h = J$): $\Delta \sim \pi/N$ (finite-size gap for OBC).
Disordered phase ($h > J$): $\Delta \approx 2(h - J)$, the paramagnetic gap.

References

P. Pfeuty, Ann. Phys. 57, 79 (1970), Eq. (3.6).
S. Sachdev, Quantum Phase Transitions (2011), §5.5.

QAtlas API

# Infinite chain — closed form Δ = 2|h − J|
QAtlas.fetch(TFIM(; J=1.0, h=0.3), MassGap(), Infinite())   # 1.4
QAtlas.fetch(TFIM(; J=1.0, h=1.0), MassGap(), Infinite())   # 0.0  (critical)

# OBC finite-N — smallest positive BdG eigenvalue
QAtlas.fetch(TFIM(; J=1.0, h=1.0), MassGap(), OBC(32))      # ≈ π/N

# PBC finite-N — smallest excitation across NS / R sectors (v0.17)
QAtlas.fetch(TFIM(; J=1.0, h=1.0), MassGap(), PBC(16))

Verification

Test file	Method	What is checked
`test_tfim_gap_closure.jl`	Dense ED ($N = 4$–$12$)	Gap shrinks with $N$ at $h = J$
`test_tfim_gap_closure.jl`	ED	Ordered-phase gap is Z₂ tunneling ($< 10^{-3}$ for $N=6$)
`test_universality_cross_check.jl`	BdG ($N = 200$)	$\Delta \approx 2\lvert h-J\rvert$; $\nu z = 1$ from log-log regression

Entanglement Entropy at OBC (Peschel)

Statement

At the critical point $h = J$, the entanglement entropy of a contiguous block of $\ell$ sites in an $N$-site OBC chain obeys the Calabrese-Cardy formula:

\[S(\ell) = \frac{c}{6}\ln\!\left[\frac{2N}{\pi}\sin\!\left(\frac{\pi \ell}{N}\right)\right] + s_1\]

with central charge $c = 1/2$ (Ising CFT). See the Calabrese-Cardy method page for OBC vs. PBC prefactors and extraction procedure.

Physical Context

The TFIM is a free-fermion system after Jordan-Wigner transformation, so the reduced density matrix on a contiguous block of $\ell$ spins is Gaussian and its von Neumann (or Rényi) entropy is computable in $O(\ell^3)$ from the Majorana covariance matrix restricted to that block (Peschel's correlation-matrix method). QAtlas exposes this directly via VonNeumannEntropy and RenyiEntropy at OBC — no Kramers-Wannier detour is needed, because the internal $\sigma^x$-string JW convention puts the Majorana pair $(\gamma_{2i-1}, \gamma_{2i})$ on spin site $i$ directly, and the JW transformation factorises across any contiguous bipartition up to a parity factor on $A$ that commutes with $\rho_A$ (Fagotti-Calabrese 2010).

Full derivation of the per-mode entropy formula $S_A = \sum_k s_2(\nu_k)$ from the Gaussian-preservation theorem, the Majorana-covariance canonical form, and the contiguous-block JW factorisation: Peschel correlation-matrix method.

References

P. Calabrese, J. Cardy, J. Stat. Mech. 0406, P06002 (2004), Eq. (19).
I. Peschel, J. Phys. A 36, L205 (2003), Eq. (9).
G. Vidal, J. I. Latorre, E. Rico, A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
M. Fagotti, P. Calabrese, Phys. Rev. Lett. 104, 227203 (2010).

QAtlas API

# Ground-state von Neumann, ℓ = N/2 at criticality
QAtlas.fetch(TFIM(; J=1.0, h=1.0), VonNeumannEntropy(), OBC(100); ℓ=50)
# ≈ 0.7256  ((c/6) log((2N/π) sin(πℓ/N)) + s_1, c = 1/2)

# Thermal von Neumann at β = 1
QAtlas.fetch(TFIM(; J=1.0, h=1.0), VonNeumannEntropy(), OBC(100); ℓ=50, beta=1.0)

# Rényi α ≠ 1 (v0.18)
QAtlas.fetch(TFIM(; J=1.0, h=1.0), RenyiEntropy(2.0), OBC(100); ℓ=50)

Verification

Test file	Method	What is checked
`test_TFIM_entanglement.jl`	Peschel vs. full ED SVD	Machine-precision agreement for every $\ell$ at $N = 10$, three $(J, h)$ points
`test_TFIM_entanglement.jl`	Peschel ($N = 100$)	Extracted $c \approx 0.5$ within 5% at criticality
`test_TFIM_entanglement.jl`	Peschel	Symmetric $S(\ell) = S(N-\ell)$, area law away from criticality
`test_TFIM_renyi.jl`	Peschel α-trace	Rényi $\alpha = 2, 3$ matches small-$N$ ED
`test_TFIM_cft_entanglement.jl`	CC at Infinite	Critical T=0/T>0 and gapped T=0 closed forms vs. analytic
`test_entanglement_central_charge.jl`	ED ($N \le 14$)	$c_{\text{extracted}} \approx 0.5$ within 10%

Coverage by Reference

Physical / methodological backing of each fetch surface:

BdG (OBC ground / thermal): Pfeuty 1970.
PBC parity projection (NS+R): Lieb-Schultz-Mattis 1961; Sachdev §4.2.
Peschel correlation matrix (entanglement, OBC): Peschel 2003; Calabrese-Cardy 2004; Fagotti-Calabrese 2010.
Calabrese-Cardy (entanglement, Infinite): Calabrese-Cardy 2004,
Pfaffian Wick (XX static / connected): Wick 1950 + free-fermion Σ contraction.
Pfeuty closed forms (Z-axis Infinite): Pfeuty 1970 (spontaneous magnetisation, correlation length).
Two-spinon DOS / dispersion helpers: standard Bogoliubov dispersion + convolution; see Calabrese-Mussardo for the form-factor programme (not yet implemented).

Connections

Universality: Ising universality class — $c = 1/2$, $\nu = 1$, $z = 1$.
Classical counterpart: IsingSquare — the 1+1D TFIM maps to the 2D classical Ising model via the quantum-classical correspondence ($\beta_{\text{classical}} \leftrightarrow$ imaginary time).
Disordered version: Random TFIM — the Fisher infinite-randomness fixed point at $[\ln J]_{\text{avg}} = [\ln h]_{\text{avg}}$.
E8 spectrum: E8 universality — perturbing the critical TFIM at $h = J$ by a longitudinal field $\lambda \sigma^z$ is the $\Phi_{(1,2)} = \sigma$ magnetic perturbation of the Ising CFT. Zamolodchikov (1989) showed the resulting massive field theory remains integrable and its eight stable particles realise the $E_8$ mass spectrum.

API

Modules = [QAtlas] at the end of index.md already pulls docstrings for the exported observable types and TFIM helpers (tfim_quasiparticle_dispersion, tfim_two_spinon_dos); no @autodocs block is needed here.

<!– ATLAS:HUBS:START – auto-generated by docs/atlas/generate.jl. Do not edit by hand; edits between these markers are overwritten on next regen. –>

Verified hubs

In the Verified Atlas, this model registers 61 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

Quantity	BC	Assurance	Cards
`CentralCharge`	`Infinite`	🟢 corroborated-at-p	5
`ConformalTower`	`OBC`	🟠 uncorroborated-but-feasible	0
`ConformalTower`	`PBC`	🟠 uncorroborated-but-feasible	0
`CorrelationLength`	`Infinite`	🟢 corroborated-at-p	3
`CriticalExponents`	`Infinite`	🟠 uncorroborated-but-feasible	0
`Energy`	`Infinite`	🟢 corroborated-at-p	12
`Energy`	`OBC`	🟢 corroborated-at-p	58
`Energy`	`PBC`	🟢 corroborated-at-p	16
`EnergyLocal`	`OBC`	🟠 uncorroborated-but-feasible	0
`FidelitySusceptibility`	`Infinite`	🟢 corroborated-at-p	2
`FidelitySusceptibility`	`OBC`	🟢 corroborated-at-p	9
`FreeEnergy`	`Infinite`	🟢 corroborated-at-p	9
`FreeEnergy`	`OBC`	🟢 corroborated-at-p	45
`FreeEnergy`	`PBC`	🟢 corroborated-at-p	30
`GGEValue`	`Infinite`	🔵 coherent	1
`LiebRobinsonBound`	`Infinite`	🟠 uncorroborated-but-feasible	0
`LiebRobinsonVelocity`	`Infinite`	🟠 uncorroborated-but-feasible	0
`LoschmidtEcho`	`Infinite`	🟢 corroborated-at-p	6
`LoschmidtEcho`	`OBC`	🟢 corroborated-at-p	12
`MagnetizationX`	`Infinite`	🟢 corroborated-at-p	6
`MagnetizationX`	`OBC`	🟢 corroborated-at-p	18
`MagnetizationX`	`PBC`	🟢 corroborated-at-p	18
`MagnetizationXLocal`	`Infinite`	🔵 coherent	2
`MagnetizationXLocal`	`OBC`	🟠 uncorroborated-but-feasible	0
`MagnetizationY`	`OBC`	🟢 corroborated-at-p	3
`MagnetizationZ`	`Infinite`	🟢 corroborated-at-p	5
`MagnetizationZLocal`	`OBC`	🟠 uncorroborated-but-feasible	0
`MassGap`	`Infinite`	🟢 corroborated-at-p	25
`MassGap`	`OBC`	🟢 corroborated-at-p	1
`MassGap`	`PBC`	🟠 uncorroborated-but-feasible	0
`NMRRelaxationExponent`	`Infinite`	🟠 uncorroborated-but-feasible	0
`NMRSpinRelaxationRate`	`Infinite`	🟠 uncorroborated-but-feasible	0
`NMRSpinRelaxationRate`	`OBC`	🟠 uncorroborated-but-feasible	0
`RenyiEntropy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`RenyiEntropy`	`OBC`	🟢 corroborated-at-p	74
`SpecificHeat`	`Infinite`	🔵 coherent	7
`SpecificHeat`	`OBC`	🟢 corroborated-at-p	40
`SpecificHeat`	`PBC`	🟢 corroborated-at-p	18
`SpontaneousMagnetization`	`Infinite`	🟢 corroborated-at-p	7
`SusceptibilityXX`	`Infinite`	🟠 uncorroborated-but-feasible	0
`SusceptibilityXX`	`OBC`	🟢 corroborated-at-p	18
`SusceptibilityXX`	`PBC`	🟢 corroborated-at-p	12
`SusceptibilityYY`	`OBC`	🟢 corroborated-at-p	18
`SusceptibilityZZ`	`Infinite`	🟠 uncorroborated-but-feasible	0
`SusceptibilityZZ`	`OBC`	🟢 corroborated-at-p	18
`ThermalEntropy`	`Infinite`	🔵 coherent	11
`ThermalEntropy`	`OBC`	🟢 corroborated-at-p	41
`ThermalEntropy`	`PBC`	🟢 corroborated-at-p	15
`UniversalityClass`	`Infinite`	🟠 uncorroborated-but-feasible	0
`VonNeumannEntropy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`VonNeumannEntropy`	`OBC`	🟢 corroborated-at-p	42
`XXCorrelation`	`Infinite`	🟢 corroborated-at-p	21
`XXCorrelation`	`OBC`	🟢 corroborated-at-p	1
`XXStructureFactor`	`Infinite`	🔵 coherent	14
`XXStructureFactor`	`OBC`	🟢 corroborated-at-p	27
`YYCorrelation`	`OBC`	🟢 corroborated-at-p	1
`YYStructureFactor`	`Infinite`	🔵 coherent	14
`YYStructureFactor`	`OBC`	🟢 corroborated-at-p	27
`ZZCorrelation`	`OBC`	🟢 corroborated-at-p	3
`ZZStructureFactor`	`Infinite`	🔵 coherent	14
`ZZStructureFactor`	`OBC`	🔵 coherent	24

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API

Every fetch(::Model, …) method registered for this model — together with the model struct(s) and exported helpers — generated directly from the source (in lock-step with @register):

QAtlas.TFIM — Type

TFIM(; J = 1.0, h = 1.0) <: AbstractQAtlasModel

The 1D transverse field Ising model with Hamiltonian

H = -J Σ_i σᶻ_i σᶻ_{i+1} - h Σ_i σˣ_i

J > 0 is ferromagnetic, h is the transverse field. The critical point sits at h = J.

Currently registered fetches:

Quantity	BC	Coverage
`Energy`	`OBC` / `Infinite`	Exact energy computed via BdG transformation
`SpecificHeat`	`Infinite`	Specific heat at finite temperature
`FreeEnergy`	`Infinite`	Free energy density at finite temperature
`ThermalEntropy`	`Infinite`	Thermal entropy density at finite temperature
`UniversalityClass`	`Infinite`	`:Ising` universality class at the critical point `h = J` (flows to `:IsingSDRG` under strong disorder)

Convention	Formula	Origin
Inverse mass gap (this fetch)	`1 / (2	h - J
Pfeuty 1970 longitudinal	`1 / log(max(J,h) / min(J,h))`	lattice JW-fermion <σᶻσᶻ> decay exact
Sachdev lattice relativistic	`min(J,h) /	h - J