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.

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:

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

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)

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:

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:

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

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

Finite-Temperature Observables

Statement

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

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

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

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

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.

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