Transverse-Field Ising Model (TFIM)

Overview

The one-dimensional transverse-field Ising model is one of the simplest quantum many-body systems exhibiting a quantum phase transition. It is exactly solvable via Jordan-Wigner transformation followed by Bogoliubov-de Gennes (BdG) diagonalization.

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

Parameters: Ising coupling $J$ (default 1.0), transverse field $h$.

Phase diagram:

• \[h/J < 1\]

  : Ferromagnetic ordered phase ($\langle \sigma^z \rangle \neq 0$)

• \[h/J = 1\]

  : 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).

Boundary Conditions

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

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

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).

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:

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

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

# Ground-state energy (β → ∞), OBC, N=16
E₀ = QAtlas.fetch(:TFIM, :energy, OBC(); N=16, J=1.0, h=0.5)

# Finite-temperature energy
E_β = QAtlas.fetch(:TFIM, :energy, OBC(); N=16, J=1.0, h=0.5, beta=2.0)

# Thermodynamic limit (PBC, N→∞)
ε = QAtlas.fetch(:TFIM, :energy, Infinite(); J=1.0, h=0.5, 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 $\{\Lambda_n\}$:

Derivation

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

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

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

References

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

QAtlas API

F = QAtlas.fetch(:TFIM, :free_energy, OBC(); N=16, J=1.0, h=0.5, beta=2.0)
S = QAtlas.fetch(:TFIM, :entropy, OBC(); N=16, J=1.0, h=0.5, beta=2.0)
Cv = QAtlas.fetch(:TFIM, :specific_heat, OBC(); N=16, J=1.0, h=0.5, beta=2.0)

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 diagonalization 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 ($h > J$): $\Delta \approx 2(h - J)$, i.e., 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: gap closes)

# OBC finite-N — smallest positive BdG eigenvalue
QAtlas.fetch(TFIM(; J=1.0, h=1.0), MassGap(), OBC(32))
# → π J / N ≈ 0.098   (finite-size CFT gap)

Symbol aliases recognised by the legacy layer include :mass_gap, :gap, :Δ, :excitation_gap, and the capitalised :MassGap.

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 | h-J | $; $\nu z = 1$ from log-log regression |

Central Charge from Entanglement Entropy

Statement

At the critical point $h = J$, the entanglement entropy of a subsystem of $l$ consecutive sites in an $N$-site OBC chain obeys the Calabrese-Cardy formula:

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

with central charge $c = 1/2$ (Ising CFT). See the Calabrese-Cardy method page for the general formula, 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 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 fetch(TFIM(; J, h), VonNeumannEntropy(), OBC(N); ℓ) — 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: TFIM entanglement entropy .

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, block of ℓ = N/2 sites at criticality:
QAtlas.fetch(TFIM(; J=1.0, h=1.0), VonNeumannEntropy(), OBC(100); ℓ=50)
# → 0.7256…  (matches (c/6) log((2N/π) sin(πℓ/N)) + s_1 with c = 1/2)

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

Symbol aliases recognised by the legacy layer: :entanglement_entropy, :ee, :EE, :S_vN, :EntanglementEntropy.

Verification

Connections

• Universality: Ising universality class — the TFIM critical point has $c = 1/2$, $\nu = 1$, $z = 1$.
• 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 IRFP 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.