---
title: Transverse Longitudinal Field Ising Model
url: https://www.emergentmind.com/topics/transverse-longitudinal-field-ising-model-tlfim
type: topic
---

# Transverse Longitudinal Field Ising Model

The transverse longitudinal field Ising model (TLFIM) denotes quantum Ising systems in which Ising exchange coexists with both a transverse field and a longitudinal field. In the one-dimensional ferromagnetic-chain form, \(H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]\) with \(J>0\), the longitudinal field \(h_z\) breaks the model’s free-fermion integrability when nonzero [2107.08287]. Across dimensions and lattice geometries, closely related Hamiltonians have been used to analyze quantum criticality, fidelity susceptibilities, operator growth, prethermalization, frustrated ordering, random-field effects, and Monte Carlo dynamics [1307.3603], [2103.12364], [2501.02643], [2409.17835].

## 1. Canonical Hamiltonians and sign conventions

Published TLFIM conventions differ by lattice, exchange sign, and field notation, but they all contain an Ising interaction in the \(z\)-channel together with longitudinal \(z\)-field and transverse \(x\)-field terms. The variation is not merely cosmetic: it distinguishes ferromagnetic from antiferromagnetic exchange, uniform from random fields, and regular from frustrated lattices.

| Setting | Hamiltonian | Distinctive feature |
|---|---|---|
| 1D ferromagnetic chain | \(H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]\) | Uniform \(h_z\neq 0\) breaks free-fermion integrability [2107.08287] |
| 2D triangular lattice | \(\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z\) | Fidelity susceptibilities define two independent critical exponents [1307.3603] |
| 2D square-lattice antiferromagnet | \(H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x\) | QMC resolves a narrow reentrant region [2103.12364] |
| 1D random chain | \(H=-\sum_{i=1}^L J_i\sigma_i^z\sigma_{i+1}^z-\sum_{i=1}^L h_i^x\sigma_i^x-\sum_{i=1}^L h_i^z\sigma_i^z\) | SDRG yields multiple fixed points and a separatrix [2501.02643] |

In the one-dimensional operator-growth study, the Pauli matrices satisfy \(\sigma_l^a\sigma_l^b=\delta_{ab}I+i\epsilon_{abc}\sigma_l^c\), and the operator dynamics is formulated in Liouville space [2107.08287]. In the two-dimensional fidelity-susceptibility study, the model is placed on a triangular lattice with ferromagnetic nearest-neighbor coupling set to unity [1307.3603]. In square-lattice antiferromagnetic realizations, \(J>0\) is the antiferromagnetic coupling, and the fields compete with Néel order rather than with a ferromagnetic order parameter [2103.12364], [1207.2119].

## 2. Heisenberg-picture operator growth and chaos diagnostics

For the one-dimensional ferromagnetic TLFIM, operator spreading can be studied directly in the Heisenberg picture by evolving an initial local operator as
\[
O(t)=e^{iHt}O(0)e^{-iHt}.
\]
Equivalently, \(O(t)\) is treated as a state \(|O(t))\) in operator Hilbert space, with dynamics generated by the Liouvillian \(\mathcal L(A)\equiv[H,A]\) and inner product \((A|B)=D^{-1}\mathrm{Tr}[A^\dagger B]\). The associated Krylov basis \(\{|O_n)\}\) is built by the Lanczos recurrence, producing Lanczos coefficients \(b_n\) and amplitudes \(\phi_n(t)=(O_n|O(t))\) satisfying
\[
\dot\phi_n(t)=b_n\phi_{n-1}(t)-b_{n+1}\phi_{n+1}(t),\qquad \phi_n(0)=\delta_{n0},
\]
while the mean depth
\[
(n)_t=\sum_n n\,|\phi_n(t)|^2
\]
defines the Krylov complexity [2107.08287].

In one dimension, Parker et al.’s upper bound implies \(b_n=O(n/\ln n)\), and maximal scrambling corresponds to the asymptotic form
\[
b_n\sim \alpha\,\frac{n}{W(n)},
\]
with \(W(n)\) the Lambert \(W\)-function. When this scaling is realized, the mean depth grows asymptotically as
\[
(n)_t\sim \frac{e^{(2\alpha+o(1))t}}{\sqrt{4\pi\alpha t}}.
\]
The numerics for the TLFIM with uniform longitudinal field show that the operator growth dynamics follows this universal scaling law for one-dimensional chaotic systems [2107.08287].

The integrable limit \(h_z=0\) exhibits two distinct Lanczos-coefficient scalings depending on the initial operator: type I with \(b_n\to\mathrm{const}\), implying linear \((n)_t\), and type II with \(b_n\sim \sqrt n\), implying \((n)_t\sim t^2\). Turning on a small uniform longitudinal field \(h_z\ll 1\) produces a crossover:
\[
1\ll n\ll n_c(h_z):\quad b_n(h_z)\approx b_n(0),\qquad
n\gg n_c(h_z):\quad b_n(h_z)\sim \mathrm{const}\times \frac{n}{\ln n}.
\]
The deviation obeys
\[
\frac{b_n(h_z)-b_n(0)}{b_n(0)}=h_z^2\,\mathcal F_O(n),
\]
with crossover depth and crossover time scaling as
\[
n_c(h_z)\sim |\ln h_z|,\qquad t_c(h_z)\sim |\ln h_z|.
\]
Because any nonzero uniform \(h_z\) ultimately drives the chain into the \(n/\ln n\) regime, the integrability-breaking threshold is \(h_z^c=0\) in this operator-growth sense. The same \(h_z^2\) scaling underlies the perturbative thermalization rate \(\Gamma\sim h_z^2\), linking Krylov-complexity crossover to prethermal-plateau timescales [2107.08287].

A strictly local longitudinal perturbation behaves differently. For \(g_l=h_z\delta_{l,0}\), the crossover is from type I to type II, but not to the \(n/\ln n\) chaotic regime; maximal scrambling requires a finite threshold \(h_z^c>0\). This establishes a distinction between uniform integrability breaking and genuinely local integrability breaking in one dimension [2107.08287].

## 3. One-dimensional phase structure and pattern-based classifications

Ground-state and low-lying-state phase structure in one-dimensional TLFIMs depends strongly on the exchange sign. In an antiferromagnetic chain with
\[
H=J_2\sum_{i=1}^L \sigma_i^z\sigma_{i+1}^z-B_x\sum_i\sigma_i^x-B_z\sum_i\sigma_i^z,
\]
exact diagonalization up to \(L=24\) and fidelity susceptibility reveal three phases in the \((B_x,B_z)\) plane rather than two: an antiferromagnetic phase for \(B_x<1\) and small \(B_z\), a paramagnetic phase for large \(B_z\), and a disordered phase for \(B_x>1\) and low \(B_z\) [1808.07591]. The AF–PM boundary passes through the exactly known points \((B_x,B_z)=(0,2)\) and \((1,0)\), while the disordered–paramagnetic boundary emerges only when the fidelity is scanned with respect to \(B_z\). At \(B_x=1.5\), the disordered–paramagnetic transition occurs at \(B_z\approx 1.788\) after extrapolation to the thermodynamic limit [1808.07591].

The same chain illustrates a methodological controversy. Earlier work had reported only two phases, but the disordered–paramagnetic boundary produces a fidelity-susceptibility peak roughly \(1/35\) the height of the AF–PM peak at comparable sizes, making it comparatively easy to miss [1808.07591]. This suggests that the longitudinal field can reorganize low-energy structure in ways that are much less visible in conventional order-parameter scans than in overlap-based diagnostics.

A complementary pattern-based treatment of the ferromagnetic chain rewrites the Hamiltonian in an operator basis and diagonalizes a \(3L\times 3L\) matrix to obtain pattern operators \(\hat A_n\) and occupancies \(p_n=\langle\Psi|\hat A_n^\dagger \hat A_n|\Psi\rangle\) [2301.02066]. In that formulation, the ground state at \(h=0\) has the standard second-order transition at \(J_c^{(0)}\simeq 0.50\), but a finite longitudinal bias smears out the ground-state singularity. By contrast, the first excited state exhibits a first-order quantum phase transition at
\[
J_c^{(1)}(h)\simeq h,
\]
with a discontinuity in \(\partial E_1/\partial J\) and a finite jump in \(m_z^{(1)}\) [2301.02066].

For the one-dimensional antiferromagnetic chain with uniform longitudinal field, the pattern picture leads to a different classification. The competition of \(J\), \(h\), and \(g\) yields three regimes: a “ferro-like” unstable phase dominated by \(\lambda_{16}\), a metastable antiferromagnetic-like phase dominated by \(\lambda_9\), and a stable antiferromagnetic phase dominated by \(\lambda_1\) [2301.05040]. The boundary at \(J_{c1}(h)\) is a continuous second-order quantum phase transition, remaining nearly fixed at \(J_c\approx 0.5\) up to \(h\sim 1\), whereas the change near \(J_{c2}(h)\approx h/2\) is a broad crossover with no singularity in derivatives of the ground-state energy. On that basis, the intermediate regime is interpreted not as a genuinely disordered or mixed-order phase, but as a metastable AF-like pattern sector [2301.05040].

## 4. Two-dimensional criticality, fidelity susceptibilities, and multicritical structure

In two dimensions, the TLFIM has been studied both as a direct quantum critical system and as a benchmark for universality diagnostics. On the triangular lattice, numerical diagonalization for \(N=14,16,18,\dots,32\) spins with Novotny’s screw-boundary construction yields two independent fidelity susceptibilities, \(\chi_F^{(t)}\) and \(\chi_F^{(h)}\), associated with transverse and longitudinal perturbations [1307.3603]. At criticality, finite-size scaling gives
\[
\alpha_F^{(t)}/\nu = 1.205(62),\qquad \alpha_F^{(h)}/\nu = 2.90(20),
\]
leading to
\[
\alpha_F^{(t)} = 0.752(24),\quad \alpha_F^{(h)} = 1.81(13),\quad \nu = 0.624(12),\quad \gamma = 1.19(13).
\]
These values agree within errors with the \(3d\) classical Ising universality class [1307.3603]. Because \(\alpha_F^{(t)}\) and \(\alpha_F^{(h)}\) are independent, fidelity data alone suffices to extract both \(\nu\) and \(\gamma\).

On the square lattice with antiferromagnetic exchange, stochastic-series-expansion QMC with directed-loop updates maps the zero-temperature AFM–disordered boundary in the \((h,\Gamma)\) plane [2103.12364]. At \(\Gamma=0\), the transition is first-order at \(h_c/J=2\). For small transverse field,
\[
h_c/J=2+c_2'(\Gamma/J)+\cdots,\qquad c_2'\simeq +0.16(2),
\]
so the critical longitudinal field initially increases rather than decreases. At \(h=0\), the critical transverse field is \(\Gamma_c/J=1.5220(4)\). The resulting phase diagram contains a narrow reentrant region near \(h/J\approx 2\), with maximum shift \(\Delta h_c/J\approx +0.0025\) at \(\Gamma/J\approx 0.075\); for all \(\Gamma>0\), the transition is continuous in the \(3D\) Ising universality class [2103.12364].

This QMC result is notable because the earlier single-site-cluster effective-field theory for the same square-lattice antiferromagnet found only second-order transitions and no reentrant behavior [1207.2119]. A distinct variational mean-field treatment of a two-dimensional transverse Ising metamagnet on an anisotropic square lattice, based on the Peierls–Bogoliubov inequality, found first-order and second-order lines together with tricritical points, but likewise no reentrance along the first-order lines at low temperature [1607.03930]. For the isotropic case \(\alpha=1\), that approach gives a classical tricritical point at \((T_t,h_t)\approx(2.667\,J_x,1.756\,J_x)\) and a persistence of tricriticality up to \(\delta\approx 2.86\,J_x\) in the \(h\)-\(T\) diagram [1607.03930]. Taken together, these results show that the two-dimensional TLFIM is sensitive not only to lattice geometry and exchange type, but also to the approximation scheme used to resolve longitudinal-field effects.

## 5. Frustration, quenched disorder, and overlap structure

Frustration introduces additional low-energy structure beyond the clean ferro/antiferro dichotomy. On the checkerboard lattice, the Hamiltonian
\[
\mathcal H=J\sum_{\langle i,j\rangle} s_i^z s_j^z+\Gamma\sum_i s_i^x+h\sum_i s_i^z,\qquad J>0,
\]
combines transverse and longitudinal fields with antiferromagnetic couplings on edges and crisscrossing diagonals of half the plaquettes [1107.4180]. In the classical limit \(\Gamma=h=0\), the crossed plaquettes obey the two-up two-down ice rule, giving residual entropy \(S_0\sim \frac N2\ln(3/2)\). Continuous-time QMC reveals three low-temperature regimes: persistent Curie-law divergences \(\chi\propto 1/T\) with \(\rho\approx 1\) at weak field, incipient Néel order with \(T_c\lesssim 0.01J\) at intermediate field, and a very slow-relaxing horizontal-stripe instability at larger longitudinal field [1107.4180]. The last regime was not anticipated by the \(T=0\) perturbative analysis that had predicted Néel order throughout \(0<h<2J\).

Randomness produces a different type of restructuring. In the one-dimensional random chain with random bonds, random transverse fields, and random longitudinal fields, the SDRG flow contains four fixed points: a trivial quantum-ordered fixed point, a trivial quantum-disordered fixed point, a classical random-field fixed point, and an infinite-disorder critical point at \(h_0=0\) [2501.02643]. At the infinite-disorder critical point, activated scaling takes the form \(\ln\epsilon\sim \ell^\psi\) with \(\psi=1/2\), the cluster moment scales as \(\mu\sim \ell^{d_f}\) with \(d_f=(1+\sqrt5)/4\), the true correlation-length exponent is \(\nu=2\), and \(z=\infty\) [2501.02643]. For any \(h_0>0\), long-range order is destroyed in \(d=1\), and all RG trajectories flow to one of two disordered fixed points separated by a separatrix. Along that \(h_0\)-direction, the correlation length diverges as \(\xi\sim h_0^{-\nu_h}\) with \(\nu_h\simeq 1\), while the low-energy variable \(u=\epsilon L^z\) has a Fréchet distribution [2501.02643].

A separate rigorous random-field result concerns replica structure rather than phase topology. For the transverse and longitudinal random-field Ising model with Gaussian random fields in both \(x\) and \(z\) directions, the variance of the spin overlap of any component vanishes in the infinite-volume limit, for any dimension and almost every choice of coupling constants [1706.09543]. In that sense there is no replica-symmetry breaking. The proof combines a Lie–Trotter–Suzuki path-integral representation, a weak FKG property, and Ghirlanda–Guerra identities in perturbed artificial models [1706.09543].

## 6. Computational and analytical methods

The TLFIM has served as a testbed for several complementary techniques. Exact diagonalization and fidelity methods remain central for small and intermediate systems. In two dimensions, the triangular-lattice fidelity-susceptibility calculation uses numerical diagonalization for \(N=14\) to \(32\) spins with screw-boundary conditions and numerical second derivatives of state overlaps [1307.3603]. In one dimension, the antiferromagnetic-chain phase diagram from fidelity susceptibility relies on a \(2^L\times 2^L\) Hamiltonian in the \(\sigma^z\) basis, Lanczos/conjugate-gradient extraction of the lowest eigenstate, and overlaps at \(\delta\lambda\approx 10^{-3}\), reaching \(L=24\) with energy accuracy \(10^{-12}\)–\(10^{-14}\) [1808.07591].

Quantum Monte Carlo methods bifurcate into continuous-time worldline and stochastic-series-expansion formulations. On the checkerboard lattice, continuous-time QMC eliminates Trotter errors, supplements single-spin worldline updates with loop-flip moves in space \(\times\) imaginary time, and uses replica exchange along constant \(\Gamma/T\) lines; equilibration and measurement were carried out with \(3\times 10^4\) and \(\sim 1.2\times 10^5\) sweeps, respectively [1107.4180]. On the square-lattice antiferromagnet, SSE QMC with directed-loop updates and \(\beta J/L=8\) supports finite-size scaling up to \(L=32\) using \(10^5\) thermalization sweeps, \(10^5\) measurement sweeps, and 64 independent runs per data point [2103.12364].

A recent algorithmic advance targets the inefficiency of standard SSE at large longitudinal field. The merge–unmerge loop algorithm for the transverse Ising model in a longitudinal field introduces a worm that merges a single-site operator with a neighbor, propagates through the operator string by local Metropolis decisions and equal-probability exits from merged operators, then unmerges visited operators after the loop terminates [2409.17835]. For a Rydberg atom chain of \(N=51\) sites at \(\beta=20\), the speed-up over the line algorithm is about \(2\times\) in the small longitudinal-field region and up to \(4\)–\(6\times\) at large \(\Delta\gg \Omega\); the off-diagonal CPU time is also about \(30\)–\(50\%\) lower. For Kagome qubit ice with \(N=24\times 24\times 3\), the autocorrelation ratio \(\tau_{mc}^{\text{line}}/\tau_{mc}^{\text{loop}}\) reaches \(\sim 3\)–\(4\times\) at large longitudinal field [2409.17835]. The algorithm remains sign-problem free for transverse-Ising-type SSE decompositions, though at extremely large \(B/\Gamma\) the diagonal pass-through acceptance still vanishes exponentially.

For strong randomness, SDRG provides a complementary asymptotic description. Its decimation rules identify the largest local gap \(\Omega=\max\{2J_i,\gamma_i\}\), combine neighboring sites into ferromagnetic clusters under bond decimation, and generate renormalized couplings and induced longitudinal fields under site decimation [2501.02643]. This framework is especially effective for locating fixed points, Griffiths behavior, and separatrices that would be difficult to resolve directly in finite-size exact diagonalization or conventional QMC.

The resulting picture is not a single universal phase diagram, but a family of related mixed-field Ising problems. Uniform longitudinal fields can destroy integrability and induce maximal operator growth in one-dimensional ferromagnets; in antiferromagnets they can preserve continuous transitions, generate additional phases, or produce narrow reentrant regions; in frustrated and random settings they reorganize manifolds of low-energy states, alter RG basins, and reshape the meaning of criticality itself [2107.08287], [2301.05040], [2103.12364], [2501.02643].

Source: https://www.emergentmind.com/topics/transverse-longitudinal-field-ising-model-tlfim