Transverse Longitudinal Field Ising Model

• TLFIM is a quantum Ising model with both transverse and longitudinal field terms that break free-fermion integrability and generate rich phase behavior.
• It employs a variety of analytical and computational methods, including fidelity susceptibilities, QMC, and SDRG, to study operator growth and critical phenomena.
• The model reveals diverse behaviors such as universal scaling in operator dynamics, reentrant phase regions, and disorder-induced critical points across different lattice geometries.

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 (Noh, 2021). 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 (Nishiyama, 2013, Kaneko et al., 2021, Petö et al., 5 Jan 2025, Xu et al., 2024).

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 (Noh, 2021)
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 (Nishiyama, 2013)
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 (Kaneko et al., 2021)
1D random chain	$J>0$0	SDRG yields multiple fixed points and a separatrix (Petö et al., 5 Jan 2025)

In the one-dimensional operator-growth study, the Pauli matrices satisfy $J>0$1, and the operator dynamics is formulated in Liouville space (Noh, 2021). In the two-dimensional fidelity-susceptibility study, the model is placed on a triangular lattice with ferromagnetic nearest-neighbor coupling set to unity (Nishiyama, 2013). In square-lattice antiferromagnetic realizations, $J>0$2 is the antiferromagnetic coupling, and the fields compete with Néel order rather than with a ferromagnetic order parameter (Kaneko et al., 2021, Neto et al., 2012).

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

$J>0$3

Equivalently, $J>0$4 is treated as a state $J>0$5 in operator Hilbert space, with dynamics generated by the Liouvillian $J>0$6 and inner product $J>0$7. The associated Krylov basis $J>0$8 is built by the Lanczos recurrence, producing Lanczos coefficients $J>0$9 and amplitudes $h_z$0 satisfying

$h_z$1

while the mean depth

$h_z$2

defines the Krylov complexity (Noh, 2021).

In one dimension, Parker et al.’s upper bound implies $h_z$3, and maximal scrambling corresponds to the asymptotic form

$h_z$4

with $h_z$5 the Lambert $h_z$6-function. When this scaling is realized, the mean depth grows asymptotically as

$h_z$7

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 (Noh, 2021).

The integrable limit $h_z$8 exhibits two distinct Lanczos-coefficient scalings depending on the initial operator: type I with $h_z$9, implying linear $z$0, and type II with $z$1, implying $z$2. Turning on a small uniform longitudinal field $z$3 produces a crossover: $z$4 The deviation obeys

$z$5

with crossover depth and crossover time scaling as

$z$6

Because any nonzero uniform $z$7 ultimately drives the chain into the $z$8 regime, the integrability-breaking threshold is $z$9 in this operator-growth sense. The same $z$0 scaling underlies the perturbative thermalization rate $z$1, linking Krylov-complexity crossover to prethermal-plateau timescales (Noh, 2021).

A strictly local longitudinal perturbation behaves differently. For $z$2, the crossover is from type I to type II, but not to the $z$3 chaotic regime; maximal scrambling requires a finite threshold $z$4. This establishes a distinction between uniform integrability breaking and genuinely local integrability breaking in one dimension (Noh, 2021).

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

$z$5

exact diagonalization up to $z$6 and fidelity susceptibility reveal three phases in the $z$7 plane rather than two: an antiferromagnetic phase for $z$8 and small $z$9, a paramagnetic phase for large $x$0, and a disordered phase for $x$1 and low $x$2 (Bonfim et al., 2018). The AF–PM boundary passes through the exactly known points $x$3 and $x$4, while the disordered–paramagnetic boundary emerges only when the fidelity is scanned with respect to $x$5. At $x$6, the disordered–paramagnetic transition occurs at $x$7 after extrapolation to the thermodynamic limit (Bonfim et al., 2018).

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 $x$8 the height of the AF–PM peak at comparable sizes, making it comparatively easy to miss (Bonfim et al., 2018). 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 $x$9 matrix to obtain pattern operators $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$0 and occupancies $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$1 (Yang et al., 2023). In that formulation, the ground state at $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$2 has the standard second-order transition at $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$3, 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

$$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$4

with a discontinuity in $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$5 and a finite jump in $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$6 (Yang et al., 2023).

For the one-dimensional antiferromagnetic chain with uniform longitudinal field, the pattern picture leads to a different classification. The competition of $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$7, $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$8, and $$H=J\sum_l[\sigma_l^z\sigma_{l+1}^z+h_x\sigma_l^x+h_z\sigma_l^z]$$9 yields three regimes: a “ferro-like” unstable phase dominated by $h_z\neq 0$0, a metastable antiferromagnetic-like phase dominated by $h_z\neq 0$1, and a stable antiferromagnetic phase dominated by $h_z\neq 0$2 (Yang et al., 2023). The boundary at $h_z\neq 0$3 is a continuous second-order quantum phase transition, remaining nearly fixed at $h_z\neq 0$4 up to $h_z\neq 0$5, whereas the change near $h_z\neq 0$6 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 (Yang et al., 2023).

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 $h_z\neq 0$7 spins with Novotny’s screw-boundary construction yields two independent fidelity susceptibilities, $h_z\neq 0$8 and $h_z\neq 0$9, associated with transverse and longitudinal perturbations (Nishiyama, 2013). At criticality, finite-size scaling gives

$$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$0

leading to

$$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$1

These values agree within errors with the $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$2 classical Ising universality class (Nishiyama, 2013). Because $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$3 and $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$4 are independent, fidelity data alone suffices to extract both $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$5 and $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$6.

On the square lattice with antiferromagnetic exchange, stochastic-series-expansion QMC with directed-loop updates maps the zero-temperature AFM–disordered boundary in the $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$7 plane (Kaneko et al., 2021). At $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$8, the transition is first-order at $$\mathcal H=-\sum_{\langle ij\rangle}\sigma_i^z\sigma_j^z-h_t\sum_i\sigma_i^x-h_l\sum_i\sigma_i^z$$9. For small transverse field,

$$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$0

so the critical longitudinal field initially increases rather than decreases. At $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$1, the critical transverse field is $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$2. The resulting phase diagram contains a narrow reentrant region near $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$3, with maximum shift $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$4 at $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$5; for all $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$6, the transition is continuous in the $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$7 Ising universality class (Kaneko et al., 2021).

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 (Neto et al., 2012). 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 (Nascimento et al., 2016). For the isotropic case $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$8, that approach gives a classical tricritical point at $$H=J\sum_{\langle i,j\rangle}\sigma_i^z\sigma_j^z-h\sum_i\sigma_i^z-\Gamma\sum_i\sigma_i^x$$9 and a persistence of tricriticality up to $J>0$00 in the $J>0$01-$J>0$02 diagram (Nascimento et al., 2016). 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

$J>0$03

combines transverse and longitudinal fields with antiferromagnetic couplings on edges and crisscrossing diagonals of half the plaquettes (Ishizuka et al., 2011). In the classical limit $J>0$04, the crossed plaquettes obey the two-up two-down ice rule, giving residual entropy $J>0$05. Continuous-time QMC reveals three low-temperature regimes: persistent Curie-law divergences $J>0$06 with $J>0$07 at weak field, incipient Néel order with $J>0$08 at intermediate field, and a very slow-relaxing horizontal-stripe instability at larger longitudinal field (Ishizuka et al., 2011). The last regime was not anticipated by the $J>0$09 perturbative analysis that had predicted Néel order throughout $J>0$10.

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 $J>0$11 (Petö et al., 5 Jan 2025). At the infinite-disorder critical point, activated scaling takes the form $J>0$12 with $J>0$13, the cluster moment scales as $J>0$14 with $J>0$15, the true correlation-length exponent is $J>0$16, and $J>0$17 (Petö et al., 5 Jan 2025). For any $J>0$18, long-range order is destroyed in $J>0$19, and all RG trajectories flow to one of two disordered fixed points separated by a separatrix. Along that $J>0$20-direction, the correlation length diverges as $J>0$21 with $J>0$22, while the low-energy variable $J>0$23 has a Fréchet distribution (Petö et al., 5 Jan 2025).

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 $J>0$24 and $J>0$25 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 (Itoi, 2017). 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 (Itoi, 2017).

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 $J>0$26 to $J>0$27 spins with screw-boundary conditions and numerical second derivatives of state overlaps (Nishiyama, 2013). In one dimension, the antiferromagnetic-chain phase diagram from fidelity susceptibility relies on a $J>0$28 Hamiltonian in the $J>0$29 basis, Lanczos/conjugate-gradient extraction of the lowest eigenstate, and overlaps at $J>0$30, reaching $J>0$31 with energy accuracy $J>0$32–$J>0$33 (Bonfim et al., 2018).

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 $J>0$34 imaginary time, and uses replica exchange along constant $J>0$35 lines; equilibration and measurement were carried out with $J>0$36 and $J>0$37 sweeps, respectively (Ishizuka et al., 2011). On the square-lattice antiferromagnet, SSE QMC with directed-loop updates and $J>0$38 supports finite-size scaling up to $J>0$39 using $J>0$40 thermalization sweeps, $J>0$41 measurement sweeps, and 64 independent runs per data point (Kaneko et al., 2021).

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 (Xu et al., 2024). For a Rydberg atom chain of $J>0$42 sites at $J>0$43, the speed-up over the line algorithm is about $J>0$44 in the small longitudinal-field region and up to $J>0$45–$J>0$46 at large $J>0$47; the off-diagonal CPU time is also about $J>0$48–$J>0$49 lower. For Kagome qubit ice with $J>0$50, the autocorrelation ratio $J>0$51 reaches $J>0$52–$J>0$53 at large longitudinal field (Xu et al., 2024). The algorithm remains sign-problem free for transverse-Ising-type SSE decompositions, though at extremely large $J>0$54 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 $J>0$55, combine neighboring sites into ferromagnetic clusters under bond decimation, and generate renormalized couplings and induced longitudinal fields under site decimation (Petö et al., 5 Jan 2025). 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 (Noh, 2021, Yang et al., 2023, Kaneko et al., 2021, Petö et al., 5 Jan 2025).