Non-Integrable Transverse Field Ising Model

• Non-integrable TFIM is a quantum spin model with transverse fields and spatially varying couplings that break integrability.
• Hyperbolic deformation introduces cosh-weighted interactions that drive a first-order quantum phase transition with a discontinuous order parameter.
• Numerical and scaling analyses reveal finite correlation lengths and robust Ising universality despite strong spatial inhomogeneities.

The non-integrable transverse field Ising model (TFIM) refers to a class of quantum Ising spin systems subject to a transverse magnetic field, with modifications—such as inhomogeneities, additional interactions, spatially varying couplings, geometric frustration, or explicit nonlocality—that preclude mapping the system to free fermions or exact analytic solution via standard integrability techniques. Whereas the uniform one-dimensional TFIM is exactly solvable by Jordan–Wigner transformation, a variety of deformations render the model non-integrable, resulting in rich and sometimes unexpected quantum, dynamical, and thermal properties.

1. Generic Structure and Hyperbolic Deformation

A prototypical non-integrable deformation arises by making the local couplings explicitly position-dependent, as in the hyperbolically deformed TFIM examined in "Transverse Field Ising Model Under Hyperbolic Deformation" (Ueda et al., 2010). The Hamiltonian is

$$H^{(c)}(\lambda) = - J \sum_j \cosh(j\lambda)\, \sigma^z_j \sigma^z_{j+1} - \Gamma \sum_j \cosh\left((j-\frac{1}{2})\lambda\right) \sigma^x_j,$$

where $\lambda$ is a deformation parameter that determines the scale of inhomogeneity; local energy scales vary exponentially away from the system center. Despite explicit breaking of spatial uniformity, numerical DMRG calculations reveal that the bulk of the ground state remains nearly uniform and exhibits finite correlations over a characteristic length $\xi \sim 1/\lambda$. The boundary effects, amplified by inhomogeneity, decay rapidly over this length scale.

This configuration fundamentally differs from the uniform TFIM in two ways:

• The explicit spatial variation in couplings destroys integrability.
• Physical observables and the ground state are strongly controlled by the imposed inhomogeneity, yet the "bulk" system retains uniformity with respect to local correlators, enabling well-defined large-scale properties even in highly non-uniform systems.

2. Nature of Phase Transitions

The non-integrable TFIM with hyperbolic deformation undergoes a first-order quantum phase transition at $\Gamma=J$ (using $J$ as the energy unit). This is sharply diagnosed by observing a discontinuity ("kink") in the local energy density at the system center and a finite jump in the order parameter $\langle \sigma^z_0 \rangle$ at the critical point. The magnitude of the discontinuity in spontaneous magnetization scales as $\lambda^{1/8}$, corresponding to the Ising universality class, via the relation

$$\Delta\Gamma \propto \xi^{-\nu} \propto \lambda, \;\;\; \langle \sigma^z_0 \rangle \propto (\Delta\Gamma)^{1/8}.$$

Unlike the second-order transition in the uniform TFIM, the hyperbolically deformed system displays clear first-order behavior, including phase coexistence and coexistence of energy minima for ordered and disordered states near $\Gamma=1$.

3. Bulk Properties, Energy Crossovers, and Uniformity

Non-integrable perturbations (particularly those rendering the Hamiltonian position-dependent) could be expected to induce strong spatial inhomogeneity in observables. However, the hyperbolic deformation reveals that, beyond a relatively narrow boundary region, the local expectation values (such as on-site transverse magnetization and near-center bond energies) are nearly position-independent. This is evidenced by uniform values of

$$\langle \Gamma \sigma_j^x \rangle, \quad \langle h_{j,j+1} \rangle = -\left[ \frac{\Gamma}{2} \sigma^x_j + J \sigma^z_j \sigma^z_{j+1} + \frac{\Gamma}{2} \sigma^x_{j+1} \right]$$

at the system center, across bulk regions of the chain—demonstrating a finite correlation length $\xi$ controls the decay of inhomogeneities produced by the boundaries.

As the transverse field $\Gamma$ is tuned across the transition, the model displays an energy crossover: the ordered and disordered states (selected by ferromagnetic or paramagnetic boundary conditions) coexist over a finite region, with the physically stable state determined by minimization of the local energy density at the center.

4. Mathematical and Scaling Structure

The hyperbolic deformation modifies both the local interactions and the field terms, with the cosh-weighted terms embedding an exponential spatial variation. The emergent length scale $\xi \propto 1/\lambda$ sets the region of bulk uniformity and controls the scaling of physical quantities at the phase transition.

Key mathematical relations include:

• The Hamiltonian with explicit position dependence (cosh weights),
• Local bond operator definitions for probing uniformity and energy crossover,
• Scaling laws consistent with Ising universality, but with effective "distance from criticality" controlled by $\lambda$ via $\Delta\Gamma \propto \lambda$.

5. Theoretical and Physical Implications for Non-Integrability

The emergence of an almost uniform, finitely correlated ground state in a strongly inhomogeneous, non-integrable TFIM is a nontrivial result. It demonstrates that non-integrable perturbations, even those breaking translational invariance, do not necessarily destroy the essential quantum critical structure in the system's bulk. This enables systematic paper of first-order quantum phase transitions, universality, and boundary-induced effects in highly inhomogeneous and non-integrable quantum chains.

Further implications include:

• Hyperbolic and other spatially varying deformations act as analytical and numerical tools to explore finite-size scaling, boundary effects, and crossovers when integrability is absent.
• Such studies suggest that universal features of phase transitions, including scaling laws and order parameter discontinuities, persist even in non-integrable models, though with modified control parameters.
• Relevant applications include quantum lattice systems with spatial inhomogeneity, quantum simulators with engineered coupling variation, and the investigation of robust quantum phases where analytical results are otherwise limited.

6. Broader Connections and Applications

The insights obtained from the paper of the non-integrable TFIM under hyperbolic deformation generalize beyond one-dimensional systems. Possible extensions include:

• Investigation of quantum phases and transitions in models with engineered spatial inhomogeneities.
• Understanding the interplay between boundary conditions, spatial disorder, and bulk phase behavior, particularly in the context of quantum simulation and cold-atom experiments.
• Providing a framework for analyzing criticality and universality in the absence of exact solvability, relevant for numerous quantum condensed matter and statistical mechanics systems.

Hyperbolically deformed models thereby constitute both a testbed for fundamental questions in non-integrable quantum criticality and a methodological advance for exploring the rich landscape of quantum many-body phenomena outside the field of integrability.