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arXiv:2601.07829

Interface roughening in the 3-D Ising model with tensor networks

Atsushi Ueda, Lander Burgelman, Luca Tagliacozzo, Laurens Vanderstraeten

Published Jan 12, 2026• 0 citations
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Abstract

Interfaces in three-dimensional many-body systems can exhibit rich phenomena beyond the corresponding bulk properties. In particular, they can fluctuate and give rise to massless low energy degrees of freedom even in the presence of a gapped bulk. In this work, we present the first tensor-network study of the paradigmatic interface roughening transition of the 3-D Ising model using highly asymmetric lattices that are infinite in the $(xy)$ direction and finite in $z$. By reducing the problem to an effective 2-D tensor network, we study how truncating the $z$ direction reshapes the physics of the interface. For a truncation based on open boundary conditions, we demonstrate that varying the interface width gives rise to either a $\mathbb{Z}_2$ symmetry breaking transition (for odd $L_z$) or a smooth crossover(for even $L_z$). For antiperiodic boundary conditions, we obtain an effective $\mathbb{Z}_q$ clock model description with $q=2L_z$ that exhibits an intermediate Luttinger liquid phase with an emergent $\U(1)$ symmetry.

DOI
10.48550/arxiv.2601.07829

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Interfaces in three-dimensional many-body systems can exhibit rich phenomena beyond the corresponding bulk properties. In particular, they can fluctuate and give rise to massless low energy degrees of freedom even in the presence of a gapped bulk. In this work, we present the first tensor-network study of the paradigmatic interface roughening transition of the 3-D Ising model using highly asymmetric lattices that are infinite in the (xy)(xy)(xy) direction and finite in zzz. By reducing the problem to an effective 2-D tensor network, we study how truncating the zzz direction reshapes the physics of the interface. For a truncation based on open boundary conditions, we demonstrate that varying the interface width gives rise to either a Z2\mathbb{Z}_2Z2​ symmetry breaking transition (for odd LzL_zLz​) or a smooth crossover(for even LzL_zLz​). For antiperiodic boundary conditions, we obtain an effective Zq\mathbb{Z}_qZq​ clock model description with q=2Lzq=2L_zq=2Lz​ that exhibits an intermediate Luttinger liquid phase with an emergent \U(1)\U(1)\U(1) symmetry.
The properties of interfaces are key to understanding the physics of matter. However, the study of quantum interface dynamics has remained an outstanding challenge. Here, we use large-scale tree tensor network simulations to identify the dynamical signature of an interface roughening transition within the ferromagnetic phase of the 2D quantum Ising model. For initial domain wall profiles we find extended prethermal plateaus for smooth interfaces, whereas above the roughening transition the domain wall decays quickly. Our results can be readily explored experimentally in Rydberg atomic systems.
In a recent paper [Phys. Rev. Lett. 129, 120601 (2022)], we have shown that the dynamics of interfaces, in the symmetry-broken phase of the two-dimensional ferromagnetic quantum Ising model, displays a robust form of ergodicity breaking. In this paper, we elaborate more on the issue. First, we discuss two classes of initial states on the square lattice, the dynamics of which is driven by complementary terms in the effective Hamiltonian and may be solved exactly: (a) Strips of consecutive neighboring spins aligned in the opposite direction of the surrounding spins and (b) a large class of initial states, characterized by the presence of a well-defined ``smooth'' interface separating two infinitely extended regions with oppositely aligned spins. The evolution of the latter states can be mapped onto that of an effective one-dimensional fermionic chain, which is integrable in the infinite-coupling limit. In this case, deep connections with noteworthy results in mathematics emerge, as well as with similar problems in classical statistical physics. We present a detailed analysis of the evolution of these interfaces both on the lattice and in a suitable continuum limit, including the interface fluctuations and the dynamics of entanglement entropy. Second, we provide analytical and numerical evidence supporting the conclusion that the observed nonergodicity---arising from Stark localization of the effective fermionic excitations---persists away from the infinite-Ising-coupling limit, and we highlight the presence of a timescale $T\ensuremath{\sim}{e}^{cLlnL}$ for the decay of a region of large linear size LLL. The implications of our work for the classic problem of the decay of a false vacuum are also discussed.
Gapless Luttinger liquid is conventionally viewed as topologically trivial, unless it hosts degenerate ground states and or entanglement spectrum, which necessitates partial bulk degree of freedom to be gapped out. Here we predict an emergent gapless topological Luttinger liquid which is beyond the conventional scenarios and is characterized by the nontrivial many-body bulk spin texture, and propose feasible scheme for experimental observation. We consider a one-dimensional spin-orbit coupled Fermi-Hubbard model with fractional filling, whose low-energy physics is effectively described by a spinless Luttinger liquid and is trivial in the conventional characterization. We show that, as being tuned by the filling factor and interaction strength, the many-body ground state may exhibit nontrivial winding in its bulk spin texture in the projected momentum space, manifesting an emergent topological phase. A topological transition occurs when the projected spin-state at a high symmetry momentum becomes fully mixed one, resulting from competing processes of particle scattering to the lower and higher subbands, for which the spin texture at such momentum point is ill-defined, but the Luttinger liquid keeps gapless through the transition. Surprisingly, at relatively small filling the Luttinger liquid remains topologically nontrivial even at infinitely strong interaction. The results can be generalized to finite temperature which facilitates the real experimental detection. This work shows a novel gapless topological Luttinger liquid whose characterization is beyond the low-energy effective theory, and can be verified based on current experiments.
Abstract Key aspects of the AdS/CFT correspondence can be captured in terms of tensor network models on hyperbolic lattices. For tensors fulfilling the matchgate constraint, these have previously been shown to produce disordered boundary states whose site-averaged ground state properties match the translation-invariant critical Ising model. In this work, we substantially sharpen this relationship by deriving disordered local Hamiltonians generalizing the critical Ising model whose ground and low-energy excited states are accurately represented by the matchgate ansatz without any averaging. We show that these Hamiltonians exhibit multi-scale quasiperiodic symmetries captured by an analytical toy model based on layers of the hyperbolic lattice, breaking the conformal symmetries of the critical Ising model in a controlled manner. We provide a direct identification of correlation functions of ground and low-energy excited states between the disordered and translation-invariant models and give numerical evidence that the former approaches the latter in the large bond dimension limit. This establishes tensor networks on regular hyperbolic tilings as an effective tool for the study of conformal field theories. Furthermore, our numerical probes of the bulk parameters corresponding to boundary excited states constitute a first step towards a tensor network bulk-boundary dictionary between regular hyperbolic geometries and critical boundary states.
Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.
Abstract We study numerically the roughening properties of an interface in a two-dimensional Ising model with either random bonds or random fields, which are representative of universality classes where disorder acts only on the interface or also away from it, in the bulk. The dynamical structure factor shows a rich crossover pattern from the form of a pure system at large wavevectors k , to a different behavior, typical of the kind of disorder, at smaller k 's. For the random field model a second crossover is observed from the typical behavior of a system where disorder is only effective on the surface, as the random bond model, to the truly large scale behavior, where bulk-disorder is important, that is observed at the smallest wavevectors.