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arXiv:2406.03426 · Physical review. E

Vertex representation of hyperbolic tensor networks

Matej Moško, Maria Polackova, Roman Krčmár, Andrej Gendiar

Published Feb 4, 2025• 1 citation
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Abstract

We propose a vertex representation of the tensor network (TN) for classical spin systems on hyperbolic lattices. The tensors form a network of regular p-sided polygons (p>4) with the coordination number 4. The response to multistate spin systems on the hyperbolic TN is analyzed for their entire parameter space. We show that entanglement entropy is sensitive to distinguish various hyperbolic geometries, whereas other thermodynamic quantities are not. We test the numerical accuracy of vertex TNs in the phase transitions of the first, second, and infinite order at the point of maximal entanglement entropy. The hyperbolic structure of TNs induces noncritical properties in the bulk, although boundary conditions significantly affect the total free energy in the thermodynamic limit. Thus a developed vertex-type TN can be used for the lowest-energy quantum states on the hyperbolic lattices.

DOI
10.1103/physreve.111.024105

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We propose a vertex representation of the tensor network (TN) for classical spin systems on hyperbolic lattices. The tensors form a network of regular p-sided polygons (p>4) with the coordination number 4. The response to multistate spin systems on the hyperbolic TN is analyzed for their entire parameter space. We show that entanglement entropy is sensitive to distinguish various hyperbolic geometries, whereas other thermodynamic quantities are not. We test the numerical accuracy of vertex TNs in the phase transitions of the first, second, and infinite order at the point of maximal entanglement entropy. The hyperbolic structure of TNs induces noncritical properties in the bulk, although boundary conditions significantly affect the total free energy in the thermodynamic limit. Thus a developed vertex-type TN can be used for the lowest-energy quantum states on the hyperbolic lattices.
One of the challenging problems in the condensed matter physics is to understand the quantum many-body systems, especially, their physical mechanisms behind. Since there are only a few complete analytical solutions of these systems, several numerical simulation methods have been proposed in recent years. Amongst all of them, the Tensor Network algorithms have become increasingly popular in recent years, especially for their adaptability to simulate strongly correlated systems. The current work focuses on the generalization of such Tensor-Network-based algorithms, which are sufficiently robust to describe critical phenomena and phase transitions of multistate spin Hamiltonians in the thermodynamic limit. We have chosen two algorithms: the Corner Transfer Matrix Renormalization Group and the Higher-Order Tensor Renormalization Group. This work, based on tensor-network analysis, opens doors for the understanding of phase transition and entanglement of the interacting systems on the non-Euclidean geometries. We focus on three main topics: A new thermodynamic model of social influence, free energy is analyzed to classify the phase transitions on an infinite set of the negatively curved geometries where a relation between the free energy and the Gaussian radius of the curvature is conjectured, a unique tensor-based algorithm is proposed to study the phase transition on fractal structures.
Magnetic properties of the transverse-field Ising model on curved (hyperbolic) lattices are studied by a tensor product variational formulation that we have generalized for this purpose.First we identify the quantum phase transition for each hyperbolic lattice by calculating the magnetization.We study the entanglement entropy at the phase transition in order to analyze the correlations of various subsystems located at the center with the rest of the lattice.We confirm that the entanglement entropy satisfies the area law at the phase transition for fixed coordination number, i.e., it scales linearly with increasing size of the subsystems.On the other hand, the entanglement entropy decreases as power-law with respect to the increasing coordination number.
The investigation of the behavior of both classical and quantum systems on non-Euclidean surfaces near the phase transition point represents an interesting research area of modern physics. In the case of classical spin systems, a generalization of the Corner Transfer Matrix Renormalization Group algorithm has been developed and successfully applied to spin models on infinitely many regular hyperbolic lattices. In this work, we extend these studies to specific types of lattices. It is important to say that no suitable algorithms for numerical analysis of ground-states of quantum systems in similar conditions have been implemented yet. In this work, we offer a particular solution by proposing a variational numerical algorithm Tensor Product Variational Formulation, which assumes a quantum ground-state written in the form of a low-dimensional uniform tensor product state. We apply the Tensor Product Variational Formulation to three typical quantum models on a variety of regular hyperbolic lattices. The main outcomes are the following: (1) We propose an algorithm for calculation and classification of the thermodynamic properties of the Ising model on triangular-tiled hyperbolic lattices. In addition, we investigate the origin of the mean-field universality on a series of weakly curved lattices. (2) We develop the Tensor Product Variational Formulation algorithm for the numerical analysis of the ground-state of the quantum systems on the hyperbolic lattices. (3) We study quantum phase transition phenomena for the three selected spin models on various types of the hyperbolic lattices including the Bethe lattice.
We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous noncritical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents β=0.4999 and δ=3.007, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multistate spin models.
Abstract The quantum XY, Heisenberg, and transverse field Ising models on hyperbolic lattices are studied by means of the tensor product variational formulation algorithm. The lattices are constructed by tessellation of congruent polygons with coordination number equal to four. The calculated ground-state energies of the XY and Heisenberg models and the phase transition magnetic field of the Ising model on the series of lattices are used to estimate the corresponding quantities of the respective models on the Bethe lattice. The hyperbolic lattice geometry induces mean-field-like behavior of the models. The ambition to obtain results on the non-Euclidean lattice geometries has been motivated by theoretical studies of the anti-de Sitter/conformal field theory correspondence.
For any D-dimensional quantum lattice system, the fidelity between two ground state many-body wave functions is mapped onto the partition function of a D-dimensional classical statistical vertex lattice model with the same lattice geometry. The fidelity per lattice site, analogous to the free energy per site, is well defined in the thermodynamic limit and can be used to characterize the phase diagram of the model. We explain how to compute the fidelity per site in the context of tensor network algorithms, and demonstrate the approach by analyzing the two-dimensional quantum Ising model with transverse and parallel magnetic fields.