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arXiv:2003.10717 · Acta Physica Polonica A

Area-Law Study of Quantum Spin System on Hyperbolic Lattice Geometries

Andrej Gendiar

Published May 1, 2020• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.12693/aphyspola.137.589

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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.
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.
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.
Recently, tight-binding models on hyperbolic lattices (discretized anti–de Sitter space) have gained significant attention, leading to hyperbolic band theory and non-Abelian Bloch states. In this paper, we investigate these quantum systems from the perspective of quantum information, focusing particularly on the scaling of entanglement entropy (EE) that has been regarded as a powerful quantum-information probe into exotic phases of matter. It is known that on a d -dimensional translation-invariant Euclidean lattice, the EE of band insulators scales as an area law ( ∼ L d − 1 , where L is the linear size of the boundary between two subsystems). Meanwhile, the EE of metals [with a finite density of state (DOS)] scales as the renowned Gioev-Klich-Widom scaling law ( ∼ L d − 1 ln L ). The appearance of logarithmic divergence, as well as the analytic form of the coefficient c , is mathematically controlled by the Widom conjecture of asymptotic behavior of Toeplitz matrices and can be physically understood via the Swingle's argument. However, the hyperbolic lattice, which generalizes translational symmetry, results in the inapplicability of these analytic approaches and the potential nontrivial behavior of the EE. Here we make an initial attempt through numerical simulation. Remarkably, we find that both cases adhere to the area law, indicating the effect of background hyperbolic geometry that influences quantum entanglement. To achieve the results, we first apply the vertex-inflation method to generate a hyperbolic lattice on the Poincaré disk, and then apply the Haydock recursion method to compute the DOS. Finally, we study the scaling of the EE for different bipartitions via exact diagonalization and perform finite-size scaling. We also investigate how the coefficient of the area law is correlated to the bulk gap in the gapped case and to the DOS in the gapless case, respectively. Future directions are discussed.
The uniform two-dimensional variational tensor product state is applied to the transverse-field Ising, XY, and Heisenberg models on a regular hyperbolic lattice surface. The lattice is constructed by tessellation of the congruent pentagons with the fixed coordination number being four. As a benchmark, the three models are studied on the flat square lattice simultaneously. The mean-field-like universality of the Ising phase transition is observed in full agreement with its classical counterpart on the hyperbolic lattice. The tensor product ground state in the thermodynamic limit has an exceptional three-parameter solution. The variational ground-state energies of the spin models are calculated.
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.
Ground state of the one-dimensional transverse field Ising model is investigated under the hyperbolic deformation, where the energy scale of j-th bond is proportional to the function \cosh ( j λ) that contains a parameter λ. Although the Hamiltonian is position dependent, the ground state is nearly uniform and finitely correlated. We observe the energy cross over between the ordered and disordered state with respect to the transverse field. The model shows first order phase transition, and the discontinuities in the magnetization and entanglement entropy at the transition point detect the Ising universality.