Title

Automatic structural optimization of tree tensor networks

Author

Toshiya Hikihara, Hiroshi Ueda, Kouichi Okunishi, Kenji Harada, and Tomotoshi Nishino

Abstract

Tree tensor network (TTN) provides an essential theoretical framework for the practical simulation of quantum many-body systems, where the network structure defined by the connectivity of the isometry tensors plays a crucial role in improving its approximation accuracy. In this paper, we propose a TTN algorithm that enables us to automatically optimize the network structure by local reconnections of isometries to suppress the bipartite entanglement entropy on their legs. The algorithm can be seamlessly implemented to such a conventional TTN approach as density-matrix renormalization group. We apply the algorithm to the inhomogeneous antiferromagnetic Heisenberg spin chain having a hierarchical spatial distribution of the interactions. We then demonstrate that the entanglement structure embedded in the ground-state of the system can be efficiently visualized as a perfect binary tree in the optimized TTN. Possible improvements and applications of the algorithm are also discussed.

Comments

11 pages, 10 figures, 2 tables

Preprint

arXiv:2209.03196

Title

Neural Network Approach to Scaling Analysis of Critical Phenomena

Author

Ryosuke Yoneda and Kenji Harada

Abstract

Determining the universality class of a system exhibiting critical phenomena is one of the central problems in physics. As methods for determining this universality class from data, polynomial regression, which is less accurate, and Gaussian process regression, which provides high accuracy and flexibility but is computationally heavy, have been proposed. In this paper, we propose a method by a regression method using a neural network. The computational complexity is only linear in the number of data points. We demonstrate the proposed method for the finite-size scaling analysis of critical phenomena on the two-dimensional Ising model and bond percolation problem to confirm the performance. This method efficiently obtains the critical values with accuracy in both cases.

Comments

10 pages, 9 figures

Preprint

arXiv:2209.01777

Date

Aug 26, 2022

Conference (invited talk)

The 15th Asia Pacific Physics Conference (APPC15), Korea (online)

Title

Tensor renormalization group study of the non-equilibrium critical fixed point of the one-dimensional contact process

Abstract

The steady-state of many stochastic systems is non-equilibrium. We studied the phase of non-equilibrium systems and the transition similar to equilibrium systems. In particular, the critical phase transition is interesting because we can define the non-equilibrium universality class. To confirm the existence of a non-equilibrium critical fixed point, we study the time evolution operator of one-dimensional contact processes by using a tensor renormalization group technique. The time evolution operators converge to universal critical tensors in the tensor renormalization group flow. The spectrums of critical tensors are strongly anisotropic but share the intrinsic structure each for the universality class. The integer structure for the universality class of compact-directed percolation in the time direction is consistent with the exact spectrum structure of the diffusion-annihilation process.

We will hold the oneline workshop, Tensor Network States: Algorithms and Applications (TNSAA) 2021-2022, from Jan. 17 to Jan. 21, 2022. The series of workshop, TNSAA, has been organized for the purposes of exchanging new developments, having discussions toward future studies, and providing intruductory talks for new generation of researchers about tensor networks.

http://bussei.gs.niigata-u.ac.jp/~okunishi/tnsaa/

Date

May 31, 2021

Seminar

StatPhys seminar at University of Tokyo, Hongo, Japan

Title

Universal spectrum structure on the nonequilibrium critical line of the one-dimensional Domany-Kinzel cellular automaton

Abstract

The Domany-Kinzel(DK) cellular automaton is a stochastic time-evolutional system with an absorbing state from which the system cannot escape and a canonical model for nonequilibrium critical phenomena[1]. We introduce the tensor network method as a new tool to study it. Estimating the entropy of the DK automaton with a matrix product state representation of distribution, we reported a new cusp of the Renyi entropy in the active phase of the DK cellular automaton[2]. We recently applied a tensor renormalization group method to transfer matrices at the nonequilibrium critical point of the DK cellular automaton, confirming a universal spectrum structure[3]. In this talk, we will report our results with a brief review of models and methods.

[1] M. Henkel, H. Hinrichsen, and S. Lübeck, Non-Equilibrium Phase Transitions. Volume 1: Absorbing Phase Transitions, Vol. 1 (Springer, 2008).
[2] K. H. and N. Kawashima, Entropy Governed by the Absorbing State of Directed Percolation, Physical Review Letters 123, 090601 (2019).
[3] K. H., Universal spectrum structure at nonequilibrium critical points in the (1+1)-dimensional directed percolation, arXiv:2008.10807.

Title

Critical exponents in coupled phase-oscillator models on small-world networks

Reference

Physical Review E 102, 062212 (2020)

DOI

10.1103/PhysRevE.102.062212

Author

Ryosuke Yoneda, Kenji Harada, and Yoshiyuki Y. Yamaguchi

Abstract

A coupled phase-oscillator model consists of phase oscillators, each of which has the natural frequency obeying a probability distribution and couples with other oscillators through a given periodic coupling function. This type of model is widely studied since it describes the synchronization transition, which emerges between the nonsynchronized state and partially synchronized states. The synchronization transition is characterized by several critical exponents, and we focus on the critical exponent defined by coupling strength dependence of the order parameter for revealing universality classes. In a typical interaction represented by the perfect graph, an infinite number of universality classes is yielded by dependency on the natural frequency distribution and the coupling function. Since the synchronization transition is also observed in a model on a small-world network, whose number of links is proportional to the number of oscillators, a natural question is whether the infinite number of universality classes remains in small-world networks irrespective of the order of links. Our numerical results suggest that the number of universality classes is reduced to one and the critical exponent is shared in the considered models having coupling functions up to second harmonics with unimodal and symmetric natural frequency distributions.

Comments

8 pages, 8 figures

Preprint

arXiv:2007.04539