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Physical review. B./Physical review. B

Scalable tensor network algorithm for thermal quantum many-body systems in two dimensions

Meng Zhang, Hao Zhang, Chao Wang, Lixin He

Published Feb 19, 2025• 2 citations
Publisher

Abstract

Simulating strongly correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state to represent the vectorization of the quantum thermal state, and we utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the two-dimensional antiferromagnetic Heisenberg model, the ${J}_{1}\ensuremath{-}{J}_{2}$ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion, exact diagonalization, and determinant quantum Monte Carlo.

DOI
10.1103/physrevb.111.075146

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•Jan 1, 2023•0 citations
Simulating strongly correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state to represent the vectorization of the quantum thermal state, and we utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the two-dimensional antiferromagnetic Heisenberg model, the J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion, exact diagonalization, and determinant quantum Monte Carlo.
Simulating strongly-correlated quantum many-body systems at finite temperatures is a significant challenge in computational physics. In this work, we present a scalable finite-temperature tensor network algorithm for two-dimensional quantum many-body systems. We employ the (fermionic) projected entangled pair state (PEPS) to represent the vectorization of the quantum thermal state and utilize a stochastic reconfiguration method to cool down the quantum states from infinite temperature. We validate our method by benchmarking it against the 2D antiferromagnetic Heisenberg model, the J1J_1J1​-J2J_2J2​ model, and the Fermi-Hubbard model, comparing physical properties such as internal energy, specific heat, and magnetic susceptibility with results obtained from stochastic series expansion (SSE), exact diagonalization, and determinant quantum Monte Carlo (DQMC).
Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of two-dimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finite-temperature regime. As a benchmark, we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultracold atoms.
Aimed at a more realistic classical description of natural quantum systems, we present a two-dimensional tensor network algorithm to study finite temperature properties of frustrated model quantum systems and real quantum materials. For this purpose, we introduce the infinite projected entangled simplex operator ansatz to study thermodynamic properties. To obtain state-of-the-art benchmarking results, we explore the highly challenging spin-1/2 Heisenberg antiferromagnet on the Kagome lattice, a system for which we investigate the melting of the magnetization plateaus at finite magnetic field and temperature. Making a close connection to actual experimental data of real quantum materials, we go on to studying the finite temperature properties of Ca10Cr7O28{\mathrm{Ca}}_{10}{\mathrm{Cr}}_{7}{\mathrm{O}}_{28}Ca10​Cr7​O28​. We compare the magnetization curve of this material in the presence of an external magnetic field at finite temperature with classically simulated data. As the first theoretical tool that incorporates both thermal fluctuations as well as quantum correlations in the study of this material, our work contributes to settling the existing controversy between the experimental data and previous theoretical works on the magnetization process.
A Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian HHH can be represented by a 3D tensor network, with the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension. The coarse graining is performed by a tree tensor network of isometries. They are optimized variationally to maximize the accuracy of the PEPO as a representation of the 2D thermal state ${e}^{\ensuremath{-}\ensuremath{\beta}H}$. The algorithm is applied to the two-dimensional Hubbard model on an infinite square lattice. Benchmark results at finite temperature are obtained that are consistent with the best cluster dynamical mean-field theory and power-series expansion in the regime of parameters where they yield mutually consistent results.
Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32\ifmmode\times\else\texttimes\fi{}32$ sites, as well as a highly frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ model up to $24\ifmmode\times\else\texttimes\fi{}24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract