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arXiv:2312.04420

Finite-Temperature Simulations of Quantum Lattice Models with Stochastic Matrix Product States

Jianxin Gao, Gao Yuan, Qiaoyi Li, Wei Li

Published Dec 7, 2023• 0 citations
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Abstract

In this work, we develop a stochastic matrix product state (stoMPS) approach that combines the MPS technique and Monte Carlo samplings and can be applied to simulate quantum lattice models down to low temperature. In particular, we exploit a procedure to unbiasedly sample the local tensors in the matrix product states, which has one physical index of dimension $d$ and two geometric indices of dimension $D$, and find the results can be continuously improved by enlarging $D$. We benchmark the methods on small system sizes and then compare the results to those obtained with minimally entangled typical thermal states, finding that stoMPS has overall better performance with finite $D$. We further exploit the MPS sampling to simulate long spin chains, as well as the triangular and square lattices with cylinder circumference $W$ up to 4. Our results showcase the accuracy and effectiveness of stochastic tensor networks in finite-temperature simulations.

DOI
10.48550/arxiv.2312.04420

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In this work, we develop a stochastic matrix product state (stoMPS) approach that combines the MPS technique and Monte Carlo samplings and can be applied to simulate quantum lattice models down to low temperature. In particular, we exploit a procedure to unbiasedly sample the local tensors in the matrix product states, which has one physical index of dimension ddd and two geometric indices of dimension DDD, and find the results can be continuously improved by enlarging DDD. We benchmark the methods on small system sizes and then compare the results to those obtained with minimally entangled typical thermal states, finding that stoMPS has overall better performance with finite DDD. We further exploit the MPS sampling to simulate long spin chains, as well as the triangular and square lattices with cylinder circumference WWW up to 4. Our results showcase the accuracy and effectiveness of stochastic tensor networks in finite-temperature simulations.
Finite temperature problems in the strong correlated systems are important but challenging tasks. Minimally entangled typical thermal states (METTS) are a powerful method in the framework of tensor network methods to simulate finite temperature systems, including Fermions and frustrated spins which introduce a sign problem in the typical Monte Carlo methods. In this work, we introduce an extension of the METTS algorithm by using a new basis, the auxiliary matrix product state. This new basis achieves the pre-summation process in the partition function, and thus improve the convergence in the Monte Carlo samplings. The method also has the advantage of simulating the grand canonical ensemble in a computationally efficient way by employing good quantum numbers. We benchmark our method on the spin-1/21/21/2 XXZ model on the triangular lattice, and show that the new method outperforms the original METTS as well as the purification methods at sufficiently low temperature, the usual range of applications of METTS. The new method also naturally connects the METTS method to the purification method.
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.
Accurate calculations of correlated fermions at low temperatures constitute a major challenge in quantum many-body physics, most notably in the studies of high-TcT_cTc​ superconductivity and ultracold atom optical lattices. Stochastic sampling of tensor-network states offers a promising route to tackle this problem; however, existing schemes have long faced a fundamental dilemma---sampling efficiency and symmetry acceleration \textit{cannot} be achieved simultaneously. Here we propose a perfect Born sampling approach for thermal tensor networks, which performs importance sampling directly from the purified density matrix via the Born rule and incorporates Abelian and non-Abelian symmetries by sampling symmetry quantum numbers. We benchmark the method, realized as both stochastic matrix product states (stoMPS) and stochastic projected entangled pair states (stoPEPS), on large-scale quantum lattice models. Using stoMPS, we accurately simulate the square-lattice Hubbard model on cylinders up to width W=10W=10W=10, and study the triangular-lattice Hubbard model down to T/t=1/64T/t = 1/64T/t=1/64. We reveal scalar chiral order at half filling and kinetic ferromagnetism upon electron doping, with the latter directly compared to ultracold atom experiments. We further extend the stoMPS method to compute finite-temperature quantum dynamics, as demonstrated by the optical conductivity of the Hubbard model. We showcase the generalization of stoPEPS on up to 20×2020\times2020×20 square-lattice quantum Ising model, where accurate results are obtained near its quantum critical point. Our method combines high sampling efficiency with full symmetry acceleration. The resulting state-of-the-art framework captures both equilibrium and dynamical properties down to ultralow temperatures, thereby providing a powerful theoretical tool for correlated quantum systems, particularly fermion Hubbard quantum gas microscopy.
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We review the basic theory of matrix product states (MPS) as a nu-merical variational ansatz for time evolution, and present two methods to simulate finite temperature systems with MPS: the ancilla method and the minimally entangled typical thermal state method. A sample calculation with the Bose-Hubbard model is provided. The dimension of the Hilbert space for a general many-body system in-creases exponentially with the system size, severely restricting the sizes which are amenable to straightforward numerical study. Several techniques have been developed to deal with this fact, such as the stochastic sam-