Thermal Tensor Renormalization Group Simulations of Square-Lattice Quantum Spin Models

Abstract

In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a thermal correlation length, which justifies the analysis of finite system size for the sake of numerical efficiency. In this paper we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width $W=10$. We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc., are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH we obtain at low temperature an energy per site $u_g^{\ast }\simeq -0.6694(4)$ and a spontaneous magnetization $m_S^{\ast }\simeq 0.30(1)$, which is already consistent with the ground state properties. We extract an exponential divergence vs. $T$ of the structure factor $S(M)$, as well as the correlation length $\xi$, at the ordering wave vector $M=(\pi ,\pi )$, which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analysing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature $T_c$.

16 pages, 16 figures

References in corpus (14)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Minimally Entangled Typical Thermal State Algorithms
• Spin liquid phase of the $S=\frac{1}{2}$ $J_1-J_2$ Heisenberg model on the triangular lattice
• Operator space entanglement entropy in transverse Ising chain
• Spectral functions in one-dimensional quantum systems at T>0
• Projected Entangled Pair States at Finite Temperature: Imaginary Time Evolution with Ancillas
• Entanglement Entropy of the Two-Dimensional Heisenberg Antiferromagnet
• Linearized Tensor Renormalization Group Algorithm for Thermodynamics of Quantum Lattice Models
• Complexity of thermal states in quantum spin chains
• Variational tensor network renormalization in imaginary time: benchmark results in the Hubbard model at finite temperature
• Ba8CoNb6O24: a spin-1/2 triangular-lattice Heisenberg antiferromagnet in the 2D limit
• Finite correlation length scaling with infinite projected entangled pair states at finite temperature
• Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries

Cited by in corpus (1)

• Universal Thermodynamics in the Kitaev Fractional Liquid