Thermal tensor renormalization group simulations of square-lattice quantum spin models
Thermal tensor renormalization group simulations of square-lattice quantum spin models
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DOI：
10.1103/physrevb.100.045110
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发表时间：
2019-04
影响因子：
3.7
通讯作者：
Han Li;Bin-Bin Chen-Bin;Ziyu Chen;J. von Delft;A. Weichselbaum;Wei Li
中科院分区：
文献类型：
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作者：
Han Li;Bin-Bin Chen-Bin;Ziyu Chen;J. von Delft;A. Weichselbaum;Wei Li
在这项工作中，我们在二维相互作用自旋模型的模拟中对控制良好且数值精确的指数热张量重正化群（XTRG）进行了基准测试。有限温度引入了热相关长度，这证明了为了数值效率而对有限系统尺寸进行分析的合理性。在本文中，我们重点关注宽度为 $W=10$ 的开放和圆柱形几何形状上的方晶格海森堡反铁磁体 (SLH) 和量子伊辛模型 (QIM)。我们探索了矩阵乘积算子（MPO）表示中的各种一维映射路径，其性能清楚地表明与几何相关。我们以量子蒙特卡罗 (QMC) 数据为基准，同时也以级数展开热张量网络结果为基准。高精度计算了包括内能、比热和自旋结构因子等在内的热性质，与QMC结果具有良好的一致性。 XTRG 还使我们能够达到非常低的温度。对于 SLH，我们在低温下获得每个位点的能量 $u_g^*\simeq -0.6694(4)$ 和自发磁化强度 $m_S^*\simeq0.30(1)$，这已经与基态特性一致。我们在有序波矢量 $M=(\pi,\pi)$ 处提取结构因子 $S(M)$ 相对于 $T$ 的指数散度，以及相关长度 $\xi$，它代表重正化的经典行为，并且可以通过 XTRG 模拟分析有限大小的数据在狭窄但可感知的温度窗口内观察到。对于具有有限温度相变的 QIM，我们采用几个热学量，包括比热、粘合剂比以及 MPO 纠缠来确定临界温度 $T_c$。
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 $u_g^*\simeq -0.6694(4)$ and a spontaneous magnetization $m_S^*\simeq0.30(1)$, 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 $T_c$.