Exponential Thermal Tensor Network Approach for Quantum Lattice Models
Exponential Thermal Tensor Network Approach for Quantum Lattice Models
复制标题
量子晶格模型的指数热张量网络方法
DOI：
10.1103/physrevx.8.031082
复制
发表时间：
2018-09-26
影响因子：
12.5
通讯作者：
Weichselbaum, Andreas
中科院分区：
文献类型：
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作者：
Chen, Bin-Bin;Chen, Lei;Weichselbaum, Andreas
我们通过迭代地将热密度矩阵$\hat\rho=e^{-\beta \hat{H}}$投影到自身上，以指数方式加速了一维和二维模型中量子多体系统的热模拟。我们将这种在虚时间演化的每一步中将$\beta$加倍的方案称为指数张量重整化群（XTRG）。这种方法是在形成鲜明对比，传统的Trotter-Suzuki型的方法，发展$\hat\rho$的线性准连续网格上的逆温度$\beta \equiv 1/T$。一般来说，XTRG可以指数快速地达到低温，因此不仅节省了计算时间，而且由于截断步骤明显减少而具有更好的精度。我们在一个（有效的）1D设置利用矩阵乘积算子（MPO），使我们能够充分和独特地实现非阿贝尔和阿贝尔对称性，大大提高数值性能。我们使用我们的XTRG机器来探索海森堡模型在一维链和二维正方形和三角形晶格上的热性质，直到接近基态性质的低温。纠缠特性，以及在MPO纠缠谱的重整化群流，进行了讨论，其中对数熵（约$\ln\beta$）显示在自旋链和正方形晶格模型与无隙塔的状态。我们还发现，XTRG可以用来准确地模拟海森堡XXZ模型的正方形晶格，经历一个热相变。我们确定其临界温度的基础上的热物理观测值，以及纠缠措施。总的来说，我们证明了XTRG提供了一种优雅，通用和极具竞争力的方法来探索1D和2D量子晶格模型中的热特性。
We speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix $\hat\rho=e^{-\beta \hat{H}}$ onto itself. We refer to this scheme of doubling $\beta$ in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve $\hat\rho$ on a linear quasi-continuous grid in inverse temperature $\beta \equiv 1/T$. In general, XTRG can reach low temperatures exponentially fast, and thus not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. We work in an (effective) 1D setting exploiting matrix product operators (MPOs) which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground state properties. The entanglement properties, as well as the renormalization group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately $\ln\beta$) are shown in both spin chains and square lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties in both 1D and 2D quantum lattice models.