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 ρ ^ = e ? β H ^ onto itself. We refer to this scheme of doubling β 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 ρ ^ on a linear quasicontinuous grid in inverse temperature β ≡ 1 / T . As an aside, the large steps in XTRG allow one to swiftly jump across finite-temperature phase transitions, i.e., without the need to resolve each singularly expensive phase-transition point right away, e.g., when interested in low-energy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. 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. For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. 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 β ) 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 X X Z 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, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.
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机译：我们通过迭代地突出热密度矩阵ρ^ = e以指数方式加速了一个（1D）和二维（2D）模型中量子多体系的热模拟βh ^本身。我们在假想时间演进的每个步骤中指代该β的加倍β的方案作为指数张量重整组（XTRG）。这种方法与传统的托特特 - 铃木型方法呈现出常规的托特特 - 铃木型方法，该方法在逆温度β1/ t中的线性逐个栅格上演化ρ^。除了旁边，XTRG中的大步骤允许人们跨越有限温度的相位转换，即，无需立即解析每个单个昂贵的相变点，例如，当对低能量行为感兴趣。然而，通过使用交织的温度网格可以获得精细温度分辨率。通常，XTRG可以呈指数快速达到低温，因此不仅可以节省计算时间，而且由于截断步骤明显较少，因此可能具有更好的准确性。出于类似的原因，我们还发现，来自对数温度设置的效率和精度，串联膨胀热张量网络接近。我们在一个（有效）的1D设置开发矩阵产品运算符（MPOS）中工作，使我们能够完全和唯一地实施非雅中和雅典的对称，以大大提高数值性能。我们使用XTRG机械设备探索1D链和2D方形的Heisenberg模型的热特性，三角形格子下降到接近地面特性的低温。讨论纠缠属性，以及MPO中的缠结光谱的重新运行 - 组流程，其中，对数熵（大约LNβ）显示在旋转链和方形晶格模型中，具有良好的状态。我们还揭示了XTRG可以用来准确地模拟Heisenberg X X Z模型在经过热相转变的方形晶格上。我们基于热物理观察品以及纠缠措施来确定其临界温度。总的来说，我们证明XTRG提供了优雅，多功能和高竞争力的方法来探索热性质，包括有限温度的热相转变以及针对沮丧的系统的各种温度尺度的不同排序趋势。
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