In this work, we exponentially speed up thermal simulations of quantummany-body systems in both one- (1D) and two-dimensional (2D) models usingmatrix product operators (MPOs). Instead of evolving the density operator{$hat{ho}=e^{-eta hat{H}}$} linearly in inverse temperature $eta equiv1/T$ as conventional Trotter-Suzuki methods do, we cool down the system {byiteratively projecting the MPO representation of $hat{ho}$ to itself, i.e.,doubling $eta$} in the process of imaginary time evolution. This exponentialtensor renormalization group (XTRG) scheme, compared to linear evolutionschemes, reaches low temperatures much faster, and thus not only savescomputational time but also merits better accuracy due to significantly fewertruncation steps. For similar reasons, we also find that the series expansionthermal tensor network (SETTN) approach benefits in both efficiency andprecision, from the logarithmic temperature scale setup. For both thermalalgorithms, XTRG as well as SETTN, we fully implement non-Abelian and Abeliansymmetries to greatly enhance their numerical performance. We employ thesecutting-edge techniques for finite temperature simulations to explorelow-temperature thermal states of both 1D and 2D Heisenberg models. Theentanglement properties, as well as the renormalization group flow ofentanglement spectra in MPOs, are discussed, where logarithmic entropies ($simln{eta}$) are shown in both spin chains and square lattice models withgapless tower of states.
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机译：在这项工作中，我们在使用Matrix产品运算符（MPOS）中的一个（1D）和二维（2D）型号中的量子型 - 身体系统的热模拟速度模拟。而不是演变密度运算符{$ hat { rho} = e ^ { - beta hat {h}} $ in逆温度$ beta Equiv1 / t $}作为传统的托洛特 - 铃木方法，我们冷却系统{才能在虚数演变过程中投射$ hat { rho} $的MPO表示，即+ beta $}。与线性进化型化相比，该展示传感器重整组（XTRG）方案达到了更快的温度，因此不仅可以节省代表时间，而且由于显着截断步骤，因此不仅可以获得更好的准确性。出于类似的原因，我们还发现，从对数温度设置的效率和施用，串联的扩展张量网络（Settn）接近效率。对于热晶体，XTRG以及塞子，我们完全实施非阿比越和雅思人对比，以大大提高其数值。我们采用了对5D和2D Heisenberg型号的有限温度模拟进行了电影为有限温度模拟的技术。讨论了MPOS的NeTontLement属性，以及MPO中的应入光谱的重新调度组流程，其中对数熵（$ sim ln { beta} $）显示在旋转链和方形格子型号中的旋转链中。
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