论文标题
低温下的张量重归其化组：不连续点
Tensor Renormalization Group at Low Temperatures: Discontinuity Fixed Point
论文作者
论文摘要
我们继续研究ARXIV：2107.11464的张量网络严格重新归一化组（RG）图。在本文中，我们为2D张量网络构建了严格的RG图，其域名包括张量，代表磁场$ H $的低温下2D ISING模型。我们证明RG图具有两个稳定的固定点，与两个基础状态相对应，一个不稳定的固定点是不连续性固定点的一个示例。对于低温下的ISING模型，如果$ H \ neq 0 $，RG地图将流向稳定的固定点之一，如果$ h = 0 $，则将RG地图和不连续点的固定点流动。除了在哈密顿量中最近的邻居和磁场项之外，我们还可以包括不需要自旋不变的小术语。在这种情况下，我们证明该字段的临界值$ h_c $（取决于这些其他小相互作用和温度），因此，如果$ h = h_c $以及否则，RG映射会流向不连续点的固定点，则否则将其流向稳定的固定点之一。我们使用RG地图为一阶过渡的先前结果提供了新的证明，即，自由能是$ h \ neq h_c $的分析性，并且磁化值在$ h = h_c $的情况下不连续。我们的低温RG图的构建，尤其是Disentangler，与高温阶段的Arxiv：2107.11464中的地图的构造非常相似。我们还对无限尺寸张量网络的一些一般严格转换进行了教学讨论，并概述了ARXIV中RG MAP的高温固定点的稳定性证明：2107.11464。
We continue our study of rigorous renormalization group (RG) maps for tensor networks that was begun in arXiv:2107.11464. In this paper we construct a rigorous RG map for 2D tensor networks whose domain includes tensors that represent the 2D Ising model at low temperatures with a magnetic field $h$. We prove that the RG map has two stable fixed points, corresponding to the two ground states, and one unstable fixed point which is an example of a discontinuity fixed point. For the Ising model at low temperatures the RG map flows to one of the stable fixed points if $h \neq 0$, and to the discontinuity fixed point if $h=0$. In addition to the nearest neighbor and magnetic field terms in the Hamiltonian, we can include small terms that need not be spin-flip invariant. In this case we prove there is a critical value $h_c$ of the field (which depends on these additional small interactions and the temperature) such that the RG map flows to the discontinuity fixed point if $h=h_c$ and to one of the stable fixed points otherwise. We use our RG map to give a new proof of previous results on the first-order transition, namely, that the free energy is analytic for $h \neq h_c$, and the magnetization is discontinuous at $h = h_c$. The construction of our low temperature RG map, in particular the disentangler, is surprisingly very similar to the construction of the map in arXiv:2107.11464 for the high temperature phase. We also give a pedagogical discussion of some general rigorous transformations for infinite dimensional tensor networks and an overview of the proof of stability of the high temperature fixed point for the RG map in arXiv:2107.11464.
