[论文解读] A Recipe for Constructing Exactly Soluble Lattice Models with Gauge and Matter Fields in One and Two Dimensions

Miguel Jorge Bernabé Ferreira, J. P. Ibieta Jimenez+2 more|arXiv (Cornell University)|Mar 26, 2015

Quantum many-body systems参考文献 23被引用 3

一句话总结

本文提出了一种在1维和2维空间中精确可解的格点模型的状态和构造方法，该模型同时包含规范场和物质场。通过将局部张量关联到三角剖分的流形，并在顶点处引入物质场作为A-模，作者推导出转移矩阵，从而得到精确可解的哈密顿量——特别是当投影算符可交换时，该方法能够建模具有玻色子物质场的短程与长程纠缠相。

ABSTRACT

State sum models can be used to obtain partition functions of physical systems in various dimensions. Several prescriptions have been given to construct such state sum models all of which proceed by associating local tensors, which can be thought of as weights, to different parts of a closed triangulated manifold. An example of this approach is the Kuperberg’s algorithm for finding 3-manifold invariants. From the physics point of view an analogous construction results in the partition functions of three dimensional lattice gauge theories based on involutory Hopf algebras A. For the familiar case of group algebras, A = C(G), we obtain the partition functions of lattice gauge theories. In this paper we extend this construction for lattice gauge theories to one with gauge and matter fields. We build the partition functions of these theories in both two and three space-time dimensions. The additional ingredients in this construction, when compared to the pure gauge case, are the matter fields located on the vertices of the triangulated manifold which are acted upon by the gauge fields living on the edges of the same manifold. The matter fields correspond to Potts spin configurations located at the vertices interacting with the gauge field. They can be described by an A-module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain the transfer matrices of the lattice theories with gauge and matter fields. These transfer matrices are written as a product of local operators acting on vertices, links and plaquettes, very much similar to the ones occurring in lattice gauge theories where they can be identified with those appearing in Kitaev’s Quantum Double Models (QDM). The transfer matrices constructed are functions of a number of parameters which come along with the initial weights associated to different parts of the triangulated manifold. In general the transfer matrices are products of local operators, each of which are sums of projectors, but do not [e-mail] [e-mail] [e-mail] [e-mail] 1 commute with each other. However for certain values of the parameters we obtain transfer matrices made up of commuting projectors. Thus the Hamiltonians obtained from these transfer matrices are exactly soluble and their ground states can mimic both long-ranged and short-ranged entangled phases. We illustrate this construction in both two and three dimensions to obtain exactly soluble quantum lattice models of gauge and matter fields in one and two dimensions. We only consider bosonic matter fields in this paper.

研究动机与目标

• 将状态和模型扩展至包含1维和2维空间中的规范场与物质场。
• 利用三角剖分的流形，为包含物质场的格点规范理论构建配分函数和转移矩阵。
• 识别出导致转移矩阵中投影算符对易的参数区域，从而实现精确可解性。
• 证明所得哈密顿量能够实现短程与长程纠缠的基态。

提出的方法

• 该构造使用带有顶点、边和面权重的三角剖分流形，推广了Kuperberg用于3-流形不变量的算法。
• 物质场以带有内积的A-模形式引入，位于顶点处，并在边上规范场的作用下变换。
• 配分函数通过三角剖分上的张量收缩构建，同时包含规范场和物质场自由度。
• 转移矩阵从带有边界的流形上的配分函数推导而来，表示为顶点、键和面处局部算符的乘积。
• 转移矩阵由初始状态和的权重参数化，特定的参数选择可导致对易投影算符。
• 所得哈密顿量由这些转移矩阵构造而成，当投影算符可交换时为精确可解。

实验结果

研究问题

• RQ1 如何将状态和模型扩展以在1维和2维空间中同时包含规范场与物质场？
• RQ2 模型参数需满足何种条件，才能使转移矩阵中的投影算符对易？
• RQ3 所得哈密顿量能否描述短程与长程纠缠的物相？
• RQ4 在此格点构造中，物质场如何在规范变换下变换？
• RQ5 带有内积的A-模在编码物质自由度中起什么作用？

主要发现

• 该构造产生的转移矩阵是作用于顶点、键和面的局部算符的乘积，类似于Kitaev的量子双模型。
• 对于特定参数值，转移矩阵由对易投影算符构成，从而导致精确可解的哈密顿量。
• 这些哈密顿量的基态可模拟长程与短程纠缠的物相。
• 物质场被描述为带有内积的A-模，确保与规范不变性的一致性。
• 该方法将纯规范状态和模型推广至包含物质场，同时在1D和2D中保持精确可解性。
• 该框架仅限于玻色子物质场，未考虑费米子的推广。