[论文解读] More Solvable 2D Quantum Models from Lattice Gauge Theories and Beyond

Ibieta Jimenez, Miguel Jorge Bernabé Ferreira+1 more|arXiv (Cornell University)|Aug 11, 2014

Quantum many-body systems参考文献 42被引用 2

一句话总结

本论文通过从规范场论转移矩阵引入广义参数，构建了新的二维精确可解量子模型，包括保持可解性和拓扑序的微扰项。识别出阿贝尔（Zn）与非阿贝尔（S3）量子相，其任意子的统计与耦合规则被修改，并将框架扩展至非规范场论的转移矩阵，从而得到扭曲量子双模型，包括Z2的双半子模型。

ABSTRACT

We construct two dimensional lattice models from transfer matrices of lattice gauge theories with discrete gauge groups. These transfer matrices are built out of local operators acting on links, vertices and plaquettes and are parametrized by the center of the gauge group algebra and its dual. They contain the familiar 2D quantum double models for a particular choice of parameters, which includes the well studied example of the toric code. However for more general choices of parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase towards a paramagnetic phase. These perturbations can be thought of as magnetic fields added to the system which destroy the exact solvability of the quantum double model. Nevertheless from the same transfer matrix with perturbations we exhibit exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. This is shown by working with the groups Zn and S3 for the Abelian and non-Abelian cases respectively. The quantum phases are found by studying the excitations of these systems. The fusion rules and the statistics of these anyons indicate the quantum phases of these models. The implementation of these models can possibly improve the use of quantum double models for fault tolerant quantum computation. We then construct theories which arise from transfer matrices that are not the transfer matrices of lattice gauge theories. In particular we show that for the Z2 case this contains the double semion model. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.

研究动机与目标

• 通过从规范群代数及其对偶中引入参数，推广二维量子双模型。
• 证明这些微扰可通过调节参数保持精确可解性与拓扑序，即使原始量子双对称性被破坏。
• 通过任意子激发及其统计性质，识别并表征新量子相——Zn为阿贝尔相，S3为非阿贝尔相。
• 将框架拓展至非格点规范场论的转移矩阵，通过引入额外参数，得到扭曲量子双模型。
• 研究广义模型中任意子的编织与耦合规则，以分类其拓扑序。

提出的方法

• 从具有离散规范群的格点规范场论构造转移矩阵，参数化为规范群代数及其对偶的元素。
• 通过代数及其对偶张量积中的中心元引入微扰，与三维格点中的顶点和体积相关联。
• 分析所得哈密顿量，识别任意子激发，并计算其耦合规则与统计性质。
• 以Zn与S3群为案例研究，区分阿贝尔与非阿贝尔拓扑相。
• 通过在转移矩阵中增加额外参数，将构造拓展至非格点规范场论转移矩阵，恢复如双半子模型与扭曲量子双模型等模型。
• 通过任意子编织与耦合表征结果模型的拓扑序，确认非平凡任意子统计的存在。

实验结果

研究问题

• RQ1 是否能以一种方式引入对量子双模型的微扰，从而保持精确可解性与拓扑序？
• RQ2 由修改的转移矩阵构造的广义模型中，任意子的耦合规则与统计性质为何？
• RQ3 广义模型的拓扑相与标准量子双相有何不同，特别是在非阿贝尔情况下？
• RQ4 是否可从非源自格点规范场论的转移矩阵仍能获得精确可解且具有非平凡拓扑序的模型？
• RQ5 代数及其对偶中的中心元在定义标准量子双框架之外的新量子相中起何作用？

主要发现

• 广义转移矩阵产生精确可解模型，即使在微扰下，通过调节规范群代数及其对偶中的参数，仍保持在拓扑相中。
• 在阿贝尔情形（Zn）中，模型被证明处于标准量子双相，任意子服从阿贝尔耦合与统计。
• 在非阿贝尔情形（S3）中，模型处于与标准量子双相不同的修正相，具有非阿贝尔任意子统计与耦合规则。
• 该构造超越格点规范场论，包含如双半子模型（Z2情形）等模型，展示了更广泛的应用性。
• 通过引入额外参数——代数及其对偶张量积中的中心元——可实现扭曲量子双模型的涌现。
• 所有模型中的任意子激发均表现出明确的编织与耦合特性，证实了拓扑序的存在，并具备容错量子计算的潜力。