[论文解读] Non-invertible symmetry-protected topological order in a group-based cluster state

Christopher Fechisin, Nathanan Tantivasadakarn+1 more|arXiv (Cornell University)|Dec 14, 2023

Topological Materials and Phenomena被引用 16

一句话总结

本文介绍了一个由基于群的Pauli算符构建的一维稳定子哈密顿量，其基态具有 G×Rep(G) 对称性，证明它实现了一种融合范畴 SPT 相，识别微观特征（边缘模态、字符串序、拓扑响应），并讨论 MBQC 应用以及一个具体非阿贝尔 G（半直接积）情形。

ABSTRACT

Despite growing interest in beyond-group symmetries in quantum condensed matter systems, there are relatively few microscopic lattice models explicitly realizing these symmetries, and many phenomena have yet to be studied at the microscopic level. We introduce a one-dimensional stabilizer Hamiltonian composed of group-based Pauli operators whose ground state is a $G imes ext{Rep}(G)$-symmetric state: the $G extit{ cluster state}$ introduced in Brell, New Journal of Physics 17, 023029 (2015) [at http://doi.org/10.1088/1367-2630/17/2/023029]. We show that this state lies in a symmetry-protected topological (SPT) phase protected by $G imes ext{Rep}(G)$ symmetry, distinct from the symmetric product state by a duality argument. We identify several signatures of SPT order, namely protected edge modes, string order parameters, and topological response. We discuss how $G$ cluster states may be used as a universal resource for measurement-based quantum computation, explicitly working out the case where $G$ is a semidirect product of abelian groups.

研究动机与目标

• Motivate the study of beyond-group symmetries in microscopic lattice models and introduce a group-based stabilizer framework for SPTs.
• Define and realize a G×Rep(G) symmetry-protected topological phase in a 1D lattice model.
• Identify microscopic signatures of SPT order in the G cluster state, including edge modes, string order, and topological response.
• Discuss how G cluster states can serve as universal resources for measurement-based quantum computation (MBQC).

提出的方法

• Construct a 1D stabilizer Hamiltonian using group-based Pauli operators acting on group-valued qudits.
• Represent the ground state as a matrix product state (MPS) and connect to matrix product operators (MPOs).
• Demonstrate that the G cluster state is protected by a fusion-category symmetry given by G×Rep(G).
• Provide duality arguments to show distinctness from the symmetric product state.
• Analyze edge modes, ground-state degeneracy, string order parameters, and topological response as signatures of SPT order.
• Develop an MBQC framework for G cluster states, including cases where G is a semidirect product of abelian groups.

实验结果

研究问题

• RQ1 How can a microscopic 1D lattice model realize fusion-category (non-invertible) symmetry protected topological order with G×Rep(G) symmetry?
• RQ2 What are the microscopic indicators (edge modes, string order, topological response) that certify SPT order in the G cluster state?
• RQ3 Can G cluster states function as universal resources for measurement-based quantum computation, especially for non-abelian G (semidirect products)?
• RQ4 How does the G cluster state relate to and extend known Z2×Z2 (cluster) SPT order and KW dualities?

主要发现

• The G cluster state realizes an SPT phase protected by a fusion-category symmetry G×Rep(G).
• Edge modes and ground-state degeneracy appear as signatures of the SPT order in the 1D model.
• String order parameters and topological response provide microscopic diagnostics of the SPT phase.
• The formalism connects group-based Pauli stabilizers to MPS/MPO descriptions, clarifying the lattice realization of fusion-category symmetries.
• For certain non-abelian groups (semidirect products of abelian groups), there is an explicit MBQC protocol using the G cluster state as a universal resource.
• The work extends known 1D SPT results (e.g., Z2×Z2 cluster state) to group-based and non-invertible symmetry contexts.