[论文解读] Symmetry-protected topological orders for interacting fermions -- Fermionic topological nonlinear $σ$ models and a special group supercohomology theory

Zheng‐Cheng Gu, Xiao-Gang Wen|DSpace@MIT (Massachusetts Institute of Technology)|Jan 12, 2012

Topological Materials and Phenomena被引用 13

一句话总结

本文引入一种特殊的群上同调理论来分类相互作用费米子对称保护拓扑（SPT）序相，推广了玻色子的群上同调理论。该理论构建了费米子SPT波函数和体哈密顿量，预测了异常边界模式，并识别出新的三维和二维SPT相——例如具有$Z_2^T \times Z_2^f$对称性的三维费米子超导体——这些相无法通过自由费米子或玻色子理论实现。

ABSTRACT

Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry $G$, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of super cocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (ie, non-on-site) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric} boundary must be gapless for 1+1D boundary, and must be gapless or topologically ordered beyond 1+1D. As an application of this general result, we construct a new SPT phase in 3D, for interacting fermionic superconductors with coplanar spin order (which have $T^2=1$ time-reversal $Z_2^T$ and fermion-number parity $Z_2^f$ symmetries described by a full symmetry group $Z_2^T imes Z_2^f$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion-pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in 2D with a full symmetry group $Z_2 imes Z_2^f$. Those 2D fermionic SPT phases all have central-charge $c=1$ gapless edge excitations, if the symmetry is not broken.

研究动机与目标

• 将玻色子SPT相的群上同调分类推广至相互作用费米子系统，采用广义的上同调框架。
• 利用上循环数据，为费米子SPT相构造显式的基态波函数和体哈密顿量。
• 证明费米子SPT相的对称边界必须是无能隙或拓扑有序的，暗示异常保护机制。
• 识别出在自由费米子K-理论和玻色子群上同调理论范围之外的新SPT相。
• 通过具有非局部对称性的费米子拓扑非线性$\sigma$模型，提供系统化的场论描述。

提出的方法

• 发展一种特殊的群上同调理论，作为标准群上同调的推广，纳入费米子宇称和时间反演对称性。
• 利用上循环定义费米子SPT态的精确波函数，编码拓扑序和对称性性质。
• 构建实现SPT相的体哈密顿量，其体态为能隙态，边界上具有异常（非局部）对称性。
• 分析边界有效场论，表明非局部对称性在1+1维导致无能隙，在更高维导致拓扑序。
• 将形式化应用于特定对称群，包括三维的$Z_2^T \times Z_2^f$和二维的$Z_2 \times Z_2^f$，以推导具体模型。
• 利用费米子非线性$\sigma$模型描述这些SPT相的场论，捕捉其拓扑性质和对称性异常。

实验结果

研究问题

• RQ1 能否发展一种通用分类框架，用于描述超越自由费米子和玻色子理论的相互作用费米子SPT相？
• RQ2 费米子统计和非阿贝尔对称性（如$Z_2^T \times Z_2^f$）如何影响SPT序的结构？
• RQ3 上循环在构建费米子SPT相的精确波函数和哈密顿量中起什么作用？
• RQ4 为何费米子SPT相的对称边界必然表现出无能隙或拓扑序？
• RQ5 能否在二维和三维中构造出未被K-理论或玻色子群上同调理论涵盖的新SPT相？

主要发现

• 提出一种新的特殊群上同调理论，可对一大类相互作用费米子SPT相进行分类。
• 该理论利用上循环数据，为费米子SPT态提供了显式的波函数和哈密顿量。
• 费米子SPT相的边界表现出异常（非局部）对称性，意味着在1+1维边界必须无能隙，或在更高维呈现拓扑序。
• 为具有$Z_2^T \times Z_2^f$对称性的相互作用费米子超导体构造了一种新的三维SPT相，该相无法通过自由费米子或玻色子配对实现。
• 识别出三种具有$Z_2 \times Z_2^f$对称性的二维费米子SPT相，当对称性未被破坏时，每种相均具有中心电导$c=1$的无能隙边界模式。
• 费米子拓扑非线性$\sigma$模型为这些SPT相提供了场论描述，准确捕捉其拓扑不变量和对称性异常。