Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants

Abstract

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. In this paper, based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups

6+11 pages, 2 tables. As published in Chin. Phys. Lett

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Cited by in corpus (8)

• Symmetric Mass Generation
• Real-space construction of crystalline topological superconductors and insulators in 2D interacting fermionic systems
• Symmetry invariants and classes of quasiparticles in magnetically ordered systems having weak spin-orbit coupling
• Exactly solvable lattice models for interacting electronic insulators in two dimensions
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• Stacking Group Structure of Fermionic Symmetry-Protected Topological Phases
• Crystallography, Group Cohomology, and Lieb-Schultz-Mattis Constraints
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases