• Open Access

Construction and Classification of Symmetry-Protected Topological Phases in Interacting Fermion Systems

Qing-Rui Wang and Zheng-Cheng Gu
Phys. Rev. X 10, 031055 – Published 10 September 2020

Abstract

The classification and lattice model construction of symmetry-protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed-point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group Gf=Z2f×ω2Gb which is a central extension of bosonic symmetry group Gb (may contain time-reversal symmetry) by the fermion parity symmetry group Z2f={1,Pf}. Our construction is based on the concept of an equivalence class of finite-depth fermionic symmetric local unitary transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We also discuss the systematical construction and classification of boundary anomalous SPT states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point and space group symmetry as well as continuum Lie group symmetry.

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  • Received 12 February 2020
  • Revised 20 June 2020
  • Accepted 7 July 2020

DOI:https://doi.org/10.1103/PhysRevX.10.031055

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Qing-Rui Wang and Zheng-Cheng Gu*

  • Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China

  • *zcgu@phy.cuhk.edu.hk

Popular Summary

The discovery of topological insulators—exotic materials in which electrons can move only along the surface—has led to a search for similar underlying phenomena in a range of systems. It turns out that topological insulators are just one example of a broader class of materials known as symmetry-protected topological phases (SPT), where properties are protected by topology and certain global symmetries. Here, we propose a way to construct and classify SPT phases in systems of interacting fermions.

Until now, the understanding of SPT phases in systems of interacting fermions has been very limited. Researchers have found much greater success in classifying SPT phases in systems of interacting bosons and free fermions. Previous efforts to extend this success to interacting fermions have largely relied on modifications to existing mathematical descriptions of interacting bosons and free fermions, but such attempts end up missing many fermionic phases. Rather than use existing math, we develop new mathematical structures that allow us to fully characterize the many SPT phases that can arise in systems of interacting fermions.

Our work describes the full landscape of potential SPT phases that may be realized when fermions interact. The challenge for future work is to detect and realize these phases in experimental setups.

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Vol. 10, Iss. 3 — July - September 2020

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