Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases

Shenghan Jiang,Meng Cheng,Yang Qi,Yuan-Ming Lu

doi:10.21468/scipostphys.11.2.024

Abstract

We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.

Highlights

We show that the symmetry enforced symmetry protected topological (SPT) phase can be understood from decorated domain wall picture: by condensing domain walls from a specific spontaneously symmetry breaking (SSB) phase, we either obtain another SSB phase, or get a non-trivial SPT phase
We focus on lattice Y which satisfies the pointwise-action condition mentioned in the last part: for cell ∆ ∈ Yn, the global symmetry for spin system on ∆ are identified as SG∆, which acts as onsite symmetry
We present a general theoretical framework for LSM-type theorems for bosonic SPT phases through a real-space construction, and describe a general approach to construct new SPT-LSM theorems from known results of more conventional LSM theorems

Summary

Introduction

Since the discovery of topological band insulators [1,2], symmetry-protected topological (SPT) phases have attracted considerable research interests both theoretically and experimentally [3,4,5]. Once such a gauge theory is identified, condensing gauge charges (binding with some symmetry charges) leads to a SPT phase and a candidate SPT-LSM system, where the gauge symmetry becomes a global symmetry We prove this SPT-LSM theorem in a more rigorous approach, based on (i) a real-space construction of crystalline SPT phases [20, 27,28,29,30,31] and (ii) in certain cases entanglement-spectrum-based argument [18, 24] to find the precise constraints on the ground state. Appendix D give an overview on how to obtain SPT phases by condensing topological excitations in fractionalized phases

A 1D example

Models and symmetry

General framework for SPT-LSM systems – real space constructions

Examples for SPT-LSM systems in higher dimension

Conclusion and future directions