[Paper Review] Lieb-Schultz-Mattis theorems for symmetry protected topological phases

Yuan-Ming Lu|arXiv (Cornell University)|May 12, 2017

Topological Materials and Phenomena1 references27 citations

TL;DR

This paper establishes a new class of Lieb-Schultz-Mattis theorems for symmetry-protected topological (SPT) phases by replacing lattice translation symmetry with magnetic translation symmetry in two-dimensional systems. It proves that a unique, symmetric, gapped, short-range-entangled ground state is impossible unless the system is a topological phase with robust gapless edge modes, particularly for fermionic systems with odd Majorana zero modes per unit cell or bosonic systems with time-reversal symmetry and fractionalized anyons, thereby providing a general framework for identifying SPT order in interacting quantum systems.

ABSTRACT

The Lieb-Schultz-Mattis (LSM) theorem and its descendants represent a class of powerful no-go theorems that rule out any short-range-entangled (SRE) symmetric ground state irrespective of the specific Hamiltonian, based only on certain microscopic inputs such as symmetries and particle filling numbers. In this work, we introduce and prove a new class of LSM-type theorems, where any symmetry-allowed SRE ground state must be a symmetry-protected topological (SPT) phase with robust gapless edge states. The key ingredient is to replace the lattice translation symmetry in usual LSM theorems by magnetic translation symmetry. These theorems provide new insights into numerical models and experimental realizations of SPT phases in interacting bosons and fermions.

Motivation & Objective

• To extend the Lieb-Schultz-Mattis (LSM) theorem framework beyond conventional lattice translation symmetry to include magnetic translation symmetry.
• To establish rigorous no-go theorems that rule out symmetric short-range-entangled (SRE) gapped ground states in 2D systems with specific filling and symmetry conditions.
• To demonstrate that such systems must instead be symmetry-protected topological (SPT) phases with robust gapless edge states.
• To provide a theoretical foundation for identifying SPT order in interacting fermions and bosons, especially in numerically and experimentally realizable models.
• To unify the understanding of SPT phases in non-interacting and strongly correlated systems via symmetry-protected topological constraints.

Proposed method

• Replace standard lattice translation symmetry in LSM theorems with magnetic translation symmetry, defined by a flux per plaquette $\phi = 2\pi p/q$.
• Use the magnetic translation algebra $\tilde{T}_1 \tilde{T}_2 \tilde{T}_1^{-1} \tilde{T}_2^{-1} = e^{i\phi \hat{F}}$, where $\hat{F}$ is the total fermion number operator.
• Apply the method of twisted boundary conditions and flux insertion to probe the response of the ground state to symmetry transformations.
• Use Schmidt decomposition of the ground state on a cylinder to analyze the symmetry quantum numbers of entanglement eigenstates.
• Analyze the transformation properties of Schmidt states under time-reversal and charge symmetries to detect topological pumping and edge mode formation.
• Prove that a change in boundary conditions by $\pi$ flux induces a shift in symmetry quantum numbers, signaling the pumping of a Kramers doublet or anyonic excitation, which is only compatible with a non-trivial SPT phase.

Experimental results

Research questions

• RQ1 Can a unique, symmetric, gapped, short-range-entangled ground state exist in a 2D fermionic system with an odd number of Majorana zero modes per unit cell under magnetic translation symmetry?
• RQ2 What constraints does magnetic translation symmetry impose on the existence of SRE symmetric ground states in interacting fermionic and bosonic systems?
• RQ3 How does the interplay between fractional filling, time-reversal symmetry, and magnetic flux lead to the emergence of topological order?
• RQ4 Under what conditions does a $\pi$-flux insertion induce a topological response that signals a non-trivial SPT phase?
• RQ5 Can the LSM theorem be generalized beyond translation symmetry to include magnetic translation symmetry in a way that protects gapless edge modes?

Key findings

• A unique symmetric gapped SRE ground state is forbidden in a 2D fermionic system with an odd number of Majorana zero modes per unit cell if magnetic translation symmetry with $\phi = \pi$ is present.
• The system must be a $\nu = $ odd topological superconductor in class D, hosting chiral Majorana edge modes with central charge $c_- = \nu/2$, as required by the topological index.
• For a system with time-reversal symmetry and $\phi = \pi$ flux, a unique gapped symmetric ground state is impossible, proving a no-go theorem for such SRE states.
• A $\pi$ flux insertion induces the pumping of a Kramers doublet across the cylinder, which is only compatible with a BQSH (bipartite quantum spin Hall) phase in the bosonic case.
• The symmetry character of Schmidt eigenstates changes sign under $\pi$ flux insertion on odd-length cylinders, signaling topological pumping and confirming the presence of a non-trivial SPT phase.
• The results hold even when $U(1)$ symmetry is reduced to a discrete $Z_2$ subgroup, as long as the time-reversal symmetry remains, proving the robustness of the SPT classification under symmetry breaking.