Protected edge modes without symmetry

TL;DR

The paper addresses when a 2D gapped system without symmetry can host a protected gapless edge, showing that a nonzero thermal Hall conductance is not the only protection mechanism: certain abelian systems with $K_H=0$ can still have robust edges if quasiparticles exhibit fractional statistics. It introduces a precise criterion based on a Lagrangian subgroup of quasiparticle types and proves it through three complementary approaches (microscopic edge analysis, braiding- statistics, and modular invariance). The results, exemplified by ν=8/9 and ν=2/3 quantum Hall edges, reveal how edge gapping is intimately tied to bulk braiding structure and topological data, and they extend to bosonic systems as well. This work provides a framework for classifying and understanding protected edge modes in 2D topological phases beyond symmetry protection, with implications for edge engineering and topological quantum computation.

Abstract

We discuss the question of when a gapped 2D electron system without any symmetry has a protected gapless edge mode. While it is well known that systems with a nonzero thermal Hall conductance, $K_H \neq 0$, support such modes, here we show that robust modes can also occur when $K_H = 0$ -- if the system has quasiparticles with fractional statistics. We show that some types of fractional statistics are compatible with a gapped edge, while others are fundamentally incompatible. More generally, we give a criterion for when an electron system with abelian statistics and $K_H = 0$ can support a gapped edge: we show that a gapped edge is possible if and only if there exists a subset of quasiparticle types $M$ such that (1) all the quasiparticles in $M$ have trivial mutual statistics, and (2) every quasiparticle that is not in $M$ has nontrivial mutual statistics with at least one quasiparticle in $M$. We derive this criterion using three different approaches: a microscopic analysis of the edge, a general argument based on braiding statistics, and finally a conformal field theory approach that uses constraints from modular invariance. We also discuss the analogous result for 2D boson systems.

Figures (9)

• Figure 1: To demonstrate the subtleties of protected edge modes without symmetry, we consider the $\nu = 2/3$ and $\nu = 8/9$ fractional quantum Hall edges, proximity-coupled to an adjacent superconductor. (a) In the $\nu = 2/3$ case, the edge has vanishing thermal Hall conductance, $K_H = 0$, since it contains two modes moving in opposite directions. Even so, we will show that the edge is protected. (b) In the $\nu = 8/9$ case, the edge also has $K_H = 0$, but in this case we will show that the edge is not protected. We argue that the two states behave differently because of the different quasiparticle braiding statistics in the bulk.
• Figure 2: The concept of "annihilating" particles at a gapped boundary. (a) Consider a thought experiment in which we create a pair of quasiparticle excitations $m, \overline{m}$ in the bulk and then bring them near to two points $a,b$ at the edge. We denote the resulting excited state by $|\Psi_{ex}\rangle$. (b) We say that $m, \overline{m}$ can be annihilated at the boundary if there exist operators $U_a, U_b$ acting in the vicinity of $a,b$, such that $U_a U_b |\Psi_{ex}\rangle = |\Psi\rangle$, where $|\Psi\rangle$ is the ground state. Otherwise we say the particles cannot be annihilated.
• Figure 3: (a) For each $m \in \mathcal{M}$, we can consider a process in which we create a pair of quasiparticles $m,\overline{m}$ in the bulk, move them along some path $\beta$ to two points on the edge, and then annihilate them. We define $\mathbb{W}_{m\beta}$ to be the operator that implements this process. (b) To establish condition (1) of the criterion, we consider two paths $\beta, \gamma$, and two associated operators $\mathbb{W}_{m\beta}$, $\mathbb{W}_{m'\gamma}$. We then make use of the relations (\ref{['wmm1']}-\ref{['wmm2']}) along with the commutation relation (\ref{['statrel']}).
• Figure 4: (a) The concept of braiding non-degeneracy in the bulk: if $l$ is a quasiparticle that cannot be annihilated in the bulk, then there must be at least one quasiparticle species $m$ that has nontrivial mutual statistics with respect to $l$, i.e. $e^{i\theta_{lm}} \neq 1$. (b) The concept of braiding non-degeneracy at a gapped edge: if $l$ cannot be annihilated at the edge, then there must be at least one quasiparticle $m$ that can be annihilated at the edge such that $e^{i\theta_{lm}} \neq 1$.
• Figure 5: We consider the system in a strip geometry with finite width $L_y$ in the $y$-direction. We assume that the lower edge is gapped, while the upper edge is gapless and described by (\ref{['genedgeth']}). The system has two types of scaling operators: (a) charge neutral operators (\ref{['scalop1']}) acting on the upper edge and (b) charged operators of the form $e^{il^T \phi}$. Operators of type (b) can only appear as a low-energy description of a tunneling process in which a quasiparticle of type $l$ tunnels from the upper edge to the lower edge and is subsequently annihilated.