Protected edge modes without symmetry (1301.7355v3)

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.

• The paper establishes a criterion for gapped edges in 2D electron systems, revealing that fractional statistics can protect edge modes even when thermal Hall conductance vanishes.
• It combines microscopic edge analysis, braiding statistics, and conformal field theory to validate the conditions for edge gapping.
• Case studies of the ν = 2/3 and ν = 8/9 quantum Hall states highlight that the selection of quasiparticle types dictates whether an edge remains gapless or can be gapped.

Protected Edge Modes Without Symmetry

The paper presented in this paper by Michael Levin explores the intricate dynamics of gapped 2D electron systems and examines conditions under which these systems can sustain protected, gapless edge modes absent any symmetries. The inquiry primarily hinges on systems with zero thermal Hall conductance, denoted by $K_H = 0$, a condition assumed to typically correspond to systems that could have a gapped edge. However, Levin challenges this assumption by showing that even when $K_H = 0$, systems can maintain robust edge modes if they host quasiparticles with certain types of fractional statistics.

The central result of the paper is a criterion that determines when a gapped edge can exist in 2D electron systems characterized by abelian statistics and having $K_H = 0$. Levin establishes that a gapped edge is feasible if and only if there exists a subset $\mathcal{M}$ of quasiparticle types that (1) mutually possess trivial statistics, and (2) interact non-trivially with at least one member of $\mathcal{M}$ if they are not part of $\mathcal{M}$. The criterion is derived through three distinct approaches: a microscopic analysis of the edge, an argument based on braiding statistics, and a verification using conformal field theory (CFT) considerations and modular invariance constraints.

Strong Numerical Results and Bold Claims

The paper scrutinizes two illustrative cases: the $\nu = 2/3$ and $\nu = 8/9$ fractional quantum Hall states. Notably, while both cases involve edge modes and a vanishing thermal Hall conductance, only the $\nu = 8/9$ state permits a gapped edge, aligning with the criterion. For the $\nu = 2/3$ state, the quasiparticle set $\mathcal{M}$ fails the necessary conditions, indicating its edge remains gapless and protected.

The multistep derivation notably engages the Chern-Simons framework to elucidate the behavior of these systems. This approach enables meticulous examination of edge states through null vector analysis, elucidating when possible perturbations can or cannot induce a gap. A significant bold claim emerges when the analysis concludes that protected edge modes arise because fractional statistics inherently inhibit the occurrence of a gapped edge in the $\nu = 2/3$ state despite $K_H=0$.

Implications and Theoretical Speculations

The implications for both practical applications and theoretical perspectives in condensed matter physics and quantum computing are considerable. Practically, this understanding can guide the engineering of materials with tailored edge conductance properties, potentially leading to developments in low-energy electronics or quantum devices. Theoretically, Levin's criterion with its reliance on the statistical properties of quasiparticles suggests novel avenues for exploring nontrivial topological order without relying on conventional symmetry constraints.

Future advancements might focus on extending these results to non-abelian statistics and beyond the simple edge-vacuum geometries considered here. Moreover, the bridge between edge protection and bulk quasiparticle interaction properties hints at deeper, yet uncovered principles governing condensed matter phases.

In conclusion, Levin’s paper advances our comprehension of topological matter’s edge phenomena by unveiling the subtleties that drive the existence of gapless modes even in the absence of symmetry and thermal Hall conductance. This adds a significant piece to the puzzle of understanding how topological phases manifest and can be manipulated, further enriching the landscape of quantum materials research.