Boundary RG Flows for Fermions and the Mod 2 Anomaly

TL;DR

This paper studies boundary RG flows for Majorana fermions in 1+1 dimensions, revealing how boundary conditions and operators influence boundary states, central charges, and anomaly cancellation, with implications for SPT phases.

Contribution

It identifies the IR boundary states for Majorana fermions under boundary RG flows and relates boundary central charge ratios to operator dimensions, advancing understanding of boundary anomalies.

Findings

Boundary conditions for Majorana fermions can flip SPT classes.

The ratio of IR to UV boundary central charges equals the operator dimension.

Relevant boundary operators have dimension less than 1, ensuring g-theorem compliance.

Abstract

Boundary conditions for Majorana fermions in d=1+1 dimensions fall into one of two SPT phases, associated to a mod 2 anomaly. Here we consider boundary conditions for 2N Majorana fermions that preserve a $U(1)^N$ symmetry. In general, the left-moving and right-moving fermions carry different charges under this symmetry, and implementation of the boundary condition requires new degrees of freedom, which manifest themselves in a boundary central charge, $g$. We follow the boundary RG flow induced by turning on relevant boundary operators. We identify the infra-red boundary state. In many cases, the boundary state flips SPT class, resulting in an emergent Majorana mode needed to cancel the anomaly. We show that the ratio of UV and IR boundary central charges is given by $g^2_{IR} / g^2_{UV} = {\rm dim}\,({\cal O})$, the dimension of the perturbing boundary operator. Any relevant operator necessarily has ${\rm dim}({\cal O}) < 1$, ensuring that the central charge decreases in accord with the g-theorem.