# Boundary RG Flows for Fermions and the Mod 2 Anomaly

### Submission summary

 As Contributors: David Tong Preprint link: scipost_202011_00013v2 Date accepted: 2020-12-28 Date submitted: 2020-12-10 13:30 Submitted by: Tong, David Submitted to: SciPost Physics Academic field: Physics Specialties: Quantum Physics Approach: Theoretical

### 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.

Published as SciPost Phys. 10, 010 (2021)

Thank you again to both referees for their very prompt replies.

### List of changes

In response to the second referee: the matrix R is rational simply because the charges Q are integer. This follows because the symmetries are U(1) rather than the real line. We've now stressed that Q is integer in a couple of places in the text.

The first referee suggests that we include the "Dirac = Majorana^2" argument for the \sqrt{2} factor. This is indeed the simplest argument, but we already give it in Hamiltonian form at the bottom of page 1, and we don't believe that repeating it in path integral language would add anything.