Boundary RG Flows for Fermions and the Mod 2 Anomaly

TL;DR

The article analyzes boundary conditions for 2N massless Majorana fermions in $d=1+1$, revealing a mod $2$ anomaly that classifies conformal boundary states into two SPT classes (vector and axial) and introduces a boundary central charge $g$. It develops the chiral boundary-state formalism, encoding boundary data in a lattice $ ext{Vol}(oldsymbol{ L})$ and showing how boundary operators are organized by a dual lattice with dimension $L_0= frac12\rho^2$, leading to a general RG-flow rule: $g_{IR}=g_{UV}\sqrt{\mathrm{dim}\,\mathcal{O}}$ for a relevant boundary operator with $\mathrm{dim}\mathcal{O}=L_0<1$. Flows may flip the SPT class and require an emergent boundary Majorana mode or a discrete symmetry summation; the analysis covers non-primitive charges and shows how Majorana modes and IR degeneracies reconcile the flows with the $g$-theorem. The results provide a precise framework linking boundary charges, IR fixed points, and topological boundary phenomena in fermionic systems, with connections to D-brane descriptions and Fidkowski–Kitaev-type gapped phases.

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.

Figures (3)

• Figure 1: The various conformal identifications used, including those which correspond to an $\mathcal{S}$ transformation of the argument of the partition function.
• Figure 2:
• Figure 3: How $(-1)^{F+\bar{F}}$ sits in $U(1)^{N-1} \times \mathbb{Z}_n \subset U(1)^N$.