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Boundary RG Flows for Fermions and the Mod 2 Anomaly
by Philip Boyle Smith, David Tong
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Submission summary
Authors (as registered SciPost users):  David Tong 
Submission information  

Preprint Link:  scipost_202011_00013v2 (pdf) 
Date accepted:  20201228 
Date submitted:  20201210 13:30 
Submitted by:  Tong, David 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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 leftmoving and rightmoving 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 infrared 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 gtheorem.
Published as SciPost Phys. 10, 010 (2021)
Author comments upon resubmission
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.