# Boundary RG Flows for Fermions and the Mod 2 Anomaly

### Submission summary

 As Contributors: David Tong Preprint link: scipost_202011_00013v1 Date submitted: 2020-11-21 18:07 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.

###### Current status:
Editor-in-charge assigned

### Author comments upon resubmission

First, we would like to thank both referees for their thorough reading of the manuscript and their detailed comments. We really appreciate the effort that it took.

I think that we've addressed all the issues the referees raised. We attach a detailed list of changes that we've made in response to the reports.

Thanks again.

David and Philip

### List of changes

-- Both referees requested that we make clearer what results are specific to the free fermion theories studied in the paper, and which hold more generally. We've added some comments in the summary section and (with regards to the instability of g>1 states), at the end of Section 2.1. We make no claim that our most striking result -- the relation between UV and IR central charges -- holds more generally although clearly it would be interesting to explore this further. We note, however, that's difficult to see how such a relation would work in, say, the Kondo problem where the flow is initiated by a marginally relevant operator. We now mention this explicitly in the summary section in the introduction.

-- We reworded the discussion in the introduction around the Majorana partition function, hopefully making it clearer. Both referees requested that we cite literature beyond Witten's talk on the calculation of the Majorana partition function. We share the referees' expectation that such literature exists, but we have been unable to find it. Moreover, Witten's talk just states the result that the partition function is \sqrt{2}, but doesn't derive it. We added an appendix which presents this calculation explicitly.

-- As the second referee pointed out, the paper does indeed hinge on the assumption of symmetry restoration in the IR. We've stressed more clearly that the results provide evidence for this assumption, since the g-theorem is always satisfied, often in a non-trivial way.

-- We addressed each of the technical issues raised in Point 3 of the Report 2. Thanks for pointing these out.

-- The "fermion vector" is entirely determined by the matrix R. We added a statement to this effect in Section 3.2 and a citation to a later paper where this is proven.

-- The first referee asks about the famous hotel referenced on page 7. We're happy to oblige:

https://www.youtube.com/watch?v=fn6Lg7Neg9I

-- We only claim that the relation between the boundary central charge g^2 and the number of
"holomorphic selection sector" holds for these particular examples where g^2 captures the amount of "chirality" of the system. There's no reason to believe that this holds more generally.

-- We slightly changed the wording of the RG flow from one Maldacena-Ludwig state to another, as described by referee 2.

-- We disagree with the second referee's statement that the paper has no conclusion. The
conclusion was in the introduction with the heading "A Summary of Our Results". For some papers, a final conclusion section is appropriate; for others not. We don't feel that anything will be
added by simply repeating what we have already said in a final section.

-- We removed the suggestion that the Majorana mode was killed on a boat. We instead propose
that it fled to Venezuela where it lived a full and happy life.

-- The referee is, of course, entirely right about the subtleties in bosonization. We wanted to duck all of these in describing the geometric D-brane picture, with the goal of focussing on how the chiral nature of the boundary conditions arises in this setting. We've now made it clearer that we're ducking! We also fixed the typo in the D-brane appendix. Thanks for pointing it out!

### Submission & Refereeing History

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Resubmission scipost_202011_00013v1 on 21 November 2020
Submission 2005.11314v2 on 6 August 2020