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Boundary RG Flows for Fermions and the Mod 2 Anomaly

by Philip Boyle Smith, David Tong

This is not the current version.

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
  • Quantum Physics
Approach: Theoretical


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:
Has been resubmitted

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:

-- 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!

Reports on this Submission

Anonymous Report 2 on 2020-12-7 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202011_00013v1, delivered 2020-12-07, doi: 10.21468/SciPost.Report.2261


I should thank the authors for addressing my concerns and making the paper easier to read.

Firstly, on the point of a conclusions section, I was guided by the acceptance criteria which say

General acceptance criteria (all required) - the paper must:
6. Contain a clear conclusion summarizing the results (with objective statements on their reach and limitations) and offering perspectives for future work.

I agree that the section "RG Flows: A Summary of Our Results" does this, I leave it to the editor to see whether it is ok not to call it "Conclusions".

I find the argument in appendix A both very cute and also completely unconvincing. How, after throwing away an unknown divergent $C$ and an infinite number of $\pi$'s, can one have any confidence in the remaining factor $\sqrt 2$? But it is ok to leave the argument for the reader to make up their own mind about its strength.

I am afraid that having read through the discussion on allowed charges $Q_{\alpha i}$, $\bar Q_{\beta i}$, I don't see why $R$ should be rational. This seems a necessary condition for $\Lambda[R]$ to be rank $n$, but is there any more fundamental reason? I would have thought that any orthogonal matrices $Q_{\alpha i}$ and $\bar Q_{\beta i}$ would satisfy equation (2.2). Sorry not to have raised this before. It should be very easy to add a few words explaining why $R$ is (taken to be) rational.

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Anonymous Report 1 on 2020-11-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202011_00013v1, delivered 2020-11-30, doi: 10.21468/SciPost.Report.2244


The improvements made by the authors are satisfactory and I consider this version publishable as is.

The authors might want to still consider the following suggestion. The new Appendix A is nice in that it presents the computation of the infamous $\sqrt2$ in an explicit manner. But I still feel somewhat uneasy about all these manipulations of infinite constants to get the final number.

Probably the authors would like to add the following, mostly Hamiltonian version of the derivation, for the sake of future readers. You take two copies of Majorana fermions. Then the path integral should equal Tr 1, which is clearly 2. Therefore the path integral of single Majorana fermion should produce $\sqrt{2}$. (It's somewhat analogous to the derivation of the Gaussian integral by squaring it.)

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