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Boundary condition and reflection anomaly in $2+1$ dimensions
by Jiunn-Wei Chen, Chang-Tse Hsieh, Ryutaro Matsudo
Submission summary
| Authors (as registered SciPost users): | Ryutaro Matsudo |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2306.10845v2 (pdf) |
| Date submitted: | 2023-11-28 05:23 |
| Submitted by: | Matsudo, Ryutaro |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
It is known that the $2+1$d single Majorana fermion theory has an anomaly of the reflection, which is canceled out when 16 copies of the theory are combined. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition that varies at a single point, and find that there is a problem with one Majorana fermion. The problem is the absence of a corresponding outgoing wave to a specific incoming wave into the boundary, which leads to the non-conservation of the energy. For 16 Majorana fermions, it is possible to connect every incoming wave to an outgoing wave without breaking the reflection symmetry. In addition, we discuss the connection with the fermion-monopole scattering in $3+1$ dimensions.
Current status:
Reports on this Submission
Report
This paper explores the possible connection between two notions. The first notion is the existence of a 't Hooft anomaly for a symmetry group G in a quantum field theory. The second notion, dubbed edge-ability, is the existence of G-preserving boundary conditions. The authors explore this theme using the specific example of Majorana fermions in 2+1 dimensions, where the symmetry G in question is reflection. By considering a boundary condition varying at a point they conclude that indeed the existence of the reflection anomaly leads to an inconsistency in reflection symmetric boundary conditions. They further make connections between this lack of edge-ability and recent progress on the fermion-monopole scattering in 3+1 dimensions.
The paper is well written, the logic is clearly explained and the main results are put front and centre away from any superfluous distractions. The question they pose is novel and unexplored in higher dimensions, and the connection they point out to fermion-monopole scattering may have further applications. I therefore think that it can be published in Scipost physics, provided the following points are addressed.
Requested changes
1) The boundary condition in eq (3.1) implies that \psi=0 at the origin (r=0). This is clearly not the case for the 0-mode of the solution they propose in (3.6) since f_0 contains 0^0=1 and moreover the Kummer's function give 1 at r=0. How is this consistent with the boundary condition?
2) Below equation (5.1) the authors claim that coupling a symmetry with a 't Hooft anomaly to a background leads to an inconsistency. This is not true, the theory is perfectly consistent i.e. there is no tension with regularisation as opposed to dynamical symmetries. I ask the authors to rephrase this paragraph appropriately.
Recommendation
Ask for minor revision
Author: Ryutaro Matsudo on 2024-06-05 [id 4539]
(in reply to Report 1 on 2024-05-21)Thank you very much for reading our manuscript carefully, and giving positive comments. The following are the answers to your questions:
(1) The equation (3.1) does not specify the value of $\psi$ at $r=0$, and only specifies the values of the upper component of $\psi$ at $x< 0$ and the lower component at $x>0$. Also, any function in the expansion (3.2) satisfies the boundary condition, as explained in the paper. You may worry about the fact that the zero mode diverges at r=0, but it does not cause any problem since the action for the $n=0$ modes is just a 2d fermion theory (on a half line), which is finite and gives a consistent theory (when the numbers of the left and right movers are matched). As we explained in the paper, we need to specify additional boundary conditions at r=0 to fix the behavior of the $n=0$ modes.
(2) The single Weyl fermion theory in the monopole background is inconsistent since the corresponding outgoing wave to the s-wave component does not exist. We refer to a theory that cannot preserve energy as an inconsistent theory throughout this paper.
In general, the theory with an 't Hooft anomaly can be inconsistent when we introduce background gauge fields. For example, in the single Weyl fermion theory, when we introduce a background U(1) gauge field with non-vanishing $\vec E\cdot \vec B$, the U(1) symmetry is broken, i.e., the gauge symmetry is broken. A theory is said to be inconsistent when a gauge symmetry is broken. Note that even when we introduce a background gauge field and not a dynamical gauge field, the corresponding symmetry becomes a gauge symmetry. In other words, we cannot introduce a background gauge field when the symmetry is broken.
In the literature, 't Hooft anomalies refer to impossibility to couple background gauge fields. See, e.g., the first paragraph of [D. Gaiotto et al., "Theta, time reversal and temperature", JHEP 05 (2017) 91].