• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,348 answers , 22,725 comments
1,470 users with positive rep
818 active unimported users
More ...

  Quantizing fermion zero modes in topological phases

+ 4 like - 0 dislike

I don't understand how the fermionic zero modes which become monopole operators are being quantized in Witten and Seiberg's paper on gapped boundary phases of TI's via weak coupling: https://arxiv.org/abs/1602.04251.

I'm having trouble with some of the statements made in Appendix B (and used throughout the paper). In Eq. B.16, they list the possible (classical) spins of the zero modes as a function of vorticity: $j'=(v-1)/2, (v-3)/2,\dots,-(v-1)/2$ where $v$ is an integer. I think I understand how this expression is derived -- what happens upon quantizing is the confusing point.

They go on to say that for $v=1$, there is a single zero mode with spin $j'=0$.

For $v=2$, the zero modes have spins $\pm1/2$. At the bottom of page 70, they simply assert that upon quantization this means that upon quantizing, this becomes pair of states with spin $\pm1/4$.

$v=3$ gives three zero modes with spins 1,0,-1. Quantizing the $\pm1$ modes, gives two states with spins $\pm1/2$. The third mode keeps its zero spin.

I've tried comparing with the discussion in Borokhov et al. (arXiv:hep-th/0206054) as well as Sec. 7.1 of Dyer et al. (arXiv:1309.1160), but I don't see how their arguments can be generalized

In particular, Witten and Seiberg give the usual definition of a monopole by performing the path integral in the presence of a Dirac singularity. With fermions present in the action, the semi-classical equations of motion return fermionic zero modes, whose number depends both on the topological charge of the monopole and the charge of the fermions.

Each zero mode is then subsequently treated as an operator which can act of the Fock vacuum $\sim \chi_{i_1}^\dagger\chi_{i_2}^\dagger\cdots\chi_{i_n}^\dagger\left|0\right>$. Enforcing the Gauss constraint places restrictions on the linear combinations which are allowed and these become the monopole operators. My understanding from the Borokhov and Dyers papers is both charge neutrality and that the ground states contain only spin singlets. What am I missing?

This post imported from StackExchange Physics at 2017-03-03 11:22 (UTC), posted by SE-user alex

asked Feb 24, 2017 in Theoretical Physics by alex_utf8 (20 points) [ revision history ]
edited Mar 3, 2017 by Dilaton

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights