Quantcast
  • 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.

News

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

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Partition function of a $1+1$ D bosonic SPT with $D_4$ symmetry

+ 1 like - 0 dislike
411 views

There should be a nontrivial $1+1$ D bosonic SPT with $D_4$ symmetry, which can be obtained by breaking the $SO(3)$ symmetry of a Haldane chain to $D_4$.

Concretely, write $D_4=Z_4\rtimes Z_2$, denote the generator of $Z_4$ by $r$, and denote the generator of $Z_2$ by $s$. Then $srs=r^{-1}$. Suppose this SPT is put on a triangulated manifold $X$. Denote the gauge connection corresponding to the $Z_4$ by $a\in C^1(X, Z_4)$, and the gauge connection corresponding to $Z_2$ by $b\in C^1(X, Z_2)$. Here $a$ and $b$ are thought of as 1-cochains that represent the flat $Z_4$ and $Z_2$ gauge connections, respectively.

I have a couple of related questions:

  1. Is the partition function of this SPT something like $\exp(i\pi\int_X a\cup b)$? I think it somehow makes sense, but I do not fully understand it. In fact, now that $Z_4$ and $Z_2$ do not commute, what does such a cup product mean?
  2. If the above (or something like it) is indeed the partition function, how should I obtain it from the known partition function of the parent $SO(3)$ SPT, $\exp(i\pi\int_X w_2(SO(3)))$, where $w_2(SO(3))$ is the second Stiefel-Whitney class of the $SO(3)$ gauge bundle that the parent Haldane chain is coupled to? I understand perhaps I need to first embed this $D_4$ group into $SO(3)$, and then pullback $w_2(SO(3))$. But how should this be done properly?
  3. Now that the $Z_4$ and $Z_2$ do not commute, I think there should be a twist if we perform a coboundary operation on $a$. But besides this, is there any other constraint or relation between $a$ and $b$? For example, does $a\cup b$ have any special property?
asked Nov 13, 2020 in Theoretical Physics by Mr. Gentleman (270 points) [ revision history ]
recategorized Feb 18, 2021 by Mr. Gentleman

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:
p$\hbar$ysicsO$\varnothing$erflow
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
...