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 ...

  Does the projected spin state of the $d+id$ mean-field Hamiltonian on a triangular lattice has time-reversal(TR) symmetry?

+ 4 like - 0 dislike
1234 views

Consider the following $d+id$ mean-field Hamiltonian for a spin-1/2 model on a triangular lattice $$H=\sum_{<ij>}(\psi_i^\dagger\chi_{ij}\psi_j+H.c.)$$, with $\chi_{ij}=\begin{pmatrix} 0 & \Delta_{ij}\\ \Delta_{ij}^* & 0 \end{pmatrix}$, fermionic spinons $\psi_i=\binom{f_{i\uparrow}}{f_{i\downarrow}^\dagger}$, and the mean-field parameters $\Delta_{ij}=\Delta_{ji}$ defined on links have the same magnitudes and their phases differ by $\frac{2\pi}{3}$ with each other referring to the three bond-direction.

My question is, does the projected spin state $\Psi=P\phi$ have the TR symmetry? Where $\phi$ is the mean-field ground state of $H$, and $P$ removes the unphysical states with empty or doubly occupied sites.

Notice that from the viewpoint of Wilson loop, you can check that the Wilson loops $W_l=tr(\chi_{12}\chi_{23}\chi_{31})=0$ on each triangle plaquette, thus all the Wilson loops are invariant under the TR transformation $W_l\rightarrow W_l^*=W_l$. Thus, the TR symmetry should be maintained.

On the other hand, from the viewpoint of $SU(2)$ gauge-transformation, if there exist $SU(2)$ matrices $G_i$ such that $\chi_{ij}\rightarrow\chi_{ij}^*=G_i\chi_{ij}G_j^\dagger$, then the projected spin state $\Psi$ is TR invariant. But so far, I can not find out those $SU(2)$ matrices $G_i$. So can anyone work out the explicit form of those $SU(2)$ matrices $G_i$? Or they do not exist at all?

Thanks in advance.

By the way, I think it would be awkward to explicitly write the form of state $\Psi$ to check the TR symmetry.

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user K-boy
asked Jan 21, 2014 in Theoretical Physics by Kai Li (980 points) [ no revision ]
This state has time reversal symmetry, but if hopping is turned on in the mean field Hamiltonian the resulting state will no longer have time reversal symmetry. You may be interested in the supplementary information to a recent paper I was involved with, at arxiv.org/abs/1307.0829

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user Jim Garrison
@Jim Garrison Yes, I agree with you. If the nearest neighbor hopping $t$ is turned on, then the triangle Wilson loop $W_l$ will take a nonzero imaginary value $\propto it\Delta^2$ and $W_l$ is changed to $-W_l$ under TR operation, thus TR symmetry would be broken.

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user K-boy
@Jim Garrison But I want to know that whether the $SU(2)$ matrices mentioned in my question exist? And from which viewpoint(Wilson loop or $SU(2)$ matrices) you infer that the projected spin state has TR symmetry?

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user K-boy
@Jim Garrison And thanks for your reference.

This post imported from StackExchange Physics at 2014-03-09 08:38 (UCT), posted by SE-user K-boy

The interplay between Wilson loops and TR symmetry can be found, e.g., in this paper http://arxiv.org/abs/1409.7820

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
...