• 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
5,064 questions , 2,215 unanswered
5,347 answers , 22,728 comments
1,470 users with positive rep
818 active unimported users
More ...

  Determining if a Hamiltonian has anti-unitary symmetries

+ 1 like - 0 dislike

I'm trying to understand the topological properties of different systems and where they fall on the Periodic Table of Topological Phases.  Such systems might include the Quantum Anomalous Hall Effect, Su-Schrieffer-Heeger model of trans-polyacetylene, Kitaev chain, or B-phase of He-3.

Once a resource gives me the right matrices, I can verify that indeed, there does indeed exist a U such that

\(U H U^{-1} = -H \qquad U U^{\dagger} = \mathbb{1}\)

(or respective for the T and C symmetries) .

But how do I go about showing that a system doesn't have any U such that this holds? Or go about finding such a U if I didn't read it in a paper?

My attempt using eigenvectors seems to be telling me that things I know aren't symmetric are.

asked Apr 1, 2019 in Theoretical Physics by anonymous [ revision history ]

1 Answer

+ 1 like - 0 dislike

You're right in the sense that only the eigenvectors of $H$ are relevant.

If you can arrange the set of eigenvectors in pairs such that the corresponding eigenvalues are opposite then you can build a unitary or anti-unitary $U$.

It is an equivalence,

What do you mean then by "things I know aren't symmetric" ?

answered Apr 1, 2019 by anonymous [ revision history ]
edited Apr 1, 2019

It is not quite an equivalence. Necessary and sufficient is that the nonzero eigenvalues come in opposite pairs, and a similar requirement (but more complex to formulate) for the continuous part of the spectrum if there is one.

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