Determining if a Hamiltonian has anti-unitary symmetries

+ 1 like - 0 dislike
241 views

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.

+ 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
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.

 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): Email me at this address if my answer is selected or commented on: 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$\hbar$ysicsOverflo$\varnothing$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). To avoid this verification in future, please log in or register.