1
$\begingroup$

If $T$ is time-reversal transformation $t\mapsto -t$, Why do $T$-invariant Bloch Hamiltonians obey $$H(-k) = T H(k) T^{-1}$$ and not $$H(k) = T H(k) T^{-1}$$ Somehow I understand the word "invariant" as being unchanged under a transformation. As the way an operator $O$ on a Hilbert space transforms as $O\mapsto P O P^{-1}$ where $P$ is the transformation, I don't see why it is usually the former and not the latter which you find in literature (for example Hasan and Kane 2010).

$\endgroup$
  • $\begingroup$ Time-reversal is an anti-unitary operator (not linear). Does it answer your question? $\endgroup$ – Prof. Legolasov Jan 13 '15 at 4:47
  • $\begingroup$ Hi, thanks for your comment. It doesn't answer my question yet, at least not explicitly so. What kind of condition is more important, from which the first equation in my original question stems? Is it the invariance of the absolute value of a transition amplitude? If so, why is the Hamiltonian called invariant? $\endgroup$ – PPR Jan 13 '15 at 10:53
  • 1
    $\begingroup$ it is the same thing... $\endgroup$ – Phoenix87 Jan 16 '15 at 23:21
  • $\begingroup$ Found a discussion that shows why this is true in Chapter 4 of a book by Bernevig: press.princeton.edu/titles/10039.html $\endgroup$ – PPR Jan 16 '15 at 23:49
2
$\begingroup$

It is the total $H = \sum_k H_k$ invariant under time reversal

$\endgroup$
  • 1
    $\begingroup$ what's wrong with this answer? I think it prfectly points out the OP's confusion! $\endgroup$ – an offer can't refuse May 11 '15 at 15:53
1
$\begingroup$

Sometimes invariant is used to mean covariant. If you have an Hamiltonian in momentum-space representation, the time inversion changes the sign of velocities, and therefore of the momentum. As a consequence of this fact it turns out that a time reversal in quantum mechanics must be implemented by an antilinear unitary operator, as shown by Wigner (1931).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.