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I was wondering if someone could give a simple explanation of what is meant by time reversal invariance. Is it analogous to spatial translational symmetry? If so, how? By spatial translational symmetry I mean the following. Suppose, for example, one has a solid consisting of an array of ions and electrons. If we pick a coordinate system we can write the Hamiltonian of the solid in terms of the coordinates of the ions and electrons. If we translate the origin of our coordinate system our Hamiltonian will be expressed using new coordinates, however, the Hamiltonian will have the same form. Is there a similar understanding for time reversal invariance?

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    $\begingroup$ Time reversal is a transformation like any other. Is there something specific you want to know about it? $\endgroup$ – ACuriousMind Apr 14 '15 at 18:33
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    $\begingroup$ Right. I mean I see what it means mathematically. That if we choose to parametrize time in the system by $-t$ instead of $t$ then the Hamiltonian will remain unchanged. But is there a physical picture associated with this in the same way that there is with translational symmetries? $\endgroup$ – Laplacian Apr 14 '15 at 18:48
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    $\begingroup$ If one had two physical systems. One that possessed time reversal invariance and one that didn't. What would the physical difference between these two systems be that result in one having time reversal invariance and one not? $\endgroup$ – Laplacian Apr 14 '15 at 18:49
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Time reversal essentially means a system looks the same if you reverse the flow of time. The only difference beeing that things like velocity go in the opposite direction. In condensed matter systems it is represented as a unitary matrix times complex conjugation $\mathcal{T} = U\mathcal{K}$. A simple system that follows T-symmetry would be a system described by a real Hamiltonian (if we're ignoring spin, anyway.) If you include spin, for spin 1/2 particles it is represented as $\mathcal{T}=i\sigma_y \mathcal{K}$ and then we can have certain complex Hamiltonians as well.

As examples, a quantum spin Hall insulator is a system that preserves T-symmetry (because of the spins and their chirality when we reverse time we see the same system as before). A system which breaks T-symmetry is a ferromagnet. In this case the spin reversal is again the culprit, but because we do not see the same system as before, it has broken the symmetry. Hope this makes it a bit more clear!

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    $\begingroup$ A Chern insulator with chiral edge states breaks time-reversal symmetry, just like quantum Hall states. A quantum spin Hall insulator preserves time-reversal symmetry (opposite spins have opposite chiralities). $\endgroup$ – Meng Cheng Apr 16 '15 at 18:42
  • $\begingroup$ You are correct, I wrote the wrong thing down. Thanks for the catch! I will edit it now. $\endgroup$ – Lost In Thought Apr 16 '15 at 21:58

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