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Specifically, I am interested in the time-reversal symmetry $(\mathcal{T})$ in quantum mechanics. The $\mathcal{T}$ is considered as an anti-unitary operator, namely: $$\mathcal{T}\psi(\vec{r},t)=\psi^*(\vec{r},-t)$$

One of the most important properties related to $\mathcal{T}$ is the Kramers' degeneracy. I have known the external magnetic will break this symmetry. However, are there any conditions that will break explicitly the $\mathcal{T}$-symmetry? Such as the intrinsic magnetic ordering (AFM/FM) and spin-orbit coupling $et.all$? Any good tutorials and suggestions are welcome.

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    $\begingroup$ There are many systems that break T.R. symmetry, starting with your Ferromagnet, Antiferromagnet and then systems like chiral spin liquids. There is a famous physics answer (physics.stackexchange.com/questions/29469/…) that explains a simple way to break T.R. symmetry in systems. This should be a good starting point for you. $\endgroup$ – Xivi76 Jan 13 at 18:55
  • $\begingroup$ @Xivi76 I'd vote for that answer if you want to explain it a little more $\endgroup$ – Cody Aldaz Jan 14 at 4:59
  • $\begingroup$ @CodyAldaz Done! $\endgroup$ – Xivi76 Jan 15 at 0:39
  • $\begingroup$ Is the question specifically about explicit symmetry breaking? Because the link and answer provided by @Xivi76 are focused on spontaneous symmetry breaking. $\endgroup$ – Anyon Jan 15 at 2:50
  • $\begingroup$ @Anyon I want to know both situations. $\endgroup$ – Jack Jan 15 at 3:03
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I will first take a generic view-point and then quote some examples in condensed matter & materials modeling. Time-reversal symmetry is one of two discrete symmetries usually discussed in the context of condensed-matter, the other being Parity(Inversion). The simplest way in which this concept is presented is a transformation : $ t \rightarrow -t $ . Think of it as a motion reversal - If you go back in time, does the system look the same?

Take a video clip for example. And let us say you start to rewind the video. Your observation will basically be the same, but in reverse. Such a system is Time-reversal invariant, meaning it has T.R. symmetry. Now try to think of cases (not just systems) where such a reversal breaks a fundamental law of nature - The simplest example being, the 2$^{nd}$ law of thermodynamics which effectively states that the total entropy in a system can only increase or stay constant. You will immediately conclude that winding back the clock (so to speak) breaks a fundamental law since you cannot remove entropy that has already been created.

In condensed matter, a variety of systems break T.R. symmetry - Ferromagnets, Anti-ferromagnets, Chiral spin liquids, some Superconductors etc. The simplest case is the ferromagnet - There are two ways to reconcile this:

a) If you relate the origin of magnetism to circulating current loops: Reversing the motion results in a current that flows in the opposite sense, giving you a magnetic field with the opposite sign. This clearly breaks T.R. symmetry.

b) A more generic viewpoint: Spontaneous magnetization. Take a virgin sample of Iron. When you apply a magnetic field, it becomes magnetized. Can you remove the magnetic ordering ('demagnetize') the sample just by removal of the magnetic field?

In the context of materials modeling, T.R. symmetry can often lead to degeneracies that improve the computational speed greatly. The simplest example is Kramer's rule, which can be proved trivially from by taking a conjugate of the time-dependent Schrodinger equation:

$ E _n (\vec k) = E _n (-\vec k) $ where $n$ is the band index and $\vec k $ runs over the brillouin zone

As far as spin-orbit coupling is concerned, it does not break time-reversal symmetry by itself. Think of SO-coupling as $(coeffiecent)* L.S $. Since both L (angular momentum) and S (spin operator) change sign, their product retains the sign.

I have not delved much into the math in my answer, but I would recommend Sakurai's Quantum Mechanics, a classic. For an insight into systems with non-trivial T.R. breaking,here's an excellent answer on physics stack exchange.

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Another way to explicitly break time-reversal symmetry is by applying circularly polarized light. Under time reversal, left circularly polarized light transforms to right circularly polarized light, and vice versa. One way to see this is to note that circularly polarized light is an eigenstate of the spin angular momentum operator, which is odd under time reversal. Hence, if such light is applied to an otherwise time-reversal symmetric system, the Hamiltonian will no longer have $\mathcal{T}$-symmetry, and Kramers' theorem no longer applies.

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