# How does time reversal symmetry work in topological insulator?

I am doing microelectronic devices with topological insulators. Can some one explain time reversal symmetry in a topological insulator to electrical engineering student like me?

I assume you refer to something like a BHZ-$\mathbb{Z}_2$-insulator (with the experimental realization of a 2d-$\text{HgTe}$-quantum well).

This model can be understood as two Chern insulators. Chern insulators have a chiral edge mode (that is, an edge mode, that propagates only in one turning sense, thus breaking time reversal). This is analogous to the way a magnetic field breaks time reversal.

The two spin channels of the BHZ model are time reversal conjugates for the spinless Chern insulator, furthermore the spin channels are flipped when applying time reversal, leading to a time reversal invariant spinful model.

I hope, I was not too technical and sufficiently detailed.

• Thank you for your explanation. I still not quit understand why magnetic field can break the time reversal symmetry? Is that means when the time reserved the spin direction will change? For example, the spin is in clockwise direction and when time reserved it should go to the counter clockwise direction. When applied magnetic field it does not go to counter clockwise direction? May 15, 2015 at 21:32
• @Michelle Classical explanation: Magnetic fields break the time reversal because on time reversal $\vec v \to -\vec v$ thus $\vec F_L = \frac q c (\vec v \times \vec B) \to - \frac q c (\vec v \times \vec B)$. Note: The magnetic field analogy in a TI does not apply to the spin degree of freedom! May 15, 2015 at 21:36
• So what you mean is the time reverse leads to the velocity reverse then the Lorenz force reversed and the electron motion reverse direction? thank you for your patience May 15, 2015 at 21:40
• Yes. Reversing the velocity and swapping initial and final position is roughly the usual definition for time reversal. May 15, 2015 at 21:45
• Thank you. So in the TI systems. The band structure is like physics.stackexchange.com/questions/175926/…. In this question, the lower one. How do we explain that the surface band structure is protected by time reversal symmetry? May 15, 2015 at 23:40

You should know Kramers_theorem, enenrgy band is at least doubly degenerate at time reversal invariant momentum (TRIM) points, which satisfies -k=k+G, G is reciprocal lattice vector. So edge states will doubly degerate at TRIM, and protected by time-reversal symmetry. It means if you add perturbation without breaking TR symmetry, you cannot gap the edge states.