# Floquet bandstructure calculation

In this paper "Photonic Floquet Topological Insulators" the authors calculate the bandstructure of a time-periodic Hamiltonian. They create a time-dependent tight-binding Hamiltonian via the Peierl's substitution.

Now my first question would be about equation (3) of this paper. Since it is in lattice site basis, would i need to FT in order to get to transverse crystal momentum basis so i can create a bandstructure? And what would happen to the phase factor aquired through the Peierls substitution?

Secondly, how would I calculate the Floquet quasi-energies from there? From what I have read so far they are usually calculated by calculating the Eigenvalues of the stroboscopic time-evolution operator $$U(T)=\mathrm{exp}\left[\int_{0}^{T}{H(t)dt}\right]$$. This has not worked for me so far. Is this even the right approach?

I hope this question is not too "do-my-homework-for-me"-y. Its a completely new topic for me and I have been trying to solve this for what feels like and eternity. Any help would be greatly appreciated.

You are correct to say that the quasienergies can be obtained from the time-evolution operator for a single period, $$U(T)$$. As $$U$$ is unitary, its eigenvalues are pure phases, which are related to the quasienergies $$\epsilon_j$$ via $$\lambda_j = \exp[-i T \epsilon_j ]$$. So a typical procedure is to prepare $$U(T)$$ by time-evolving the identity matrix over a single period of driving, finding its eigenvalues, and then taking their logarithm (and dividing by a factor of $$T$$).
• M = [0,exp(-1i*((kx+R*sin(omega*t))*a*sin(120/360*2*pi)-(ky-R*cos(omega*t))*a*cos(120/360*2*pi)))+exp(-1i*(-(kx+R*sin(omega*t))*a*sin(120/360*2*pi)-(ky-R*cos(omega*t))*a*cos(120/360*2*pi)))+exp(+1i*(ky-R*cos(omega*t))*a);exp(1i*((kx+R*sin(omega*t))*a*sin(120/360*2*pi)-(ky-R*cos(omega*t))*a*cos(120/360*2*pi)))+exp(1i*(-(kx+R*sin(omega*t))*a*sin(120/360*2*pi)-(ky-R*cos(omega*t))*a*cos(120/360*2*pi)))+exp(-1i*(ky-R*cos(omega*t))*a),0]; So I've tried using this Hamiltonian, but it seems it's still not quite right. Now I'm not sure if the Hamiltonian is incorrect or my time-evolution is faulty – JanJasper May 5 at 13:23
• It's a bit hard to read, but is $M$ the Hamiltonian matrix? Are you trying to get the topological edge states, as shown in Fig. 2 of the paper, or do you want the bandstructure (like Fig. 1)? If you are trying to get the full bandstructure, I would expect the Hamiltonian to be a 2x2 matrix, which is a function of $k_x$ and $k_y$. But if you want the edge states, then you should work in real-space using a tight-binding Hamiltonian, with periodic boundary conditions in$x$ and closed boundaries in $y$. And how are you doing the time-evolution? – Clara Diaz Sanchez May 5 at 17:17