Introduction

Today's topic, Floquet topological insulators, is introduced by Mark Rudner from the Niels Bohr Institute at Copenhagen.

Periodically driven systems

The new generalization of the topology we will consider now is considering quantum evolution of systems with a time-dependent Hamiltonian. If you remember we've already encountered time dependence when we considered quantum pumps. However then we have assumed that the time evolution is very slow, and the system stays in the ground state at all times. But can we relax adiabaticity constraint? Can we find any analog of topology in systems driven so fast that the energy isn't conserved?

For the same reasons as before, we'll consider periodic driving

$$H(t + T) = H(t).$$

This is once again necessary because otherwise any system can be continuously deformed into any other, and there is no way to define a gap.

Before we get to topology, let's refresh our knowledge of time-dependent systems.

The Schrodinger equation gives us:

$$i\frac{d \psi}{dt} = H(t) \psi$$

It's linear, so we can write its solution as

$$\psi(t_2) = U(t_2, t_1) \psi(t_1),$$

with $U$ being a unitary time evolution operator. It solves the same Schrodinger equation as the wave function and it is equal to identity matrix at the initial time. It is commonly written as

$$U(t_2, t_1) = \mathcal{T} \exp\,\left[-i\int_{t_1}^{t_2} H(t) dt\right]$$

where $\mathcal{T}$ represents time-ordering (and not time-reversal symmetry). The time-ordering is just a short-hand notation for the need to solve the full differential equation, and it is necessary if Hamiltonians evaluated at different times in the integral do not commute.

The evolution operator safisfies a very simple multiplication rule:

$$U(t_3, t_1) = U(t_3, t_2) U(t_2, t_1),$$

which just says that time evolution from $t_1$ to $t_3$ is a product of time evolutions from $t_1$ to $t_2$ and then from $t_2$ to $t_3$. Of course an immediate consequence of this is the equality $U(t_2, t_1)^\dagger = U(t_1, t_2)$.

Floquet theory

The central object for the study of driven systems is the evolution operator over one period of the drive:

$$U(t + T, t) \equiv U.$$

It is important because it allows us to find the wave functions that do not change if we wait for an arbitrary number of drive periods. These are the stationary states of a driven system, and they are given by the eigenvalues of the Floquet operator:

$$U \psi = e^{i \alpha} \psi$$

the stationary states are extremely similar to the eigenstates of a stationary Hamiltonian, with the only difference that they are only stationary if we look at fixed times $t + nT$. That's why the Floquet time evolution operator is also called stroboscopic time evolution operator.

We can very easily construct a Hermitian matrix from $U$, the Floquet Hamiltonian:

$$H_\textrm{eff} = i T^{-1} \,\log U.$$

Its eigenvalues $\varepsilon = \alpha / T$ are called quasi-energies, and they are always belonging to the interval $-\pi < \alpha \leq \pi$.

If the system is translationally invariant, we can study the effective band structure of $H_\textrm{eff}(\mathbf{k})$, find an energy in which the bulk Hamiltonian has no states, and study the topological properties of such a Hamiltonian: most of the things we already know still apply.

Of course, selecting a single quasi-energy as the Fermi level is arbitrary, since the equilibrium state of driven systems doesn't correspond to a Fermi distribution of filling factors, but at least it seems close enough for us to try applying topological ideas.

Conclusions