We will now learn about a new generalization of topology, namely how it applies to the quantum evolution of systems with a time-dependent Hamiltonian. As you may recall, we’ve already encountered time dependence, back when we considered quantum pumps. However, back then we assumed that the time evolution was very slow, such that the system stayed in the ground state at all times, i.e. it was adiabatic. Can we relax the adiabaticity constraint? Can we find an analog of topology in systems that are driven so fast that energy isn’t conserved?
For the same reasons as before, we’ll consider periodic driving
\[ H(t + T) = H(t). \]
Once again, this is 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 Schrödinger equation is:
\[ i\frac{d \psi}{dt} = H(t) \psi. \]
It is a linear equation, so we can write its solution as
\[ \psi(t_2) = U(t_2, t_1) \psi(t_1), \]
where \(U\) is a unitary time evolution operator. It solves the same Schrödinger equation as the wave function, and is equal to the 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 (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 the Hamiltonians evaluated at different times in the integral do not commute.
The time evolution operator satisfies 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 \(U(t_2, t_1)^\dagger = U(t_2, t_1)^{-1} = U(t_1, t_2)\).
The central object for the study of driven systems is the evolution operator over one period of the driving,
\[ U(t + T, t) \equiv U, \]
which is called the Floquet time evolution operator. It is important because it allows us to identify the wave functions that are the same if an integer number of drive periods passes. 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 very similar to the eigenstates of a stationary Hamiltonian, except that they are only stationary if we look at fixed times \(t + nT\). That’s why the Floquet time evolution operator is also called a stroboscopic time evolution operator.
We can very easily construct a Hermitian matrix from \(U\), the Floquet Hamiltonian:
\[ H_\textrm{eff} = i T^{-1} \,\ln U. \]
Its eigenvalues \(\varepsilon = \alpha / T\) are called quasi-energies, and they always belong 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 to apply topological ideas.