Thouless pumps and winding invariant

Thouless pumps

Dganit Meidan from Ben Gurion University will introduce Thouless pumps,.

Hamiltonians with parameters

Previously, when studying the topology of systems supporting Majoranas (both the Kitaev chain and the nanowire), we were able to calculate topological properties by studying the bulk Hamiltonian \(H(k)\).

There are two points of view on this Hamiltonian. We could either consider it a Hamiltonian of an infinite system with momentum conservation

\[ H = H(k) |k\rangle\langle k|, \]

or we could equivalently study a finite system with only a small number of degrees of freedom (corresponding to a single unit cell), and a Hamiltonian which depends on some continuous periodic parameter \(k\).

Of course, without specifying that \(k\) is the real space momentum, there is no meaning in bulk-edge correspondence (since the edge is an edge in real space), but the topological properties are still well-defined.

Sometimes we want to know how a physical system changes if we slowly vary some parameters of the system, for example a bias voltage or a magnetic field. Because the parameters change with time, the Hamiltonian becomes time-dependent, namely

\[ H = H(t). \]

The slow adiabatic change of parameters ensures that if the system was initially in the ground state, it will stay in the ground state, so that the topological properties are useful.

A further requirement for topology to be useful is the periodicity of time evolution:

\[ H(t) = H(t+T). \]

The period can even go to \(\infty\), in which case \(H(-\infty) = H(+\infty)\). The reasons for the requirement of periodicity are somewhat abstract. If the Hamiltonian has parameters, we’re studying the topology of a mapping from the space of parameter values to the space of all possible gapped Hamiltonians. This mapping has nontrivial topological properties only if the space of parameter values is compact.

For us, this simply means that the Hamiltonian has to be periodic in time.

Of course, if we want systems with bulk-edge correspondence, then in addition to \(t\) our Hamiltonian must still depend on the real space coordinate, or the momentum \(k\).

Quantum pumps

In the image below (source: Chambers’s Encyclopedia, 1875, via Wikipedia) you see a very simple periodic time-dependent system, an Archimedes screw pump.

The changes to the system are clearly periodic, and the pump works the same no matter how slowly we use it (that is, change the parameters), so it is an adiabatic tool.

What about a quantum analog of this pump?

Let’s take a one-dimensional region, coupled to two electrodes on both sides, and apply a strong sine-shaped confining potential in this region. As we move the confining potential, we drag the electrons captured in it.

So our system now looks like this:


It is described by the Hamiltonian

\[ H(t) = \frac{k^2}{2m} + A [1 - \cos(x/\lambda + 2\pi t/T)]. \]

As we discussed, if we change \(t\) very slowly, the solution will not depend on how fast \(t\) varies.

When \(A \gg 1 /m \lambda^2\) the confining potential is strong, and additionally if the chemical potential \(\mu \ll A\), the states bound in the separate minima of the potential have very small overlap.

The potential near the bottom of each minimum is approximately quadratic, so the Hamiltonian is that of a simple Harmonic oscillator. This gives us discrete levels of the electrons with energies \(E_n = (n + \tfrac{1}{2})\omega_c\), with \(\omega_c = \sqrt{A/m\lambda^2}\) the oscillator frequency. In the large A limit, the states in the different minima are completely isolated so that the energy bands are flat with vanishing (group) velocity \(d E_n(k)/d k=0\) of propagation.

We can numerically check how continuous bands in the wire become discrete evenly spaced bands as we increase \(A\):

So unless \(\mu = E_n\) for some \(n\), each minimum of the potential contains an integer number of electrons \(N\).There are a large number of states at this energy and almost no states at \(\mu\) away from \(E_n\).

Since electrons do not move between neighboring potential minima, so when we change the potential by one time period, we move exactly \(N\) electrons.

Why are some levels in the band structure flat while some are not?

The flat levels are the ones whose energies are not sensitive to the offset of confining potential.
Destructive interference of the wave functions in neighboring minima suppresses the dispersion.
The flat levels are localized deep in the potential minima, so the bandwidth is exponentially small.
The flat levels correspond to filled states, and the rest to empty states.

Quantization of pumped charge

As we already learned, integers are important, and they could indicate that something topological is happening.

At this point we should ask ourselves these questions: Is the discreteness of the number of electrons \(N\) pumped per cycle limited to the deep potential limit, or is the discreteness a more general consequence of topology?

Thought experiment

Let us consider the reservoirs to be closed finite (but large) boxes. When the potential in the wire is shifted the electrons clearly move from the left to the right reservoir. How do the reservoirs accomodate these electrons?

Since the Hamiltonian is periodic in time, the Hamiltonian together with all its eigenstates return to the initial values at the end of the period. The adiabatic theorem guarantees that when the Hamiltonian changes slowly the eigenstates evolve to an eigenstate that is adjacent in energy.

We indeed see that the levels move up and down in energies. The states that don’t shift in energy are the ones trapped in the minima of the periodic potential.

To understand better what is happening, let us color each state according to the position of its center of mass, with red corresponding to the left reservoir, blue to the right one, and white to the middle of the system.