Localizing Transitions via Interaction-Induced Flat Bands

In a flat band the kinetic energy stops counting. Every other scale in the problem — interaction, disorder, whatever is left — becomes infinitely more important than the one that usually decides everything, which is why flat bands are where strongly correlated physics happens. The usual way to get one is to build it in from outside: twist a bilayer, tune a field. This Letter asks whether a system can do it to itself.

The answer is yes, under a condition that can be stated in one line, and the route there is a counting argument rather than a perturbative one — which matters, because a diverging effective mass is exactly the kind of singularity perturbation theory handles badly.

Count the zero modes

If the two halves of a Hamiltonian are different sizes, the larger one has to make up the difference in zeros.

Take any Hamiltonian whose square splits into two blocks. Every non-zero eigenvalue of one block has a partner in the other — the map between them is explicit — so if the two blocks are not the same size, the extra dimensions of the larger one have nowhere to go but zero. Zero modes appear, and their number is a difference of dimensions rather than a solution of anything.

For physicists

with a regulator, one below its argument and falling smoothly to zero above it. The non-zero eigenvalues pair up and cancel, so what is left counts zeros. More generally, any operator that maps the zero-mode subspace to itself and has zero trace on the complement will do, and . If rather than , the trace refines into a character sum — which is what a material with symmetry wants.

Eqs. (1)–(5) · equation numbers throughout are those of arXiv:2308.16440v1

Then repeat the cell

A zero mode confined to a region, repeated across space, is a flat band.

That is the whole construction. Confinement gives you the mode; periodicity gives you the extra quantum number — the lattice momentum — and since the count of zeros does not depend on that momentum, the band sits at zero energy across the whole zone. Nothing in the argument depends on the details, so a smooth change of the model cannot remove the band. That is what makes it topological.

Two realizations follow: a bipartite lattice whose two sublattices have different numbers of sites per cell, and — the one that matters here — an electron in a periodic magnetic field, where the count is the flux.

For physicists

Worked out from the regulated trace, with separating the blocks. The integral can be restricted to the regions where , so the zero modes are confined to those regions; tile the plane with them, one flux quantum each, and a periodic magnetic field with an integer flux through every cell yields a flat band. The band is exactly solvable: with and ,

with elliptic and quasi-periodic, — which is where the lattice momentum comes from. If the field has equally strong positive and negative patches the band is two-fold degenerate; add a constant background and becomes unbounded, leaving only one normalizable branch.

Eqs. (6)–(11)

The same integer condition can be read as an anomaly rather than an index. A flat band makes transition amplitudes time-independent, so time drops out of the path integral; a full chiral rotation then multiplies it by a phase equal to the flux, and the only way the path integral can equal itself is for that flux to be a whole number. The two readings are the same statement.

So far the field has been given. Now let the electrons make it.

The field the interactions supply

Decouple an interaction and you get a field that is not imposed but chosen — a Hubbard–Stratonovich field whose value is set by the electrons themselves. Its on-shell value is the current. And the construction above already tells us which configurations of that field would produce a flat band: any periodic one carrying an integer flux per cell.

Such a configuration is a phase: a periodic texture of loop currents, with a flat band of Bogoliubov excitations on top of it. Whether a system actually goes there is a question of energetics, and nothing so far has answered it.

For physicists

from current–current interactions, with on shell. Setting and demanding is a phase of spontaneously generated periodic loop-current textures whose Bogoliubov excitations are flat.

Eq. (12)

Why a nearly flat band flattens itself

The order parameter only has to make up the difference — and making up a small difference is cheap while the reward for making it up is not.

Put the system in an external field that is close to a magic configuration but not quite at it. Then the interaction-generated field does not have to build the whole texture; it only has to supply the shortfall. The cost of that shortfall is quadratic — it is the Hubbard–Stratonovich term, a square. The benefit of arriving at a flat band is linear. For a small enough shortfall the square loses.

distance from magic, δ 0.35 energy at the minimum — —

The shape of the argument. The Letter’s claim is that the cost of the texture is — the term in Eq. (12), quadratic — while the gain from flattening the band is linear in , and stops once the field has been corrected. Drawn at those shapes, the total has a minimum away from zero whenever the shortfall is small: the system pays to build the texture. This is the argument plotted, not a calculated energy; the Letter states the two scalings and leaves the coefficients to the model.

For physicists

With an external reintroduced and the magic configuration, what the order parameter must supply is only the difference:

so the quadratic cost is negligible against the linear benefit of flattening, and the transition can happen for close enough to . The argument does not care what the gauge field physically is.

Eq. (13)

Where to look for it

Moiré systemsThe moiré pattern is itself the background gauge field, and at magic angles it already makes flat bands. Keep the twist a little off magic, or break the chiral symmetry that makes them exact — as happens in real samples — and the error is small enough to be corrected spontaneously. Lattice relaxation is another way to perturb the quantisation condition slightly.

The superconducting channelThe same construction in Nambu space gives a Bogoliubov flat band. One magic pairing texture maps the zero-mode problem onto the chiral model of twisted bilayer graphene, with the first Bogoliubov flat band at — magic without a lattice.

Heavy bandsThe scenario is not generic: it becomes competitive when the electrons are heavy to begin with, where it competes with other channels as the rectifying mechanism. More than one order parameter may be needed, which is where coexisting phases come in.

The mechanism is stated as a seed rather than a finished theory: a general way of constructing periodic textures that flatten bands, a condition tied to the index theorem, and a proof of principle that the condition can be met from the inside. The exactly solvable version of the patching is in Zero-Flux Localization; the moiré case is in A Generic Topological Criterion for Flat Bands.