Two-body problem in a multiband lattice and the role of quantum geometry

TL;DR

This paper analyzes the two-body bound states in a multiband lattice, revealing how quantum geometry influences their dispersion and effective mass, especially in bipartite lattices with flat bands.

Contribution

It provides an exact variational solution for bound-state dispersion in multiband lattices, highlighting the impact of quantum metric on effective mass, with applications to bipartite and flat-band systems.

Findings

Bound states disperse quadratically with momentum.

Quantum metric partially controls the effective mass.

Interband processes induce finite effective mass in flat bands.

Abstract

We consider the two-body problem in a periodic potential, and study the bound-state dispersion of a spin-$\uparrow$ fermion that is interacting with a spin-$\downarrow$ fermion through a short-range attractive interaction. Based on a variational approach, we obtain the exact solution of the dispersion in the form of a set of self-consistency equations, and apply it to tight-binding Hamiltonians with onsite interactions. We pay special attention to the bipartite lattices with a two-point basis that exhibit time-reversal symmetry, and show that the lowest-energy bound states disperse quadratically with momentum, whose effective-mass tensor is partially controlled by the quantum metric tensor of the underlying Bloch states. In particular, we apply our theory to the Mielke checkerboard lattice, and study the special role played by the interband processes in producing a finite effective mass for the bound states in a non-isolated flat band.