Effective-mass tensor of the two-body bound states and the quantum-metric tensor of the underlying Bloch states

TL;DR

This paper derives an exact relation linking the effective-mass tensor of two-body bound states to the quantum-metric tensor of Bloch states in multiband lattices, with applications demonstrated on a Kagome lattice.

Contribution

It introduces a generalized relation that includes intraband and interband quantum metric contributions for multiband systems with time-reversal symmetry.

Findings

Derived an exact relation between effective-mass tensor and quantum metric.

Validated the relation using a Kagome lattice example.

Provided analytical expressions applicable to specific multiband lattices.

Abstract

By considering an onsite attraction between a spin-$\uparrow$ and a spin-$\downarrow$ fermion in a multiband tight-binding lattice, here we study the two-body spectrum, and derive an exact relation between the inverse of the effective-mass tensor of the lowest bound states and the quantum-metric tensor of the underlying Bloch states. In addition to the intraband (or the so-called conventional) contribution that depends only on the single-particle spectrum and the interband (or the so-called geometric) contribution that is controlled by the quantum metric, our generalized relation has an additional interband contribution that depends on the so-called band-resolved quantum metric. All of our analytical expressions are applicable to those multiband lattices that simultaneously exhibit time-reversal symmetry and fulfill the condition on spatially-uniform pairing. As a nontrivial illustration we analyze the two-body problem in a Kagome lattice with nearest-neighbor hoppings, and show that the exact relation provides a perfect benchmark.