Two-particle bound states on a lattice

TL;DR

This paper systematically investigates two-particle bound states on lattices across dimensions, deriving conditions, phase diagrams, and physical properties, with implications for superconductivity and many-pair stability.

Contribution

It provides a comprehensive analysis of lattice-bound pairs, including analytical derivations and physical insights, combining original research with pedagogical explanations.

Findings

Derived pairing conditions and phase diagrams for lattice pairs

Analyzed effects like light pairs and momentum dependence of binding

Discussed implications for superconductivity and phase stability

Abstract

Two-particle lattice states are important for physics of magnetism, superconducting oxides, and cold quantum gases. The quantum-mechanical lattice problem is exactly solvable for finite-range interaction potentials. A two-body Schroedinder equation can be reduced to a system of linear equations whose numbers scale with the number of interacting sites. For the simplest cases such as on-site or nearest-neighbor attractions, many pair properties can be derived analytically, although final expressions can be quite complicated. In this work, we systematically investigate bound pairs in one-, two-, and three-dimensional lattices. We derive pairing conditions, plot phase diagrams, and compute effective masses, radii, and energies. Along the way, we analyze nontrivial physical effects such as light pairs and the dependence of binding thresholds on pair momenta. At the end, we discuss the preformed-pair mechanism of superconductivity and stability of many-pair systems against phase separation. The paper is a combination of original work and pedagogical tutorial.