Novel correlations in two dimensions: Two-body problem

TL;DR

This paper investigates a two-dimensional many-body Hamiltonian with two- and three-body interactions, focusing on the exactly solvable two-body problem, revealing simplified spectra at strong interactions and connections to known differential equations.

Contribution

It provides a detailed analysis of the exactly solvable two-body problem within a complex many-body Hamiltonian, linking it to Heun's differential equation and elucidating spectral properties.

Findings

Two-body spectrum simplifies at large interaction strength

Spectrum resembles Landau Levels

Ultraviolet regularization clarifies inverse-square potential handling

Abstract

We discuss a many-body Hamiltonian with two- and three-body interactions in two dimensions introduced recently by Murthy, Bhaduri and Sen. Apart from an analysis of some exact solutions in the many-body system, we analyze in detail the two-body problem which is completely solvable. We show that the solution of the two-body problem reduces to solving a known differential equation due to Heun. We show that the two-body spectrum becomes remarkably simple for large interaction strength and the level structure resembles that of the Landau Levels. We also clarify the "ultraviolet" regularization which is needed to define an inverse-square potential properly and discuss its implications for our model.