Multi-Particle Pseudopotentials for Multi-Component Quantum Hall Systems

TL;DR

This paper extends the Haldane pseudopotential framework to multi-particle, multi-component quantum Hall systems, providing a basis for understanding complex interactions involving internal degrees of freedom like spin and valley.

Contribution

It generalizes pseudopotential construction to N-body interactions and internal degrees of freedom, offering a comprehensive basis for such quantum Hall systems.

Findings

Developed a basis of wavefunctions for N particles with fixed angular momentum

Decomposed wavefunctions into SU(n) representations for internal degrees of freedom

Analyzed specific cases including spin-1/2, spin-1, and graphene systems

Abstract

The Haldane pseudopotential construction has been an extremely powerful concept in quantum Hall physics --- it not only gives a minimal description of the space of Hamiltonians but also suggests special model Hamiltonians (those where certain pseudopotential are set to zero) that may have exactly solvable ground states with interesting properties. The purpose of this paper is to generalize the pseudopotential construction to situations where interactions are N-body and where the particles may have internal degrees of freedom such as spin or valley index. Assuming a rotationally invariant Hamiltonian, the essence of the problem is to obtain a full basis of wavefunctions for N particles with fixed relative angular momentum L. This basis decomposes into representations of SU(n) with n the number of internal degrees of freedom. We give special attention to the case where the internal degree of freedom has n=2 states, which encompasses the important cases of spin-1/2 particles and quantum Hall bilayers. We also discuss in some detail the cases of spin-1 particles (n=3) and graphene (n=4, including two spin and two valley degrees of freedom).