Multi-Particle Pseudopotentials for Multi-Component Quantum Hall 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).

46 pages ; 9 tables ; no figures. (The revision fixes a number of typos and updates the formatting)

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• Landau Level Mixing in the Perturbative Limit
• Floquet Mechanism for Non-Abelian Fractional Quantum Hall States
• Geometric construction of Quantum Hall clustering Hamiltonians
• Entangled Pauli Principles: the DNA of Quantum Hall Fluids
• Non-Abelian phases in two-component $\nu =2/3$ fractional quantum Hall states: Emergence of Fibonacci anyons
• SU(N) fractional quantum Hall effects in topological flat bands
• Josephson Coupled Moore-Read States
• Landau level quantization for massless Dirac fermions in the spherical geometry: graphene fractional quantum Hall effect on the Haldane sphere
• Fusion mechanism for quasiparticles and topological quantum order in the lowest Landau level
• Spin-singlet Gaffnian wave function for fractional quantum Hall systems
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• Exact Mappings in Condensed Matter Physics