Search for exact local Hamiltonians for general fractional quantum Hall states

TL;DR

This paper investigates local Hamiltonians for fractional quantum Hall states, finding that most do not produce the desired ground states, but identifying a specific interaction that yields a candidate state with potential topological equivalence to known states.

Contribution

The study introduces a novel interaction term that produces a unique zero-energy state, advancing the understanding of Hamiltonians for fractional quantum Hall states.

Findings

No local Hamiltonian up to four particles reproduces the state.

A specific four-particle interaction yields a candidate state with topological features.

The candidate state shows high overlap and similar quantum numbers to the composite-fermion state.

Abstract

We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $\nu=\frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6\hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same root partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $\lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.