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arXiv:2305.12400 · Physical review. B./Physical review. B

Candidate local parent Hamiltonian for the 3/7 fractional quantum Hall effect

Koji Kudo, Anirban Sharma, G. J. Sreejith, Jainendra K. Jain

Published Aug 21, 2023• 0 citations
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Abstract

Although a parent Hamiltonian for the Laughlin $1/3$ wave function has been long known in terms of the Haldane pseudopotentials, no parent Hamiltonians are known for the lowest-Landau-level projected wave functions of the composite fermion theory at $n/(2n+1)$ with $n\ensuremath{\ge}2$. If one takes the two lowest Landau levels to be degenerate, the Trugman-Kivelson interaction produces the unprojected 2/5 wave function as the unique zero energy solution. If the lowest three Landau levels are assumed to be degenerate, the Trugman-Kivelson interaction produces a large number of zero energy states at $\ensuremath{\nu}=3/7$. We propose that adding an appropriately constructed three-body interaction yields the unprojected $3/7$ wave function as the unique zero energy solution and report extensive exact diagonalization studies that provide strong support to this proposal.

DOI
10.1103/physrevb.108.085130

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Although a parent Hamiltonian for the Laughlin 1/31/31/3 wave function has been long known in terms of the Haldane pseudopotentials, no parent Hamiltonians are known for the lowest-Landau-level projected wave functions of the composite fermion theory at n/(2n+1)n/(2n+1)n/(2n+1) with n\ensuremath≥2n\ensuremath{\ge}2n\ensuremath≥2. If one takes the two lowest Landau levels to be degenerate, the Trugman-Kivelson interaction produces the unprojected 2/5 wave function as the unique zero energy solution. If the lowest three Landau levels are assumed to be degenerate, the Trugman-Kivelson interaction produces a large number of zero energy states at \ensuremathν=3/7\ensuremath{\nu}=3/7\ensuremathν=3/7. We propose that adding an appropriately constructed three-body interaction yields the unprojected 3/73/73/7 wave function as the unique zero energy solution and report extensive exact diagonalization studies that provide strong support to this proposal.
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 nontrivial system beyond the Laughlin states, namely, bosons at filling factor $\ensuremath{\nu}=2/3$, 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 a 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 $6\ensuremath{\hbar}$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $\ensuremath{\lambda}$ state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two states 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 $\ensuremath{\lambda}$ 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.
Recent advances in engineered quantum platforms have enabled the realization of bosonic Laughlin states at ν=1/2ν=1/2ν=1/2 and brought non-Abelian topological phases within experimental reach. A central theoretical challenge is to develop a unified framework connecting lattice fractional quantum Hall (FQH) model states, exact parent Hamiltonians, and experimentally accessible interactions. We systematically construct Hermitian parent Hamiltonians for which continuum lowest Landau level (LLL) model states sampled on lattice sites are exact zero modes. Using these model manifolds as quantitative references, we find that, within the projected lattice LLL at the flux densities studied, short-range density interactions stabilize Laughlin ground-state manifolds at ν=1/3ν=1/3ν=1/3 and 1/41/41/4, while two-body onsite repulsion supports a Moore--Read triplet at ν=1ν=1ν=1. Full multiband calculations reveal a sharp contrast: the Laughlin manifolds remain robust against interband mixing on the studied tori, whereas the Moore--Read triplet becomes less spectrally isolated and eventually undergoes a level crossing with competing states near the interband scale. Our framework provides a theoretical foundation for using lattice model states to guide the search for experimentally accessible Abelian and non-Abelian FQH states.
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