Abstract
Many fractional quantum Hall wave functions are known to be unique highest-density zero modes of certain “pseudopotential” Hamiltonians. While a systematic method to construct such parent Hamiltonians has been available for the infinite plane and sphere geometries, the generalization to manifolds where relative angular momentum is not an exact quantum number, i.e., the cylinder or torus, remains an open problem. This is particularly true for non-Abelian states, such as the Read-Rezayi series (in particular, the Moore-Read and Read-Rezayi states) and more exotic nonunitary (Haldane-Rezayi and Gaffnian) or irrational (Haffnian) states, whose parent Hamiltonians involve complicated many-body interactions. Here, we develop a universal geometric approach for constructing pseudopotential Hamiltonians that is applicable to all geometries. Our method straightforwardly generalizes to the multicomponent cases with a combination of spin or pseudospin (layer, subband, or valley) degrees of freedom. We demonstrate the utility of our approach through several examples, some of which involve non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and we verify the results by numerically computing their entanglement properties.
4 More- Received 19 February 2015
DOI:https://doi.org/10.1103/PhysRevX.5.041003
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Published by the American Physical Society
Popular Summary
The fractional quantum Hall effect (FQHE), discovered in 1982, refers to the Hall resistance of a two-dimensional interacting electron gas becoming fractionally quantized in the presence of a strong magnetic field. The FQHE has triggered enormous interest in the community of strongly correlated electron systems and influences the fields of topological quantum computation, entanglement, and quantum spin liquids. In 1983, Laughlin proposed an idea of a wave function to describe the interacting many-body state in a FQHE electronic system at magnetic filling in the lowest Landau level. This idea was soon complemented by the development of an analogous Hamiltonian description for which the Laughlin state turns out to be the exact ground state. Crucial numerical evidence revealed that the Laughlin state accurately describes the physics of the generic Coulomb state investigated in experiments. Here, we develop a geometric approach to derive many-body Haldane pseudopotentials for arbitrary geometries, in particular, for cylinders and tori that lack continuous rotational symmetry and are characterized by boundary conditions.
Understanding a quantum Hall state often requires understanding the class of parent Hamiltonians associated with it. These Hamiltonians, composed of different pseudopotentials, can inform studies of excitations, stability, and topological character of a given quantum Hall state that otherwise might not be easy to access. Here, we assume a two-dimensional electron gas with a perpendicular magnetic field; the particles have no degrees of freedom other than spin, which is fully polarized. We consider two-, three-, and four-body interactions for both bosons and fermions. We construct potentials based on symmetry principles and, in particular, calculate parent Hamiltonians for all geometries. Our approach can be used for systems with different spin types and colors, and our work informs previous approaches by simplifying them. We use entanglement spectra to check our results, as well as analytical and numerical analyses.
Our findings establish a promising tool to numerically look for unprecedented topological quantum states of matter in the FQHE. We expect that our results will motivate studies of new zero-mode ground states, which can be found by modulating the combination of pseudopotential terms.