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arXiv:1609.02435 · Physical review. B./Physical review. B

Lattice effects on Laughlin wave functions and parent Hamiltonians

Ivan Glasser, J. I. Cirac, Germán Sierra, Anne E. B. Nielsen

Published Dec 1, 2016• 26 citations
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Abstract

We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux $\ensuremath{\nu}=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\ensuremath{\mu}$ particles per lattice site, where $\ensuremath{\mu}$ may be different than $\ensuremath{\nu}$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of $q$ and for different fillings $\ensuremath{\mu}$. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for $q\ensuremath{\le}4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\ensuremath{\nu}=1/q$, while for larger $q$ there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor $1/q$. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\ensuremath{\mu}\ensuremath{\le}1/q$ or $\ensuremath{\mu}\ensuremath{\ge}1\ensuremath{-}1/q$ and when $q=4$ also at half filling.

DOI
10.1103/physrevb.94.245104

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We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux \ensuremathν=1/q\ensuremath{\nu}=1/q\ensuremathν=1/q in the Landau levels. These wave functions are defined analytically on lattices with \ensuremathμ\ensuremath{\mu}\ensuremathμ particles per lattice site, where \ensuremathμ\ensuremath{\mu}\ensuremathμ may be different than \ensuremathν\ensuremath{\nu}\ensuremathν. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of qqq and for different fillings \ensuremathμ\ensuremath{\mu}\ensuremathμ. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for q\ensuremath≤4q\ensuremath{\le}4q\ensuremath≤4 this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction \ensuremathν=1/q\ensuremath{\nu}=1/q\ensuremathν=1/q, while for larger qqq there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor 1/q1/q1/q. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when \ensuremathμ\ensuremath≤1/q\ensuremath{\mu}\ensuremath{\le}1/q\ensuremathμ\ensuremath≤1/q or \ensuremathμ\ensuremath≥1\ensuremath−1/q\ensuremath{\mu}\ensuremath{\ge}1\ensuremath{-}1/q\ensuremathμ\ensuremath≥1\ensuremath−1/q and when q=4q=4q=4 also at half filling.
We introduce a two-parameter family of strongly-correlated wave functions for bosons and fermions in lattices. One parameter, q , is connected to the filling fraction. The other one, η , allows us to interpolate between the lattice limit ( ) and the continuum limit ( ) of families of states appearing in the context of the fractional quantum Hall effect or the Calogero–Sutherland model. We give evidence that the main physical properties along the interpolation remain the same. Finally, in the lattice limit, we derive parent Hamiltonians for those wave functions and in 1D, we determine part of the low-energy spectrum.
We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.
We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2{Z}_{2}Z2​ properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.
Recently, it was shown that fractional quantum Hall states can be defined on fractal lattices. Proposed exact parent Hamiltonians for these states are nonlocal and contain three-site terms. In this work, we look for simpler, approximate parent Hamiltonians for bosonic Laughlin states at half filling, which contain only onsite potentials and two-site hopping with the interaction generated implicitly by hardcore constraints (as in the Hofstadter and Kapit-Mueller models on periodic lattices). We use an ``inverse method'' to determine such Hamiltonians on finite-generation Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski carpet and triangle lattices. The ground states of some of the resulting models display relatively high overlap with the model states if up to third-neighbor hopping terms are considered, and by increasing the maximum hopping distance one can achieve nearly perfect overlaps. When the number of particles is reduced and additional potentials are introduced to trap quasiholes, the overlap with a model quasihole wave function is also high in some cases, especially for the nonlocal Hamiltonians. We also study how the small system size affects the braiding properties for the model quasihole wave functions and perform analogous computations for Hamiltonian models.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
We introduce a family of strongly-correlated spin wave functions on arbitrary spin- and spin-1 lattices in one and two dimensions. These states are lattice analogues of Moore–Read states of particles at filling fraction , which are non-Abelian fractional quantum Hall states in 2D. One parameter enables us to perform an interpolation between the continuum limit, where the states become continuum Moore–Read states of bosons (odd q ) and fermions (even q ), and the lattice limit. We show numerical evidence that the topological entanglement entropy stays the same along the interpolation for some of the states we introduce in 2D, which suggests that the topological properties of the lattice states are the same as in the continuum, while the 1D states are critical states. We then derive exact parent Hamiltonians for these states on lattices of arbitrary size. By deforming these parent Hamiltonians, we construct local Hamiltonians that stabilize some of the states we introduce in 1D and in 2D.