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arXiv:1611.01333 · Nuclear Physics B

Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories

Anna Hackenbroich, Hong-Hao Tu

Published Jan 5, 2017• 9 citations
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Abstract

We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction ν = 2 / ( 2 m + 1 ) are derived from deformations of the Wess–Zumino–Witten model su ( 3 ) 1 and are related to the ( m + 1 , m + 1 , m ) Halperin fractional quantum Hall states. We derive long-range SU(2) invariant parent Hamiltonians for these states which in two dimensions are chiral t – J – V models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square t – J – V model proposed in [25] . We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified as the low-energy effective theory for the periodic inverse-square t – J – V model.

DOI
10.1016/j.nuclphysb.2016.12.025

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We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction ν = 2 / ( 2 m + 1 ) are derived from deformations of the Wess–Zumino–Witten model su ( 3 ) 1 and are related to the ( m + 1 , m + 1 , m ) Halperin fractional quantum Hall states. We derive long-range SU(2) invariant parent Hamiltonians for these states which in two dimensions are chiral t – J – V models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square t – J – V model proposed in [25] . We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified as the low-energy effective theory for the periodic inverse-square t – J – V model.
We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane–Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.
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.
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 suggest a class of two-dimensional lattice spin Hamiltonians describing non-Abelian SU(2) chiral spin liquids---spin analogs of fractional non-Abelian quantum Hall states---with gapped bulk and gapless chiral edge excitations described by the SU(2)n{}_{n}n​ Wess-Zumino-Novikov-Witten conformal field theory. The models are constructed from an array of generalized spin-n/2n/2n/2 ladders with multi-spin-exchange interactions which are coupled by isolated spins. Such models allow a controllable analytic treatment starting from the one-dimensional limit and are characterized by a bulk gap and non-Abelian SU(2)n{}_{n}n​ gapless edge excitations.
We employ the SU(n)k\mathrm{SU}{(n)}_{k}SU(n)k​ Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the SU(n)\mathrm{SU}(n)SU(n) irreducible representation with a single row and kkk boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ series are derived. The SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the k=2k=2k=2 member hosts Fibonacci anyons.
I present a family of one-dimensional time reversal lattice Hamiltonians whose exact ground states are bosonic non-Abelian fractional quantum Hall liquids with well defined total chiral momentum. These Hamiltonians describe k-hard-core bosons and involve long-range tunneling and interaction processes with an amplitude decaying with the chord distance. For k=2 the ground state is a p-wave paired phase corresponding to the Pfaffian quantum Hall state. This family of non-Abelian liquids is shown to be in one-to-one correspondence with a family of non-Abelian spin-k/2 liquids exhibiting hidden non-local order. They are total singlets made out of k indistinguishable Resonating Valence Bond states. The corresponding spin parent Hamiltonians are obtained.