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arXiv:1210.1481 · Physical Review B

Projected BCS states and spin Hamiltonians for the SO( n ) 1 Wess-Zumino-Witten model

Hong-Hao Tu

Published Jan 14, 2013• 41 citations
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Abstract

We propose a class of projected BCS wave functions and derive their parent spin Hamiltonians. These wave functions can be formulated as infinite matrix product states constructed by chiral correlators of Majorana fermions. In one dimension, the spin Hamiltonians can be viewed as SO($n$) generalizations of Haldane-Shastry models. We numerically compute the spin-spin correlation functions and R\'enyi entropies for $n=5$ and 6. Together with the results for $n=3$ and 4, we conclude that these states are critical and their low-energy effective theory is the SO($n$)${}_{1}$ Wess-Zumino-Witten model. In two dimensions, we show that the projected BCS states are chiral spin liquids, which support non-Abelian anyons for odd $n$ and Abelian anyons for even $n$.

DOI
10.1103/physrevb.87.041103

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We propose a class of projected BCS wave functions and derive their parent spin Hamiltonians. These wave functions can be formulated as infinite matrix product states constructed by chiral correlators of Majorana fermions. In one dimension, the spin Hamiltonians can be viewed as SO(nnn) generalizations of Haldane-Shastry models. We numerically compute the spin-spin correlation functions and R'enyi entropies for n=5n=5n=5 and 6. Together with the results for n=3n=3n=3 and 4, we conclude that these states are critical and their low-energy effective theory is the SO(nnn)1{}_{1}1​ Wess-Zumino-Witten model. In two dimensions, we show that the projected BCS states are chiral spin liquids, which support non-Abelian anyons for odd nnn and Abelian anyons for even nnn.
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 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.
We investigate a family of spin-SSS chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For S=1/2S=1/2S=1/2, it corresponds to the Haldane-Shastry model. For general spin SSS, we find indication that the low-energy theory of these spin chains is described by the SU(2)k{}_{k}k​ Wess-Zumino-Witten model with coupling k=2Sk=2Sk=2S. In particular, we investigate the S=1S=1S=1 model whose ground state is given by a Pfaffian for even number of sites NNN. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary NNN with trial states obtained by Schwinger projection of two Haldane-Shastry chains.
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 propose resonating valence bond wave functions for a spin-1 system on the torus that realize a non-Abelian chiral spin liquid. The wave functions take the form of infinite-dimensional matrix product states constructed from conformal blocks of the SO(3)1\mathrm{SO}{(3)}_{1}SO(3)1​ Wess-Zumino-Witten model. This means that they are lattice analogs of the bosonic Moore-Read state introduced in fractional quantum Hall systems. The topological order of this system is revealed by explicit construction of threefold-degenerate ground states and analytical computation of the modular SSS and TTT matrices.