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arXiv:1201.3096 · Physical Review Letters

Laughlin Spin-Liquid States on Lattices Obtained from Conformal Field Theory

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

Published Jun 20, 2012• 91 citations
arXivHTMLPDF

Abstract

We propose a set of spin system wave functions that is very similar to lattice versions of the Laughlin states. The wave functions are conformal blocks of conformal field theories and for a filling factor of ν = 1/2 we provide a parent Hamiltonian, which is valid for any even number of spins and is at the same time a 2D generalization of the Haldane-Shastry model. We also demonstrate that the Kalmeyer-Laughlin state is reproduced as a particular case of this model. Finally, we discuss various properties of the spin states and point out several analogies to known results for the Laughlin states.

DOI
10.1103/physrevlett.108.257206

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We propose a set of spin system wave functions that is very similar to lattice versions of the Laughlin states. The wave functions are conformal blocks of conformal field theories and for a filling factor of ν = 1/2 we provide a parent Hamiltonian, which is valid for any even number of spins and is at the same time a 2D generalization of the Haldane-Shastry model. We also demonstrate that the Kalmeyer-Laughlin state is reproduced as a particular case of this model. Finally, we discuss various properties of the spin states and point out several analogies to known results for the Laughlin states.
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The Kalmeyer-Laughlin state, which is a lattice version of the bosonic Laughlin state at filling factor one half, has attracted much attention due to its topological and chiral spin liquid properties. Here we show that the Kalmeyer-Laughlin state on the torus can be expressed in terms of a correlator of conformal fields from the SU(2)1SU(2)_1SU(2)1​ Wess-Zumino-Witten model. This reveals an interesting underlying mathematical structure and provides a natural way to generalize the Kalmeyer-Laughlin state to arbitrary lattices on the torus. We find that the many-body Chern number of the states is unity for more different lattices, which suggests that the topological properties of the states are preserved when the lattice is changed. Finally, we analyze the symmetry properties of the states on square lattices.
We study lattice wave functions obtained from the SU(2)1{}_{1}1​ Wess-Zumino-Witten conformal field theory. Following Moore and Read's construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function with each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
We develop a method of constructing excited states in one-dimensional spin chains which are derived from the SU(2)1\text{SU}{(2)}_{1}SU(2)1​ Wess-Zumino-Witten conformal field theory (CFT) using a parent Hamiltonian approach. The resulting systems are equivalent to the Haldane-Shastry model. In our ansatz, correlation functions between primary fields correspond to the ground state of the spin system, whereas excited states are obtained by insertion of descendant fields. Our construction is based on the current algebra of the CFT and emphasizes the close relation between the spectrum of the spin system and the underlying CFT. This general structure might imply that the method could be applied to a wider range of model systems.
We define a class of tensor network states for spin systems where the individual tensors are functionals of fields. The construction is based on the path-integral representation of correlators of operators in quantum field theory. These tensor network states are infinite-dimensional versions of matrix product states and projected entangled pair states. We find the field tensor that generates the Haldane-Shastry wave function and extend it to two dimensions. We give evidence that the latter underlies the topological chiral state described by the Kalmeyer-Laughlin wave function.
Abstract We construct a family of 1D and 2D long-range SU (2) spin models as parent Hamiltonians associated with infinite dimensional matrix product states that arise from simple current correlation functions in SU (2) k WZW models. Our results provide a conformal field theory foundation for recent proposals by Greiter and coauthors regarding the realization of non-abelian chiral spin liquids. We explain, in particular, how the symmetrization procedure for the amplitudes of the ground state wave function suggested by Greiter M and Thomale R (2009 Phys. Rev. Lett. 102 207203) originates from the conformal field theory description.