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arXiv:1312.5134 · Journal of Statistical Mechanics Theory and Experiment

Bosonic fractional quantum Hall states on the torus from conformal field theory

Anne E. B. Nielsen, Germán Sierra

Published Apr 8, 2014• 13 citations
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Abstract

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 $SU(2)_1$ 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.

DOI
10.1088/1742-5468/2014/04/p04007

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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.
The relation between the chiral-spin-liquid state and the correlators of the Wess-Zumino-Witten model is shown. I argue that in the Kalmeyer-Laughlin state there is a hidden symmetry, associated with the conformal invariance in the continuum limit. The fractional statistics of the spinons naturally follows from the conformal weight for the primary fields. The relation between the Kalmeyer-Laughlin and the spin-singlet quantum-Hall-effect (QHE) wave functions is shown and leads to the possibility of the spinon excitations in the spin-singlet QHE.
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.
Abstract Conformal field theory has turned out to be a powerful tool to derive two-dimensional lattice models displaying fractional quantum Hall physics. So far most of the work has been for lattices with open boundary conditions in at least one of the two directions, but it is desirable to also be able to handle the case of periodic boundary conditions. Here, we take steps in this direction by deriving analytical expressions for a family of conformal field theory states on the torus that is closely related to the family of bosonic and fermionic Laughlin states. We compute how the states transform when a particle is moved around the torus and when the states are translated or rotated, and we provide numerical evidence in particular cases that the states become orthonormal up to a common factor for large lattices. We use these results to find the S -matrix of the states, which turns out to be the same as for the continuum Laughlin states. Finally, we show that when the states are defined on a square lattice with suitable lattice spacing they practically coincide with the Laughlin states restricted to a lattice.
Conformal field theory has turned out to be a powerful tool to derive two-dimensional lattice models displaying fractional quantum Hall physics. So far most of the work has been for lattices with open boundary conditions in at least one of the two directions, but it is desirable to also be able to handle the case of periodic boundary conditions. Here, we take steps in this direction by deriving analytical expressions for a family of conformal field theory states on the torus that is closely related to the family of bosonic and fermionic Laughlin states. We compute how the states transform when a particle is moved around the torus and when the states are translated or rotated, and we provide numerical evidence in particular cases that the states become orthonormal up to a common factor for large lattices. We use these results to find the S-matrix of the states, which turns out to be the same as for the continuum Laughlin states. Finally, we show that when the states are defined on a square lattice with suitable lattice spacing they practically coincide with the Laughlin states restricted to a lattice.
Gauge theories form a large class of interacting conformal field theories in 3D, among which critical Chern-Simons-matter theories play a prominent role. Here, we study a simple instance: a gapless complex scalar coupled to a U(1)_{2} Chern-Simons gauge field. This theory is conjectured to be dual to three other Chern-Simons-matter theories and to describe the transition between a Kalmeyer-Laughlin chiral spin liquid (a ν=1/2 bosonic Laughlin state) and a trivially gapped phase. Using the fuzzy-sphere regularization, we realize the corresponding quantum phase transition between a ν_{f}=2 fermionic integer quantum Hall state and a ν_{b}=1/2 bosonic fractional quantum Hall state. We show that the transition is continuous and governed by a conformal field theory with SO(3) global symmetry. The operator spectrum contains only one relevant Lorentz scalar, the SO(3) singlet with scaling dimension Δ_{S}=1.52(18), which tunes the transition. Our nonperturbative results identify the universality class naturally associated with the U(1)_{2} Chern-Simons-Higgs theory and show that the conjectured dualities remain self-consistent.
The Kalmeyer-Laughlin chiral spin liquids (CSL) and the Z2 spin liquids are two of the simplest topologically ordered states. Here I develop a theory of a direct quantum phase transition between them. Each CSL is characterized by an integer n and is topologically equivalent to the 1/2n Laughlin fractional quantum Hall (FQH) state. Depending on the parity of n, the transition from the CSL is either to a "twisted" version of the Z2 spin liquid, the "doubled semion" model, or the conventional Z_2 spin liquid ("toric code"). In the presence of SU(2) spin symmetry, the triplet gap remains open through the transition and only singlet operators acquire algebraic correlations. An essential observation is that the CSL/Laughlin FQH states can be understood in terms of bosonic integer quantum Hall (BIQH) states of Schwinger bosons or vortices, respectively. I propose several novel many-body wave functions that can interpolate through the transition.