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

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 $\nu =1/2$ 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.

Figure 1

(a) Comparison between $-\ln [|f_5(z)|]$ and $-\ln [|f_{\infty }(z)|]=\pi |z{|}^2/(2b^2)+\mathrm{constant}$. The points almost fall on a straight line with unit slope (solid line). (b) Plot of $|\prod _{m(\ne n)}{\rho }_{nm}|$ as a function of $n$ for 100 spins on a sphere.

Figure 2

(a) Low-lying part of the energy spectrum for a $2\times 3$ ($+$) and a $4\times 3$ ($\times$) square lattice centered at the origin. (The horizontal axis shows the degeneracy.) (b) Averaged spin–spin correlation function for 200 spins on the sphere as a function of the distance between the spins. The error bars are of order a few times ${10}^{-5}$, and so the results are only converged for $d\lesssim 1.2$. The symbols encode the sign of the correlations (plus for positive and circle for negative).

Figure 3

(a) Inverse correlation length as a function of $\alpha$ for 100 spins on a sphere. (b–d) Averaged spin–spin correlation function as a function of the distance between the spins for (b) $\alpha =0.24$, (c) $\alpha =0.25$, and (d) $\alpha =0.26$. The error bars in (b) and (d) are of order ${10}^{-5}$ for all points, while the results in (c) are exact. The solid curve in (c) is minus the expression for the correlation function of the continuous $\nu =1$ Laughlin state on the sphere found in 22.

Figure 4

(a) Rényi entropy $S_L^{(2)}=-\ln [\mathrm{Tr}({\rho }_A^2)]$ for 200 spins on a sphere for $\alpha =1/2$ (blue circles) and $\alpha =1/4$ (green crosses) when region $A$ is chosen to be the $L$ spins that are closest to the North Pole. The solid and dashed lines are linear fits. (b) The coefficients in (6) for $N=12$.