Projected BCS states and spin Hamiltonians for the SO($n$)${}_1$ Wess-Zumino-Witten model

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 Rényi 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 $n$.

Figure 1

Schematic of a valence-bond configuration in the projected BCS state (1). The valence bonds (blue) are SO($n$) singles formed by two SO($n$) vectors. In the uniform case $z_j=\exp \left(i\frac{2\pi }{N}j\right)$, the periodic chain is viewed as a unit circle and $|z_i-z_j|$ is the chord distance between two sites.

Figure 2

Spin-spin correlation function (logarithmic scale)$\ln \left[{(-1)}^x\langle L_j^{12}{L}_{j+x}^{12}\rangle \right]$ as a function of $\ln \left[\sin \right(\pi x/N\left)\right]$ in the projected BCS state (1) for $N=200$ and (a) $n=5$ and (b) $n=6$. The solid lines (red) are fits of the form $\ln \left[{(-1)}^x\langle L_j^{12}{L}_{j+x}^{12}\rangle \right]=\eta \ln \left[\sin (\pi x/N)\right]+A$, where $\eta$ and $A$ are fitting parameters. The dotted lines are also fits of this formula but with the field theory prediction $\eta =n/4$ (Ref. 21).

Figure 3

Rényi entropy $S_L^{\left(2\right)}$ as a function of $\ln \left[\sin \right(\pi L/N\left)\right]/4$ in the projected BCS state (1) for $N=200$ and (a) $n=5$ and (b) $n=6$. The solid lines (red) are fits of the form $S_L^{\left(2\right)}=c\ln \left[\sin (\pi L/N)\right]/4+c_2^{\prime }$, where $c$ and $c_2^{\prime }$ are fitting parameters. The dotted lines are also fits of this formula but the central charge $c$ is fixed to $c=n/2$ of the SO($n$)${}_1$ WZW model.