Family of spin-$S$ chain representations of SU(2)${}_k$ Wess-Zumino-Witten models

We investigate a family of spin-$S$ chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For $S=1/2$, it corresponds to the Haldane-Shastry model. For general spin $S$, we find indication that the low-energy theory of these spin chains is described by the SU(2)${}_k$ Wess-Zumino-Witten model with coupling $k=2S$. In particular, we investigate the $S=1$ model whose ground state is given by a Pfaffian for even number of sites $N$. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary $N$ with trial states obtained by Schwinger projection of two Haldane-Shastry chains.

Figure 1

Entanglement entropy $S\left(x\right)$ of ${\mathcal{H}}_S$ for $S=1$ and $S=3/2$ vs the conformal length $d\left(x\right)=L/\pi \sin \pi x/L$. Data points are given for $L=20$ (red), $L=24$ (blue), and $L=30$ (green). The thermodynamic WZW results are sketched by dashed lines. The fitted central charges (solid lines) agree well with SU(2) WZW${}_{2S}$ field theory within reasonable error bars.

Figure 2

Scaling dimensions for the Haldane-Shastry model (left) and ${\mathcal{H}}_1$ (right). Left: The finite size data (crosses) exactly matches the estimated CFT result $x=\frac{1}{2}$ (dashed line). Right: The finite size linear fit (solid line) shows a mismatch with the estimated CFT result $x=\frac{3}{8}$.

Figure 3

$S=1$ energy spectra scaled to units of unity for the $S^z=0$ sector for $N=16$ sites. Left: $S=1$ Takhtajan-Babudjan Hamiltonian. Right: ${\mathcal{H}}_S$ for $S=1$.

Figure 4

(a) Construction of the $S=1$ trial state ${\psi }_{\mathrm{trial}}$ ($k=0$) for $N$ odd out of two Haldane-Shastry one-spinon states with momenta $k=\pm \pi /2$. (b) Spectrum of the $S=1$ chain for $N=15$. The ground state at $k=0$ is ${\psi }_0^{\mathrm{odd}}$. (c) Overlap $\langle {\psi }_{\mathrm{trial}}|{\psi }_0^{\mathrm{odd}}\rangle$ vs $1/N$. For $N\to \infty$, the overlap is 0.9983(1).