Realization of higher Wess-Zumino-Witten models in spin chains

Building on the generalization of the exactly dimerized Majumdar-Ghosh ground state to arbitrary spin $S$ for the Heisenberg chain with a three-site term $({\mathbf{S}}_{i-1}·{\mathbf{S}}_i)({\mathbf{S}}_i·{\mathbf{S}}_{i+1})+\text{H.c.}$, we use density-matrix renormalization group simulations and exact diagonalizations to determine the nature of the dimerization transition for $S=1$, $3/2$, and 2. The resulting central charge and critical exponent are in good agreement with the $\text{SU}{\left(2\right)}_{k=2S}$ Wess-Zumino-Witten values $c=3k/(2+k)$ and $\eta =3/(2+k)$. Since the three-site term that induces dimerization appears naturally if exchange interactions are calculated beyond second order, these results suggest that $\text{SU}{\left(2\right)}_{k>1}$ Wess-Zumino-Witten models might finally be realized in actual spin chains.

Figure 1

Dimerization as a function of $J_3/J_1$ for $S=3/2$ for different system sizes with up to $L=150$ sites in the vicinity of the phase transition $J_3/J_1\approx 0.063$. The left insets show the size dependence at $J_3/J_1=0.0625$, 0.063, and $0.0635$, respectively. The right inset shows the location of the first excited state crossing point as a function of the system size.

Figure 2

Dimerization as a function of $J_3/J_1$ for $S=2$ for different system sizes with up to $L=150$ sites in the vicinity of the phase transition $J_3/J_1\approx 0.0403$. The left insets show the size dependence at $J_3/J_1=0.0402$, 0.0403, and $0.0404$, respectively. The right inset shows the location of the first excited state crossing point as a function of the system size.

Figure 3

Extrapolation to the thermodynamic limit of the central charge for $S=3/2$. The data points shown are results from an extrapolation as a function of the discarded weight, which is displayed in the inset.

Figure 4

The same as in Fig. 3 for $S=2$.

Figure 5

Decay of the spin correlation function $\langle S_i^z·S_j^z\rangle$ with distance $|i-j|$ for systems with $L=150$ and spin $S=3/2$ and $S=2$. The solid lines are fits to power laws with exponents indicated in the plot.