Entanglement entropy of the $Q\ge 4$ quantum Potts chain

The entanglement entropy $\mathcal{S}$ is an indicator of quantum correlations in the ground state of a many-body quantum system. At a second-order quantum phase-transition point in one dimension $\mathcal{S}$ generally has a logarithmic singularity. Here we consider quantum spin chains with a first-order quantum phase transition, the prototype being the $Q$-state quantum Potts chain for $Q>4$ and calculate $\mathcal{S}$ across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point $\mathcal{S}$ shows a jump, which is expected to vanish for $Q\to 4^{+}$. This jump is calculated in leading order as $\mathrm{\Delta }\mathcal{S}=lnQ[1-4/Q-2/(QlnQ)+O(1/Q^2)]$.

Figure 1

Ground-state energy per site of the $Q=4$-state model at $h_c$ calculated by DMRG in finite chains, ${\tilde{e}}_0\left(L\right)$ (see text), as a function of $L^{-2}$. The arrow indicates the exact value. In the inset the effective central charge is shown versus ${ln}^{-2}\left(L\right)$. Top points: calculated with the use of the exact asymptotic value $e_0=4ln4-2$; bottom points: calculated from two-point fits. The dashed and full lines are guide to the eye.

Figure 2

Ground-state energy per site of the $Q=6$-state model at $h_c$ calculated by DMRG in finite chains, ${\tilde{e}}_0\left(L\right)$ (see text), as a function of $L^{-2}$. The arrow indicates the exact value. In the inset the effective correlation length calculated through Eq. (13) is shown as a function of $1/L$. (Bottom points: calculated by two-point fits with the use of the exact asymptotic value of $e_0$; top points: calculated from three-point fits not using the exact asymptotic value of $e_0$.) The analytical result in Eq. (11) is indicated by an arrow, the dashed lines are guide to the eye.

Figure 3

Estimates for the effective latent heat as defined in Eq. (14). (a) $Q=4$. In the inset the latent heat vs ($1-h$) is shown in log-log plot. The slope of the dashed line is 0.3. (b) $Q=6$. The arrow indicates the analytical result in Eq. (10). The dashed and full lines are guide to the eye.

Figure 4

Entanglement entropy of the Potts chain with $L=4$ sites for different values of $Q$. The curves for $h>1(h<1)$ satisfy: $S\left(Q_1\right)/ln{Q}_1>S\left(Q_2\right)/ln{Q}_2$ for $Q_1<Q_2$($Q_1>Q_2$).

Figure 5

Eigenvalues of the reduced density matrix calculated in chains of length $L=60$ for $h=1.02$ (paramagnetic phase), $h=0.98$ (ferromagnetic phase) and at the transition point, $h=1$. (a) $Q=4$, (b) $Q=6$.

Figure 6

Size dependence of the Schmidt gap at the transition point for $Q=4,6$, and 8. For $Q=4$ the slope of the dashed line is $-0.118$.

Figure 7

Entanglement entropy of the $Q=4$ model in the vicinity of the phase transition calculated by DMRG. The arrow indicates the limiting value at $h=0$ and the dotted lines are guide to the eye. In the inset the inverse slope is plotted versus $(1-h)$, see Eq. (25). Here the slope of the dashed straight line is 9, which corresponds to the analytical prediction.

Figure 8

Entanglement entropy of the $Q=6$ model in the vicinity of the phase transition calculated by DMRG. The arrow indicates the limiting value at $h=0$.

Figure 9

The same as in Fig. 8 for the $Q=8$ model.