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

The entanglement entropy, ${\cal S}$, 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 ${\cal S}$ 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 ${\cal S}$ across the transition point. According to numerical, density matrix renormalization group results at the first-order quantum phase transition point ${\cal S}$ shows a jump, which is expected to vanish for $Q \to 4^+$. This jump is calculated in leading order as $\Delta {\cal S}=\ln Q[1-4/Q-2/(Q \ln Q)+{\cal O}(1/Q^2)]$.