Quantum Potts chain in alternating field

The $q$-state Potts chain with ferromagnetic couplings, $J=1$, in the presence of a transverse field, $\Gamma$, has a quantum phase transition at $\Gamma/q=1$, which is continuous for $q \le 4$ and of first order for $q>4$. Here we introduce a $q$-periodic alternating longitudinal field of strength, $h$, and study the phase diagram and the critical properties of the model. For $h<q/(q-1)$ there is a ferromagnetic ordered phase, for $\Gamma<\Gamma_c(h)$ and at $h=q/(q-1)$ there is a classical endpoint at $\Gamma=0$, with finite entropy at $T=0$. We considered the $q=3$ model and using DMRG techniques we calculated the low-laying spectrum of the Hamiltonian, the transverse magnetisation and the spin-spin correlation function, all of which signalled a diverging correlation length at the transition point with the exponent of the three-state Potts model. In the vicinity of the classical endpoint the model is mapped to a quantum hard rod model, which belongs also to the universality class of the three-state Potts model. Also the spectrum of the critical Hamiltonian is found in agreement with conformal invariance. At the same time the correlation function shows a jump at the transition point, thus the transition is of mixed order for $h<q/(q-1)$.