Disorder line and incommensurate floating phases in the quantum Ising model on an anisotropic triangular lattice

We present a quantum Monte Carlo study of the Ising model in a transverse field on a square lattice with nearest-neighbor antiferromagnetic exchange interaction J and one diagonal second-neighbor interaction J', interpolating between square-lattice (J' = 0) and triangular-lattice (J' = J) limits. At a transverse field of B-x = J, the disorder line first introduced by Stephenson [J. Math. Phys. 5, 1009 (1964)], where the correlations go from Neel to incommensurate, meets the zero-temperature axis at J' approximate to 0.7 J. Strong evidence is provided that the incommensurate phase at larger J', at finite temperatures, is a floating phase with power-law decaying correlations. We sketch a general phase-diagram for such a system and discuss how our work connects with the previous quantum Monte Carlo work by Isakov and Moessner [Phys. Rev. B 68, 104409 (2003)] for the isotropic triangular lattice (J' = J). For the isotropic triangular lattice, we also obtain the entropy function and constant entropy contours using a mix of quantum Monte Carlo, high-temperature series expansions and high-field expansion methods and show that phase transitions in the model in the presence of a transverse field occur at very low entropy.