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We present a Quantum Monte Carlo study of the Ising model in a transversefield on a square lattice with nearest-neighbor antiferromagnetic exchangeinteraction J and one diagonal second-neighbor interaction $J'$, interpolatingbetween square-lattice ($J'=0$) and triangular-lattice ($J'=J$) limits. At atransverse-field of $B_x=J$, the disorder-line first introduced by Stephenson,where the correlations go from Neel to incommensurate, meets the zerotemperature axis at $J'\approx 0.7 J$. Strong evidence is provided that theincommensurate phase at larger $J'$, at finite temperatures, is a floatingphase with power-law decaying correlations. We sketch a general phase-diagramfor such a system and discuss how our work connects with the previous QuantumMonte Carlo work by Isakov and Moessner for the isotropic triangular lattice($J'=J$). For the isotropic triangular-lattice, we also obtain the entropyfunction and constant entropy contours using a mix of Quantum Monte Carlo,high-temperature series expansions and high-field expansion methods and showthat phase transitions in the model in presence of a transverse field occur atvery low entropy.

Disorder Line and Incommensurate Floating Phases in the Quantum Ising Model on an Anisotropic Triangular Lattice