Phase transitions in the quantum Ising and rotor models with a long-range interaction

Abstract

We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as 1/rd+σ, where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when σ=2. The dynamical exponent varies continuously as the parameter σ and is unity for σ=2. The one-dimensional long-range quantum Ising system shows a phase transition at T=0 for all values of σ. The most interesting observation is that the phase diagram for σ=d=1 shows a line of Kosterlitz-Thouless transition at finite temperature even though the T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter σ. We explore the phase diagram extending the short-range quantum nonlinear σ model renormalisation group equations to the present case.