Quantum phase transitions in the Ising model in a spatially modulated field

The phase transitions in the transverse field Ising model in a competing spatially modulated (periodic and oscillatory) longitudinal field are studied numerically. There is a multiphase point in absence of the transverse field where the degeneracy for a longitudinal field of wavelength lambda is [(1+root5)/2](2N/lambda) for a system with N spins, an exact result obtained from the known result for lambda = 2. The phase transitions in the Gamma (transverse field) versus h(o) (amplitude of the longitudinal field! phase diagram are obtained from the vanishing of the mass gap Delta. We find that for all the phase transition points obtained in this way, Delta shows finite size scaling behavior signifying a continuous phase transition everywhere. The values of the critical exponents show that the model belongs to the universality class of the two dimensional Ising model. The longitudinal field is found to have the same scaling behavior as that of the transverse field, which seems to be a unique feature for the competing field. The phase boundaries for two different wavelengths of the modulated field are obtained. Close to the multiphase point at h(c), the phase boundary behaves as (h(c) - h(o))(b), where b is also lambda dependent.