Orthogonal transforms and lattice field theories

Abstract

In a previous paper we showed that a class of real-space renormalization-group calculations could be reinterpreted as an orthogonal transformation of coordinates followed by an approximate (variational) evaluation of the resulting Hamiltonian. Here we shall discuss the orthogonal-transform method and concentrate on analytic rather than numerical results. The transform method may ultimately be more accurate because even though similar approximations are made in coupling the lattice oscillators, at no time is the system split into noninteracting finite-sized blocks. Approximate variational solutions to a ${\ensuremath{\varphi}}^{4}$ lattice field theory in 2, 3, and 4 spacetime dimensions are constructed using an anharmonic-oscillator basis. The solutions exhibit an ordered and a disordered phase. Analytic expressions are obtained for certain critical surfaces of these theories in regions of parameter space where a spin approximation would be invalid. The continuum limit of our solutions is discussed. The resultant perturbation expansion is also discussed and a method for its evaluation is described.