A Field Theoretical Study of a Lattice Gas in Two Planes

Abstract

Appealing to universality the critical exponents of a lattice system can be calculated considering a continuum version easier to deal with. Algebraic manipulations and renormalization group arguments enables one to find out such a simpler model. However, this is not a straightforward procedure, in general, and technical problems can arise. In this spirit we have performed a field-theoretic approach to the two-layer lattice gas system with total particle density, ϱ, conserved (Achahbar 1993). Particles can hop from one lattice to the other, but there is no interaction between particles in different planes. The system undergoes a second order Ising like phase transition for ϱ = 1/2, and a discontinuous one if ϱ ≠ 1/2. After a Gaussian transformation of the lattice partition function and a renormalization group discussion of the operators relevance, one gets an O(M) continuous symmetric Lagrangian with an additional term with cubic symmetry, so it possesses two coupling constants which are known functions of the temperature and ϱ. A one-loop expansion for the free energy gives us a flow pattern of the couplings constants with four fixed points and runaway trajectories, i.e. trajectories that do not flow to any of the fixed points and which are a reflection of a first order phase transition. In order to match these results with our problem we have to set g and the temperature values as initial conditions for the flow trajectories to find out which critical theory corresponds to our original parameters. We thus find for ϱ = 1/2 that the system belongs to the Ising universality class, as expected. If ϱ ≠ 1/2, there is no trace of a first-order phase transition, but a Heisenberg like continuous transition occurs. This is in disagreement with the exact solution, and some explanation is required. We think the irrelevant operators that were neglected on the basis of renormalization group arguments can produce a shift of the bare parameters. Finally we discuss the use of this formalism to study first-order phase transitions.