MICROSCOPIC CALCULATION OF CRITICAL EXPONENTS WITHOUT THE 1 / n OR EXPANSIONS

Critical exponents of a Bose system are calculated microscopically without an expansion in 1/n ore. As expected, quantum corrections are found to be absent and the results to agree with the 1/n expansion result, for n = 2, to 0(1/n). The purpose of this letter is to discuss the critical behavior of a quantum system, which exhibits a phase transition , without the commonly employed techniques of 1/n or expansions [1] (n is the number of components of the order parameter and = 4—d, where d is the spatial dimension of the system). Our motivations for such an investigation are threefold: First, to present an alternative calculation of critical indices not based on an expansion in n or d, in order to break away from the a priori assumption of " universal " significance given to these quantities ; second, to treat a quantum mechanical system strictly within a quantum statistical formulation to test the universality assumption that the critical exponents are independent of quantum effects; third, to establish a closer connection between the standard microscopic approaches to many-body theory 121 and the newly developed 1/n and expansions [11. As our model, we consider a system of spinless bosons of mass m at temperature Tabove Tc and at a fixed density in a unit volume. In order to avoid arbitrary assumptions about the strength of the potential and introduction of cutoffs [3, 4], we consider only a specific potential and assume that the particles are interacting with the Coulomb potential, V(q) = 4ire 2/q2 and the system is placed in a rigid background of opposite charge to ensure overall charge neutrality. Certain static critical exponents are defined by the asymptotic form of the relevant correlation functions for small k at Tc. For example, the order parameter correlation function G(k) and (in a charged system) 131 the irre-ducible density correlation function 11(k) for small k at Tc behave like G(k) '-~ k2~ and 11(k)-~ k1~if X <0, defining the exponents ~ and X ~'hichwe now proceed to calculate for our model. We employ the usual diagrammatic perturbation theory techniques [21 and seek the lowest order correction to the properties of the non-interacting system (i.e., ideal Bose gas). The simplest self-consistent approximation is the well-known Hartree-Fock approximation, which for a charged Bose gas in the static limit takes the form [2] G(k)—— (k)—~(k)+~(O)—r, (rOatTc) (1) where ~(k) = [V(p)/1-V(p)11(p)] n(p+k), (2) (2ir)~ …