Smooth cutoff formulation of hierarchical reference theory for a scalarϕ4field theory

Abstract

The φ 4 scalar field theory in three dimensions, prototype for the study of phase transitions, is investigated by means of the Hierarchical Reference Theory (HRT) in its smooth cut-off formulation. The critical behavior is described by scaling laws and critical exponents which compare favorably with the known values of the Ising universality class. The inverse susceptibility vanishes identically inside the coexistence curve, providing a first principle implementation of the Maxwell construction, and shows the expected discontinuity across the phase boundary, at variance with the usual sharp cut-off implementation of HRT. The correct description of first and second order phase transitions within a microscopic, non-perturbative approach is thus achieved in the smooth cut-off HRT.

Recent developments of the hierarchical reference theory of fluids and its relation to the renormalization group

“…[31] it was suggested that the smooth cut-off formulation of RG qualitatively improves on the description of the system close to the phase boundary. This expectation was later confirmed both within HRT [26] and NPRG [32]. Following a procedure similar to that adopted in the sharp cut-off case, we first define the auxiliary quantity…”

“…Moreover we will sketch the argument leading to the conclusion that already at the level of this simple approximation, HRT always gives a convex free energy, even at coexistence. The unphysical divergence of the compressibility at the phase boundary predicted by the sharp cut-off closure [25] is circumvented by the smooth cut-off formulation which then provides a satisfactory description of first order transitions [26]. [24] obtained by extrapolation of high-temperature series expansions.…”

Liquid-Vapor Transition from a Microscopic Theory: Beyond the Maxwell Construction

“…The precise form of the rescaled equation actually depends on the specific form of the attractive part of the interaction r (6), which in HRT plays the role of the smooth cutoff of the RG approach. A fixed point analysis, similar to that performed for a 4 theory [9], shows that the HRT equa- …”

Critical properties of the hierarchical reference theory: Further investigations

“…From numerical work on HRT accurate results for the equation of state of continuum fluids came out. Especially, the critical region was described well, and very good results for the critical indices were found [2,[5][6][7][8][9].…”