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arXiv:0709.2024 · Physical Review E

Smooth cutoff formulation of hierarchical reference theory for a scalar ϕ4 field theory

C. Ionescu, Alberto Parola, Davide Pini, L. Reatto

Published Sep 12, 2007• 13 citations
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Abstract

The phi4 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 cutoff 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 cutoff implementation of HRT. The correct description of first and second order phase transitions within a microscopic, nonperturbative approach is thus achieved in the smooth cutoff HRT.

DOI
10.1103/physreve.76.031113

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•Jan 1, 2001•1 citation
The phi4 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 cutoff 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 cutoff implementation of HRT. The correct description of first and second order phase transitions within a microscopic, nonperturbative approach is thus achieved in the smooth cutoff HRT.
We study how the formalism of the hierarchical reference theory (HRT) can be extended to inhomogeneous systems. HRT is a liquid-state theory which implements the basic ideas of the Wilson momentum-shell renormalization group (RG) to microscopic Hamiltonians. In the case of homogeneous systems, HRT provides accurate results even in the critical region, where it reproduces scaling and nonclassical critical exponents. We applied the HRT to study wetting critical phenomena in a planar geometry. Our formalism avoids the explicit definition of effective surface Hamiltonians but leads, close to the wetting transition, to the same renormalization group equation already studied by RG techiques. However, HRT also provides information on the nonuniversal quantities because it does not require any preliminary coarse graining procedure. A simple approximation to the infinite HRT set of equations is discussed. The HRT evolution equation for the surface free energy is numerically integrated in a semi-infinite three-dimensional Ising model and the complete wetting phase transition is analyzed. A renormalization of the adsorption critical amplitude and of the wetting parameter is observed. Our results are compared to available Monte Carlo simulations.
The asymptotic and preasymptotic critical behavior in fluids, mixtures, and uniaxial magnets is believed to be described by an effective straight phi(4) scalar field theory with suitable, nonuniversal, coupling constants. The critical parameters as well as the extent of crossovers and corrections to the leading critical behavior in physical systems, crucially depends on the choice of these couplings. Here we propose a new method for deriving the effective scalar field theory appropriate to a microscopic model in this universality class. Use is made of the hierarchical reference theory, which implements the basic ideas of Wilson momentum space renormalization group to microscopic Hamiltonians. The effective low-energy field theory is then analyzed by the minimal subtraction scheme of Schloms and Dohm. We discuss the application of this method to the three-dimensional Ising model and to the liquid-vapor phase transition. We make comparison with high-temperature expansion results and with experimental data for rare gas.
The critical region of the hierarchical reference theory (HRT) is investigated further. This extends an earlier work by us where the critical properties of the HRT were concluded indirectly via another accurate but somewhat different theory, the self-consistent Ornstein-Zernike approximation (SCOZA), and numerical work. In the present work we perform our analysis directly upon the HRT partial differential equation to establish ordinary differential equations for the subleading scaling contributions. Again we find that the HRT critical indices in three dimensions are simple rational numbers.
Following the same strategy adopted in the hierarchical reference theory of fluids the author has derived a new hierarchy of differential equations for the free energy and the direct correlation functions of a classical fluid. In the critical region the equations resemble the renormalisation group in a smooth cut-off formulation and in fact the theory gives the usual epsilon =4-d expansion for the critical exponents. In this theory the universality of the critical behaviour is not manifest and it is generally lost in an approximate solution of the hierarchy. They have studied a very simple approximation in d=3 giving a non-classical value of the compressibility critical exponent and they have checked that the violation of universality is small.
The lambda phi4 scalar field model can be applied to interpret pion-pion scattering and properties of hadrons. In this work, the mathematical basis, phase transitions and singularities of a (3+1)-dimensional (i.e., (3+1)D) phi4 scalar field model are investigated. It is found that as a specific example of topological quantum field theories, the (3+1)D phi4 scalar field model must be set up on the Jordan-von Neumann-Wigner framework and dealt with the parameter space of complex time (or complex temperature). The use of the time average and the topologic Lorentz transformation representing Reidemeister moves ensure the integrability, which takes into account for the contributions of nontrivial topological structures to physical properties of the many-body interacting system. The ergodic hypothesis is violated at finite temperatures in the (3+1)D phi4 scalar field model. Because the quantum field theories with ultraviolet cutoff can be mapped to the models in statistical mechanics, the (3+1)D phi4 scalar field model with ultraviolet cutoff is studied by inspecting its relation with the three-dimensional (3D) Ising model. Furthermore, the direct relation between the coupling K in the 3D Ising model and the bare coupling lambda0 in the (3+1)D phi4 scalar field model is determined in the strong coupling limit. The results obtained in the present work can be utilized to investigate thermodynamic physical properties and critical phenomena of quantum (scalar) field theories.
The goal of this thesis is to provide a practical method to calculate, in scalar field theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric and broken phases to perform high precision calculations of the Green's functions at zero momentum. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting β reach a critical value βc. We show that the round-off errors on the magnetic susceptibility grow like (βc − β) −1 near criticality. We show that the systematic errors (finite truncation and volume) can be controlled with an exponential precision and reduced to a level lower than numerical errors. We probe the numerical errors made in Renormalization Group (RG) calculations by varying slightly the resealing factor of the fields and resealing back in order to get the same (if there were no round-off errors) magnetic susceptibility. We find a significant departure from Gaussian behavior. In addition, the average value differs from the exact answer by a quantity which is of the same order as the standard deviation. We calculate the critical exponents γ and the subleading exponent Δ of Dyson's model by direct fits of the zero-momentum two-point function calculated with an Ising and a Landau-Ginzburg measure and by linearization about the non-trivial fixed point for O(N) model up to N = 5. We extend the method to the low temperature phase and show that accurate values of the connected one-, two- and three- point functions can be obtained at large volume and in a limited range of constant external field coupled linearly to the field variable. We show that the leading exponents obey the scaling and hyperscaling relations. We discuss the concept of universality in terms of critical exponents, correction amplitudes and the universal coupling constant for different Landau-Ginzburg measures. We compare our accurate numerical results with those obtained from perturbative methods.