On the Large N Limit of 3D and 4D Hermitian Matrix Models

The large N limit of the hermitian matrix model in three and four Euclidean space-time dimensions is studied with the help of the approximate Renormalization Group recursion formula. The planar graphs contributing to wave function, mass and coupling constant renormalization are identified and summed in this approximation. In four dimensions the model fails to have an interacting continuum limit, but in three dimensions there is a non trivial fixed point for the approximate RG relations. The critical exponents of the three dimensional model at this fixed point are $\nu = 0.665069$ and $\eta=0.19882$. The existence (or non existence) of the fixed point and the critical exponents display a fairly high degree of universality since they do not seem to depend on the specific (non universal) assumptions made in the approximation.Comment: Number 0.519689 in eqs (36) and (37) should be replaced by 0.303152. The critical exponents were computed with the correct number entry and are therefore UNCHANGE

A Guide to Precision Calculations in Dyson's Hierarchical Scalar Field Theory

The goal of this article is to provide a practical method to calculate, in a scalar 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 phase to perform high-precision calculations of the unsubtracted Green's functions at zero momentum in dimension 3, 4, and 5. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting beta reach a critical value beta_c (with up to 16 significant digits in our actual calculations). We show that the round-off errors on the magnetic susceptibility grow like (beta_c -beta)^{-1} near criticality. We show that the systematic errors (finite truncations and volume) can be controlled with an exponential precision and reduced to a level lower than the numerical errors. We justify the use of the truncation for calculations of the high-temperature expansion. We calculate the dimensionless renormalized coupling constant corresponding to the 4-point function and show that when beta -> beta_c, this quantity tends to a fixed value which can be determined accurately when D=3 (hyperscaling holds), and goes to zero like (Ln(beta_c -beta))^{-1} when D=4.Comment: Uses revtex with psfig, 31 pages including 15 figure

Algebraic Computation of the Hierarchical Renormalization Group Fixed Points and their ϵ\epsilon-Expansions

Nontrivial fixed points of the hierarchical renormalization group are computed by numerically solving a system of quadratic equations for the coupling constants. This approach avoids a fine tuning of relevant parameters. We study the eigenvalues of the renormalization group transformation, linearized around the non-trivial fixed points. The numerical results are compared with $\epsilon$-expansion.Comment: LaTex file, 24 pages, 5 figures appended as 1 PostScript file, preprint MS-TPI-94-

Analysis of a three-component model phase diagram by Catastrophe Theory: Potentials with two Order Parameters

In this work we classify the singularities obtained from the Gibbs potential of a lattice gas model with three components, two order parameters and five control parameters applying the general theorems provided by Catastrophe Theory. In particular, we clearly establish the existence of Landau potentials in two variables or, in other words, corank 2 canonical forms that are associated to the hyperbolic umbilic, D_{+4}, its dual the elliptic umbilic, D_{-4}, and the parabolic umbilic, D_5, catastrophes. The transversality of the potential with two order parameters is explicitely shown for each case. Thus we complete the Catastrophe Theory analysis of the three-component lattice model, initiated in a previous paper.Comment: 17 pages, 3 EPS figures, Latex file, continuation of Phys. Rev. B57, 13527 (1998) (cond-mat/9707015), submitted to Phys. Rev.

Wilson-Polchinski exact renormalization group equation for O(N) systems: Leading and next-to-leading orders in the derivative expansion

With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson-Polchinski equation in the case of the $N$-vector model with the symmetry $\mathrm{O}(N)$. As a test, the critical exponents $$ and $\nu$ as well as the subcritical exponent $\omega$ (and higher ones) are estimated in three dimensions for values of $N$ ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other $\mathrm{O}(N)$ exact RG equations at second order. The possibility of varying $N$ allows to size up the derivative expansion method. The values obtained from the resummation of high orders of perturbative field theory are used as standards to illustrate the eventual convergence in each case. A peculiar attention is drawn on the preservation (or not) of the reparametrisation invariance.Comment: Dedicated to Lothar Sch\"afer on the occasion of his 60th birthday. Final versio

Buckling Instabilities of a Confined Colloid Crystal Layer

A model predicting the structure of repulsive, spherically symmetric, monodisperse particles confined between two walls is presented. We study the buckling transition of a single flat layer as the double layer state develops. Experimental realizations of this model are suspensions of stabilized colloidal particles squeezed between glass plates. By expanding the thermodynamic potential about a flat state of $N$ confined colloidal particles, we derive a free energy as a functional of in-plane and out-of-plane displacements. The wavevectors of these first buckling instabilities correspond to three different ordered structures. Landau theory predicts that the symmetry of these phases allows for second order phase transitions. This possibility exists even in the presence of gravity or plate asymmetry. These transitions lead to critical behavior and phases with the symmetry of the three-state and four-state Potts models, the X-Y model with 6-fold anisotropy, and the Heisenberg model with cubic interactions. Experimental detection of these structures is discussed.Comment: 24 pages, 8 figures on request. EF508

Crossover phenomena in spin models with medium-range interactions and self-avoiding walks with medium-range jumps

We study crossover phenomena in a model of self-avoiding walks with medium-range jumps, that corresponds to the limit $N\to 0$ of an $N$-vector spin system with medium-range interactions. In particular, we consider the critical crossover limit that interpolates between the Gaussian and the Wilson-Fisher fixed point. The corresponding crossover functions are computed using field-theoretical methods and an appropriate mean-field expansion. The critical crossover limit is accurately studied by numerical Monte Carlo simulations, which are much more efficient for walk models than for spin systems. Monte Carlo data are compared with the field-theoretical predictions concerning the critical crossover functions, finding a good agreement. We also verify the predictions for the scaling behavior of the leading nonuniversal corrections. We determine phenomenological parametrizations that are exact in the critical crossover limit, have the correct scaling behavior for the leading correction, and describe the nonuniversal lscrossover behavior of our data for any finite range.Comment: 43 pages, revte

Thermodynamic characteristics of the classical n-vector magnetic model in three dimensions

The method of calculating the free energy and thermodynamic characteristics of the classical n-vector three-dimensional (3D) magnetic model at the microscopic level without any adjustable parameters is proposed. Mathematical description is perfomed using the collective variables (CV) method in the framework of the $\rho^4$ model approximation. The exponentially decreasing function of the distance between the particles situated at the N sites of a simple cubic lattice is used as the interaction potential. Explicit and rigorous analytical expressions for entropy,internal energy, specific heat near the phase transition point as functions of the temperature are obtained. The dependence of the amplitudes of the thermodynamic characteristics of the system for $T>T_c$ and $T<T_c$ on the microscopic parameters of the interaction potential are studied for the cases $n=1,2,3$ and $n\to\infty$. The obtained results provide the basis for accurate analysis of the critical behaviour in three dimensions including the nonuniversal characteristics of the system.Comment: 25 pages, 5 figure

Exact multilocal renormalization on the effective action : application to the random sine Gordon model statics and non-equilibrium dynamics

We extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the $\eta$ exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent $z$ are recovered and several new results are obtained. The equilibrium two-point scaling functions are obtained. The nonequilibrium, finite momentum, two-time $t,t'$ response and correlations are computed. They are shown to exhibit scaling forms, characterized by novel exponents $\lambda_R \neq \lambda_C$, as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be non trivial and of the form $X(q^z (t-t'), t/t')$. Analogies and differences with pure critical models are discussed.Comment: 33 pages, RevTe

Improved high-temperature expansion and critical equation of state of three-dimensional Ising-like systems

High-temperature series are computed for a generalized $3d$ Ising model with arbitrary potential. Two specific ``improved'' potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are: $\gamma=1.2371(4)$, $\nu=0.63002(23)$, $\alpha=0.1099(7)$, $\eta=0.0364(4)$, $\beta=0.32648(18)$. By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.Comment: 65 pages, 1 figure, revtex. New Monte Carlo data by Hasenbusch enabled us to improve the determination of the critical exponents and of the equation of state. The discussion of several topics was improved and the bibliography was update