Generalized numerical renormalization group for dynamical quantities

In this paper we introduce a new approach for calculating dynamical properties within the numerical renormalization group. It is demonstrated that the method previously used fails for the Anderson impurity in a magnetic field due to the absence of energy scale separation. The problem is solved by evaluating the Green function with respect to the reduced density matrix of the full system, leading to accurate spectra in agreement with the static magnetization. The new procedure provides a unifying framework for calculating dynamics at any temperature and represents the correct extension of Wilson's original thermodynamic calculation.     

Variational approximations for renormalization group transformations

Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by expansion in agreement with the results of Wilson and Fisher to leading order in . The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.Supported in part by the National Science Foundation under Grants Nos. MPS73-04886A01 and GH-41512 and by the Brown University Materials Research Laboratory supported by the National Science Foundation. M.C.Y. was supported by a grant from the Scientific and Technical Research Council of Turkey.     

A Markov chain approach to renormalization group transformations

We aim at an explicit characterization of the renormalized Hamiltonian after decimation transformation of a one-dimensional Ising-type Hamiltonian with a nearest-neighbor interaction and a magnetic field term. To facilitate a deeper understanding of the decimation effect, we translate the renormalization flow on the Ising Hamiltonian into a flow on the associated Markov chains through the Markov–Gibbs equivalence. Two different methods are used to verify the well-known conjecture that the eigenvalues of the linearization of this renormalization transformation about the fixed point bear important information about all six of the critical exponents. This illustrates the universality property of the renormalization group map in this case.     

Analogs of renormalization group transformations in random processes

We review the properties of a real-space renormalization group transformation of the free energy, including the existence of oscillatory terms multiplying the non-analytic part of the free energy. We then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation. (The essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary.) These random processes are shown to be generated directly from Cantor sets. In a spatial representation, the ensuing random process exhibits a transition between Gaussian and fractal behavior. In the fractal regime, the trajectories will, in an average sense, form self-similar clusters. In a temporal representation, the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior. The fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses.     

Inverse Monte Carlo renormalization group transformations for critical phenomena

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.     

Consistency of blocking transformations in the finite-temperature renormalization group

The finite-temperature renormalization group is formulated via the Wilson-Kadanoff blocking transformation. Momentum modes and the Matsubara frequencies are coupled by constraints from a smearing function which plays the role of an infrared cutoff regulator. Using the scalar λφ⁴ theory as an example, we consider four general types of smearing functions and show that, to zeroth order in the derivative expansion, they yield qualitatively the same temperature dependence of the running constants and the same critical exponents within numerical accuracy.     

Singular renormalization group transformations and first order phase transitions (II). Monte Carlo renormalization group results

The first order phase transitions in the two-dimensional 10-state Potts model and in the two-dimensional Ising model with magnetic field are studied with Monte Carlo renormalization group methods. The deconfining phase transition of the four-dimensional U(1) lattice gauge theory is treated similarly. The results are not consistent with the standard discontinuity fixed point picture of first order phase transitions. In the U(1) case, where this possibility exists, they are not consistent with a second order phase transition either. The results show a discontinuous flow on the first order transition surface, which is a Monte Carlo renormalization group signal of singular RG transformations.     

Random field singularities in renormalization group transformations on Van der Waals-Ising models

Griffiths-Pearce “peculiarities” of renormalization transformations acting on nonrandom mean field Ising models are studied exactly and shown to occur in connection with critical singularities in the quenched free energies of systems with random magnetic fields. The result applies to the transformations of the full hamiltonian, considered as a function of spin configurations, and not only to the recurrence relations for its interaction constants.     

Some rigorous results on majority rule renormalization group transformations near the critical point

We consider the majority rule renormalization group transformation with two-by-two blocks for the Ising model on a two-dimensional square lattice. For three particular choices of the block spin configuration we prove that the model conditioned on the block spin configuration remains in the high-temperature phase even when the temperature is slightly below the critical temperature of the ordinary Ising model with no conditioning. We take as the definition of the infinite-volume limit an equation introduced in earlier work by the author. We use a computer to find an approximate solution of this equation and verify a condition which implies the existence of an exact solution.     

Rapidity renormalization group

We introduce a systematic approach for the resummation of perturbative series which involves large logarithms not only due to large invariant mass ratios but large rapidities as well. A series of this form can appear in a variety of gauge theory observables. The formalism is utilized to calculate the jet broadening event shape in a systematic fashion to next-to-leading logarithmic order. An operator definition of the factorized cross section as well as a closed form of the next-to-leading-log cross section are presented. The result agrees with the data to within errors.     

Quarkonia from charmonium and renormalization group equations

A prediction of the upsilon and strangeonium spectra is made from the charmonium spectrum by solving the Salpeter equation using an identical potential to that used in charmonium. Effective quark masses and coupling parameters α_s are functions of the inter-quark distance according to the renormalization group equations. The use of the Fermi-Breit-Hamiltonian for obtaining the charmonium hyperfine splitting is criticized.     

Relativistic transformations of thermodynamic quantities

A unique solution is proposed to the problem of how thermodynamic processes between thermodynamic systems at relative rest appear to a moving observer. Assuming only transformations for entropy, pressure, and volume and the invariance of the fundamental thermodynamic equation, one can derive transformations for (thermodynamic) energy and temperature. The invariance of the first and second laws entails transformations for work and heat. All thermodynamic relations become Lorentz-invariant. The transformations thus derived are in principle equivalent to those of Einstein and Planck, except that our expressions for energy and work do not include the mass motion energy. This results in a simpler formulation and endows our transformations (especially that of temperature) with a straightforward physical interpretation.     

Polyakov effective action from functional renormalization group equation

We discuss the Polyakov effective action for a minimally coupled scalar field on a two dimensional curved space by considering a non-local covariant truncation of the effective average action. We derive the flow equation for the form factor in , and we show how the standard result is obtained when we integrate the flow from the ultraviolet to the infrared.     

Ambiguities from quark masses in the renormalization group

Apparent discrepancies between calculations of the quark mass dependence of the lowest order β function in gauge theories are due to an inherent convention dependence of that function. Observable, physical quantities are, however, unambiguous.     

Supersymmetric renormalization group flows

The functional renormalization group equation for the quantum effective action is a powerful tool to investigate non-perturbative phenomena in quantum field theories. We discuss the application of manifest supersymmetric flow equations to the N = 1 Wess-Zumino model in two and three dimensions and the linear O(N) sigma model in three dimensions in the large-N limit. The former is a toy model for dynamical supersymmetry breaking, the latter for an exactly solvable field theory.     

Analyticity and renormalization group

We prove the inequalities ψ(y, α) ?α, |α_s(d/dα_s)(β(α_s)/α_s| ? 1 (for the Paterman-Stueckelberg-Gell-Mann-Low functions in QED and QCD) and γ₀(α_s ? 1 (for the anomalous dimension of the gauge-invariant operator O(x)). The consequences of the inequalities are discussed: for modern energies, comparison of theoretical and experimental moments of deep inelastic structure functions has a meaning only for N ? 7 (singlet case) and N ? 50 (non-singlet case); perturbation theory in QCD has a meaning only for α_rms ? 0.45.     

The eigenvalue-matching renormalization group

The eigenvalue-matching renormalization-group method is extended to the computation of the universal rescaling factors in addition to the universal bifurcation rate. Both the one-dimensional quadratic map and the two-dimensional area-preserving Hénon map are studied. The computation has been carried to very high orders: eleventh in the one-dimensional case and eighth in the two-dimensional case. The accuracy is so high and the algorithm so efficient that it may be used as an alternative to the direct numerical procedure.     

Monte Carlo renormalization group

Monte Carlo computer simulations have long been used to obtain information on the behavior of thermodynamic systems. The method has the advantages of being applicable to a very large class of models and of using only systematically improvable approximations (finite size of system, statistical errors, etc.). However, in the critical region, finite-size effects mask the critical singularities, and put severe practical limits onto the accuracy to which the true critical behavior can be determined. By combining Monte Carlo simulations with a real-space renormalization-group analysis, a large increase in efficiency and accuracy can be achieved—without the uncertainties of the usual truncation approximations. The methods are illustrated by explicit calculations on models exhibiting critical and tricritical behavior.