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Journal of Physics A Mathematical and General

Critical properties of Dyson's hierarchical model. III. The n-vector and Heisenberg models

D Kim, Colin J. Thompson

Published Feb 1, 1978• 14 citations
Publisher

Abstract

For pt.II see ibid., vol.11, p.375 (1978). Exact renormalisation group equations (RGE) are obtained for the d-dimensional n-vector and spin-S Heisenberg versions of Dyson's hierarchical model. In the quantum mechanical case the fixed point and neighbourhood properties of the RGE are shown to be independent of S indicating universality of critical behaviour in spin magnitude. Expansions for critical exponents, analogous to the usual epsilon =4-d expansions, are obtained and the RGE solved numerically to obtain relevant scaling indices. The n to infinity or spherical limit is also discussed and 1/n corrections to critical exponents are calculated to first order in 1/n.

DOI
10.1088/0305-4470/11/2/015

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For pt.II see ibid., vol.11, p.375 (1978). Exact renormalisation group equations (RGE) are obtained for the d-dimensional n-vector and spin-S Heisenberg versions of Dyson's hierarchical model. In the quantum mechanical case the fixed point and neighbourhood properties of the RGE are shown to be independent of S indicating universality of critical behaviour in spin magnitude. Expansions for critical exponents, analogous to the usual epsilon =4-d expansions, are obtained and the RGE solved numerically to obtain relevant scaling indices. The n to infinity or spherical limit is also discussed and 1/n corrections to critical exponents are calculated to first order in 1/n.
Critical properties of Dyson's hierarchical Ising model in one dimension with 'potential' falling off like r -(1+ sigma ) are examined in the range 0< sigma <1 where a phase transition is known to occur. A new exact renormalisation group recursion relation is derived for a 'dual' spin probability density function. Together with a scaling-type assumption the authors obtain eta =2- sigma and delta =(1+ sigma )/(1- sigma ) for all 0< sigma <1. Independent evidence, however, suggests that delta =3 for 0< sigma <or=1/2, along with other classical critical exponents in this region. In the non-classical region 1/2< sigma <1 accurate numerical values are obtained for the critical exponents nu and gamma and various Delta sigma = sigma -1/2 expansions to third order in Delta sigma .
We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigourous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and functional RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large- N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review is to present a configuration space counterpart, suitable for lattice formulations, of functional RG equations formulated in momentum space (often called exact RG equations and abbreviated ERGE).
The application of renormalisation group recursion relations to the calculation of critical exponents in d dimensions to second order in epsilon =4-d is discussed. It is shown that in the limit of a large scale factor b, the fixed point value, u* of the four-spin coupling constant is correctly given to second order by a renormalisation group equation in which all irrelevant variables, mass renormalisation graphs, and mass terms in the propagators are ignored. Related approximations in the calculation of the critical exponents are justified. The unexpected equality to order epsilon 2 of u* and u 0c , the Feynman graph coupling constant, is discussed.
In this work we present a new theoretical approach for the study of the phase diagram of interacting quantum particles: bosons, fermions or spins. Starting from the microscopic Hamiltonian of a quantum model, we first derive a set of exact differential equations for the free energy and the correlation functions describing the effects of fluctuations on the thermodynamics of the system. These equations reproduce the full renormalization group structure in the neighbourhood of a critical point keeping, at the same time, fuli information on the non universal properties of the model. A simple approximation to our formally exact equations is studied for the spin-S Heisenberg model where explicit results for critical exponents, critical temperature and coexistence curve are obtained. In three dimensions the results are in good agreement with available quantum Monte Carlo simulations and series expansions.
The fixed points of the exact renormalisation group equations with smooth cut-off are determined in the epsilon -expansion for systems characterised by mn-component vector order-parameters. The critical behaviour of quenched random n-component spin systems is obtained by taking the limit m=O. The critical exponent is obtained to order epsilon 2 , and the other exponents to order epsilon . A derivation of Nelson's trajectory-integral expression for the free energy, appropriate for the case of smooth cut-off renormalisation group equations, is presented.
We review recent results concerning the renormalization group (RG) transformation of Dyson’s hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigorous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson’s approximate recursion formula and Polchinski’s equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and other RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large-N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review article is to present a configuration space counterpart, suitable for lattice formulations, of other RG equations formulated in momentum space (sometimes abbreviated as ERGE). § To whom correspondence should be addressed (yannick-meurice@uiowa.edu) Nonlinear Aspects of the RG Flows of Dyson’s Hierarchical Model 2