Journal of Physics C Solid State Physics

Perturbation theory, scaling and the spherical model

Dennis C. Rapaport

Published May 10, 1972• 3 citations

Abstract

The four dimensional spherical model is treated as a perturbation of the three dimensional model. An exact resummation of the expansion is possible and the perturbed critical properties are compared with the predictions of the scaling theoretical approach to the problem. Both methods are found to yield the correct dependence on the strength of the perturbation, which can also be found from the exact solution of the model. A more general method of resummation is discussed and applied to the expansion of both the four and five dimensional models. The effects of perturbations which do not alter the critical exponents are also investigated.

DOI: 10.1088/0022-3719/5/9/008

Related papers

Critical Exponents from Other Expansions

It is useful to compare the results obtained so far with other approaches to the critical exponents. One is a similar field-theoretic approach based on perturbation expansions of φ 4-theories. But instead of working in D = 4−ε dimensions and continuing the results to ε = 1 to obtain critical exponents in the physical dimension D = 3, perturbation expansions can be derived directly in three dimensions, and studied in the regime of small masses. Fundamentally, this approach is less satisfactory than the previous one since there is no small parameter analogous to ε which permits us to prove the existence of a scaling limit at least for small ε. Another disadvantage is that it is impossible to calculate the expansion coefficients analytically, except for one- and two loop-diagrams. In one respect the three-dimensional approach has, however, an advantage: the power series have long been available up to six loops for ω, η, and ν, and there exist recent results up to seven loops for the critical exponents η and ν. Although numerical expansions are not as esthetic as the exact expansions derived in this book, their increased order leads to a higher accuracy in the critical exponents, as we shall see below in Section 20.4. Another approach is based on high-temperature expansions for lattice models of the O(N)symmetric classical Heisenberg model with the energy of Eq. (1.52). The critical exponents will be derived from such expansions in Section 20.8. Finally we take advantage of the fact that the O(N)-symmetric classical Heisenberg model is exactly solvable in the limit of large N, where it reduces to the spherical model [recall the remark after Eq. (1.69)]. It is therefore possible to expand the φ 4-theory around this limit in powers of 1/N. The results will be compared with those from the perturbation expansions in

WORLD SCIENTIFIC eBooksJul 1, 20010 citations

Critical exponents predicted by grouping of Feynman diagrams in 4 model

J. Kaupužs

Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.

Annalen der PhysikApr 1, 20011 citation

Critical exponents predicted by grouping of Feynman diagrams inϕ4 model

J. Kaupužs

arXiv:cond-mat/0104183Annalen der PhysikApr 1, 200127 citations

Perturbation theory methods applied to critical phenomena

J. Kaupužs

arXiv:cond-mat/0004255Apr 15, 20000 citations

Spherical model and quantum phase transitions

V. N. Udodov

Abstract The spherical Berlin-Katz model is considered in the framework of the epsilon expansion in one-dimensional and two-dimensional space. For the two-dimensional and threedimensional cases in this model, an exact solution was previously obtained in the presence of a field, and for the two-dimensional case the critical temperature is zero, that is, a “quantum” phase transition is observed. On the other hand, the epsilon expansion of critical exponents with an arbitrary number of order parameter components is known. This approach is consistent with the scaling paradigm. Some critical exponents are found for the spherical model in one-and twodimensional space in accordance with the generalized scaling paradigm and the ideas of quantum phase transitions. A new formula is proposed for the critical heat capacity exponent, which depends on the dynamic index z, at a critical temperature equal to zero. An expression is proposed for the order of phase transition with a change in temperature (developing the approach of R. Baxter), which also depends on the z index. An interpolation formula is presented for the effective dimension of space, which is valid for both a positive critical temperature and a critical temperature equal to zero. This formula is general. Transitions with a change in the field in a spherical model at absolute zero are also considered.

Journal of Physics Conference SeriesNov 1, 20210 citations

The derivative expansion of the exact renormalization group

Michael Turner

We formulate a method of performing non-perturbative calculations in quantum field theory, based upon a derivative expansion of the exact renormalization group. We then proceed to apply this method to the calculation of critical exponents for three dimensional O(N) symmetric theory. Finally we discuss how the approximation scheme manages to reproduce some exactly known solutions in critical phenomena.

arXiv:hep-th/0108031Aug 6, 20010 citations