Progress of Theoretical Physics

Quantum Effect, Critical Dynamics and One-Particle Excitation in 1/n Expansion. II

R. Abe, S. Hikami

Published Nov 1, 1974• 9 citations

Abstract

The dynamical scaling for the one-particle Green function up to O(1/m) is discussed on the basis of the same model and formulation as in I. A characteristic frequency ωq is studied, and the result is shown to be consistent with that obtained in I. The dynamic critical exponents ^z and λ are calculated up to O(1/m).

DOI: 10.1143/ptp.52.1463

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The classical 1/n expansion for critical phenomena is extended to a quantum system by introducing the Bose field with m(=n/2) independent components. The case just at the critical point is considered. It is shown that there is no quantum effect on r; up to 0(1/m). The energy spectrum E. of one-particle excitation is expressed as E.ocq•-c and t; is calcu-lated exactly up to 0 (1/m). The result turns out to be in accordance with the one ob-tained recently by Kondor and Szepfalusy. The dynamic scaling for the present model will be discussed in a subsequent paper.

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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

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