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Physics Letters A

Universal ratios of critical amplitudes near four dimensions

E. Brézin, J. C. Le Guillou, Jean Zinn‐Justin

Published Apr 1, 1974• 130 citations
Publisher
DOI
10.1016/0375-9601(74)90168-6

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We present an analytic four-loop calculation of the free energy in three dimensions within the O(n) symmetric phi(4) theory at infinite cutoff for general n above and below T(c). It is shown that Goldstone singularities arising at intermediate stages of the calculation cancel among themselves. The correlation length above T(c) and an appropriately defined pseudocorrelation length below T(c) are calculated analytically up to four-loop order for general n. The method of minimal renormalization at fixed dimension d=3 is used to determine the analytic expressions for the four-loop series of the amplitude functions of the free energy, correlation length, and specific heat above and below T(c) in terms of the renormalized coupling. These expressions provide the basis for future accurate Borel resummations of universal amplitude ratios characterizing the asymptotic critical behavior and of crossover functions describing the nonasymptotic critical behavior. A brief application is given by a variational calculation of the universal specific-heat amplitude ratios A+/A- and P=alpha(-1)(1-A+/A-) and of the universal quantity R+(xi)=xi+0(A+)(1/d) for general n.
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.