Critical exponents from seven-loop strong-coupling φ 4 theory in three dimensions
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The accurate calculation of critical exponents from field theory presents a theoretical challenge, since the relevant information is available only from divergent power series expansions. The results are also of practical relevance, since they predict the outcome of many possible future experiments on many second-order phase transitions. In recent work @1# we have developed a novel method for extracting these exponents from such expansions via a strong-coupling theory of scalar fields with a f 4 interaction. The fields are assumed to have n components with an action which is O(n) symmetric. As an application, we have used available sixloop perturbation expansions of the renormalization constants in three dimensions @2‐4# to calculate the critical exponents for all O( n) universality classes with high precision. Strong-coupling theory works also in 42e dimensions @6#, and is capable of interpolating between the expansions in 42e with those in 21e dimensions of the nonlinear s model @7#. The purpose of this note is to improve significantly the accuracy of our earlier results in three dimensions @1# by making use of new seven-loop expansion coefficients for the critical exponents n and h @8# and, most importantly, by applying a more powerful extrapolation method to infinite order than before. The latter makes our results as accurate as those obtained by Guida and Zinn-Justin @9# via a more sophisticated resummation technique based on analytic mapping and Borel transformations, which in addition takes into account information on the large-order growth of the expansion coefficients. We reach this accuracy without using that information which, as we shall demonstrate at the end in Sec. V, has practically no influence on the results, except for lowering v slightly ~by less than ;0.2%). The reason for the little importance of the large-order information in our approach is that the critical exponents are obtained from evaluations of expansions at infinite bare couplings. The information on the large-order behavior, on the other hand, specifies the discontinuity at the tip of the left-hand cut which starts at the origin of the complex-coupling constant plane @10#. This is too far from the infinite-coupling limit to be of relevance. In our resummation scheme for expansion in powers of the bare coupling constant, an important role is played by the critical exponent of approach to scaling v, whose precise calculation by the same scheme is crucial for obtaining high accuracies in all other critical exponents. It is determined by the condition that the renormalized coupling strength g goes against a constant g* in the strong-coupling limit. The knowledge of v is more yielding than the large-order information in previous resummation schemes in which the critical exponents are determined as a function of the renormalized coupling constant g near g* which is of order unity, thus lying a finite distance away from the left-hand cut in the complex g plane. Although these determinations are sensitive to the discontinuity at the top of the cut, it must be realized that the influence of the cut is very small due to the smallness of the fugacity of the leading instanton, which carries a Boltzmann factor e 2const/ g . We briefly recall the available expansions @4# of the renormalized coupling g[g/m in terms of the bare coupling g ¯ 0[g0 /m for all O( n),
[1] J. Zinn-Justin,et al. Critical exponents of the N-vector model , 1998, cond-mat/9803240.
[2] H. Kleinert. Variational interpolation algorithm between weak- and strong-coupling expansions — application to the polaron , 1995, quant-ph/9507005.
[3] H. Kleinert. Path Integrals in Quantum Mechanics Statistics and Polymer Physics , 1990 .