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arXiv:hep-th/9604192

Effective Critical Exponents from Finite Temperature Renormalization Group

Michael T. Strickland, Сен-Бен Ляо

Published Apr 30, 1996• 1 citation
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Abstract

Effective critical exponents for the λϕ^4 scalar field theory are calculated as a function of the renormalization group block size k_o^{-1} and inverse critical temperature β_c. Exact renormalization group equations are presented up to first order in the derivative expansion and numerical solutions are obtained with and without polynomial expansion of the blocked potential. For a finite temperature system in d dimensions, it is shown that \barβ_c = β_c k_o determines whether the d-dimensional (\barβ_c << 1) or (d+1)-dimensional (\barβ_c >> 1) fixed point governs the phase transition. The validity of a polynomial expansion of the blocked potential near criticality is also addressed.

DOI
10.48550/arxiv.hep-th/9604192

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Effective critical exponents for the λϕ^4 scalar field theory are calculated as a function of the renormalization group block size k_o^{-1} and inverse critical temperature β_c. Exact renormalization group equations are presented up to first order in the derivative expansion and numerical solutions are obtained with and without polynomial expansion of the blocked potential. For a finite temperature system in d dimensions, it is shown that \barβ_c = β_c k_o determines whether the d-dimensional (\barβ_c << 1) or (d+1)-dimensional (\barβ_c >> 1) fixed point governs the phase transition. The validity of a polynomial expansion of the blocked potential near criticality is also addressed.
The critical behavior of the lambda phi^4 scalar field theory is investigated as a function of the final renormalization group transformation block size k_o^{-1} and critical temperature beta_c^{-1}. We demonstrate how the dimensionless finite-size scaling variable betabar_c = beta_c k_o determines whether the three-dimensional (betabar_c 1) or four-dimensional (betabar_c 1) fixed point governs the phase transition. The effective betabar_c-dependent critical exponents are shown to interpolate between the three- and four-dimensional values. In the intermediate range of betabar_c finite-size effects on critical exponents are examined by solving the renormalization group equations numerically, both with and without polynomial truncation of the blocked potential U_{beta,k}(Phi). Wavefunction renormalization is explicitly taken into account, and the failure of polynomial expansion of U_{beta,k}(Phi) in the neighborhood of the transition is also discussed.
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Through a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ 3 theory, we show that the instability fixed points of the theory, together with their associated instability exponents, are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar ϕ 4 model, below its critical temperature and near the instability points. Finitetime scaling and leading corrections to the scaling are considered. We also show that the instability exponents of the first-order phase transitions are equivalent to those of the Yang–Lee edge singularity, and employ the latter to improve our estimates of the former. The outcomes agree well with existing numerical results.
We show by a detailed study of the mean-field approximation, the Gaussian approximation, the perturbation expansion, and the field-theoretic renormalization-group analysis of a φ3φ^{3}φ3 theory that its instability fixed points with their associated instability exponents are quite probably relevant to the scaling and universality behavior exhibited by the first-order phase transitions in a field-driven scalar ϕ4ϕ^4ϕ4 model below its critical temperature near their instability points. Finite-time scaling and leading corrections to scaling are considered. We also show that the instability exponents of the first-order phase transitions equal those of the Yang--Lee edge singularity and employ the latter to improve our estimates of the former. The outcomes agree well with existent numerical results.
An isolated critical point on a line of first-order transitions (Landau point) is studied by means of the low-temperature renormalization group and 1N\frac{1}{N}N1​ expansion. Critical exponents are computed in dimension d=3d=3d=3, using polynomials fitting the first two terms of the expansion about d=2d=2d=2 and the previously calculated first three terms of the $\ensuremath{\epsilon}$ expansion about d=4d=4d=4. The results are ${\ensuremath{\lambda}}_{3}=1.081$ and ${\ensuremath{\lambda}}_{6}=\ensuremath{-}1.617$. The conclusions for the shape of the phase diagram are drawn from these values.
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.