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Physical review. B, Condensed matter

Theory of the Landau critical point. II. Low-temperature renormalization group and 1N expansion

Chester A. Vause, Josef Sak

Published Nov 1, 1978• 3 citations
Publisher

Abstract

An isolated critical point on a line of first-order transitions (Landau point) is studied by means of the low-temperature renormalization group and $\frac{1}{N}$ expansion. Critical exponents are computed in dimension $d=3$, using polynomials fitting the first two terms of the expansion about $d=2$ and the previously calculated first three terms of the $\ensuremath{\epsilon}$ expansion about $d=4$. 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.

DOI
10.1103/physrevb.18.4842

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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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion about d=4d=4d=4. The results are \ensuremathλ3=1.081{\ensuremath{\lambda}}_{3}=1.081\ensuremathλ3​=1.081 and \ensuremathλ6=\ensuremath−1.617{\ensuremath{\lambda}}_{6}=\ensuremath{-}1.617\ensuremathλ6​=\ensuremath−1.617. The conclusions for the shape of the phase diagram are drawn from these values.
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Using the Landau Hamiltonian for the description of second-order phase transitions, we give a proof of scaling for any continuous number of dimensions below four. The proof is based on a summation of diagrams having a power-law divergence and standard renormalization-group methods. The proof is constructive in that it leads to an unambiguous calculation for the critical exponents $\ensuremath{\eta}$ and $\ensuremath{\gamma}$. We present in this paper a detailed discussion of the proof; we also compare our method with the $\ensuremath{\epsilon}$ expansion leading to an interesting aspect of that theory: We find that the contribution to the critical exponents of order $\ensuremath{\epsilon}$ can be gotten without any calculation of diagrams. In this paper we have only made a lowest-order calculation in three dimensions. To this order we are of course unable to locate the relevant fixed point, but it leads to a relation between $\ensuremath{\eta}$ and $\ensuremath{\gamma}$---also to lowest order---which is such that if $\ensuremath{\gamma}$ is fixed to be 1.25, then $\ensuremath{\eta}$ turns out to be 0.12.
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.
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.
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