Asymptotic structure of the $O(n>1)$ scaling equation of state near the coexistence curve below $T_c$: Goldstone singularities beyond the leading term

Brézin, Wallace, and Wilson (1973) computed the scaling equation of state of the general $O(n)$ model in the $\epsilon = 4-d$ expansion. Below $T_c$ near the coexistence curve, for $n>1$ the expansion is infrared-singular because of massless Goldstone modes: loop diagrams generate logarithms such as $\epsilon \ln r_T$ in the transverse mass variable $r_T \propto H/|M|$, which they conjectured to exponentiate into a nonanalytic additive term $r_L^{-1} \sim \mathrm{const} + r_T^{-\epsilon/2}$ in the longitudinal response, and the coexistence region involves two distinct length scales. The leading-order content of this conjecture is now settled: the coexistence singularity is controlled by the zero-temperature infrared-stable Gaussian fixed point, giving $f(x) \approx c_f(1+x)^{2/(d-2)}$ and $\chi_L \propto |H|^{(d-4)/2}$, a result checked in the large-$n$ limit and verified in lattice simulations of three-dimensional $O(2)$ and $O(4)$ models. The analytic structure beyond leading order, however, remains unresolved: the conjectured exponentiated double series $1+x = c_1 y + c_2 y^{1-\epsilon/2} + d_1 y^2 + \cdots$ (analytic in $y^{1/2}$ in $d=3$) was shown at $O(1/n)$ in three dimensions to be violated by logarithmic corrections of unclear origin, and it is not known whether logarithms, half-integer powers, or $n$-dependent exponents govern the subleading asymptotics at finite $n$; correspondingly, the exact nonanalytic dependence of the longitudinal correlation length on the transverse scale and the infrared behavior of derivative thermodynamic quantities are not derived. The problem is to determine this structure rigorously for $n>1$.

Background

Critical equation of state. Near a continuous phase transition at temperature $T_c$ and ordering (magnetic) field $H$, the order parameter $M$ of an $n$-component system obeys a universal scaling equation of state $H = M^{\delta} f(x)$ with $x = t M^{-1/\beta}$ and $t = (T-T_c)/T_c$, where $\beta$ and $\delta$ are critical exponents. The $O(n)$-symmetric Landau-Ginzburg-Wilson ($\phi^4$) field theory is the continuum model for this universality class, and the $\epsilon$ expansion ($\epsilon = 4-d$, with $d$ the spatial dimension) is Wilson's controlled perturbation scheme about the upper critical dimension (here $d=4$, above which mean-field theory becomes exact), organized around the Wilson-Fisher renormalization-group fixed point (the nontrivial fixed point of the flow that controls critical behavior for $d<4$). Brézin, Wallace, and Wilson computed this scaling equation of state to $O(\epsilon^2)$ by a Feynman-graph expansion, for general $n$; their companion paper treats the Ising-like case $n=1$.

Broken symmetry and Goldstone modes. For $n>1$ below $T_c$, the order parameter selects a direction and the $n-1$ orthogonal (transverse) fluctuations are Goldstone modes: they are massless at $H=0$, so the transverse susceptibility diverges along the whole coexistence curve ($T<T_c$, $H \to 0$) - in fact $\chi_T = |M|/|H|$ exactly, by rotational invariance (a Ward identity, i.e. an exact relation among response functions imposed by the symmetry). The longitudinal response $\chi_L = \partial M/\partial H$ is then dominated by transverse fluctuations, and the longitudinal and transverse correlation lengths are two distinct scales near coexistence. Perturbation theory below $T_c$ is infrared-singular: loop integrals of Goldstone modes generate logarithms such as $\epsilon \ln r_T$ (with $r_T \propto H/|M|$ the transverse mass variable vanishing at coexistence), and the naive Feynman-graph expansion produces the correct nonanalytic behavior only after these logarithms are resummed. Brézin, Wallace, and Wilson conjectured the exponentiated additive form $r_L^{-1} \sim \mathrm{const} + r_T^{-\epsilon/2}$ (with $r_L$ the longitudinal mass variable, $r_L^{-1}$ essentially $\chi_L$), observed that the coexistence region involves two length scales, and posed the question of deriving this structure from a renormalization-group treatment below $T_c$ or from the nonlinear realizations of $O(n)$ symmetry (the nonlinear $\sigma$-model, the field theory of the unit-length order parameter). They also noted that the ability of the graph expansion to reproduce the exponentiated forms, including transient terms in the associated recursion relations (crossover contributions that decay along the renormalization-group flow), must be examined carefully.

What is established. Wallace and Zia formalized the exponentiation conjecture as a double expansion $1+x = c_1 y + c_2 y^{1-\epsilon/2} + d_1 y^2 + d_2 y^{2-\epsilon/2} + d_3 y^{2-\epsilon} + \cdots$ in the scaling variable $y = H/M^{\delta}$, inferred from the Brézin-Wallace-Wilson $\epsilon$ expansion. Renormalization-group analyses of the ordered phase (Schäfer and Horner for isotropic magnets; Lawrie for $N$-vector coexistence; Brézin and Zinn-Justin near two dimensions; Nalimov within the $4-\epsilon$ expansion) established that the leading coexistence singularity is governed by the zero-temperature infrared-stable Gaussian fixed point - the free-field fixed point reached along the coexistence curve, where Goldstone interactions die out at long wavelengths - which yields $f(x) \approx c_f(1+x)^{2/(d-2)}$ as $x \to -1$ and $\chi_L \propto |H|^{(d-4)/2}$ - i.e. the conjectured $\epsilon/2$-power term at leading order. This leading behavior is confirmed within the large-$n$ expansion (a systematic expansion in inverse powers of the number of field components $n$) via the $O(1/n)$ equation of state of Brézin and Wallace, and verified numerically: lattice simulations of the three-dimensional $O(2)$ and $O(4)$ models observe the $H^{-1/2}$ Goldstone divergences of $\chi_L$ and $\xi_T$, the exact $1/H$ divergence of $\chi_T = |M|/|H|$, and the $H^{1/2}$ correction to the magnetization $M = M(T,0) + c\,H^{1/2}$ below $T_c$, and are well described by the exponentiated equation of state with nonperturbatively fixed coefficients. Parametric representations of the equation of state - the standard device enforcing regularity of $f(x)$ inside the physical region (Griffiths analyticity) - encode the leading coexistence behavior as a double zero of the parametric function $h(\theta)$ at the coexistence angle in $d=3$.

What remains unresolved. At subleading order the analytic structure is open. Pelissetto and Vicari computed the coexistence-curve expansion at $O(1/n)$ in general dimension: it is consistent with the double expansion for generic $d$, but for the special dimensions $d = 2+2m/n$ (which include $d=3$) logarithmic terms appear; in three dimensions $f(x) = \omega^2[1 + \frac{1}{n}(f_1(\omega) + f_2(\omega)\ln\omega) + O(n^{-2})]$ with $\omega = 1+x$ and explicit $f_2$, which is incompatible with an expansion analytic in $y^{1/2}$. The origin of these logarithms is unclear (they may signal $n$-dependent exponents or a breakdown of the naive exponentiation), and they are numerically elusive because they first enter at order $\omega^2$ relative to the leading term. As summarized in the 2002 review of Pelissetto and Vicari and stated in the 2008-2009 renormalization-group treatment of O'Connor, Santiago, and Stephens, there are no rigorous arguments fixing the analyticity properties of $f(x)$ near the coexistence curve beyond the leading term, and the nature of the corrections is not well understood.

Problem Statement

For the $O(n)$ universality class with $n>1$ (in $d=3$, and in general dimension $2<d<4$), determine the asymptotic form of the scaling equation of state near the coexistence curve ($T<T_c$, $H \to 0^+$, i.e. $y = H/M^{\delta} \to 0$ or $x \to -1$) beyond the established leading Goldstone singularity $f(x) \approx c_f(1+x)^{2/(d-2)}$, by a controlled derivation. Concretely: (i) decide whether $(1+x)$ admits the exponentiated double series $1+x = c_1 y + c_2 y^{1-\epsilon/2} + d_1 y^2 + d_2 y^{2-\epsilon/2} + d_3 y^{2-\epsilon} + \cdots$ (analytic in $y^{1/2}$ in $d=3$), which is the modern form of the Brézin-Wallace-Wilson exponentiation conjecture $r_L^{-1} \sim \mathrm{const} + r_T^{-\epsilon/2}$, or whether logarithmic corrections such as $\omega^2 \ln \omega$ (with $\omega = 1+x$) appear as the $O(1/n)$ analysis finds in $d=3$ - and if they do, at which orders, with what origin, whether they persist at finite $n$, whether they indicate $n$-dependent exponents replacing the half-integer powers, and how they are compatible with control of the leading behavior by the zero-temperature infrared-stable Gaussian fixed point; (ii) derive correspondingly the nonanalytic dependence of the longitudinal correlation length (equivalently the function $r_L(r_T)$ between the longitudinal and transverse mass variables, with $r_T \propto H/|M|$) on the transverse scale near coexistence, including its corrections, and the infrared asymptotics on and near the coexistence curve of derivative thermodynamic response functions (e.g. the longitudinal susceptibility $\chi_L$, the specific heat at fixed field, and mixed derivatives of the free energy); and (iii) examine whether, and in what sense, the Feynman-graph expansion (the $\epsilon$ expansion, or fixed-dimension perturbation theory) reproduces after resummation the subleading asymptotic structure identified in (i) - including the possibility that it does not reproduce the exponentiated double series - clarifying the role of the crossover (transient) contributions generated by the renormalization-group flow between the Wilson-Fisher fixed point and the zero-temperature fixed point. Sub-question (i) is the central question: at finite $n$, whether the exponentiated double series holds or is instead violated by logarithmic corrections, and in the latter case at which orders they appear and with what origin. Sub-question (ii) follows from the structure fixed in (i), and (iii) tests the resummed Feynman-graph expansion against it. The 1973-era numerical question of improving the value of $\gamma$ for $n=3$ is excluded: it has been superseded by resummed high-order calculations in per-mille agreement with experiment and simulation.

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