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Physical review. B, Condensed matter

Fluid-magnet universality: Renormalization-group analysis of φ5 operators

Jeffrey F. Nicoll, Royce K. P. Zia

Published Jun 1, 1981• 50 citations
Publisher

Abstract

The question of a possible difference between the universality classes of fluids and Ising-like magnets is addressed by perturbation theory and the renormalization group. The most dangerous possibility is that of an ${\ensuremath{\varphi}}^{5}$ addition to the usual ${\ensuremath{\varphi}}^{4}$ theory. We show that no ${\ensuremath{\varphi}}^{5}$ fixed point exists in the framework of an expansion around $d=\frac{10}{3}$. Further we show that to $O({\ensuremath{\epsilon}}_{4}^{2})$, ${\ensuremath{\epsilon}}_{4}\ensuremath{\equiv}4\ensuremath{-}d$, the ordinary ${\ensuremath{\varphi}}^{4}$ fixed point is stable against the perturbations that mix with ${\ensuremath{\varphi}}^{5}$. Two new correction-to-scaling exponents are found. One of the exponents, ${\ensuremath{\Delta}}_{5}$, is poorly determined with a range of values from 0.5 to 1.0 compatible with the $O({\ensuremath{\epsilon}}_{4}^{2})$ result. However, its positivity rules out a separate fluid fixed point, indicating fluid-magnet asymptotic universality. The second exponent, ${\ensuremath{\Delta}}_{3}$, can be determined exactly: ${\ensuremath{\Delta}}_{3}=1\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}$. This implies the universal existence of a contribution to the fluid diameter scaling like the internal energy.

DOI
10.1103/physrevb.23.6157

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The question of a possible difference between the universality classes of fluids and Ising-like magnets is addressed by perturbation theory and the renormalization group. The most dangerous possibility is that of an \ensuremathφ5{\ensuremath{\varphi}}^{5}\ensuremathφ5 addition to the usual \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 theory. We show that no \ensuremathφ5{\ensuremath{\varphi}}^{5}\ensuremathφ5 fixed point exists in the framework of an expansion around d=103d=\frac{10}{3}d=310​. Further we show that to O(\ensuremathϵ42)O({\ensuremath{\epsilon}}_{4}^{2})O(\ensuremathϵ42​), \ensuremathϵ4\ensuremath≡4\ensuremath−d{\ensuremath{\epsilon}}_{4}\ensuremath{\equiv}4\ensuremath{-}d\ensuremathϵ4​\ensuremath≡4\ensuremath−d, the ordinary \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 fixed point is stable against the perturbations that mix with \ensuremathφ5{\ensuremath{\varphi}}^{5}\ensuremathφ5. Two new correction-to-scaling exponents are found. One of the exponents, \ensuremathΔ5{\ensuremath{\Delta}}_{5}\ensuremathΔ5​, is poorly determined with a range of values from 0.5 to 1.0 compatible with the O(\ensuremathϵ42)O({\ensuremath{\epsilon}}_{4}^{2})O(\ensuremathϵ42​) result. However, its positivity rules out a separate fluid fixed point, indicating fluid-magnet asymptotic universality. The second exponent, \ensuremathΔ3{\ensuremath{\Delta}}_{3}\ensuremathΔ3​, can be determined exactly: \ensuremathΔ3=1\ensuremath−\ensuremathα\ensuremath−\ensuremathβ{\ensuremath{\Delta}}_{3}=1\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}\ensuremath{\beta}\ensuremathΔ3​=1\ensuremath−\ensuremathα\ensuremath−\ensuremathβ. This implies the universal existence of a contribution to the fluid diameter scaling like the internal energy.
The critical behavior of simple single-component fluids near the liquid-vapor critical point (and of other systems in the same universality class) is examined via an asymmetric spin Hamiltonian in Landau-Ginzburg-Wilson form. The leading important odd terms to be added to an Ising-type system are a five-spin interaction $\ensuremath{\int}\ensuremath{\phi}^{5}(x)$ and a nonlocal cubic interaction $\ensuremath{-}\ensuremath{\int}\ensuremath{\phi}^{2}\ensuremath{\nabla}^{2}\ensuremath{\phi}$. These two operators can be combined into two different eigenoperators with distinct physical interpretations. One combination can be shown to be exactly equivalent to the mixing of field variables (e.g., chemical potential and temperature) in the temperaturelike variable of an Ising or Ising-type system. This mixing has been used phenomenologically, found in certain models, and justified on general geometric grounds. This is the first proof that all asymmetric models, assuming universality and barring accidents, will have such mixing; this implies the universal occurrence of a ${t}^{1\ensuremath{-}a}$ fluid diameter. The second combination is treated by renormalization-group methods and its contributions to the Helmholtz free energy, magnetic equation of state, and correlation length are calculated to $O({\ensuremath{\epsilon}}^{2}), \ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$. There are several novel features of this interaction which persist to all orders in perturbation theory. In addition to the expected renormalized M5{M}^{5}M5 contribution to the free energy, there are also terms linear in hhh (the magnetic field) and MMM which could be described as nonanalytic fluctuation-induced shifts in the order parameter and field. These might appear experimentally as apparent shifts in the background terms for hhh and MMM.
The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations
Experimental data for critical exponents in some magnetic materials are compared with recent theoretical results on the three-dimensional (3D) Ising model, as derived by one of us (ZDZ) based on two conjectures [Z.D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag. 87 (2007), pp. 5309–5419]. It is found that critical exponents in some bulk magnetic materials indeed form a 3D Ising universality. Our attention is then focused on the critical indices at fluid–fluid phase transition. We suggest to use Zhang's exponent β = 3/8 to fit the experimental data over the wider asymptotic region near the critical point of a fluid–fluid phase transition. The 3D Ising universality should exist for critical indices in a certain class of magnets and at fluid–fluid phase transition.
Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order O\mathcal{O}O, O\mathcal{O}O phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (O=2\mathcal{O}=2O=2) and Gaussian tricritical points (O=3\mathcal{O}=3O=3) to critical points of arbitrary order O\mathcal{O}O. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for O=3,4\mathcal{O}=3,4O=3,4 are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for O=3\mathcal{O}=3O=3, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.
Wegner's paradigm for constructing renormalization groups in statistical mechanics is applied to fluids. The coordinates of the fraction ($1\ensuremath{-}y$) of the molecules are integrated over and a system of interactions describing the remaining fraction yyy is constructed with the aid of the McMillan-Mayer solution theory. Renormalized Ursell functions are defined in terms of which the group operations are shown to be linear. The group operations commute with a scaling operation and it is found that a series of renormalization groups depending on a parameter xxx can be constructed by combining the operation of integrating over coordinates with the scaling operation. By comparing the renormalized cluster functions with the form of the critical cluster functions given by scaling it is shown that a fixed point of one of the series of groups corresponds to the critical point. The parameter xxx turns out to be the scaling exponent for the chemical potential. The question of the universality of critical phenomena is expressed as the question of the discreteness or continuity of the spectrum of values of the parameter xxx for which the linear fixed-point equations have a nontrivial solution. It is suggested that this question can be answered definitively only when a precise formulation is made of the function space in which the sequence of renormalized Ursell functions must lie.
Liquid-gas phase transitions are usually assumed to belong to the same universality class as Ising magnets. However, in the liquid-gas case, the free-energy expansion contains odd powers of the order parameter. We show in this work that, even though the odd terms are irrelevant near four dimensions, they become relevant at dimensionality ${d}_{c}=\frac{(10\ensuremath{-}5\ensuremath{\eta})}{3}\ensuremath{\approx}3.33$. A new, stable, fixed point may appear. We approximately calculate, to order ${\ensuremath{\epsilon}}^{\ensuremath{'}}={d}_{c}\ensuremath{-}d$, the difference between the critical exponents for this new fixed point and the Ising critical exponents. This difference turns out to be numerically small. We conclude that, even though in three dimensions the liquid-gas and Ising-magnet phase transition do not belong to the same universality class, the effect on the critical exponents is probably too small to be observed.