Using an Environmentally Friendly Renormalization Group we derive an ab initio universal scaling form for the equation of state for the O(N) model, y=f(x), that exhibits all required analyticity properties in the limits $x\to 0$, $x\to\infty$ and $x\to -1$. Unlike current methodologies based on a phenomenological scaling ansatz the scaling function is derived solely from the underlying Landau-Ginzburg-Wilson Hamiltonian and depends only on the three Wilson functions $γ_λ$, $γ_ϕ$ and $γ_{ϕ^2}$ which exhibit a non-trivial crossover between the Wilson-Fisher fixed point and the strong coupling fixed point associated with the Goldstone modes on the coexistence curve. We give explicit results for N=2, 3 and 4 to one-loop order and compare with known results.
Using an Environmentally Friendly Renormalization Group we derive an ab initio universal scaling form for the equation of state for the O(N) model, y=f(x), that exhibits all required analyticity properties in the limits x→0, x→∞ and x→−1. Unlike current methodologies based on a phenomenological scaling ansatz the scaling function is derived solely from the underlying Landau-Ginzburg-Wilson Hamiltonian and depends only on the three Wilson functions γλ, γϕ and γϕ2 which exhibit a non-trivial crossover between the Wilson-Fisher fixed point and the strong coupling fixed point associated with the Goldstone modes on the coexistence curve. We give explicit results for N=2, 3 and 4 to one-loop order and compare with known results.
Using an Environmentally Friendly Renormalization Group we derive an ab initio universal scaling form for the equation of state for the O(N) model, y=f(x), that exhibits all required analyticity properties in the limits x→0, x→∞ and x→−1. Unlike current methodologies based on a phenomenological scaling ansatz the scaling function is derived solely from the underlying Landau-Ginzburg-Wilson Hamiltonian and depends only on the three Wilson functions γλ, γϕ and γϕ2 which exhibit a non-trivial crossover between the Wilson-Fisher fixed point and the strong coupling fixed point associated with the Goldstone modes on the coexistence curve. We give explicit results for N=2, 3 and 4 to one-loop order and compare with known results.
Using an Environmentally Friendly Renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, y=f(x), that exhibits all desired asymptotic and analyticity properties in the three limits x→0, x→∞ and x→−1. The only necessary inputs are the Wilson functions γλ, γϕ and γϕ2, associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson-Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N=1, we derive a one-loop equation of state for 2<d<4 naturally parameterized by a ratio of non-linear scaling fields. For d=3 we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.
A renormalization-group scheme is developed for the three-dimensional O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.
As discussed in the previous chapter, the phenomenological theory of critical phenomena finds a sound basis in the logical sequence of universality, scaling, relevant and irrelevant variables. The difficult problem of a large number of degrees of freedom strongly correlated within the coherence distance, which diverges at criticality, is thereby, in principle, reduced to the determination of the homogeneous form of the relevant response functions and of the evaluation of a few critical indices related to the relevant parameters, whereas the details specifying each system do not matter. In order to give a microscopic foundation to universality and to the scaling theory, an exact transformation is needed, which changes into a scaling transformation asymptotically near the critical point. This is the renormalization group (RG) approach, recently presented in a huge number of reviewpapers and books.Historically, the first comprehensive overview was given by the sixth volume of the series edited by Domb and Green (1976) and by the classic books by Patashinskij and Pokrovskij (1979) and by Ma (1976). Along with scaling theory, the field-theoretic renormalization group (RG) approach (Gell- Mann and Low (1954); Bogoliubov and Shirkov (1959); Bonch-Bruevich and Tyablikov (1962)) was introduced in critical phenomena (Di Castro and Jona-Lasinio (1969)) by showing how the RG equations generalize the universality relations in the sense that they relate one model system to another by varying the coupling and suitably rescaling the other variables and the correlation functions. At the same time a detailed analysis of the pertubative diagrammatic structure of the correlation functions together with the use of Ward identities yielded a renormalized theory (Migdal (1969); Polyakov (1968, 1969)). By iterating the RG transformation the coupling disappears from the equations near to the critical point, the homogeneous form of the static order-parameter correlation function is obtained and a microscopic definition of the critical indices is given. The scaling picture then acquires a theoretical basis. Wilson (Wilson (1971a, b); Wilson and Kogut (1974)) gave a great contribution to the understanding of the physics underlying the RG procedure and to the explicit calculation of the critical indices. He proposed a simple mathematical realization of the other very physical idea of universality due to Kadanoff of grouping together degrees of freedom on the scale s associated with larger and larger cells.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].
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