The $\epsilon$ expansion and Universality in three dimensions

TL;DR

This paper discusses how the classification of universality classes in critical phenomena, established via epsilon expansion near four or six dimensions, remains valid in three dimensions due to eigenvalue repulsion, despite failures of perturbative renormalization group.

Contribution

It provides a theoretical argument that universality classification persists in lower dimensions through eigenvalue repulsion, beyond perturbative renormalization group methods.

Findings

Universality classification remains valid in three dimensions.

Eigenvalue repulsion explains the persistence of universality.

Perturbative renormalization group fails in three dimensions, but universality persists.

Abstract

It has been observed that the clasification into universality classes of critical behaviour, as established by perturbative renormalization group in the viscinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when purturbative renormalisation group fails in three dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when purturbative renormalisation group fails, because of the well known phenomenon of eigenvalue repulsion.

Full text

**The $\epsilon$ expansion and Universality in three dimensions

** Nicolas Sourlas

Laboratoire de Physique Théorique de l’ENS, École Normale Supérieure, PSL Research University, Sorbonne Universités, UPMC Univ. Paris 06, CNRS, 75005 Paris, France.

ABSTRACT

It has been observed that the clasification into universality classes of critical behaviour, as established by perturbative renormalization group in the viscinity of four or six dimensions of space by the epsilon expansion, remains valid down to three dimensions in all known cases, even when purturbative renormalisation group fails in three dimensions. In this paper we argue that this classification into universality classes remains true in lower dimensions of space, even when purturbative renormalisation group fails, because of the well known phenomenon of eigenvalue repulsion.

PACS numbers: 05.10Cc,05.70.Jk

One of the big triumphs of the renormalization group(RG) is the explanation of universality near a second order phase transition[1, 2]. Very different physical systems like uniaxial magnets, binary mixtures, liquid-vapor transitions, share the same values of critical exponents, critical amplitude ratios etc. We say they belong to the same universality class. Few important parameters like the dimensionality of space or the symmetry of the system, determine the universality class. Details of the interaction are irrelevant.

Near a second order phase transition, the hamiltonian of the system flows under renormalization group transformations, toward a fixed point. In the viscinity of this fixed point one can use Wilson’s operator product expansion[3] and analyze the dimensions of the operators appearing in the Hamiltonian. The operators are classified according to their scaling dimension into relevant, marginal and irrelevant.

Physical systems whose hamiltonians differ by irrelevant operators belong to the same universality class.

In general it is not easy to find the RG fixed point. But above a certain dimension of space $D_{u}$, called the upper critical dimension, the renormalization group flows to the gaussian fixed point and mean field theory is valid. $D_{u}=4$ for systems with $Z_{2}$ symmetry like Ising ferromagnets and $D_{u}=6$ for systems where cubic operators are allowed, like Potts models.

Near the upper critical dimension the computation of the dimensions of operators can easily be performed in the framework of the $\epsilon=D_{u}-D$ expansion[4]. $D$ is the dimension of space. This allows for the classification into universality classes of physical systems for infinitesimal $\epsilon$.

What is quit remarquable is that this classification remains valid at $D=3$, i.e. for $\epsilon=1$ or $\epsilon=2$. Indeed there are not known cases of the classification into universality classes, based on the epsilon expansion, not being valid at D=3.

This fact is even more remarquable in the case of disordered systems. The $\epsilon$ expansion is based on the perturbative renormalization group. The straightforward application of field theoretic methods and the renormalization group (RG) is not possible because the disorder breaks the translation symmetry of the Hamiltonian. The standard procedure is then to average over disorder using the replica method [5]. One starts with $n$ noninteracting copies of the system (replicas) and averages over the disorder distribution. This produces an effective Hamiltonian with $n$ interacting fields which is translation invariant and enables the use of the RG. In the end, the $n\to 0$ limit has to be taken. The replica method is mathematically unorthodox. Its combination with the perturbative renormalization group (PRG) has been shown to produce incorrect results in 3D systems. An example is provided by the random-field Ising model (RFIM) where the combination of the replica method with the PRG predicts dimensional reduction [6, 7], which does not hold neither in three [8, 9] nor in four dimensions [12]. Mean field and the replica method are believed to be correct at infinite dimensions.

Despite this failure of PRG it has beeng shown that, contrary to previous believes, universality holds for the RFIM in three dimensions. PRG predicts that different RFIM models, where the random fields are drawn from different probability distributions of the random fields, belong to the same universality classes. Also, more surprisenly, diluted antiferromagnets in a field are predicted to belong to the same universality class. These two predictions have been recently shown numerically to be true in three and four dimensions [10, 11], despite the failure of PRG

Why even when perturbative renormalization group fails, its predictions about universality classes still remain true at low dimensions?

In this note we will argue that if perturbative renormalization group is valid in the viscinity of the upper critical dimension, i.e. if the epsilon expansion is valid for infinitesimaly small $\epsilon$, physical systems which according to the $\epsilon$ expansion belong to the same universality class, continue to belong to the same universality class also in lower dimensions, even when PRG is not valid anymore at these lower dimensions.

The reason has to do with the scaling dimensions of the operators which appear in Wilson’s operator product expansion and is the following. The renormalization group fixed point and the scaling dimension $d_{O}$ of the operator $O$’s are a function of the dimension of space $D$. The $d_{O}(D)$’s change when the dimension of space is changed. A necessary and sufficiant condition for non changing universality classes as the dimension of space $D$ varies is for the scaling dimensions $d_{O}(D)$ of the leading operators not to cross when the dimension of space is lowered from $D=D_{u}$ down to $D=3$ as this is illustrated in the figure. If this is the case, the classification of the operators into relevant, marginal and irrelevant remains unchanged when the dimension of space is lowered. The essential observation is that the scaling dimensions $d_{O}(D)$’s of the operators are eigenvalues of the scaling transformations, i.e. of the group of dilatations of space. It is well known from the early days of quantum mechanics that in the generic case the eigenvalues of operators, i.e. the eigenvalues of the matrices in the matrix representation of the operators, in the case of quantum mechanics the Hamiltonian of the system, do not cross if one changes a single parameter[13]. This is easily verified for a two by two hermitian matrix, which depends on a parameter $s$. The condition for the two eigenvalues of the matrix to be equal is that a sum of two perfect squares $a(s)^{2}+b(s)^{2}$ is zero, i.e. $a(s)=0$ and $b(s)=0$. In the generic case it is not possible to satisfy simultaneously these two equations by fixing a single parameter $s$. The general mathematical argument is due to von Neumann and Wigner[14]. This phenomenon of eigenvalue repulsion is the reason for which universality classes do not change when the dimension of space is lowered. It does not depend on the validity of perturbative renormalization group at lower dimensions. There is a regularity asumption in the previous argument, i.e. that one can follow continuously the change of the scaling dimensions of the operators when the dimension of space is lowered.

The previous argument can be inverted. If it is found by other means, like experiments or numerical simulations, that the epsilon expansion classification of universality classes is valid at three dimensions, it means that perturbative renormalization group is valid near the upper critical dimension $D_{U}$, i.e. the epsilon expansion is valid. This is a non trivial information in the case of disordered systems.

It has been shown recently[15] that dimensional reduction as predicted by the perturbative renormalization group for the random field Ising model[6, 7], is indeed valid at five dimensions. The breaking of perturbative renormalization group is a low dimensional phenomenon and does not affect universality.

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Useful and stimulating discussions with Ed Witten are gratefully aknowledged.