Critical Phenomena for Systems under Constraint

It is well known that the imposition of a constraint can transform the properties of critical systems. Early work on this phenomenon by Essam and Garelick, Fisher, and others, focused on the effects of constraints on the leading critical exponents describing phase transitions. Recent work extended these considerations to critical amplitudes and to exponents governing logarithmic corrections in certain marginal scenarios. Here these old and new results are gathered and summarised. The involutory nature of the transformations between the critical parameters describing ideal and constrained systems are also discussed, paying particular attention to matters relating to universality.


Introduction
The study of thermodynamic systems subject to constraints has a long history. In 1966, Syozi and Miyazima produced a diluted version of the Ising model and observed that annealed non-magnetic impurities affect the critical behaviour of the model [1]. In particular, the usual infinite critical peak in the specific heat is replaced by a finite cusp. In 1967, Essam and Garelick quantified the nature of this change as [2,3] α X = − α 1 − α . (1.1) Here, α represents the specific heat critical exponent for the ideal (non-diluted) system and α X is its counterpart for the diluted system. If β and γ similarly represent the magnetisation and susceptibility exponents, Essam and Garelick further showed that these transform to [2,3] In 1968, Fisher produced a general theory for critical systems under constraint and the general process linking the ideal critical exponents to those for the constrained system became known as Fisher renormalisation [4]. Because of their continued academic importance and relevance to real systems, phase transitions in constrained systems remained a focus of study [5][6][7][8][9]. In recent years the transformation has been extended to deal with other aspects of critical phenomena [10,11]. Because of their experimental accessibility, amplitude terms are important for the description of critical phenomena. Unsurprisingly, these also change when a constraint is imposed. Perhaps surprisingly, however, the precise nature of this transformation has only recently been studied [11]. Furthermore, in certain marginal circmstances, multiplicative logarithmic corrections also enter into the scaling description at continuous phase transitions. Examples include at the upper critical dimension of spin systems and at the border to regimes where the transition becomes first-order. The exponents of such logarithmic corrections also transform when the system is subjected to a constraint [11].
To give a compact description of all of these various aspects (leading critical exponents, logarithmic corrections and amplitudes), we express the scaling behaviour of the ideal system as follows.
C (t , 0) = A ± |t | −α | ln |t ||α , (1.3) m(t , 0) = B|t | β | ln |t ||β for t < 0, (1.4) Here, t and h refer to the reduced temperature and magnetic field respectively. The correlation length in the absence of external field is ξ(t , 0). The subscripts + and − refer to amplitudes for t > 0 and t < 0, respectively. In principle we could employ subscripts for the critical exponents and their logarithmic counterparts corresponding to those used for the amplitudes, but we suppress these here for simplicity and because the exponents generally coincide on either side of the transition. Note that Eq.(1.3) for the specific heat corresponds to an internal energy of leading form (1.8) Finally, and for completeness, we mention that the leading form for the critical correlation function is In what follows, we give a comprehensive overview of the effects of the presence of a constraint on the critical exponents (including those of the logarithmic corrections, when present) and the amplitudes. The critical exponents are universal quantities while the amplitudes are not. However, certain combinations of amplitudes are universal. We show that the renormalisation process (Fisher renormalisation) which transforms the universal critical paramenters is involutary in the sense that applying it twice results in the identity transformation. However, quantities which are not universal do not transform as involutions. We also show that the various scaling relations between the critical parameters (exponents and amplitudes) also hold for the transformed quantities.
In the next section, we summarise the scaling relations for the leading exponents, their logarithmic counterparts and the universal amplitude combinations. In Section 3 we apply the renormalisation process and study its effects in Section 4. We conclude in Section 5.

Scaling Relations and Universal Amplitude Combinations
The four standard scaling relations are (see, e.g., Ref. [12] and references therein) where d represents the dimensionality of the system. The corresponding scaling relations for the logarithmiccorrection exponents areα + dν = dϙ , (2.5) α +γ = 2β , (2.6) (δ − 1)β +γ = δδ , (2.7) (2 − η)ν +η =γ , (2.8) whereα is augmented by unity in certain special circumstances described in Ref. [13]. The exponentϙ ("koppa-hat") characterises the leading logarithmic correction to the finite-size scaling of the correlation length ξ L (0, 0) ∼ L(ln L)ϙ, where L is the finite extent of the system [14]. It is the logarithmic counterpart of the exponent ϙ, recently introduced to characterise the finite-size correlation length above the upper critical dimension: ξ L (0, 0) ∼ L ϙ [14]. The relations (2.1)-(2.4) for the leading exponents are derived in the appendix where it is also shown that they correspond to the following universal ratios [15]: For the derivation of the logarithmic scaling relations (2.5)-(2.8), the reader is referred to Refs. [13] In the next section, we examine the effects of constraints on the critical exponents and amplitudes. It will turn out that the renormalised critical exponents obey the same set of scaling relations as their original counterparts and that, when applied to universal quantities, Fisher renormalisation is involutory.

Fisher Renormalisation
We consider a thermodynamic variable x conjugate to a field u, so that Here f X (t , h, u) represents the free energy of the system under constraint and u represents a quantity such as the chemical potential with x representing the density of annealed non-magnetic impurities. The constraint is then expressed in terms of an analytic function as One may further assume that the singular part of the free energy of the constrained system is structured analogously to its ideal counterpart f , so that up to a regular background term and in which t * and h * are analytic functions [4]. The ideal free energy For simplicity, we also assume h → −h symmetry so that t * is a function of h 2 . In that case, which vanishes at h = 0. ?????-3

The critical point
To identify the critical point of the constrained system, one first writes the magnetization from Eq.
From Eq. (3.8), if the dependency on h is even, the first term on the right hand side of Eq.(3.9) vanishes at where u c is the critical value of u for the constrained system. The critical point therefore has J (0, 0, u c ) = J 0 .

The relation between t * and t
The constraint (3.2) determines the relation between t * and t . Eq.(3.1) firstly gives (3.13) At h = 0, the second term on the right vanishes after Eq. (3.5). Therefore x(t , 0, u) = e(t * , 0) ∂t * (t , 0, u) ∂u . (3.14) This will give a non-trivial relationship between t * and t . Expanding t * (t , 0, u), one has t * (t , 0, u) = a 1 (u − u c ) + . . . , (3.15) where u c and the coefficients of the expansion are non-universal. Therefore x(t , 0, u) = a 1 e(t * , 0) + . . . , (3.16) which, from Eq.(1.8), is On the other hand, Taylor expansion of the constraining function gives  If α < 0, t * ∼ t and the renormalisation is trivial. In the case where α > 0, however, t renormalises to t * in a non-trivial manner. To describe this, define Then the central result is that the constraint renormalises the reduced temperature from t to t * , whereby
We write ξ X (t , 0) = N X ± |t | −ν X | ln |t || −ν X , (3.33) where Finally, the correlation function is obtainable by differentiating the free energy with respect to two local fields h 1 = h(x 1 ) and h 2 = h(x 2 ). One obtains We have observed that neither the in-field magnetisation nor the correlation function exhibit nontrivial renormalisation of the critical exponents. The former is the case by construction and the latter is so because it is defined at the critical point. Likewise, the exponents ϙ andϙ governing finite-size scaling of the correlaton length do not change under Fisher renormalisation, so that ϙ X = ϙ andϙ x =ϙ.

Properties of Renormalised Scaling Parameters
It is straightforward to verify that if the critical exponents for the ideal system satisfy the scaling relations (2.1)-(2.4), the renormalised exponents for the constrained system do likewise. (This observation for the Essam-Fisher relation (2.2) was already made in Ref. [2].) The same statement applies to the scaling relations for logarithmic corrections (2.5)-(2.8).
Of course, the critical exponents, for which the transformation is involutory, are universal, whereas the critical amplitudes are not. This observation prompts one to investigate the nature of the universal combinations (2.9)-(2.12) under Fisher renormalisation. The non-universal terms J 0 and a, which characterise the transformations of the individual amplitude terms, drop out of the transformations of the universal combinations through the scaling relations (2.1)-(2.4). The universal amplitude combinations transform as  Two successive applications of these transformations confirm the involutory nature of these universal combinations. ?????-6

Conclusions
Fisher renormalization, which generalises an earlier theory of Essam and Garelick is a staple of the established theory of critical phenomena. The early work by these authors was extended in recent years to encompass critical amplitudes and the exponents which govern logarithmic corrections to scaling, when present. Here, a comprehensive treatment of all of these various elements has been given. we also observe that the involutory nature of the renormalisation process is intrinsically linked to universality.

A. Appendix: Universal amplitude Combinations
To identify the universal amplitude combinations, we begin with the standard scaling form for the free energy and correlation length [12,15] The scaling functions Y and X are universal and all the non-universality is contained in the metric factors K t and K h .