Link between New Versions of the Hierarchical Reference Theory of Liquids and of the Non Perturbative Renormalization Group in Statistical Field Theory
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Link between New Versions of the HierarchicalReference Theory of Liquids and of the NonPerturbative Renormalization Group inStatistical Field Theory
Jean-Michel CaillolLPT (UMR 8627 )Bˆat. 210Universit´e de Paris-SudF-91405 Orsay CedexFranceandCNRS, Orsay, F-91405, FranceJune 22, 2018
Abstract
I propose a new version of the Hierarchical Reference Theory ofliquids. Two formalisms, one in the grand canonical ensemble, theother in the framework of statistical field theory are given in parallel.In the latter the theory is an avatar of a new version of the nonperturbative renormalization group (J. Phys. A : Math. Gen. ,225004 (2009)). The flow of the Wilsonian action as well as that of theeffective average action of Wetterich are derived and a simple relationbetween the two functionals is established. The standard HierarchicalReference Theory for liquids ( Adv. Phys. , 211 (1995)) is recoveredfor a sharp infra-red cut-off of the propagator key-words : statistical field theory; non-perturbative renormalization group;theory of liquids; hierarchical reference theory.1 ONTENTS Contents W Λ k . . . . . . . . . . . . . . . 114.2 RG flow of the Wilsonian action S Λ k . . . . . . . . . . . . . . . 124.3 RG flow of the effective average Kohn-Sham free energy β A Λ k . 13 W Λ k
18B Mean field theory at scale “k” 18
INTRODUCTION The first theory of liquids which incorporates the ideas of the renormal-ization group (RG) of Wilson [1, 2] was proposed more than twenty-five yearsago in a series of outstanding papers by Reatto, Parola and co-workers whodeveloped the so-called Hierarchical Reference Theory (HRT) which gives anaccurate description of the thermodynamics and the structure of a wide classof fluids close to and away from the critical region [3, 4, 5, 6, 7]. The theorywas developed and improved over years by these authors until very recently.During roughly the same period the Wilson approach to the (RG) hasbeen the subject of a revival in both statistical physics and field theory. Twomain formulations of the non perturbative renormalization group (NPRG)have been developed in parallel. In the first one, the RG transformationsconcern the Hamiltonian (or action) of the model [1, 2, 8, 9, 10] while, inthe second, the RG flow of the free energy is under scrutinity, or rather, tobe more precise, the so-called ”effective average action” [11, 12]. In bothapproaches, at scale “k” (in momentum space), a cut-off is introduced toseparate the slow ( q < k ) and the fast ( q > k ) modes of the field. In thefirst approach the Wilsonian action is defined to be the effective action ofthe low energy modes yielding the same physics a long distances than thebare action. In this case the cut-off acts as an ultra-violet (UV) cut-off. Inthe formalism of Wetterich these ideas are implemented at the level of theHelmholtz free energy of the fast modes (in fact one is rather interested in itsLegendre transform, the effective average action) and thus, in this approach,the cut-off acts conversely as an infra-rouge (IR) cut-off. The two points ofview can be reconciliated formally [10, 13].It turns out that many of new results obtained recently for the NPRGhad been anticipated by the seminal works of Reatto ‘et al.’. Since the twocommunities have few contacts and use a different language many things havebeen discovered or rediscovered independently by both sides in the ignoranceof the results obtained by the other camp. Today field theorists would saythat HRT is a sharp cut-off version of Wetterich effective average actionapproach to the RG, while specialists of liquids would claim that Wetterichtheory is nothing but a field theoretical version of HRT with smooth cut-offand strange notations.A precise comparison between the two worlds however requires a fieldtheoretical representation of simple classical liquids. Technically, a simpleHubbard-Stratonovich transform does the job [14, 15]. This has been realizedby many authors [16, 17, 18, 19]. The more achieved scalar field theoreticalrepresentation of liquids at equilibrium has been christened the KSSHE the-ory after the names of Kac [20], Siegert [21], Stratonovich [14], Hubbard [15]
THE MODEL inter alia . I showin the latter reference that the implementation of Wetterich effective averageaction method in the framework of KSSHE theory yields indeed HRT in theultra-sharp cut-off limit.In general the field theory obtained via a Hubbard-Stratonovich transformyields a non-canonical theory in the sense that the coupling between the field ϕ and the source, here the chemical potential ν , is non linear (see section 2.2below). An additional transformation is needed to obtain a standard fieldtheory with a linear coupling. I adhered to this method in my previous workson liquids [18, 19]. However it turns out that the NPRG ideas can easily beextended to the non-canonical theory itself and it has then the advantage toyield extremely simple functional relationships between the Wilsonian actionand Wetterich’s effective average action [13]. In the present paper I will showthat applying this new field theoretical formalism to the KSSHE theory leadsto a slightly different version of HRT with some advantages however. Forinstance the flow equations of Reatto ‘et al’ are reobtained by taking thelimit of an ordinary sharp cut-off. Moreover the grand-canonical functionalscorresponding to the Wilsonian and effective average actions of Wetterichare clearly identified which allows a deeper understanding of the NPRG forliquids.The paper is organized as follows. In next section 2, I circumscribe thetype of liquids under study (the simplest!) and give their representations inthe grand-canonical ensemble and in field theory. Then, in section 3, thecoarse-graining operations of the various functionals of interest are detailedand their flow equations are derived in section 4. in the last paragraph 4.3,the usual HRT equation is obtained by taking the limit of a sharp cut-off. Wesummarize and conclude in section 5. Two appendices explain some technicalpoints. I consider a classical simple fluid made of identical hard spheres (HS) ofdiameter σ with, in addition, isotropic pair interactions v ( r ij ) ( r ij = | x i − x j | , x i position of particle ” i ”). Since v ( r ) can be chosen arbitrarily in the core,we assume that v ( r ) has been regularized for 0 < r < σ in such a way that itsFourier transform e v q is a well behaved function of q and that v (0) is a finitequantity. For convenience the domain Ω ⊂ R d occupied by the molecules ischosen to be a d − dimensional cube of side L within periodic boundary (PB) THE MODEL V = L d . The fluid is at equilibrium inthe grand canonical (GC) ensemble, β = 1 /k B T is the inverse temperature( k B Boltzmann’s constant), and µ the chemical potential. In addition theparticles are subject to an external potential ψ ( x ) and we will denote by ν ( x ) = β ( µ − ψ ( x )) the dimensionless local chemical potential. I adhere tonotations usually adopted in standard textbooks on liquids (see e.g. [23]) andthus denote by w ( r ) = − βv ( r ) minus the dimensionless pair interaction.Moreover we restrict ourselves to the case of attractive interactions, i.e.such that e w ( q ) > q . The model admits a thermodynamic limit if w ( r ) decays faster than 1 /r d + ǫ as r → ∞ , where d is the space dimensionsand ǫ > C ≡ ( N ; x . . . x N ) the microscopic densityof particles at point x reads b ρ ( x |C ) = N X i =1 δ ( d ) ( x − x i ) , (1)and the grand canonical partition function (GCPF) Ξ [ ν ] which encodes allthe physics of the model at equilibrium is defined as [23]Ξ [ ν ] = Tr [ exp ( − β H [ C ])] , − β H = 12 b ρ · w · b ρ + ( ν − w (0)2 ) · b ρ , Tr [ . . . ] = ∞ X N =0 N ! Z Ω d . . . dn exp[ − βV (HS) [ C ]] . . . , (2)where i ≡ x i and di ≡ d d x i . In equation (2) βV (HS) [ C ] denotes the HScontribution to the configurational integral. Its Boltzmann factor is either 0if configuration C involves overlaps of spheres, or 1 otherwise. Note that infact Ξ [ ν ] does not depend on the self energy − w (0) /
2. For a given volume V and a given inverse temperature β , Ξ [ ν ] is a log-convex functional of the localchemical potential ν ( x ) [25, 26]. I have employed here convenient matricialnotations. ν · b ρ ≡ Z Ω d d x ν ( x ) b ρ ( x |C ) (3) b ρ · w · b ρ ≡ Z Ω d d x d d y b ρ ( x |C ) w ( x, y ) b ρ ( y |C ) . (4) It is possible to express Ξ[ ν ] as a functional integral making thus a linkbetween the theory of liquids and statistical field theory [16, 17, 18]. With THE MODEL ν ] = N − w Z D ϕ exp (cid:18) − ϕ · R · ϕ + W (HS) [ ν − w (0) / ϕ ] (cid:19) (5)where W (HS) ≡ ln Ξ (HS) , ϕ is a real random scalar field, R ≡ w − is theinverse of w (in the sense of operators, i.e. w (1 , · R (3 ,
2) = δ ( d ) (1 , (HS) [ ν − w (0) / ϕ ] denotes the GCPF of bare hard spheres in the presenceof a local chemical potential ν ( x ) − w (0) / ϕ ( x ). We optimistically supposethat the functional Ξ (HS) [ ν ] as well as its functional derivatives, i. e. thedensity-density correlation functions, are known exactly. This is nearly truesince the HS fluid has been under scrutinity since so many years [23].I have noted N w the normalization constant N w = Z D ϕ exp (cid:18) − ϕ · R · ϕ (cid:19) . (6)The functional integrals which enter equations (5) and (6) can be given aprecise meaning in the case where the domain Ω is a cube of side L withperiodic boundary conditions, in this case the measure D ϕ reads [2] D ϕ ≡ Y q ∈ Λ d e ϕ q √ πV (7a) d e ϕ q d e ϕ − q = 2 d ℜ e ϕ q d ℑ e ϕ q for q = 0 . (7b)where Λ = (2 π/L ) Z D ( Z ring of integers) is the reciprocal cubic lattice.Note that the reality of ϕ implies that, for q = 0, e ϕ − q = e ϕ ⋆q , where the starmeans complex conjugation. With this slick normalization of the functionalmeasure one has exactly N w = exp (cid:18) V Z q ln e w ( q ) (cid:19) (8)in the limit of large systems ( L → ∞ ).The field theoretical representation (5) of the GCPF Ξ[ ν ] can be extendedto arbitrary pair potentials. If w (1 ,
2) contains a definite negative part cor-responding to a repulsive interaction then a second scalar field, purely imagi-nary must be introduced via an additional Hubbard-Stratonovich transform,see e.g. ref [18]. Handling these two fields makes the algebra a little bitmore complicated but by no means intractable; we lazily skip this compli-cation in this paper and restrict ourselves to attractive w ( r ). Alternatively,one can gather the HS and repulsive pair potentials contributions in a singlefunctional W (Ref) which will play the role devoted to W (HS) . COARSE GRAINING S [ ϕ ] = ϕ · w − · ϕ − W (HS) [ ν − w (0) / ϕ ],with w (1 ,
2) playing the role of the bare propagator and the non local term W (HS) that of the interaction. However purists will note that this is not a“canonical” field theory stricto sensu since the coupling between the field ϕ and the external source ν is non-linear. As detailed at length in refs. [19, 13]this point can be circumvented by the change of variables χ = ϕ + ν − w (0) / χ and ν . Moreover one can establish a rigorous mappingbetween these two field theories. Then all the knowledge amassed in sta-tistical field theory can be injected in the theory of liquids, notably wellestablished perturbative technics and/or more recent non-perturbative ap-proaches. Recently however we proposed rather to consider the action S [ ϕ ]as a well-educated one and work directly with it rather than to try to schol-arly adhere to a canonical action via the mapping ϕ ↔ χ . As discussedin reference [13], the adoption of this “non-canonical” point of view leadsto a remarkably simple reconciliation between the points of view of Wilson-Polchinski on the one hand and that of Wetterich in the other hand. Appliedto liquids, as will be done here, it yields a new version of the smooth HRTtheory with some advantages on my previous attempts [19]. The coarse graining procedure involved to build families of actions withthe same physics at long distances is the core of the RG theory. In this sectionwe introduce this procedure step-by -step following recent developments infield theory [10, 13]
I noted in section 2.1 that the pair potential (propagator) w ( r ) can beregularized at short distances 0 < r < σ , i. e. in the ultra-violet (UV) regime,without changing the GCPF Ξ[ ν ] of the system. I introduce a potential w Λ0 ( r )defined in Fourier space by e w Λ0 ( q ) = C ( q, Λ) e w ( q ) , (9)where C ( q, Λ) is an UV cut-off which is equal to 1 for q <
Λ and to 0 for q >
Λ. Typically C ( q, Λ) = 1 − Θ ǫ ( q, Λ) where Θ ǫ ( q, Λ) is a function whichlooks like the step function Θ( q − Λ). It could be precisely this functionand we would then speak of a sharp cut-off; or it could be a smooth version
COARSE GRAINING ǫ ( q, Λ) varying smoothly from 0 to 1 in a small interval (Λ − ǫ, Λ + ǫ )about the UV cut-off; in this case we would speak of a smooth UV cut-off.It is sometimes convenient to choose C ( q, Λ) = C ( x = q/ Λ) but by no meanscompulsory.In any case if Λ σ ≫ w ( r ) and w Λ0 ( r ) coincide outside the coreand moreover we have e w Λ0 (0) = e w (0) and | w Λ0 ( r = 0) | < ∞ . Although thisUV regularization does not change the GCPF Ξ [ ν ] we shall henceforth notethis functional Ξ Λ0 [ ν ] to emphasize that it depends functionally on w Λ0 . Forfurther convenience, we also need a special notation for the inverse of theregularized propagator w Λ0 which will be christened R Λ0 = [ w Λ0 ] − . We canthus rewrite (2) and (5) asΞ Λ0 [ ν ] = Tr (cid:20) exp (cid:18) b ρ · e w Λ0 · b ρ + b ρ · [ ν − w Λ0 (0) / (cid:19)(cid:21) , (10a)= 1 N w Λ0 Z D ϕ exp (cid:18) − ϕ · R Λ0 · ϕ + W (HS) (cid:2) ν − w Λ0 (0) / ϕ (cid:3)(cid:19) . (10b)Of course Ξ Λ0 [ ν ] ≡ Ξ [ ν ] as soon as Λ σ ≫ I prefer, for the convenience of the reader, to give here two lemma nec-essary for subsequent developments rather than to postpone them in an ap-pendix. Let ∆ some definite positive operator and F [ ϕ ] an arbitrary func-tional of real scalar field ϕ , then1 N ∆ Z D ϕ exp( − ϕ · ∆ − · ϕ ) F [ ϕ + ϕ ] = exp( D ) F ( ϕ ] , (11)where the operator D reads D . . . = 12 Z Ω d d x d d y ∆( x, y ) δ . . .δϕ ( x ) δϕ ( y ) . (12)An interesting application of lemma (11) is to reconsider equation (10b) asΞ Λ0 [ ν + w Λ0 (0) /
2] = e D Λ0 Ξ (HS) [ ν ] , (13)therefore the operator e D Λ0 constructs the full GCPF of the system from theGCPF of the reference HS system.The second lemma is sometimes referred to as Bogolioubov theorem : COARSE GRAINING i , 1 ≤ i ≤ n , be n definite positive operators and F [ ϕ ] an arbitraryfunctional of real scalar field ϕ , then Z n Y i =1 (cid:26) D ϕ i N ∆ i exp( − ϕ i · ∆ − i · ϕ i ) (cid:27) F ( n X i =1 ϕ i ) = Z D ϕ N ∆ exp( − ϕ · ∆ − · ϕ ) ×× F ( ϕ ) , (14)where ∆ = P ni =1 ∆ i .The two lemma (11) and (14) are easy consequences of Wick’s theorem. I now apply the exact RG approach of Tim Morris [10, 13] to our non-canonical field theory. As a consequence of Bogolioubov theorem (14) theGCPF Ξ Λ0 [ ν ] can be rewritten in terms of two propagators and two fields asΞ Λ0 [ ν ] = 1 N w k Z D ϕ < exp (cid:18) − ϕ < · R k · ϕ < (cid:19) Ξ Λ k (cid:2) ν − w k (0) / , ϕ < (cid:3) , (15a)Ξ Λ k [ ν, ϕ < ] = 1 N w Λ k Z D ϕ > exp (cid:18) − ϕ > · R Λ k · ϕ > ++ W (HS) (cid:20) ν − w Λ k (0)2 + ϕ < + ϕ > (cid:21)(cid:19) , (15b)where 0 ≤ k ≤ Λ is the running scale of the RG and where ϕ = ϕ < + ϕ > and w Λ0 = w Λ k + w k . (16)In (15)-(16) I have separated the field ϕ into “rapid” ( ϕ > ) and “slow” modes( ϕ < ). The low-energy modes are associated to the propagator w k (withinverse R k ) which is cut off from above by k , while the high-energy modes areassociated to the propagator w Λ k (with inverse R Λ k ) which is cut off from belowby k and from above by Λ. I demand that e w Λ k ( q ) = e w ( q )( C ( q, Λ) − C ( q, k ))should be positive which will be assumed henceforth. In the popular casewhere C ( q, Λ) = C ( q/ Λ) it is sufficient for the cut-off function C ( x ) to bea monotonous decreasing function of its argument. Some comments are inorder.(i) In order to establish equations (15) I used w Λ0 (0) = w Λ k (0) + w k (0) asimplied by (16). COARSE GRAINING ϕ < and ϕ > are or-dinary scalar fields, in particular they have a full spectrum of Fouriercomponents e ϕ < ( q ) and e ϕ > ( q ) (even the condition 0 ≤ q ≤ Λ is notcompulsory if the bare propagator is UV regularized). The cut-off atscale “k” acts only on the propagators.(iii) The functional Ξ Λ k [ ν, ϕ ] noted in this way by Morris [10] is the cruxof the whole matter since it allows, as I will show, to make explicitthe link between the Wilsonian action and the effective average action.However here this link proves trivial since, as apparent in formula (15b),Ξ Λ k [ ν, ϕ ], depends functionally only on the sole variable ν + ϕ .Let me first set ϕ < = 0 in (15b). It yieldsΞ Λ k [ ν, ϕ < = 0] , Ξ Λ k [ ν ] ( , exp (cid:0) W Λ k [ ν ] (cid:1) ) (17a)= 1 N w Λ k Z D ϕ exp (cid:18) − ϕ · R Λ k · ϕ + W (HS) (cid:2) ν − w Λ k (0) / ϕ (cid:3)(cid:19) . (17b)From the point of view of field theory this shows shows that W Λ k [ ν ] =ln Ξ Λ k [ ν ] is the Helmholtz free energy of the rapid modes ϕ > in the presence ofthe source ν ( x ); W Λ k is thus the generating functional of connected correlationfunctions with UV regularization (at Λ) and an infra-red (IR) cut-off ( i. e. at k). This functional is related by a Legendre transformation to the effectiveaverage action of Wetterich, as it willl be discussed in section 4.From the point of view of the theory of liquids clearly, (see e. g. equa-tion (5)) Ξ Λ k [ ν ] is precisely the GCPF of a system of hard spheres withadditional pairwise potentials w Λ k [ r ], i. e. , the k-system, to adopt theterminology of Reatto ‘et al.’ [6]. Therefore one also has :Ξ Λ k [ ν ] = Tr (cid:20) exp (cid:18) b ρ · e w Λ k · b ρ + b ρ · [ ν − w Λ k (0) / (cid:19)(cid:21) . (18)Note that equation (18) is valid provided the propagator e w Λ k ( x, y ) (pair po-tential) is definite positive (attractive) so that a Hubbard-Stratonovich trans-form is licit; that is why I imposed earlier that the cut-off function x → C ( x )should be a decreasing function. Comparing equations (15b) and (17b) onethus has synthetically Ξ Λ k [ ν, ϕ ] = Ξ Λ k [ ν + ϕ ] . (19)I now introduce the Wilsonian action S Λ k [ ϕ ] , − W Λ k [ ϕ ] , (20) RG FLOW EQUATIONS Λ0 (cid:2) ν + w k (0) / (cid:3) = 1 N w k Z D ϕ < exp (cid:18) − ϕ < · R k · ϕ < − S Λ k [ ν + ϕ < ] . (cid:19) (21)Indeed, in equation (21) S Λ k [ ϕ < ] play the role of the effective action of theslow modes at scale k [1, 2, 10]. Here k plays the role of an UV cut-off.Note that the functional identity S Λ k = − W Λ k is not true for a canonical fieldtheory [10].We end this section by applying lemma (11) to the 2 equations (15) whichcan thus be rewritten asΞ Λ0 [ ν + w k (0) /
2] = e D k Ξ Λ k [ ν ] (22a)Ξ Λ k [ ν + w Λ k (0) /
2] = e D Λ k Ξ (HS) [ ν ] (22b)from which it follows thatΞ Λ0 [ ν + w k (0) /
2] = e D k e D Λ k Ξ (HS) [ ν ] . (23)Comparing with equation (13) yields e D Λ0 = e D k e D Λ k . (24)This is the semi-group law of the RG. W Λ k From the definitions (16) when k → Λ then w Λ k ( r ) → k = Λ is a fluid of hard spheres. When k decreases from Λ to k = 0more and more Fourier components are included in the potential w Λ k ( r ) andfinally precisely at k = 0 the k-system coincides with the full model since w Λ k ( r ) tends to w Λ0 ( r ), i. e. essentially w ( r ) since both potentials differ onlyin the hard cores with no effect on the physics. In this section I establishthe equations which govern the flow of the grand potential W Λ k introduced inSec 3 and its Legendre transform. The flow equations of the Wilsonian action S Λ k follows trivially. Note that the formal solution of the not yet establishedflow equation for W Λ k is already known, it is given by equation (22b).The flow equation for W Λ k can be obtained in the framework of statisticalfield theory and, apart some tricks due to the self-energies, along essentially RG FLOW EQUATIONS ν ( x ). It gives readily ∂ k W Λ k [ ν ] ν = 12 Z Ω d d x d d y ∂ k w Λ k ( x, y ) (cid:8) G Λ k ( x, y ) − ρ Λ k ( x ) δ ( d ) ( x − y ) (cid:9) (25)where ρ Λ k ( x ) = < b ρ ( x |C ) > GC denotes the mean density of the k-system inthe GC ensemble and G Λ k ( x, y ) = < b ρ ( x |C ) b ρ ( y |C ) > GC its pair correlationfunction at chemical potential ν ( x ). Since one has ρ Λ k ( x ) = δW Λ k /δν ( x ) , W (1) Λ k ( x ) and also G ( T ) Λ k ( x, y ) = G ( T ) Λ k ( x, y ) − ρ Λ k ( x ) ρ Λ k ( y ) = δ (2) W Λ k /δν ( x ) δν ( y ) , W (2) Λ k ( x, y ) one can rewrite equation (25) under the closed form ∂ k W Λ k [ ν + w Λ k (0) / ν = 12 Z Ω d d x d d y ∂ k w Λ k ( x, y ) { W (2) Λ k ( x, y )+ W (1) Λ k ( x ) W (1) Λ k ( y ) } . (26)This partial differential equation (PDE) must be supplemented by theinitial condition W Λ k ≡ W (HS) . This equation is closed only superficiallysince the pair correlation function W (1) Λ k ( x ) and W (2) Λ k ( x, y ) depends func-tionally on the chemical potential ν ( x ). Differentiating functionally succes-sively both members of equation (26) with respect to the field ν ( x ) oneobtains a hierarchy, or tower, of equations for the W ( n ) Λ k ( x , . . . , x n ) = δ n W Λ k /δν ( x ) . . . δν ( x n ), i. e. the Green’s -or connected correlation- func-tions. S Λ k The flow equation (26) has the structure of Wilson-Polchinski’s equationfor the Wilsonian action; indeed with S Λ k = − W Λ k it takes the usual form[8, 9, 10] ∂ k S Λ k [ ϕ + w Λ k (0) / ϕ = 12 Z Ω d d x d d y ∂ k w Λ k ( x, y ) { S (2) Λ k ( x, y ) − S (1) Λ k ( x ) S (1) Λ k ( y ) } . (27)where S ( n ) Λ k ( x , . . . , x n ) = δ n S Λ k /δϕ ( x ) . . . δϕ ( x n ). This equation, whichmust be supplemented with the initial condition S ΛΛ = − W (HS) , does notappear to have yet been considered in the theory of liquids contrary to theabundant literature devoted to it in statistical field theory (see e. g. [9] andreferences quoted herein). RG FLOW EQUATIONS β A Λ k Since the seminal works of Reatto ‘et al.’ [3, 4, 5, 6] it is usual to considerrather the flow for the Kohn-Sham free energy A Λ k [ ρ ] of the k-system whichis defined as the Legendre transform of W Λ k [ ν ]. One has the usual couple ofrelations W Λ k [ ν ] = sup ρ n ν · ρ − β A Λ k [ ρ ] o ∀ ν , (28a) β A Λ k [ ρ ] = sup ν (cid:8) ρ · ν − W Λ k [ ν ] (cid:9) ∀ ρ . (28b)Obviously the flow equation for β A Λ k [ ρ ] should be deduced from that of W Λ k [ ν ]. Let me consider for instance equation (28a). I have for all νW Λ k [ ν ] = ρ ⋆ · ν − β A Λ k [ ρ ⋆ ] , (29)where ρ ⋆ ( x ) is, if it does exists, the unique solution of the implicit stationnaryequation ν = δβ A Λ k [ ρ ] δρ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ρ = ρ ⋆ (30)Differentiating (29) at fixed ν gives ∂ k W Λ k [ ν ] (cid:12)(cid:12) ν = ∂ k ρ ⋆ · ν − δβ A Λ k [ ρ ⋆ ] δρ ⋆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k · ∂ k ρ ⋆ − ∂ k β A Λ k [ ρ ⋆ ] (cid:12)(cid:12)(cid:12) ρ ⋆ , (31)which further simplifies thanks to the stationnary condition (30) with thefinal result : ∂ k A Λ k [ ρ ⋆ ] ρ ⋆ = − ∂ k W Λ k [ ν ] ν . (32)Similarly, starting instead from equation (28b) one finds that ∂ k A Λ k [ ρ ] ρ = − ∂ k W Λ k [ ν ⋆ ] ν ⋆ , (33)where ν ⋆ is the unique chemical potential which is, if it exists, solution ofequation (28b). I shall drop the subscript ” ⋆ ” in the sequel.At this point I introduce the direct correlation functions C ( n ) Λ k ( x , . . . , x n ) = − δ n A Λ k /δρ ( x ) . . . δρ ( x n ) , (34) RG FLOW EQUATIONS ρ ( x ). The C ( n ) Λ k ( x , . . . , x n )and the W ( n ) Λ k ( x , . . . , x n ) are linked by generalized Ornstein-Zernike (OZ)equations and in particular one has the usual OZ equation : W (2) Λ k ( x, y ) = − [ C (2) Λ k ] − ( x, y ) [6]. This last property allows to rewrite equation (32) in aclosed form involving A Λ k , its functional derivatives and the propagator. Onefinds : ∂ k A Λ k [ ρ ] (cid:12)(cid:12)(cid:12) ρ = − Z Ω d d x d d y ∂ k w k ( x, y )[ C (2) Λ k ] − ( x, y )+ 12 Z Ω d d x d d y ∂ k w k ( x, y ) ρ ( x ) ρ ( y ) − Z Ω d d x ∂ k w k ( x, x ) ρ ( x ) . (35)Note that, in order to obtain the result I made use of the trick ∂ k w k ( x, y ) = − ∂ k w Λ k ( x, y ). However, the resulting equation (35) is quite awkward and itis convenient, by adapting the ideas of Reatto ‘et al.’ [3, 4, 5, 6] to our case,to introduce a modified Kohn-Sham free energy : β A Λ k [ ρ ] = β A Λ k [ ρ ] − ρ · w k · ρ + 12 ρ · w k (0) . (36)This new functional was introduced by Wetterich ‘et al.’ [11] in statisticalfield theory under the name of “the effective average action” independentlyfrom Reatto ‘et al.’; it differs from β A Λ k [ ρ ] by a simple quadratic form andsatisfies obviously the following flow equation ∂ k A Λ k [ ρ ] = − Z Ω d d x d d y ∂ k w k ( x, y ) n C (2) Λ k − w k o − ( x, y )(37a) C ( n ) Λ k ( x , . . . , x n ) = − δ n A Λ k [ ρ ] δρ ( x ) . . . δρ ( x n ) (37b)I already obtained this equation in my first version of the RG group forliquids but with the help of a canonical field theory [19]. Note that the directcorrelation functions C ( n ) Λ k ( x , . . . , x n ) differ from the C ( n ) Λ k ( x , . . . , x n ) onlyfor n ≥
3; in particular one has C (2) Λ k ( x, y ) = C (2) Λ k ( x, y ) + w k ( x, y ) . (38)I stress that although β A Λ k [ ρ ] is a convex functional of the profile ρ ( x ) this isnot the case in general for β A Λ k [ ρ ] except of course at k = 0 since β A Λ k =0 [ ρ ] = RG FLOW EQUATIONS β A Λ k =0 [ ρ ] as apparent in formula (36). Indeed although C (2) Λ k < C (2) Λ k could become positive due tothe addition of the positive operator w k .Equation (37a) must be supplemented with an initial condition at k = Λ.From W ΛΛ = W (HS) as follows from equation (22b) and D ΛΛ = 0 one concludesthat β A ΛΛ = β A (HS) and thus from (36) β A ΛΛ [ ρ ] = β A (HS) [ ρ ] − ρ · w Λ0 · ρ + 12 ρ · w Λ0 (0) . (39)In KSSHE theory the initial conditions are identical but are obtained in amore complicated way which requires the divergence of the regulator at k = Λ[19].It turns out that the initial Kohn-Sham free energy of the k-system, i. e. at k = Λ, coincides with the mean-field Kohn-Sham free energy at any scale“k”, i. e. one has β A Λ (MF) k [ ρ ] = β A ΛΛ [ ρ ] for all k. Moreover this MF freeenergy constitutes a rigorous upper bound to the exact free energy and thus β A ΛΛ [ ρ ] ≥ β A Λ k [ ρ ]. This result already obtained in references [18, 19] for theKSSHE theory is proved for this new version of the theory in appendix Awhere a precise definition of the MF approximation is provided.For a homogeneous system β A Λ k [ ρ ] = V f Λ k ( ρ ) where β times the freeenergy per unit volume f Λ k ( ρ ) is a function (not a functional) of ρ and theflow equation for f Λ k ( ρ ) is easily deduced from equation (37a) and reads ∂ k f Λ k = − Z q ∂ k e w k ( q ) e C (2) Λ k ( q ) − e w k , (40)where R q ≡ R d d q/ (2 π ) d and e C (2) Λ k ( q ) is the Fourier transform of C (2) Λ k ( c, y ) ≡ C (2) Λ k ( r = k x − y k ) for a translationally invariant fluid. Note that the righthand side is non singular since the denominator is negative definite. It is ageneral property that the flow of f Λ k has neither UV or IR singularities, inparticular IR singularities (near a critical point) are smoothened by k andthey build up progressively as the scale-k is lowered[11, 12]. .To make contact with the work of Reatto ‘et al.’ let me consider now thecase of a sharp cut-off, i. e. C ( q, k ) = 1 − Θ( q − k ). Then equation (40)becomes [13]. ∂ k f Λ k = k d − S d (2 π ) d ln { − e w Λ0 ( k ) e C (2) Λ k ( k ) } , (41)where S d = 2 π d/ / Γ( d/ q − k ) by a smooth one, CONCLUSION ǫ ( q, k ), denotes δ ǫ ( q, k ) = − ∂ k Θ ǫ ( q, k ) the smooth δ -function and finallymake use of Morris lemma which states that, for ǫ → δ ǫ ( q, k ) f (Θ ǫ ( q, k ) , k ) → δ ( q − k ) Z dt f ( t, q ) , (42)provided that the function f (Θ ǫ ( q, k ) , k ) is continuous at k = q in the limit ǫ →
0, which is the case here. In the canonical field KSSHE theory for liquidsone needs to consider rather an ultra-sharp cut-off regulator [13].Whatever the considered version of HRT, smooth or sharp cut-off, i.e. respec. equations (40) and (41), both equations are of a formidablecomplexity despite their apparent simplicity. Indeed the kernels of these PDEinvolve the two-body C (2) Λ k [ ρ ] which depends functionally on the density.Clearly one can deduce from these equations an infinite tower of equationsfor the C ( n ) Λ k [ ρ ] (1 , . . . , n ) by differentiating them functionnaly with respectto the density profile ρ ( x ). A discussion of this point is however out the scopeof the present paper, it has been developed in detail in a general context inreference [13]. In this paper I have developed a new version of the “smooth” hierarchicalreference theory for liquids. It can be seen as an effort to conciliate the pointof view of the theory of liquids and that of statistical field theory. Technicallyit is an application to a peculiar field theory (KSSHE), aimed at representingliquids, of the more general theory developed in reference [13]. The equiv-alence pair potentials ↔ propagators is the crux of the whole matter. Inaddition, the interplay between the points of view of Wilson and Wetterichin their application to the theory of liquids yields a better understanding ofthe work of Reatto and his collaborators.From a prosaic point of view the differences with the first version ofthis work [19] are small but with interesting consequences. In practice, if oneleaves aside philosophical considerations on Wilsonian and Wetterich actions,the main changes concern essentially the expression of the pair potential w Λ k of the k-system. In the first version, following Wetterich, I add a mass term ϕ · R k · ϕ/ R k ( q ) ∼ Zk (1 − Θ ǫ ( q, k )), Z large. This results in an ugly pair potential for the k-system, of the form e w Λ k ( q ) = e w Λ0 ( q ) / (1 + e w Λ0 ( q ) e R k ( q )). In order to recover Reatto’s equation (41)one then needs to consider an ultra-sharp cut-off limit, i. e. : ǫ → Z → ∞ . I recall that, here, only the simple sharp cut-off limit ǫ → Z = 1) is required. CONCLUSION C ( x ) =exp( − x /
2) yields analytical w Λ k ( r ) in many cases, for instance w ( r ) = exp( − ar ) /r (Yukawa), w ( r ) = 1 /r d + n , etc. Indeed one then obtains for w Λ k ( r ) ≡ w Ewald ( r )nothing but the Ewald potential in direct space, the guy of numerical simula-tions. The lacking contribution in Fourier space corresponds to interactionsbetween “blocks” of size 1 /k . This potential w Ewald ( r ) is monotonous in gen-eral. In the old version the potential w Λ k ( r ) is not analytical and oscillates indirect space.A last difference can be emphasized : in the old version the initial condi-tion for the flow equation of β A Λ k requires that the cut-off function R Λ = ∞ ,which is difficult to implement rigorously in practice and could lead to unex-pected errors in numerical applications. In the present version, as discussedin section 4.3 there is no such problem and the initial condition is easilyimplemented.Despite the beauty and success of Reatto’s HRT some drawbacks of thetheory appeal a smooth cut-off version of it, such as the one exposed here.A sharp cut-off regulator induces singularities in the pair potential w Λ k ( r )which decays slowly as ∼ cos( kr ) /r for r → ∞ . This circumstance makes itdifficult to solve exactly the integral equations involved in the most widelyused closures of the Hierarchy; one has to resort to approximations which,as discussed by Reiner [27] are sometimes not easy to controll. A seconddrawback of the sharp cut-off version is the value of some critical exponents,notably that of the specific heat, α ∼ − .
05, which is negative, while positiveas it should in the smooth cut-off version [28].It seems to me that solving exactly the smooth cut-off version of HRT ispossible; that is coupling the numerical solution of a PDE (flow equation)and integral equations (closure of the hierarchy). Some of the drawbacks ofthe Reatto’s ‘et al’ version of HRT could then be cured.
Acknowledgments
This paper was written in honour of L. Reatto.
ALTERNATIVE DERIVATION OF THE RG FLOW OF W Λ K A Alternative derivation of the RG flow of W Λ k I derive the flow of W Λ k in the framework of field theory (see also reference[13]).I will start from equation (22b) L = R Ξ Λ k [ ν k = ν + w Λ k (0) /
2] = e D Λ k Ξ (HS) [ ν ] , (43)and take the partial derivatives of the both sides “ L ” and “ R ” of EQ. (43)with respect to the scale ”k”. One thus has for the l.h.s. ∂ k L | ν = ∂ k Ξ Λ k [ ν k ] + Z Ω d d x δ Ξ Λ k δν k ∂ k ν k , = Ξ Λ k [ ν k ] × { ∂ k W Λ k [ ν k ] + 12 Z Ω d d x W (1) Λ k [ ν k ]( x ) ∂w Λ k (0) / } , (44)and, for the r.h.s. ∂ k R | ν = ∂ k D Λ k e D Λ k Ξ (HS) [ ν ]= 12 Z Ω d d x d d y ∂ k w Λ k ( x, y ) δ Ξ Λ k [ ν k ] δν k ( x ) δν k ( y )= Ξ Λ k [ ν k ] × Z Ω d d x d d y ∂ k w Λ k ( x, y ) G Λ k [ ν k ]( x, y ) , (45)where I used the expression : D Λ k . . . = 12 Z Ω d d x d d y w Λ k ( x, y ) δ . . .δν k ( x ) δν k ( y ) . (46)Equating equations (44) and (45) and noting that the equality, obtained forarbitrary ν k , is then valid in the change ν k → ν , ν arbitrary, one obtainsfinally ∂ k W Λ k [ ν ] ν = 12 Z Ω d d x d d y ∂ k w Λ k ( x, y ) (cid:8) G Λ k ( x, y ) − ρ Λ k ( x ) δ ( d ) ( x − y ) (cid:9) , (47)which indeed coincides with equation(25). B Mean field theory at scale “k”
The Mean-Field approximation of the GCPF of the k-system will be de-fined as Ξ
Λ (MF) k [ ν ] , exp (cid:26) − ϕ ⋆ · R Λ k · ϕ ⋆ + W (HS) [ ν k + ϕ ⋆ ] (cid:27) , (48) MEAN FIELD THEORY AT SCALE “K” ν k = ν − w Λ k (0) / ϕ ⋆ is the location of the saddle point inte-grandl (17b). ϕ ⋆ satisfies the implicit relation R Λ k · ϕ ⋆ = δδϕ ⋆ W (HS) [ ν k + ϕ ⋆ ]= ρ (HS) [ ν k + ϕ ⋆ ]( x ) . (49)For a given chemical potential ν ( x ) the MF profile of the k-system is givenby ρ Λ (MF) k ( x ) = δ ln Ξ Λ (MF) k [ ν ] /δν ( x ) = ρ (HS) [ ν k + ϕ ⋆ ]( x ) where I made use ofthe stationary condition (49). Therefore the “true” Kohn-Sham free energyof the k-system is given by β A Λ (MF) k = − ln Ξ Λ (MF) k + ν · ρ Λ (MF) k . (50)A short calculation will show that [19, 13] β A Λ (MF) k [ ρ ] = β A (HS) [ ρ ] − ρ · w Λ k · ρ + 12 ρ · w Λ k (0) . (51)where β A (HS) [ ρ ] is the free energy functional of the HS fluid at same density.Therefore one finds for the average effective Kohn-Sham free energy β A Λ (MF) k [ ρ ] = β A (HS) [ ρ ] − ρ · w Λ0 · ρ + 12 ρ · w Λ0 (0) . (52)Note that β A Λ (MF) k [ ρ ] is independent of scale “k” and thus equal to its initialvalue β A Λ (MF)Λ [ ρ ].I prove now that β A Λ (MF) k [ ρ ] is a rigorous upper bound to the Kohn-Sham free energy of the k-system [26, 18]. We consider the Legendre-Fenchelrelation for the reference system of hard spheres : W (HS) [ ν ] = sup ρ (cid:8) ν · ρ − β A (HS) [ ρ ] (cid:9) ( ∀ ν ) (53)which implies the Young inequalities( ∀ ν , ∀ ρ ) W (HS) [ ν ] + β A (HS) [ ρ ] ≥ ρ · ν . (54)Injecting this inequality in the espression (17b) of Ξ Λ k [ ν ] one gets ( ∀ ν , ∀ ρ )Ξ Λ k [ ν ] ≥ exp( − β A (HS) [ ρ ] + ρ · ν k ) Z D ϕ N w Λ k exp( − ϕ · R Λ k · ϕ + ρ · ϕ ) ≥ exp( − β A (HS) [ ρ ] + ρ · ν k + 12 ρ · w Λ k · ρ ) , (55) MEAN FIELD THEORY AT SCALE “K” ∀ ν , ∀ ρ ) ρ · ν − W Λ k [ ν ] ≤ β A Λ (MF) k [ ρ ] , (56)and thus ( ∀ ρ ) β A Λ (MF) k [ ρ ] ≥ sup ν { ρ · ν − W Λ k [ ν ] } . (57)Hence, from the very definition of the Legendre transform (28b)( ∀ ρ ) β A Λ (MF) k [ ρ ] ≥ β A Λ k [ ρ ] (58)Turning now our attention to the effective average Kohn-Sham free energy Ifind the exact upper bound :( ∀ ρ ) β A Λ (MF) k [ ρ ] = β A ΛΛ [ ρ ] ≥ β A Λ k [ ρ ] . (59) EFERENCES References [1] K. G. Wilson and J. Kogut,
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