International audienceWe study the Coulombic criticality of ionic fluids within the restricted primitive model (RPM). We indicate that for the RPM, analysed in terms of the field of charge density, the corresponding Landau-Ginzburg-Wilson effective Hamiltonian has a negative $\varphi ^{4}$-coefficient. In that case, solving the ``exact'' renormalization group equation in the local potential approximation, we show that close initial Hamiltonians may lead either to a first order transition or to an Ising-like critical behavior, the partition being formed by the tri-critical surface. This situation apparently illustrates the theoretical wavering encountered in the literature concerning the nature of the Coulombic criticality. Nevertheless, it is most probable that, in terms of the field considered, the model does not display any criticality

Bagnuls, C.

Bervillier, C.

Brilliantov, N. V.

A field-theoretical approach to the restricted primitive model of charged hard spheres is proposed. It uses a scalar field q~, similar to that of the sine-Gordon theory of Coulomb gas, but allows taking into account the hard-core interactions. We observe that near criticality the effective Landau-Ginzburg-Wilson Hamiltonian obtained has a negative q4 coefficient. A non-perturbative RG analysis of such a Hamiltonian shows that this may lead either to a first-order transition or to an Ising-like critical behavior, the partition being formed by the tricritical surface. This may explain the theoretical wavering encountered in the literature relative to the nature of the Coulombic criticality. @ 1998 Published by Elsevier Science B.V. PACS: 61.2O.Qg; 05.4O.+j; 05.70.Jk; 64.60.Fr The Coulombic criticality is often studied theoreti-cally in terms of the restricted primitive model (RPM: equal number N+ and N _ of positive and negative hard spheres of equal diameter, d, with charges, fze, im-mersed in a structureless solvent of dielectric permit-tivity E; z2/e = 1 in what follows). Although there is general agreement on the existence of a liquid-gas-lik

N. V. Brilliantov A*b

C. Bagnuls

C. Bervillier D

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Peculiarity of the Coulombic criticality?

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Peculiarity of the Coulombic criticality ?N. V. Brilliantov, C. Bagnuls, C. BervillierTo cite this version:N. V. Brilliantov, C. Bagnuls, C. Bervillier. Peculiarity of the Coulombic criticality ?. PhysicsLetters A, Elsevier, 1998, 245, pp.274. <hal-00218242>HAL Id: hal-00218242https://hal.archives-ouvertes.fr/hal-00218242Submitted on 25 Jan 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Peculiarity of the Coulombic Criticality ?N. V. Brilliantov1,2, C. Bagnuls3 and C. Bervillier41Department of Chemistry, University ofToronto, Toronto, Canada M5S 3H62 Moscow State University, Physics Department,Moscow 119899, Russia3 Service de Physique de l’Etat Condense´C.E. Saclay, F91191 Gif-sur-Yvette cedex, France4 Service de Physique The´oriqueC.E. Saclay, F91191 Gif-sur-Yvette cedex, FranceJanuary 22, 1998AbstractWe study the Coulombic criticality of ionic fluids within the re-stricted primitive model (RPM).We indicate that for the RPM, analysedin terms of the field of charge density, the corresponding Landau-Ginzburg-Wilson effective Hamiltonian has a negative ϕ4-coefficient.In that case, solving the “exact” renormalization group equation inthe local potential approximation, we show that close initial Hamil-tonians may lead either to a first order transition or to an Ising-likecritical behavior, the partition being formed by the tri-critical sur-face. This situation apparently illustrates the theoretical waveringencountered in the literature concerning the nature of the Coulombiccriticality. Nevertheless, it is most probable that, in terms of the fieldconsidered, the model does not display any criticality.PACS numbers: 61.20.Qg,05.40.+j,05.70.Jk,64.60FrCoulombic criticality is often studied theoretically in terms of the Re-stricted Primitive Model (RPM: equal number N+ and N− of positive and1negative hard spheres of equal diameter, d, with charges, ±ze, immersed in astructureless solvent of dielectric permittivity ; z2/ = 1 in what follows). Ifone generally agrees on the existence of a liquid-gas-like transition at low con-centration and low temperature in the RPM, it is still not evident what kindof critical behavior the model actually possesses. Mean-field-like or Ising-likecritical behavior, crossover from mean-field to Ising behavior, tricriticalityor first order transition: all these conclusions have been expressed and dis-cussed [1]. In principle, a renormalization group (RG) treatment could yieldthe actual critical behavior of the model provided that an adequate Landau-Ginzburg-Wilson (LGW) effective Hamiltonian be known [2]. With this aimin view, we have used the Hubbard-Schofield method (previously applied tothe liquid-gas systems) [3] to get a LGW Hamiltonian for the RPM. The fieldarising for the RPM in this approach is conjugate to the field of charge-densityand corresponds to that of the Sine-Gordon representation of the Coulombgas model [4]. This field was used in studies of the metal-insulator transi-tion (see e.g. [5]) and of the RPM criticality [6]. Unfortunately, in all thesestudies the hard-core terms eventually disappeared, since the Sine-Gordontheory implies point particles. Irrespectively of that, there is a debate inparticular to know whether or not one can refer to tricriticality within theRPM [7, 8]. In this letter we show that the LGW Hamiltonian of the RPM,expressed in terms of the above field and with all the hard-core contributionstaken into account, is similar to that of the Ising-like systems but with a neg-ative ϕ4-coefficient u4. We then emphasize, for the first time, all the possiblescenari given by the RG analysis: an initial Hamiltonian with u4 < 0 maylead either to a first order transition or to an Ising-like critical behavior, thepartition being formed by the tri-critical surface. The large negative valuesfound for the Hamiltonian coefficients for the RPM (expressed in terms ofthe field, conjugate to the charge density), suggest that the transition so de-scribed would rather be of the first order. Let us summarize the main linesof the calculation of the effective Hamiltonian: the details will be publishedelsewhere [9].The Hamiltonian of the RPM (in units of kBT ) is a sum of the ideal part,Hid, hard-core part, Hhc, and the coulombic part, Hc:H = Hid +Hhc +12∑k′ν(k) (ρkρ−k − ρ) (1)2The coulombic part, Hc, in Eq.(1) is written in terms of the charge densityfluctuation amplitudes, ρk = nak−nbk, which are expressed in terms of the am-plitudes of density fluctuations of positive (a), nak= Ω−12∑N+j=1 exp{−ikraj},and negative (b), nbk= Ω−12∑N−j=1 exp{−ikrbj}, ions, ra,bj denote coordinatesof the particles, ρ = 2N+/Ω is the density, and Ω is the volume of the system.In Eq.(1) ν(k) ≡ (4pie2/kBT ) /k2, and the prime over the sum denotes thatterms with k = 0 are excluded.We write the free energy F of the RPM (in units of kBT ), as a sum,F = Fid + Fhc − log〈eHc〉hc(2)of the ideal part, Fid, the “direct” hard-core part, Fhc, and the coulombicpart, with Hc depending on the amplitudes, ρk; 〈· · ·〉hc denotes the aver-age over the reference hard-sphere system. Following the Hubbard-Schofieldscheme [3], we perform “Gaussian” transformation from variables ρk to vari-ables ϕk and after some algebra we arrive at the following result [10]F = − log[∫Dϕ(r)e−∫drHLGW(ϕ)]+ C (3)where the real field ϕ(r) is the Fourier transform of ϕk, C is a field indepen-dent constant (unimportant in the subsequent RG analysis) and HLGW(ϕ) isan effective LGW Hamiltonian which, discarding the derivatives of the field(local potential approximation), may be written asHLGW(ϕ) =12(∇ϕ(r))2 + V (ϕ(r)) + fid + fhc (4)with corresponding free energy densities, fid = Fid/Ω, fhc = Fhc/Ω and“potential function”:V (ϕ) =19pi2∞∑n=1b2n(2n)!u2nϕ2n (5)here b2 = 4pi (3pi)2/3 ρ∗1/3/T ∗ (ρ∗ = ρd3 is the reduced density and T ∗ =kBTd3/e2 is the reduced temperature). As for the ordinary fluid [3], the co-efficients u2n are expressed in terms of the cumulant averages, 〈ρk1 . . . ρkn〉c,hc,performed over the reference system having only hard—core interactions. Letus stress that the terms fid and fhc do not depend on the field ϕ, which3is conjugate to the charge-density fluctuations [11], and thus play a role ofadditive constants.It is straightforward to show that u2 = 1, and that neglecting in (5)all terms with n ≥ 2, (i.e. in the approximation V (ϕ) ∼ ϕ2), the usualDebye-Hu¨ckel result for the coulombic part of the free energy is obtained.Longer computations are required to find u2n for n ≥ 2. First we calculatethese quantities for the lattice—gas model and find that u2n = (−1)n+1 which,together with fid, yield the usual Sine-Gordon Hamiltonian for the RPM [12].To perform evaluation of the off—lattice coefficients u2n we use a symmetryof the RPM with respect to the hard-core interactions, and the definitions ofthe correlation functions, [13, 14]. Finally, we express these coefficients interms of the Fourier transforms (taken at zero wave—vectors) of the “cluster”functions of the reference hard—sphere system. In obvious notations [14]these read: h2(1, 2) ≡ g2(1, 2)−1, h3(1, 2, 3) ≡ g3(1, 2, 3)−g2(1, 2)−g2(1, 3)−g2(2, 3) + 2, etc., where gn(1, . . . , n) are n-particle correlation functions. Wethus conclude that the high-order terms in (5) mirror cluster-cluster (i.e.ion-dipole, dipole-dipole, etc.) interactions in the system. Using the relationbetween the gn+1 and gn [14],χρ2∂∂ρρngn = βρn[n gn + ρ∫drn+1 (gn+1 − gn)], (6)where χ = ρ−1∂ρ/∂P is the compressibility, we iteratively express the Fouriertransforms of hn at zero wave—vectors, h˜n(0), in terms of h˜n−1(0) and its den-sity derivative, and ultimately in terms of h˜2(0) and its density derivatives.Then we use the relation [14] ρh˜2(0) = ρkBTχ − 1 ≡ z0, and obtain co-efficients for the off—lattice effective Hamiltonian. In particular, u2 = 1,u4 = − (1 + 3z0), u6 = 1 + 15(z20 + z0z1 + z1), . . . where z1 ≡ ρ (∂z0/∂ρ) , . . .To obtain z0 one can use the virial expansion for the hard-sphere pressure,P/ρkBT = 1+∑kBkρk with the coefficients B1, . . . B6 known [13] (for smalldensities), or the Carnahan-Starling equation of state [13, 14]. Therefore,applying the above scheme, all the coefficients of the effective potential V (ϕ)may in principle be found.We have studied the density dependence of u2n up to 2n = 14 and ob-served that all the coefficients are negative in the density interval ∼ 0.07 ∼0.09 where the critical density of the RPM is expected to be. We have thenperformed an empirical analysis and found that the boundaries of the den-sity interval where the coefficients u2n are negative depend fairly linearly on41/n (see Fig. 1). Extrapolating this dependence we have found that all thecoefficients become positive for n > 22 (see Fig. 1). Being secure in theknowledge that the effective Hamiltonian is bounded from below, we canenvisage a RG analysis.We thus consider the “exact” RG equation in the local potential approx-imation [15] in three dimensions which, using the same notations as in [16],reads:f˙ =14pi2f ′′1 + f ′−12yf ′ +52f (7)in which y stands for the dimensionless field and f(y, l) = ∂V (y, l)/∂y, f ′ =∂f/∂y, f ′′ = ∂2f/∂y2, f˙ = ∂f/∂l with l the RG scale parameter (that relatestwo different “momentum” scales of reference such that Λl = e−lΛ0).Ideally, the question raised may be formulated as follows: considering theeffective Hamiltonian for the RPM at its assumed critical point (taken e.g.from Monte Carlo data [17, 18]) as an initial Hamiltonian (l = 0) for Eq.(7), will the solution of Eq. (7) flow toward the Wilson-Fisher (Ising) fixedpoint [the unique non-trivial fixed point of Eq.(7)] or not?Unfortunately, considering the function f(l = 0, y) which gives the initialconditions of the RPM for Eq.(7) (at ρ∗c = 0.0857, T∗c = 0.052 [17], orρ∗c = 0.080, T∗c = 0.0488 [18]), we have observed that the denominator1 + f ′(y) in (7) has singularities in the interval, 0.1 < y < 0.12. Moreover,Eq.(7) does not allow us to handle values of Hamiltonian coefficients as largeas those of Eq.(5).We have thus simply considered that the negative value of u4 found abovecould be a feature of qualitative importance for (some) ionic system. Thus weturn our attention to the solutions of Eq.(7) with initial functions involvingnegative values of u4.In [16] a detailed study of the approach to the Ising fixed point usingEq. (7) has been presented. However all the initial Hamiltonians consideredwere taken with u4 > 0. To our knowledge, the few studies based on RGtechniques that have, up to now, considered negative values of u4, eitherperturbatively [19] in three dimensions or, more recently, non-perturbatively[20] in four dimensions, have concluded that there is no stable fixed point (theIsing fixed point cannot be reached starting with u4 < 0) and thus to the lackof “true” criticality (there would be no divergent correlation length, like in afirst order transition). Let us show that these conclusions are not completely5true (in particular that the Ising fixed point can be reached starting withu4 < 0).To be short, we consider the following simple functions as initial con-ditions to Eq. (7) (a detailed study of the case u4 < 0 will be publishedelsewhere [21]):f(y, 0) = u2(0)y + u4(0)y3 + u6(0)y5 (8)corresponding to a point of coordinates (u2(0), u4(0), u6(0), 0, 0, · · ·) in thespace S of Hamiltonian coefficients (the dimension of S is infinite). Sincewe want to set u4(0) < 0, and at least one positive higher term is needed tomake the Hamiltonian bounded from below, we choose to set u6(0) positive.Having chosen a (negative) value for u4(0) and a (positive) value for u6(0),we use the “shooting” method [16] to determine the critical value uc2(0) ofu2(0) which brings f(y, 0) in the critical subspace Sc of S. The “shooting”method is based on the fact that, for sufficiently large values of l, the RGtrajectories go away from Sc in two opposite directions according to the signof u2(0)− uc2(0).Let us summarize our results by considering the case u4(0) = −6 as anexample (see Fig. 2).A If u6(0) = 16, we find 0.3836174 > uc2(0) > 0.3836151. The associated RGtrajectory goes away from the Gaussian fixed point PG and remains inthe sector u4 < 0 of Sc. Hence it never reaches the Ising fixed pointthat lies in the sector u4 > 0 . Instead the trajectory is attracted toa stable submanifold of dimension one (an infra-red stable trajectory)that, apparently, emerges from PG. Let us denote that trajectory byTu4<0. The lack of any fixed point ending Tu4<0 means that the corre-lation length remains finite at the assumed transition. This situationcould be compared to the fact that no heat—capacity divergency hasbeen observed in a Monte Carlo study of the RPM [17] or to theconclusion that “there is no criticality” [22] for the RPM expressed interms of the field of charge density.B If u6(0) = 20, we find 0.30131 > uc2(0) > 0.30122 and the associated RGtrajectory goes toward the Ising fixed point and approaches it alongthe usual renormalized trajectory associated with the continuum limitof the scalar field theory in three dimension usually called the ϕ43-field6theory. This renormalized trajectory (denoted by T1 in [16]) interpo-lates between the Gaussian and the Ising fixed points. Hence thereexist initial Hamiltonians with u4 < 0 that, nevertheless, belong to thebasin of attraction of the Ising-like fixed point.C Between the two preceding cases, we find a trajectory (with u6(0) =18.3125 · · · and 0.3324573 > uc2(0) > 0.3324549) that directly flowstowards PG (it is neither attracted to Tu4<0 nor to T1). That kind ofinitial Hamiltonian obtained by adjusting two coefficients (u2(0) andu6(0)) lies on the tri-critical subspace St of S. Any trajectory on Stapproaches PG along a unique (attractive) trajectory that imposes therequired very slow (logarithmic) flow in the vicinity of PG.So, it appears that for u4(0) < 0, very close Hamiltonians may leadto very different behaviors and this feature is due to the vicinity of thetricritical subspace. Hence, in any treatment of the RPM which uses thefield conjugate to the charge density, as an order parameter, it is justifiedto mention a possible vicinity of a tricritical point. The eventuality of anIsing behavior with a long crossover from an almost classical behavior isnot excluded. However the very large negative values that we have foundfor many coefficients of the effective LGW Hamiltonian suggest that thetransition studied with the use of the above order parameter correspondsrather to the case A described above (no criticality) in agreement with [22].Valuable discussions with John Valleau, his help and suggestions arehighly appreciated. Financial support of NSERC of Canada is acknowledged.References[1] For a review see: M. E. Fisher, J. Stat. Phys. 75, 1 (1994); J. Phys.:Cond. Matt., 8, 9103 (1996); G. Stell, J. Phys.: Cond. Matt., 8, 9329(1996); J.Stat.Phys., 78, 197 (1995).[2] Y. Levin and M. E. Fisher, Physica A225, 164 (1996).[3] J. Hubbard and P. Schofield, Phys. Lett., A40, 245 (1972).[4] J. Glimm and A. Jaffe, Quantum Physics (Springer-Verlag, New York,1981).7[5] Y.Saito, Progr.Theor.Phys. 62, 927 (1979).[6] A. L. Kholodenko and A. L. Beyerlein, Phys. Rev. A34, 3309 (1986).[7] M. E. Fisher, J. Stat. Phys. 75, 1 (1994).[8] A. L. Kholodenko and A. L. Beyerlein, Phys. Lett. A175, 366 (1993).[9] N. V. Brilliantov, J. P.Valleau, in preparation.[10] For convenience we rescale the length and the field to get a unit ultravi-olet cutoff for the wave vectors and 1/2 for the coefficient at the gradientterm. Coefficients appearing at this rescaling as well as coefficients fromthe transformation {ϕk} → ϕ(r) are adsorbed in constant C.[11] An attempt has been done by M.E.Fisher and B.P.Lee, Phys.Rev.Lett.,77, 3561 (1996), to derive an effective LGW Hamiltonian for the RPMwith the field ϕ corresponding to the overall-density fluctuations (andalso with the coefficients at ϕn determined from approximate theorieswith the ion-dipole and dipole-dipole interactions neglected).[12] Some (unimportant) difference of our result from the standard [4] ap-pears since the present calculations are performed for the canonical en-semble.[13] J. A.Barker and D. Henderson, Rev. Mod. Phys., 48, 587 (1976).[14] C. G.Gray and K. E.Gubbins, Theory of molecular fluids, ClarendonPress, Oxford, 1984.[15] A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B270 [FS16], 687 (1986).[16] C. Bagnuls and C. Bervillier, J. Phys. Stud. 1, 366 (1997); see also Phys.Rev. B41, 402 (1990).[17] J. P.Valleau and G. M.Torrie, J.Chem.Phys., submitted.[18] J. M.Caillol, D. Levesque and J. J. Weis, J. Chem. Phys., 107, 1565(1997)[19] E. K.Riedel and F. J. Wegner, Phys. Rev. Lett., 29, 349 (1972); Phys.Rev. B9, 294 (1974); A. I.Sokolov, Sov. Phys. JETP, 50, 802 (1979).8[20] K. Langfeld and H. Reinhardt, “Asymptotic freedom in a scalar fieldtheory on the lattice”, UNITU-THEP-18/1996 hep-ph/9702271.[21] C. Bagnuls and C. Bervillier, in preparation.[22] J. Stafiej and J. P. Badiali, J. Chem. Phys. 106, 8579 (1997).FIGURE CAPTIONSFigure 1 Dependence of the boundaries of the density interval where thecoefficients u2n of the effective Hamiltonian are negative as a functionof 1/n. Extrapolation suggests that all the coefficients with n > 22are positive. (Note that the density at which the “negative” intervalshrinks to zero, ρ = 0.0856 is very close to the critical density from theMonte Carlo data [17, 18]).Figure 2 Projection onto the plane {u2, u4} of various RG trajectories (inthe critical subspace Sc) obtained by solving Eq. (7). Black circlesrepresent the Gaussian (PG) and Ising (IFP) fixed points. The ar-rows indicate the directions of the RG flows on the trajectories. Theideal trajectory (dot line) which interpolates between these two fixedpoints represents an (attractive) infra-red stable trajectory (IRST) cor-responding to the so-called φ43 renormalized field theory in three dimen-sions (usual IRST for u4 > 0). White circles represent the projectionsonto the plane of initial critical Hamiltonians. For u4(0) > 0, the ef-fective Hamiltonians run toward the Ising fixed point asymptoticallyalong the usual IRST (simple fluid). Instead, for u4(0) < 0 and accord-ing to the values of Hamiltonian coefficients of higher order (u6, u8,etc.), the RG trajectories either (A) meet an endless IRST emergingfrom PG (dashed curve) and lying entirely in the sector u4 < 0 or (B)meet the usual IRST to reach the Ising fixed point. The frontier whichseparates these two very different cases (A and B) corresponds to initialHamiltonians lying on the tri-critical subspace (white square C) thatare source of RG trajectories flowing toward PG asymptotically alongthe tricritical IRST. Notice that the coincidence of the initial point Bwith the RG trajectory starting at point A is not real (it is accidentallydue to the projection onto a plane of the trajectories lying in a space ofinfinite dimension). The restricted primitive model, expressed in terms9of the field of charge density would correspond rather to the case Acharacterized by the lack of criticality (no fixed point may be reached).10

Peculiarity of the Coulombic criticality ?

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Peculiarity of the Coulombic criticality ?

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