Absence of Debye Screening in the QuantumCoulomb

We present an approximation to the quantum Coulomb plasma at equilibrium which captures the power-law violations of Debye screening which have been reported in recent papers. The objectives are: (1) to produce a simpler model which we will study in forthcoming papers, (2) to develop a strategy by which the absence of screening can be proven for the low density quantum Coulomb plasma itself. 1 The Classical Coulomb Gas The partition function for a (charge symmetric) classical Coulomb gas in three dimensions is Z =X ~ zN N ! Z dNp dN e H (1) where = (x; e) and d unites an integral over x 2 container with a sum over charges e = 1.H =X p2i 2m + 12 Z vl + i Z ext (2) where we de ne the charge density observable by (x) =X ei (x xi) (3) so that Z vl =X eiejvl(xi xj): (4) i ext is an external eld. We put in the strange factor i because it will lead to simpler expressions when we explain the sine-Gordon transformation. vl(x y) =\1r". We have put the quotes around the 1=r because it is necessary to place a cuto on the singularity of the Coulomb potential at short distances in order to have a stable interaction. This cuto will be characterised by a length l. In particular vl(0) = 1l . Having enforced a cuto the self energies of the particles are nite and we have included them in the interaction energy. The natural choice for this length l is the thermal wavelength which is the size of the typical one particle wavefunction in a corresponding quantum ideal gas l = s h2 m (5) since it is the Pauli exclusion principle and quantum mechanics that give rise to a stable system which we are approximating classically. The other lengths which naturally arise are and the Debye length lD = 1 p2z (6) where z = ~ z Z dpe 2m p2e 2l : (7) 1 z is the physical activity in the sense that the expectation of the density of particles is asymptotic to 2z as z ! 0. The factor e =l accounts for the inclusion of the self{energies in the interaction. For this system the following theorem has been proved [4], [5], [9], [10]. Theorem 1 For zl3 e 2l ; zl3 D 1; (8) all charge charge correlations decay exponentially, i.e., there are constants C1 and L > 0 such that jh (x) (y)ij C1e jx yj L (9) and higher truncated charge correlations decay exponentially as the length of the shortest tree on the positions of the observables. Also L ' lD when zl3 and zl3 D are as in the theorem. 2 Discussion In [5], p.428, it was claimed that screening of observables in the sense of exponential decay as in the theorem above will not hold for the quantum plasma. The argument was strengthened by some lower bounds (but only on time-dependent observables) given by Brydges and Seiler [3]. Since then Alastuey and Martin [1], [2] have made detailed calculations which state that within perturbation theory (the WignerKirkwood expansion) screening is destroyed by e ects due to diagrams with power-law decay at order h4 and higher. They show, for example, that for NaCl ions in water at room temperature there will be screening out to about 60 Debye lengths at which point there is a cross-over to a power law tail. According to their analysis the typical power-law is r 6 but it can be higher depending on the correlation and the system. This violation of screening has nothing to do with statistics. It is similar in mechanism to Van der Waals forces, but it occurs [2] even for one component plasmas in which there are no atoms or molecules. Similar comments appear in a paper by Ashcroft et al. [11]. In this paper we exhibit an appproximation to the quantum Coulomb plasma that captures the mechanism by which quantum uctuations destroy 2 the screening. The present paper will motivate conclusions which we will obtain by a complete mathematical analysis of this model to appear shortly [6]. The approximations we present are a possible strategy by which the conclusions of Alastuey and Martin could be established nonperturbatively, but this appears to be an unreasonably lengthy enterprise at the moment. To motivate the choice of our model we rst review some aspects of the proof of screening in the classical case. We introduce the Gaussian measure d 1 vl( ) on functions (x) which by de nition satis es e 2 R vl = Z d 1 vl( )e i R (10) If l were zero, no cuto , then formally d 1 v0( ) = D[ ]e 2 R (r )2 : (11) By substituting (10) into the partition functon (1) and interchanging the integral over d with theP and R dp d we are led to the well known sine{Gordon representation of the partition function. This represents the interacting gas as a superposition over all external elds of ideal gas partition functions for particles in external elds, Z = Z d 1 vl( )Zideal(i + i ext) (12) Zideal(i ) = e~ z R dp d e h(i ) (13) = e2ze 2l R dx cos (x) (14) where h(i ) = p2 2m + ie : (15) It is tempting to make the approximation cos ' 1 1 2 2 2 but this is not quite right in cases where =l 1. Instead the rst step in the proof of Theorem 1 is to integrate out uctuations of the eld on all scales up to the Debye length lD. Under the hypotheses of Theorem 1 this is done (exactly) by a Mayer expansion which is convergent because the hypotheses say that the plasma inside a Debye sphere is close to an ideal gas. The result 3 is that the short distance cuto l in the Gaussian measure d 1 vl in (12) is replaced by lD while the exponent 2ze 2l R dx cos (x) becomes a convergent series of non-local monomials in exp [ie (x)] but this series is still dominated by the leading term which is local and has the form 2z R dx cos (x). In other words, the e ect of a renormalization group transformation is, to a controllable approximation, to replace vl by v vlD (16) in the measure and to drop the constant e 2l . In fact there are also renormalizations of parameters, e.g. the dominant term is prefaced by a constant which tends to one as zl3e 2l ! 0, zl3 D ! 1 but we shall pretend these are not there throughout this paper. Choice of units of length: Set lD = 1: With this choice of units the hypotheses of Theorem 1 imply that 1 and 2z = 1 : (17) Having removed all scales up to lD = 1 the next step in the proof of Theorem 1 is to control the approximation 1 cos ' 1 1 2 2 2 +O( 4): (18) Within this approximation the partition function becomes, up to constants which cancel in correlations, a Gaussian integral Z ' Z d 1 v e 12 R ( + ext)2 (19) Z d 1 v e 12 R 2 e 1 2 R ext[1 u] ext (20) where u is the exponentially decaying kernel of u (v 1 + 1) 1 (21) in terms of which one can compute correlations of charge observables by functional derivatives with respect to ext and obtain the results of Debye{ H uckel theory, in particular exponential decay of correlations. 1actually e 1 cos (x) Pn e 2 [ 2 n]2 4 3 The Quantum coulomb Gas Now we turn to the analogous representation in the quantum case. For simplicity we discuss the case of Boltzmann statistics, but there are similar representations for Fermion and Bose statistics. This simpli cation is reasonable since we are discussing a regime in which the gas is very far from degenerate, l interparticle distance by the hypotheses of the theorem. The correct way to take into account the failure of commutativity [p; x] 6= 0 is to replace R dp d by the trace over the one particle Hilbert space and use time-ordered exponentials in Zideal(i ) so that Zideal(i ) = e~ zTr(e R 0 d h(i )) (22) where is now an imaginary-time dependent external eld ( ; x) which is integrated over using the Gaussian measure d vl I whose covariance is vl(x y) ( ). With these substitutions in (12) the sine{Gordon representation (12) is still valid. To understand this, set ext = 0, and consider P ~ zN N !TrNe H where H is the many body quantum Hamiltonian obtained from (2) by p! hi @ @x and TrN is the N body trace. Then TrNe H = TrN lim n!1 n Y1 he nH0e 2n R vl i : (23) We use the representation (10) for each factor of e 2n R vl , each requiring its own auxiliary eld i(x), i = 1; : : : ; n. Then TrNe H = lim n!1 Z dn n vl( )TrN n Y1 he nH0e n i R j i (24) = lim n!1 Z dn n vl( )TrNe R 0 d [H0+i R ] (25) where the exponential is time ordered and the collection of elds j(x) is united into one time dependent eld ( ; x) j(x) when 2 [ (j 1) n ; j n ). Finally we note that the trace over the many body Hilbert space factors into a product of one body traces so that X ~ zN N !TrNe H = Z d vl I( )Zideal(i + i ext) (26) 5 where we have put the external eld back in. We write R d vl I but we mean limn!1 R dn ( ). Following [8] Zideal(i ) can be written as a sum over all continuous closed paths X( ); 2 [0; ], using the FeynmanKac formula, Tr(e R 0 d h(i )) =Xe Z dW (X)e ieR 0 d ( ;X( )): (27) dW is the Wiener measure associated with the kernel of exp[t( h2 2m )]. The combination of (12) and (27) is a representation for the quantum partition function which appears in [7]. It is also derived and used in [2]. Notice that there is a Goldstone mode: ( ; x)! ( ; x) + f( ) where f is any function such that R 0 d f( ) = 0. This will be the origin of the long range forces. The intuition is that the Feynman-Kac formula represents the quantum gas as a classical gas of closed charge loops with instantaneous Coulomb interactions. Each loop represents the quantum uncertainty around a classical position. This leads to a time-dependent dipole force superimposed on the Coulomb force for the classical system. A dipole can polarise other dipoles leading to induced dipole-dipole or multipole-multipole forces which are power laws. The standard textbook discussions do not see this e ect because they make a static approximation which loses these time dependences. The mechanism is very similar to the Van der Waals forces, except that it takes place without any need for neutral objects such as atoms or molecules. To bypass some terrible technicalities we now alter the Wiener measure to obtain a simple model which exhibits destruction of screening by the same mechanism that we claim will occur in the complete model. 4 The Semiquantum Simpli cation We replace the integration R dW over all Wiener paths by a new integration concentrated on just one kind of path which oscillates about the initial point by a distance O(l) (the size of the wavepacket) in a random direction: let d (~e) be a spherically symmetric measure on vectors ~e. Then dW ! dx d (~e): (28) 6 The right hand side is a measure on paths because (x;~e) labels the path: X( ) = x+ l~ef( ) (29) f( ) = sin [2  ]: We dont claim that this is a controllable approximation in the sense that there is a physically natural parameter that can be driven to some limit to obtain it, but it is one of the simplest ways to put a little quantum mechanics into a classical model. We shall choose d (~e) = 1 2[ (~e) + (2 ) 32 e k~ek2 2 ]: (30) This choice perhaps would look more natural if there were no delta function: the delta function has the interpretation that half our particles are truly classical whilst the other half are semiquantum. The choice of proportions is not essential, indeed one could allow all the particles to be semiquantum but the resulting model is harder to analyse rigorously. We now make some more changes, but these ones, we claim, are on a di erent footing from the last change. They are attempts to extract an e ective Lagrangian which, we believe, can be justi ed by rigorous mathematics. Approximation 1 Z 0 d ( ;X( )) = Z 0 d ( ;X(0)) + Z 0 d Z t= t=0 r ( ;X(t)) dX(t) (31) = ([0; ]; x) + Z t= t=0 r ([t; ]; X(t)) dX(t) ([0; ]; x) + Z t= t=0 r ([t; ]; x) dX(t) (32) where r acts on the spatial variables and ([t; ]; x) Z t d ( ; x): (33) The consequence of these approximations is that the dependence of Zideal(i + i ext) on ( ; x) is only through 1(x) 1 p ([0; ]; x) 7 2(x) s 2 Z 0 d ( ; x)f( ): In fact, in terms of these elds we nd by the integration by parts R t= t=0 r ([t; ]; x) dX(t) = R 0 dtr (t; x) [X(t) x] that Z 0 d ( ;X( )) q 1(x) +s 2 l~e r 2(x): (34) Since the elds 1, 2 are Gaussian and R d vl I i(x) j(y) = vl(x y) ij, 1, 2 are independently distributed according to the massless Gaussian measure d vl encountered above in the classical model. Thus the partition function becomes Z = Z d vl( 2) Z d vl( 1) exp[2~ z Z dx Z d (~e) cos (q 1(x) +q ext +s 2 l~e r 2)]: (35) Notice that if d (~e) is set to (~e) we revert to the classical Coulomb gas. If 1 is set to zero then by reversing the Sine-Gordon transformation we obtain the partition function of a classical dipole gas with dipole moments ~e distributed according to d . Approximation 2 This is the same step as discussed above for the classical model in which the uctuations on scales up to lD = 1 are integrated out by a Mayer expansion, of which we keep only the leading term. Thus vl becomes v1 v and 2~ z becomes 1 . Z Z d v( 2) Z d v( 1) exp[ 1 Z dx Z d (~e) cos (q 1(x) +q ext +s 2~e lr 2)]: (36) The integration over d (~e) can be performed explicitly and the partition function becomes Z = Z d v( 2) Z d v( 1) exp[ 1 Z dxw2 cos (q 1(x) +q ext)] w2(x) 12[1 + e (lr 2(x))2 ]: (37) 8 The next approximation is of the same nature as the quadratic approximation of the cosine used to prove Theorem 1. We have by the hypotheses of Theorem 1 that 1 so that Approximation 3 1 cos (q 1 +q ext) ' 1 1 2( 1 + ext)2: Now we can integrate out the 1 eld: let uw2(x; y) kernel of the operator [v 1 + w2] 1: (38) Then Z d v( 1) e 1 2 R dxw2( 1+ ext)2 = Z d v( 1) e 1 2 R dxw2 21 e 1 2 R ext[w2 uw2 ] ext: (39) 5 Conclusions These approximations have led us to the following model of the quantum Coulomb gas: Z = Z d v( 2)e 1 R dxw2 Z d v( 1) e 1 2 R dxw2 21 e 1 2 R ext[w2 uw2 ] ext: (40) We will give a complete nonperturbative analysis of this model in a forthcoming paper [6]. Let expectations of static charge densities be obtained by functional derivatives h r Y i=1 (xi)i " 1 Z r Y i=1  ext (xi)Z( ext)# ext=0 : (41) Here are the conclusions we expect: 1. The two point correlations decay exponentially: h (x1) (x2)i Const: u(x1 x2) (42) as jx1 x2j ! 1. u is given in (21). 9 2. The higher correlations do not decay exponentially, for example:h (x) ( x) (x + y) ( x+ y)ih (x) ( x)ih (x + y) ( x+ y)iConst:[ 1u( x)u(x)]2[ l2]2 1jyj6(43)as y !1 with jxj large.Justi cation Notice that in equation (40) the kernel uw2(x; y) decaysuniformly in w2 because w2(x) is smooth in x and w2(x)12 for all 2 2.Therefore functional derivatives with respect to ext are linked in pairs byexponentially decaying propagators uw2. However the propagators uw2 de-pend on the eld r 2 through w2. This eld is distributed according to amassless Gaussian measure d v with a small perturbation by the termsZ d v( 1) e 12 R dxw221eR [O(1)+O(1) (lr2(x))2] dxe 1R dxw2eR [ 1+O(1)(lr 2))2] dx:(44)Since the perturbation is a function of r 2 it will not make the measuremassive.Comments: It is an artifact of this particular approximation and chargesymmetry that the action is separately even in 1 and 2, which is the reasonfor the di erent types of decay. Alastuey and Martin [2] also found thatthere are di erent decay rates for the correlations of two versus four chargeobservables, but the di erences were in the exponent of the power law ratherthan the drastic exponential versus power law that we obtain.If in equation (35) the two elds 1; 2 were the same eld then thatmodel would have exponential decay in all correlations. This would be aclassical system consisting of particles which carry both a charge and a smalldipole moment.The quadratic approximationcos (q 1 +q ext +s 2~e lr 2) ' 1 2 ( 1 + ext+s12~e lr2)22This is why we decided to put a function into the measure d10 in equation (36) would lose some higher order terms which are responsiblefor the destruction of the screening.For the full quantum Coulomb partition function (27) equation (31) isreplaced by:Z 0 d ( ;X( )) = ([0; ]; x) + Z t=t=0 r ([t; ]; X(t)) dX(t) +Z0 dtl2 ([t; ]; X(t)):(45)The extra term arises by the Ito calculus dX(t)2 = dtl2 and the dX(t) integralis an Ito integral. This formula has the good feature that the very singulareld ( ; x) which is a white noise in its dependence on has been tradedin for the continuous eld ([t; ]; x). It is possible that a nonperturbativeproof of the absence of screening for the quantum Coulomb system can beconstructed by integrating out the (massive) time averaged eld 1([0; ]; x)as we did (approximately) in obtaining the model (40).ACKNOWLEDGEMENTWe would like to thank Erhard Seiler and Philippe Martin for many help-ful conversations. To our knowledge the statement that Coulomb quantumplasma have these particular power law corrections to screening was rstmade by Paul Federbush.References[1] A. Alastuey and Ph. A. Martin. Absence of exponential clustering forthe static quantum correlations and the time-displaced correlations incharged uids. Europhys. Lett., 6:385{390, 1988.[2] A. Alastuey and Ph. A. Martin. Absence of exponential clustering inquantum Coulomb uids. Phys. Rev. A, 40:6485{6520, 1989.[3] D. C. Brydges and Seiler E. Absence of screening in certain lattice gaugeand plasma models. J. Stat. Phys., 42:405{424, 1986.11 [4] D. C. Brydges and P. Federbush. Debye screening. 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Frohlich, editor, Scaling and Self Similarity in Physics.Renormalization in Statistical Mechanics and Dynamics. Birkhauser,Boston, 1983.[11] A.C. Maggs and N.W. Ashcroft. Electronic uctuation and cohesion inmetals. Phys. Rev. Lett., 59:113{116, 1987.12