Correlations and sum rules in a half-space for a quantum two-dimensional one-component plasma
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Correlations and sum rules in a half-space for aquantum two-dimensional one-component plasma
B. Jancovici † and L. ˇSamaj †‡ † Laboratoire de Physique Th´eorique, Universit´e Paris-Sud (Unit´e Mixte deRecherche no. 8627 - CNRS), 91405 Orsay, France ‡ Institute of Physics, Slovak Academy of Sciences, D´ubravsk´a cesta 9, 845 11Bratislava, SlovakiaE-mail:
[email protected] , [email protected]
Abstract.
This paper is the continuation of a previous one [L. ˇSamaj and B.Jancovici, 2007
J. Stat. Mech.
P02002]; for a nearly classical quantum fluid in ahalf-space bounded by a plain plane hard wall (no image forces), we had generalizedthe Wigner-Kirkwood expansion of the equilibrium statistical quantities in powers ofPlanck’s constant ¯ h . As a model system for a more detailed study, we consider thequantum two-dimensional one-component plasma: a system of charged particles of onespecies, interacting through the logarithmic Coulomb potential in two dimensions, ina uniformly charged background of opposite sign, such that the total charge vanishes.The corresponding classical system is exactly solvable in a variety of geometries,including the present one of a half-plane, when βe = 2, where β is the inversetemperature and e is the charge of a particle: all the classical n -body densities areknown. In the present paper, we have calculated the expansions of the quantumdensity profile and truncated two-body density up to order ¯ h (instead of only to order¯ h in the previous paper). These expansions involve the classical n -body densities up to n = 4, thus we obtain exact expressions for these quantum expansions in this specialcase.For the quantum one-component plasma, two sum rules involving the truncatedtwo-body density (and, for one of them, the density profile) have been derived, a longtime ago, by heuristic macroscopic arguments: one sum rule is about the asymptoticform along the wall of the truncated two-body density, the other one is about thedipole moment of the structure factor. In the two-dimensional case at βe = 2, wehave now explicit expressions up to order ¯ h of these two quantum densities, thuswe can microscopically check the sum rules at this order. The checks are positive,reinforcing the idea that the sum rules are correct.PACS numbers: 05.30.-d, 03.65.Sq, 52.25.Kn, 05.70.Np Keywords:
Charged fluids (Theory) orrelations and sum rules in a half-space quantum plasma
1. Introduction
The model under consideration is the two-dimensional (2D) one-component plasma (alsocalled jellium). This model consists of one species of charged particles in a plane. Eachparticle has a charge e and a mass m . Two particles, at a distance r from each other,interact through the 2D Coulomb interaction v ( r ). This interaction is determined by the2D Poisson equation r v ( r ) = − πe δ ( r ), the solution of which is v ( r ) = − e ln( r/r ),where r is an arbitrary length which only fixes the zero of this potential. In addition,there is a charged uniform background of charge density opposite to the particle charge,so that the total system is neutral.When the inverse temperature β is such that the dimensionless coupling constant βe = 2, the equilibrium statistical mechanics of the classical (i.e. non-quantum) systemis completely solvable in a variety of geometries, in particular when the background andparticles are confined into a half-space by an impenetrable rectilinear plain hard wall(there are no image forces): all the classical n -body densities are known [1, 2].In its three-dimensional version, the quantum one-component plasma is not onlyof academic interest. It has been used, in first approximation, as a model for theelectrons of a metal [3]. In the present geometry of a half-space, this model might beused for describing what happens near the surface of the metal. Since the logarithmicinteraction v ( r ) is the Coulomb potential in two dimensions, the 2D one-componentplasma is expected to have general features which mimic those of the 3D one.Under certain conditions, the equilibrium properties of an infinite quantum fluid,in the nearly classical regime, can be expanded in powers of Planck’s constant ¯ h : thisis the Wigner-Kirkwood expansion [4, 5]. In a previous paper [6], we have generalizedthe Wigner-Kirkwood expansion to the case of a quantum fluid occupying a half space.We have obtained expressions for the first quantum correction of order ¯ h to the densityprofile and to the two-body density, in terms of some n -body densities of the classicalfluid. In section 3 of the present paper we extend these calculations to order ¯ h . Actually,instead of ¯ h , we use the thermal de Broglie wavelength proportional to it, λ = ¯ h q β/m .For the quantum one-component plasma in a half space, by heuristic macroscopicmethods, some sum rules involving the one-body density profile and the two-bodydensity have been derived. The purpose of the present paper is to check these sumrules at order ¯ h , in the special case of the 2D one-component plasma at βe = 2, usingthe generalized Wigner-Kirkwood expansion, which is a microscopic approach.The paper is organized as follows. Section 2 brings a recapitulation of the methodfor constructing the expansion of the quantum Boltzmann density in configuration ~ r -space for fluids constrained to a half-space [6]. This expansion is subsequently used insection 3 to compute the quantum one-body profile and two-body density to order ¯ h for the studied 2D one-component plasma. In section 4, three sum rules are reviewed:a perfect screening rule, a sum rule about the asymptotic form of the two-body densityalong the wall, and a dipole sum rule; their expansions to order ¯ h are given. In section5, some classical n -body correlation functions, which are needed, are studied. In section orrelations and sum rules in a half-space quantum plasma
36, the perfect screening sum rule is shown to hold, at order ¯ h , for the quantum 2D one-component plasma at β = 2. In sections 7 and 8, the same is done for the asymptoticform and the dipole sum rules, respectively. Section 9 is a Conclusion.
2. Boltzmann density for the half-space geometry
We first consider a general quantum system of N identical particles j = 1 , , . . . , N ofmass m , formulated in ν space dimensions. Particle position vectors r , r , . . . , r N areconfined to the half-space Λ defined by Cartesian coordinates r = ( x > , r ⊥ ), where r ⊥ ∈ R ν − denotes the set of ( ν −
1) unbounded coordinates normal to x . As usual,we start with a finite N and a finite volume | Λ | , and later we take the thermodynamiclimit N and | Λ | going to infinity (when Λ becomes a half space) with a finite meannumber density n = N/ | Λ | . The hard wall in the complementary half-space ¯Λ of points r = ( x < , r ⊥ ) is considered to be impenetrable to particles, i.e. the wavefunctionsof the particle system vanish as soon as one of the particles lies at the wall. For thesake of brevity, we denote the νN -dimensional position vector in configuration space by ~ r = ( r , r , . . . , r N ) and the corresponding gradient by ~ r = ( r , r , . . . , r N ). In theabsence of a magnetic field, the Hamiltonian of the particle system is given by H = 12 m (cid:16) − i¯ h ~ r (cid:17) + V ( ~ r ) , (2.1)where ¯ h stands for Planck’s constant and V ( ~ r ) is the total interaction potential.For infinite (bulk) quantum fluids of particles interacting via pairwise sufficientlysmooth interactions with neglected fermion/boson exchange effects, Wigner [4] andKirkwood [5] constructed a semiclassical expansion of the Boltzmann density inconfiguration space (at inverse temperature β ), B β ( ~ r ) = h ~ r | e − βH | ~ r i , in even powersof the thermal de Broglie wavelength λ = ¯ h ( β/m ) / . Recently [6], we have generalizedthe Wigner-Kirkwood method to quantum fluids constrained to the above defined half-space Λ. The final result for the Boltzmann density in configuration space was obtainedas a series B β ( ~ r ) = ∞ X n =0 B ( n ) β ( ~ r ) , (2.2)where the terms B ( n ) β ( ~ r ) with n = 0 , , , . . . can be calculated systematically with theaid of an operator technique.The result for B (0) β ( ~ r ) was found in the form B (0) β ( ~ r ) = 1( √ πλ ) νN e − βV N Y j =1 (cid:16) − e − x j /λ (cid:17) . (2.3)The bulk counterpart of this term corresponds to the classical Boltzmann density ∝ e − βV . Here, each particle gets an additional “boundary” factor 1 − exp( − x /λ )which goes from 0 at the x = 0 boundary to 1 in the bulk interior x → ∞ on thelength scale ∼ λ . The product of boundary factors then ensures that the quantumBoltzmann density vanishes as soon as one of the particles lies on the boundary. The orrelations and sum rules in a half-space quantum plasma λ is non-analytic; thisfact prevents one from a simple classification of contributions to the Boltzmann densityaccording to integer powers of λ like it is in the bulk case. However, when in thecalculation of statistical averages the exponential part exp( − x /λ ) of the boundaryfactor is integrated over the x -coordinate, the analyticity of the result in the parameter λ is restored. At this stage we only notice that when the product of boundary factorsis expanded as follows N Y j =1 (cid:16) − e − x j /λ (cid:17) = 1 − N X j =1 e − x j /λ + 12! N X j,k =1( j = k ) e − x j /λ e − x k /λ + · · · , (2.4)the integration of each exponential term exp( − x /λ ) over x produces one λ -factor asthe result of the substitution of variables x = λx ′ .The result for B (1) β ( ~ r ) reads B (1) β ( ~ r ) = 1( √ πλ ) νN ( N X k =1 N Y j =1( j = k ) (cid:16) − e − x j /λ (cid:17) x k e − x k /λ ∂∂x k e − βV + e − βV N Y j =1 (cid:16) − e − x j /λ (cid:17) λ " − β ~ r V + β (cid:16) ~ r V (cid:17) . (2.5)Here, the dependence on λ appears also via the combination x exp( − x /λ ). Thisfunction has a maximum of order λ and therefore it is a legitimate expansion parameter.When integrated over the particle coordinate x , it gives a contribution of order λ ,“weaker” than λ .Keeping all contributions up to the order λ in the Boltzmann term B (2) β ( ~ r ), onehas B (2) β ( ~ r ) = 1( √ πλ ) νN e − βV N Y j =1 (cid:16) − e − x j /λ (cid:17) λ " β ~ r V − β (cid:16) ~ r V (cid:17) + o ( λ ) . (2.6)Note that, with regard to the equality ~ r e − βV = e − βV (cid:20) β (cid:16) ~ r V (cid:17) − β ~ r V (cid:21) , (2.7)one can eliminate the squared gradient term in favour of the Laplacian term in equations(2.5) and (2.6).The first three Boltzmann terms (2.3), (2.5) and (2.6) exhibit properties analogousto their bulk counterparts: the maximum of B ( n ) β ( ~ r ) is of order λ n . We anticipatethat this formal structure is also maintained on higher levels. As a consequence, theknowledge of the first three Boltzmann terms B (0) β , B (1) β and B (2) β is sufficient in orderto obtain the expansion of the quantum Boltzmann density up to the λ order: B β ( ~ r ) = B (0) β ( ~ r ) + B (1) β ( ~ r ) + B (2) β ( ~ r ) + o ( λ ) . (2.8)The quantum fluid of present interest is the one-component plasma (jellium) in ν = 2 space dimensions. The system is composed of N mobile pointlike charges e ,neutralized by a uniform oppositely charged fixed background. The total interaction orrelations and sum rules in a half-space quantum plasma V ( ~ r ) satisfies for each of the particle coordinates the Poisson differentialequation r j V ( ~ r ) = − πe N X k =1( k = j ) δ ( r j − r k ) + 2 πe n, j = 1 , , . . . , N. (2.9)Here, the second term on the right-hand side (rhs) comes from the particle-backgroundinteraction and n = N/ | Λ | is the mean number density of the mobile charges. Thesummation over the particle index j of the set of N Poisson equations (2.9) results in ~ r V ( ~ r ) = − πe N X j,k =1( j = k ) δ ( r j − r k ) + 2 πe N n. (2.10)The first term on the rhs of (2.10), when weighted by the classical Boltzmannfactor ∝ Q j 24 2 πβe N n ! N Y j =1 (cid:16) − e − x j /λ (cid:17) + N X k =1 N Y j =1( j = k ) (cid:16) − e − x j /λ (cid:17) x k e − x k /λ ∂∂x k e − βV + N Y j =1 (cid:16) − e − x j /λ (cid:17) λ ~ r e − βV ) + o ( λ ) . (2.11) 3. Statistical quantities for the half-space geometry According to the standard formalism of statistical quantum mechanics, the partitionfunction of the N -particle fluid (with ignored exchange effects) is given by the integrationof the quantum Boltzmann density over configuration space: Z qu = 1 N ! Z Λ d ~ r B β ( ~ r ) . (3.1)The quantum average of a function f ( ~ r ) is defined as follows h f i qu = 1 Z qu N ! Z Λ d ~ r B β ( ~ r ) f ( ~ r ) . (3.2)At the one-particle level, one introduces the particle number density n qu ( r ) = * N X j =1 δ ( r − r j ) + qu . (3.3)At the two-particle level, one considers the two-body density n (2)qu ( r , r ′ ) = * N X j,k =1( j = k ) δ ( r − r j ) δ ( r ′ − r k ) + qu . (3.4) orrelations and sum rules in a half-space quantum plasma n (2)Tqu ( r , r ) = n (2)qu ( r , r ) − n qu ( r ) n qu ( r ) (3.5)vanishing at asymptotically large distances | r − r | → ∞ . The general multiparticledensities are defined in analogy with (3.4), i.e. the corresponding product of δ -functionsis summed out over all possible multiplets of different particles.The classical partition function Z and the classical average of a function f ( ~ r ) aredefined as follows Z = 1 N ! Z Λ d ~ r ( √ πλ ) νN e − βV ( ~ r ) , (3.6) h f i = 1 Z N ! Z Λ d ~ r ( √ πλ ) νN e − βV ( ~ r ) f ( ~ r ) . (3.7)The classical values of statistical quantities will be written without a subscript, like n ( r ), n (2) ( r , r ′ ), etc. In the calculations which follow, we shall need explicitly truncatedforms of the classical three-body density n (3)T ( r , r , r ) = n (3) ( r , r , r ) − n (2)T ( r , r ) n ( r ) − n (2)T ( r , r ) n ( r ) − n (2)T ( r , r ) n ( r ) − n ( r ) n ( r ) n ( r ) (3.8)and of the classical four-body density n (4)T ( r , r , r , r ) = n (4) ( r , r , r , r ) − n (3)T ( r , r , r ) n ( r ) − n (3)T ( r , r , r ) n ( r ) − n (3)T ( r , r , r ) n ( r ) − n (3)T ( r , r , r ) n ( r ) − n (2)T ( r , r ) n (2)T ( r , r ) − n (2)T ( r , r ) n (2)T ( r , r ) − n (2)T ( r , r ) n (2)T ( r , r ) − n (2)T ( r , r ) n ( r ) n ( r ) − n (2)T ( r , r ) n ( r ) n ( r ) − n (2)T ( r , r ) n ( r ) n ( r ) − n (2)T ( r , r ) n ( r ) n ( r ) − n (2)T ( r , r ) n ( r ) n ( r ) − n (2)T ( r , r ) n ( r ) n ( r ) − n ( r ) n ( r ) n ( r ) n ( r ) . (3.9)In what follows, we shall restrict ourselves to the model system of the one-component plasma constrained to the two-dimensional half-space Λ. Now, the Cartesiancoordinates of r become ( x, y ), with the origin on the rectilinear plain hard wall, the y axis along the wall, and the system occupying the x > Substituting the expansion of the Boltzmann density (2.11) into the definition (3.1) ofthe quantum partition function Z qu and performing expansions of type (2.4) for theproducts of boundary factors, we obtain Z qu Z = 1 − λ 24 2 πβe N n − Z Λ d r e − x /λ n ( r )+ 12! Z Λ d r Z Λ d r e − x /λ e − x /λ n (2) ( r , r ) orrelations and sum rules in a half-space quantum plasma Z Λ d r e − x /λ x ∂∂x n ( r ) + λ Z Λ d r ∂ ∂x n ( r ) + o ( λ ) . (3.10)Here, we keep in mind that the integration of an exponential term exp( − x /λ ) over x produces one λ -factor. To calculate the quantum one-body density (3.3), we take advantage of the invarianceof the Boltzmann density (2.11) with respect to permutations of the particle indices andwrite down n qu ( r ) = 1 Z qu NN ! Z Λ d ~ r B β ( ~ r ) δ ( r − r ) . (3.11)In each term of the Boltzmann density, we separate the “reference” r -dependent part,which is kept unchanged, and expand in analogy with (2.4) the remaining part dependenton ( r , . . . , r N ) coordinates. Like for instance, N Y j =1 (cid:16) − e − x j /λ (cid:17) = (cid:16) − e − x /λ (cid:17) ( − N X j =2 e − x j /λ + 12! N X j,k =2( j = k ) e − x j /λ e − x k /λ + · · · ) . (3.12)The quantum partition function Z qu in the denominator on the rhs of (3.11) issubstituted by the expansion (3.10) and subsequently expanded in “virtual” λ powers.After simple but lengthy algebra, one obtains n qu ( r ) = (cid:16) − e − x /λ (cid:17) ( n ( r ) − Z Λ d r e − x /λ n (2)T ( r , r )+ 12! Z Λ d r Z Λ d r e − x /λ e − x /λ n (3)T ( r , r , r )+ Z Λ d r e − x /λ x ∂∂x n (2)T ( r , r )+ λ " ∂ ∂x n ( r ) + Z Λ d r ∂ ∂x n (2)T ( r , r ) + e − x /λ x ∂∂x (cid:20) n ( r ) − Z Λ d r e − x /λ n (2)T ( r , r ) (cid:21) . (3.13)Let us assume that the classical averages under integrations in (3.13) are analyticfunctions of the x -coordinate at the boundary x = 0. For instance, the classicaldensity profile n ( r ) ≡ n ( x ) is supposed to exhibit the Taylor expansion n ( x ) = n (0) + n ′ (0) x + n ′′ (0) x / 2! + · · · . Then, the integral Z ∞ d x e − x /λ n ( x ) = λ Z ∞ d x ′ e − x ′ n ( λx ′ )= λ r π n (0) + λ n ′ (0) + O ( λ ) . (3.14) orrelations and sum rules in a half-space quantum plasma λ order: n qu ( r ) = (cid:16) − e − x /λ (cid:17) ( n ( r ) − λ r π Z d y n (2)T [ r , (0 , y )]+ λ (cid:18) π (cid:19) Z d y Z d y n (3)T [ r , (0 , y ) , (0 , y )]+ λ " ∂ ∂x n ( r ) − Z d y ∂n (2)T [ r , ( x , y )] ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + e − x /λ x ∂∂x (cid:26) n ( r ) − λ r π Z d y n (2)T [ r , (0 , y )] (cid:27) . (3.15) To calculate the λ -expansion of the quantum truncated two-body density given byrelations (3.4) and (3.5), we proceed as in the previous subsection. We first takeadvantage of the permutation invariance of the Boltzmann density (2.11) to write down n (2)qu ( r , r ′ ) = 1 Z qu N ( N − N ! Z Λ d ~ r B β ( ~ r ) δ ( r − r ) δ ( r ′ − r ) . (3.16)Then we separate the reference part in the Boltzmann density (2.11) which depends oncoordinates r , r and expand in analogy with (3.12) the remaining part. Finally, usingthe integration procedure of type (3.14) in all integrals we arrive at the expansion ofthe two-body density up to the λ order: n (2)Tqu ( r , r ) = (cid:16) − e − x /λ (cid:17) (cid:16) − e − x /λ (cid:17) ( n (2)T ( r , r ) − λ r π Z d y n (3)T [ r , r , (0 , y )]+ λ (cid:18) π (cid:19) Z d y Z d y n (4)T [ r , r , (0 , y ) , (0 , y )]+ λ " ∂ ∂x + ∂ ∂x ! n (2)T ( r , r ) − Z d y ∂n (3)T [ r , r , ( x , y )] ∂x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 + (cid:16) − e − x /λ (cid:17) x e − x /λ ∂∂x ( n (2)T ( r , r ) − λ r π Z d y n (3)T [ r , r , (0 , y )] ) + (cid:16) − e − x /λ (cid:17) x e − x /λ ∂∂x ( n (2)T ( r , r ) − λ r π Z d y n (3)T [ r , r , (0 , y )] ) . (3.17) orrelations and sum rules in a half-space quantum plasma 4. Review of the sum rules The following sum rules [except the perfect-screening one (4.1)] apply only to the one-component plasma (not to many-component ones); they rely on the facts that, forthe one-component plasma, the mass and charge fluctuations are proportional to oneanother and the static resistivity vanishes. Although the sum rules can be written fora quantum one-component plasma in a ν -dimensional half-space with any ν , here weconsider only the case ν = 2. Some of the sum rules that we review were originallywritten more generally for the time-displaced correlations, but here we consider onlytheir static limits.The perfect screening sum rule expresses that the charge cloud around a particle ofthe system has a charge opposite to the charge of this particle. This sum rule has thesame form as in the classical case: Z d r n (2)Tqu ( r , r ) = − n qu ( x ) . (4.1)Although we have no doubt about the validity of (4.1), for a check of the calculations insection 3 and of their application to the present 2D one-component plasma at βe = 2,we shall verify (4.1) at order λ in section 6.Macroscopic arguments gave the asymptotic form of the quantum truncated two-body density along the wall [7] n (2)Tqu ( r , r ) ∼ | y − y |→∞ f ( x , x )( y − y ) , (4.2)(after perhaps an averaging on local oscillations in y − y ) with the sum rule Z ∞ d x Z ∞ d x f ( x , x ) = − π e [2¯ hω s coth( β ¯ hω s / − ¯ hω p coth( β ¯ hω p / βe = 2), where the bulk and surface plasma frequencies, for two dimensions anda plain hard wall, are, respectively, ω p = πne m ! / , ω s = πne m ! / . (4.4)Any microscopic check of (4.3) would be welcome. The expansion of the r.h.s. of (4.3)in powers of ¯ h starts with the classical value − / (4 π ) and the next term is of order¯ h . Therefore, at order λ , in section 7 we shall only be able to check that there is noquantum correction.Finally, another macroscopic argument gave the quantum form of the dipole sumrule [8] Z ∞ d x (cid:20)Z ∞ d x x Z d y n (2)Tqu ( r , r ) + x n qu ( x ) (cid:21) = − ¯ hω p πe coth β ¯ hω p βe = 2). The quantity between square brackets in (4.5) is the dipole moment ofthe structure factor. Here too, a check would be welcome. Now, the expansion of therhs of (4.5) in powers of ¯ h starts with the classical value − / (4 π ) and the next term is − λ / (12 πa ), where a is the average interparticle distance defined by n = 1 / ( πa ). Insection 8 we can check this sum rule at order λ . orrelations and sum rules in a half-space quantum plasma 5. The classical densities We shall need some information about the classical densities of the 2D one-componentplasma at βe = 2. We express all lengths in units of the average interparticle distance a , thus n = 1 /π . The density profile is [1] n ( x ) = n √ π Z ∞ d t exp[ − ( t − x √ ]1 + Φ( t ) , (5.1)where Φ( t ) is the probability-integral function. The two-body truncated density is n (2)T ( r , r ) = − n exp[ − x + x )] k ( r , r ) k ( r , r ) , (5.2)where k ( r , r ) = 2 √ π Z ∞ d t exp h − t + t ( x + x ) √ − i t ( y − y ) √ i t ) . (5.3)Higher-order n -body truncated densities contain a sum of product of n factors k , the sumrunning on all oriented cycles built with r , r , ..., r n [2]. One finds for the three-bodytruncated density n (3)T ( r , r , r ) = n exp[ − x + x + x )][ k ( r , r ) k ( r , r ) k ( r , r )+ k ( r , r ) k ( r , r ) k ( r , r )] , (5.4)and for the four-body truncated density n (4)T ( r , r , r , r ) = − n exp[ − x + x + x + x )][ k ( r , r ) k ( r , r ) k ( r , r ) k ( r , r )+ k ( r , r ) k ( r , r ) k ( r , r ) k ( r , r )+ k ( r , r ) k ( r , r ) k ( r , r ) k ( r , r )]+ complex conjugate . (5.5)We shall also need some integrals of these densities. In (5.2), the product of the k functions contains R ∞ d t exp( − i ty √ R ∞ d t ′ exp(i t ′ y √ y is π √ δ ( t − t ′ ). One obtains Z d y n (2)T ( r , r ) = − n √ − x + x )] Z ∞ d t exp h − t + t ( x + x )2 √ i [1 + Φ( t )] . (5.6)By the same method, one finds Z d y n (3)T ( r , r , r ) = n √ √ π exp[ − x + x + x )] × Z ∞ d t exp[ − t + t ( x + x ) √ − i t ( y − y ) √ t ) × Z ∞ d t ′ exp[ − t ′ + t ′ ( x + x + 2 x ) √ − i t ′ ( y − y ) √ t ′ )] + complex conjugate , (5.7) Z d y Z d y n (3)T ( r , r , r ) = n √ π exp[ − x + x + x )] × Z ∞ d t exp[ − t + t ( x + x + x )2 √ t )] , (5.8) orrelations and sum rules in a half-space quantum plasma Z d y Z d y n (4)T [ r , r , (0 , y ) , (0 , y )] = − n 32 exp[ − x + x )] × ( Z ∞ d t exp[ − t + t ( x + x ) √ − i t ( y − y ) √ t ) × Z ∞ d t ′ exp[ − t ′ + t ′ ( x + x ) √ − i t ′ ( y − y ) √ t ′ )] + Z ∞ d t exp[ − t + t ( x + x ) √ − i t ( y − y ) √ t )] × Z ∞ d t ′ exp[ − t ′ + t ′ ( x + x ) √ − i t ′ ( y − y ) √ t ′ )] ) +complex conjugate , (5.9) Z d y Z d y Z d y n (4)T [ r , r , (0 , y ) , (0 , y )] = − n π √ − x + x )] × Z ∞ d t exp[ − t + t ( x + x )2 √ t )] . (5.10) 6. Perfect screening Omitting some terms which do not contribute to the sum rules at order λ , and usingthe results in section 5, we find from (3.17) Z d y n (2)Tqu ( r , r ) = [1 − exp( − x /λ )][1 − exp( − x /λ )] × exp[ − x + x )] ( − n √ Z ∞ d t exp[ − t + t ( x + x )2 √ t )] − λn √ π Z ∞ d t exp[ − t + t ( x + x )2 √ t )] − λ n √ π Z ∞ d t exp[ − t + t ( x + x )2 √ t )] − λ n √ π Z ∞ d t t exp[ − t + t ( x + x )2 √ t )] ) + ( − λ n √ ∂ ∂x + ∂ ∂x ! − n √ − x /λ ) x ∂∂x − n √ − x /λ ) x ∂∂x ) × ( exp[ − x + x )] Z ∞ d t exp[ − t + t ( x + x )2 √ t )] ) + · · · . (6.1)On the other hand, we find from (3.15) n qu ( x ) = [1 − exp( − x /λ )] ( √ π n exp( − x ) Z ∞ d t exp( − t + tx √ t )+ λn √ π exp( − x ) Z ∞ d t exp( − t + tx √ t )] orrelations and sum rules in a half-space quantum plasma λ n π / exp( − x ) Z ∞ d t exp( − t + tx √ t )] + λ n √ π ∂ ∂x " exp( − x ) Z ∞ d t exp( − t + tx √ t ) + λ n 23 exp( − x ) Z ∞ d t t exp( − t + tx √ t )] ) + exp( − x /λ ) x ∂∂x ( n √ π exp( − x ) Z ∞ d t exp( − t + tx √ t )+ λn √ π exp( − x ) Z ∞ d t exp( − t + tx √ t )] ) + · · · . (6.2)Using [9] Z ∞ d x exp( − x + tx √ 2) = r π t )[1 + Φ( t )] , (6.3)we can compute the integral on x of (6.1) at order λ and check that it is equal to theopposite of (6.2).The sum rule (4.1) is verified. 7. The sum rule about an asymptotic form For obtaining the asymptotic forms of the classical n -body densities as | y − y | → ∞ ,one uses the integration per partes for the Fourier transform of a function F ( t ) Z ∞ d tF ( t ) exp[ − i ty √ ∼ F (0)i y √ . (7.1)The different classical n -body truncated densities or their integrals which appear in(3.17) are all found to have an asymptotic form ∝ exp[ − x + x )] / ( y − y ) . Thisgives the asymptotic form (4.2), where (some terms which do not contribute to the sumrule at order λ have not been kept) f ( x , x ) = [1 − exp( − x /λ )][1 − exp( − x /λ )] × exp[ − x + x )] (cid:18) − n π − λn − λ n π (cid:19) + · · · . (7.2)Now, we note from equation (3.14) that, at order λ , an integral of the form R ∞ d x exp( − x /λ ) F ( x ) becomes λ q π/ F (0) + λ (1 / F ′ (0). Then, from (7.2), takinginto account that in our units n = 1 /π , Z ∞ d x Z ∞ d x f ( x , x ) = − π + λ (cid:18) π + 14 π − π (cid:19) + λ (cid:18) − π + 12 π + 12 π − π (cid:19) + o ( λ )= − π + o ( λ ) (7.3)where λ is in units of a .At order λ , there are no quantum corrections, in agreement with section 4. orrelations and sum rules in a half-space quantum plasma 8. Dipole sum rule Omitting some terms which do not contribute to the sum rule at order λ , using [9] Z ∞ d x x exp( − x + 2 √ tx ) = 14 + √ π t exp( t )[1 + Φ( t )] , (8.1)and noting that, at order λ , an integral of the form R ∞ d x x exp( − x /λ ) F ( x ) becomes(1 / λ F (0), one finds from (6.1), contributing to the sum rule at order λ , Z ∞ d x x Z d y n (2)Tqu ( r , r ) = [1 − exp( − x /λ )] exp( − x ) × ( − n √ Z ∞ d t " exp( − t + tx √ t )] + √ π t exp( − t + tx √ t ) + λ n √ Z ∞ d t exp( − t + tx √ t )] − λn √ π Z ∞ d t " exp( − t + tx √ t )] + √ π t exp( − t + tx √ t )] − λ n √ π Z ∞ d t " exp( − t + tx √ t )] + √ π t exp( − t + tx √ t )] − λ n √ π Z ∞ d t t " exp( − t + tx √ t )] + √ π t exp( − t + tx √ t )] − λ n √ 26 exp( − x ) Z ∞ d t exp( − t + tx √ t )] + ( − λ n √ ∂ ∂x − n √ − x /λ ) x ∂∂x ) × ( exp( − x ) Z ∞ d t " exp( − t + tx √ t )] + √ π t exp( − t + tx √ t ) + · · · . (8.2)On the other hand, n qu ( x ) is given by (6.2).The classical part of the sum rule comes from the second line of (8.2) and the firstline of (6.2). Since exp( − t ) / [1 + Φ( t )] = − ( √ π/ / d t )[1 + Φ( t )] − , an integrationper partes gives, with n = 1 /π , Z ∞ d x x Z d y n (2)T ( r , r ) + x n ( x ) = − √ π / exp( − x ) . (8.3)Integrating on x gives the classical result − / (4 π ).Using (6.3) and (8.1), it is straightforward to integrate on x most of the terms of(8.2) and x times (6.2). The term in (8.2), line − 3, of the form ∂ F ( x ) /∂x requiressome care for evaluating its integral on x , which is ∂F ( x ) /∂x | ∞ . The point is that F ( x ) has a term which is not zero at infinity. Indeedexp( − x ) Z ∞ d t t exp( − t + tx √ t ) = Z ∞ d t t exp[ − ( t − x √ ]1 + Φ( t ) orrelations and sum rules in a half-space quantum plasma ∼ x →∞ x √ Z ∞−∞ d t ′ exp( − t ′ )2 = q π/ x (8.4)where t ′ = t − x √ 2, and the derivative with respect to x of (8.4) is q π/ − λ / (24 π ).Similarly, the term in (6.2), line − 2, gives a contribution of the form x [ ∂ G ( x ) /∂x ];when integrated per partes on x , in particular it gives a term − G ( ∞ ). Since G ( x ) = Z ∞ d t exp[ − ( t − x √ ]1 + Φ( t ) ∼ x →∞ Z ∞−∞ d t ′ exp( − t ′ )2 = √ π/ , (8.5)the corresponding contribution to the sum rule is − λ / (24 π ). All other quantumcontributions to the sum rule are found to cancel each other, at order λ . Finally, Z ∞ d x (cid:20)Z ∞ d x x Z d y n (2)Tqu ( r , r ) + x n qu ( x ) (cid:21) = − π − λ π + o ( λ ) . (8.6)Since λ is in units of a , the sum rule (4.5) does have a quantum correction − λ / (12 πa ) at order λ . 9. Conclusion That the perfect screening sum rule (4.1) is satisfied is no surprise. This is rathera check of our calculations. The other sum rules are less straightforward. In theirheuristic macroscopic derivations [7, 8], an essential feature of the quantum systems,that the n -body densities have to vanish on the wall, was not explicitly taken intoaccount. Thus a check that they are indeed correct is welcome.The case of the asymptotic-form sum rule (4.3) is not entirely satisfactory. Althoughwe have checked it at order λ , at this order we could only verify that there is no quantumcorrection, in agreement with the expansion of the rhs of (4.3). It would have been moresatisfactory to verify the quantum correction of order λ of this rhs. Unfortunately, thiswould involve pushing the Wigner-Kirkwood expansion to that order, which would bestraightforward but so tedious that we cannot hope for its feasibility in a reasonabletime.The dipole sum rule (4.5) is more tractable, and we have indeed checked a finitequantum correction at order λ . Moreover, another derivation of this sum rule is feasible.Indeed, the generalization of the classical Stillinger-Lovett sum rule [10] for a bulkquantum one-component plasma has been microscopically done [11]; adapting the resultto two dimensions gives, with r = r − r , Z d r r n (2)Tqu , bulk ( r , r ) = − πe ¯ hω p coth β ¯ hω p . (9.1)Then, we can use the same kind of argument as the one in pages 965 and 966 of ref. [8]for deriving the dipole sum rule (4.5). orrelations and sum rules in a half-space quantum plasma Acknowledgments We gratefully acknowledge the support received from the European Science Foundation(ESF “Methods of Integrable Systems, Geometry, Applied Mathematics”) and from theVEGA grant 2/6071/2006 of the Slovak Grant Agency. References [1] Jancovici B, 1982 J. Stat. Phys. J. Stat. Phys. The theory of quantum liquids. Vol. I : Normal Fermi liquids (NewYork: Benjamin)[4] Wigner E P, 1932 Phys. Rev. Phys. Rev. Phys. Rev. J. Stat. Mech. P02002[7] Jancovici B, 1985 J. Stat. Phys. J. Stat. Phys. Tables of integrals, series, and products (London: Academic)[10] Stillinger F and Lovett R, 1968 J. Chem. Phys. J. Phys. A18