Residual Coulomb interaction fluctuations in chaotic systems: the boundary, random plane waves, and semiclassical theory
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Residual Coulomb interaction fluctuations in chaotic systems: the boundary, random plane waves,and semiclassical theory
Steven Tomsovic , ∗ , Denis Ullmo and Arnd B¨acker Max-Planck-Institut f¨ur Physik komplexer Systeme, D-01187 Dresden, Germany CNRS; Univ. Paris-Sud; LPTMS UMR 8626, 91405 Orsay Cedex, France and Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, 01062 Dresden, Germany (Dated: November 7, 2018)New fluctuation properties arise in problems where both spatial integration and energy summation are nec-essary ingredients. The quintessential example is given by the short-range approximation to the first orderground state contribution of the residual Coulomb interaction. The dominant features come from the regionnear the boundary where there is an interplay between Friedel oscillations and fluctuations in the eigenstates.Quite naturally, the fluctuation scale is significantly enhanced for Neumann boundary conditions as comparedto Dirichlet. Elements missing from random plane wave modeling of chaotic eigenstates lead surprisingly tosignificant errors, which can be corrected within a purely semiclassical approach.
PACS numbers: 03.65.Sq, 05.45.Mt, 71.10.Ay, 73.21.La, 03.75.Ss
The characterization of quantum systems with any of avariety of underlying classical dynamics, ranging from dif-fusive to chaotic to regular, has often demonstrated that thestudy of their statistical properties is of primary importance.Spectral fluctuations are a principle example as they gave thefirst support to one of the main results linking classical chaosand random matrix theory [1], the Bohigas-Giannoni-Schmitconjecture [2, 3]. Needless to say, the statistical propertiesof eigenfunctions are also a subject of paramount interest[4, 5, 6, 7, 8, 9, 10, 11, 12].For chaotic systems, a widely accepted starting point forthe treatment of eigenfunction fluctuations locally , such asthe amplitude distribution or the two-point correlation func-tion c ( | r − r ′ | ) = h ψ ( r ) ψ ( r ′ ) i of a given eigenfunction ψ , isa modeling in terms of a random superposition of plane waves(RPW) [4, 5]. For two-degree-of-freedom systems, c ( r ) canbe understood as being given approximately by a Bessel func-tion. For distances | r − r ′ | short compared to the system size,and in the absence of effects related to classical dynamics[8, 9, 13], this is roughly observed in numerical [7, 14, 15]and experimental [16] studies. Our interest here is in statisti-cal properties of eigenfunctions going beyond local quantitiessuch as c ( r ) .One motivation for the introduction of these new statisti-cal measures is to study the interplay between interferencesand interactions in mesoscopic systems. For typical elec-tronic densities, the screening length is close to the Fermiwavelength λ F , and the screened Coulomb interaction can beapproximated by the short range expression V sc ( r − r ′ ) =(2 ν ) − F a δ ( r − r ′ ) with ν the mean local density of states,including spin degeneracy, ( ν = m/ π ~ for d = 2 ) and F a the dimensionless Fermi liquid parameter [17], a constant oforder one. ∗ permanent address: Department of Physics and Astronomy, WashingtonState University, Pullman, WA 99164-2814 To this level of approximation, the first order ground stateenergy contribution of the residual interactions can be ex-pressed in the form δE RI = (2 ν ) − F a R d r N ↑ ( r ) N ↓ ( r ) , with N σ the unperturbed ground state density of particles withspin σ . From this expression, it is seen that the increase of in-teraction energy associated with the addition of an extra elec-tron is related to S i = A Z d r | Ψ i ( r ) | N ( r ; E + i ) , (1)with N ( r ; E ) ≡ P ∞ i =1 | Ψ i ( r ) | θ ( E − E i ) and the under-standing that E i < E + i < E i +1 . Our goal in this letter isto study the fluctuation properties of the S i , concentrating onthe case of two dimensional billiards with either Dirichlet orNeumann boundary conditions. For them, A is the billiardarea.The dominant contributions to S i (and its fluctuations) orig-inate from the Friedel oscillations of the density of particlesnear the boundary. For billiards systems, they can be ex-pressed as N Friedel ( r ; E ) = N W ( E ) A h ± J (2 kx ) kx i [18], with x the distance from the boundary, the + and − sign corre-sponding respectively to Neumann and Dirichlet boundaryconditions, and N W ( E ) refers to leading term of the Weylformula, N W ( E ) = ν A E . To leading order we can thereforeuse the approximation S i = i ± i Z d r (cid:20) J (2 k F x ) k F x | Ψ i ( r ) | (cid:21) . (2)To proceed, a description of the fluctuations of | Ψ i ( r ) | isalso required. These are obtained ahead using a semiclassi-cal approach closely related to the Gutzwiller trace formula.However, it is useful first to consider the oft-employed RPWdescription, which very interestingly turns out to lack a cou-ple of crucial ingredients. Nevertheless, it sheds light on themechanism governing the fluctuations under study.Within RPW [4, 5] eigenstates are represented, in the ab-sence of any symmetry, by a random superposition of planewaves P l a l exp( i k l · r ) with wave-vectors of fixed modulus | k l | = k F distributed isotropically. Time reversal invarianceintroduces a correlation between time reversed plane wavessuch that the eigenfunctions are real. Similarly, the presenceof a planar boundary imposes a constraint between the coeffi-cients of plane waves related by a sign change of the normalcomponent of the wave-vector k l [19, 20]. Near a boundary,and using a system of coordinates r = ˆ x + ˆ y with ( ˆ x , ˆ y ) thevectors respectively parallel and perpendicular to the bound-ary, eigenfunctions are mimicked statistically by a superposi-tion, ψ i ( r ) = 1 N eff N eff X l =1 a l cs ( k l · ˆ x ) cos ( k l · ˆ y + ϕ l ) (3)where cs ( · ) def = sin( · ) for Dirichlet and cos( · ) for Neumannboundary conditions. The phase angle ϕ l , the orientation ofthe wave vector k l , and the real amplitude a l with h a l a l ′ i = δ ll ′ σ are all chosen randomly. Normalization of the wave-functions fixes the variance σ through the relation Z d r D | ψ i ( r ) | E = A σ N eff (cid:18) ± L k F A (cid:19) , (4)where L is the perimeter.The variance is a natural measure of the fluctuations. Toleading order Var ( S i ) = i A Z d r d r J (2 k F x ) k F x J (2 k F x ) k F x × (cid:2)(cid:10) | Ψ i ( r ) | | Ψ i ( r ) | (cid:11) − (cid:10) | Ψ i ( r ) | (cid:11) (cid:10) | Ψ i ( r ) | (cid:11)(cid:3) . The fluctuations are thus given by pair-wise correlating therandom plane wave coefficients a n such that h a l a l ′ a m a m ′ i = h a l a l ′ ih a m a m ′ i + h a l a m ih a l ′ a m ′ i + h a l a m ′ ih a l ′ a m i = σ ( δ ll ′ δ mm ′ + δ lm δ l ′ m ′ + δ lm ′ δ l ′ m ) . One obtains in this way Var [ S i ] = 8 i A N N eff X l,m =1 Z d r d r J (2 k F x ) k F x J (2 k F x ) k F x cs ( k l · x ) cs ( k l · x ) cs ( k m · x ) cs ( k m · x )cos [ k l · ( y − y )] cos [ k m · ( y − y )] . (5)Performing the integral gives Var[ S i ] = k F L π h λ ( θ ) i θ (6)where we have introduced the function λ ( θ ) def = (cid:2) ± | sin(¯ θ ) | (cid:3) / | cos(¯ θ ) | and the expression for h λ ( θ ) i θ ≡ R d (sin θ ) λ ( θ ) is given ahead in Eq. (15).In Fig. 1, the variance of S i as a function of i is repre-sented for a chaotic system, the cardioid billiard with Dirichletboundary conditions [21, 22]. In this case, quite surprisinglygiven the history of modeling chaotic eigenstates within the
012 0 500 1000 1500 2000 i Var( S i ) FIG. 1: Variance of S i for the cardioid billiard with Dirichlet bound-ary conditions. The solid line is the RPW expression, Eq. (6), thedashed line is the semiclassical prediction, Eq. (14), and the discretepoints are for the cardioid billiard (shown in inset). RPW framework, its predictions significantly overestimate thefluctuations. In fact, two important elements are missing fromthis approach, both of which are addressed properly within apurely semiclassical approach.One difficulty immediately encountered with a semiclas-sical treatment of the S i is that their computation im-plies addressing the fluctuations of individual wave-functions,whereas the semiclassical approximations valid for chaoticsystems of use here converge only for (locally) smoothedquantities. This difficulty may be overcome by following thespirit of Bogomolny’s calculation [23], and introducing a localenergy averaging, hSi ∆ N def = 1∆ N X E − ∆ E