Semiclassical theory of non-local statistical measures: residual Coulomb interactions

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1 of non-local statistical measures: residual Coulomb interactions Steve Tomsovic 1, Denis Ullmo 2, and Arnd Bäcker 3 1 Washington State University, Pullman 2 Laboratoire de Physique Théorique et Modèles Statistiques, Orsay 3 Technische Universität Dresden supported by the US National Science Foundation and Deutsche Forschungsgemeinschaft PRL 2008, PRE 2009

2 Today s thread of logic Motivating Quantum dots Ground state energy shifts from residual interactions Normalization Averages Fluctuations Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems

Quantum dots Residual interaction energy shifts Quantum dot illustration Christian Schönenberger, University of Basel http://pages.unibas.ch/phys-meso/pictures/pictures.

3 Quantum dots Residual interaction energy shifts Quantum dot illustration Christian Schönenberger, University of Basel Interested in a quantum dot holding on the scale of a hundred to a few hundreds of electrons Also interested in the entrance and exit leads being so narrow as to require tunneling on and off, i.e. a nearly isolated dot Can speak of the many-body energy levels, ground state, etc... E gr [N] = e2 N 2 2C + i,σ f σ i ǫ i + E ri

4 Quantum dots Residual interaction energy shifts An aside Peak heights are correlated! Narimanov et al, PRL 1999 & PRB periodic orbits induce correlations Folk et al., PRL 1996

5 Quantum dots Residual interaction energy shifts A comment i=70 i=69 i=68 i=67 i=66 i=65 i=64 i=63 i=62 N = Ullmo et al., PRL 2003

6 Quantum dots Residual interaction energy shifts Chaotic billiard models D cardioid billiard purely chaotic Dirichlet boundary conditions, entire boundary N N D stadium billiard purely chaotic, but has bouncing ball modes partial Dirichlet and Neumann boundary conditions

7 Quantum dots Residual interaction energy shifts Consider the short-range screened Coulomb interaction: V sc (r r ) = F a 0 ν δ(r r ) First order perturbation theory corrects the ground state energy by an amount: E RI = F 0 a dr n (r)n (r) = F 0 aa f ν 2 i,(+) f j,( ) dr ψ i (r) 2 ψ j (r) 2 But, the ground state is the minimum of the various configurations of occupied levels and the quantity i S i = A dr ψ i (r) 2 ψ j (r) 2 j=1 determines roughly whether one configuration or another with one particle displaced has the lower energy i,j

8 Quantum dots Residual interaction energy shifts S i component parts S i think of as being comprised of two component parts S i = A dr Ψ i (r) 2 Ψ j (r) 2 = A dr Ψ i (r) 2 N(r;E + i ) j i N(r,E + i ) = (Friedel-like osc.) + tiny fluctuations The other component, Ψ i (r) 2, is often taken statistically to be a random wave field

9 Normalization Averages Fluctuations Random plane waves in the neighborhood of the boundary (using locally defined coordinates): 1 = dr ψ i (r) = 1 N eff N eff a l cs (k l ˆx)cos (k l ŷ + ϕ l ) l=1 ψ i (r) 2 = σ2 4N eff ( dr [1 ± J 0 (2k F x)] = Aσ2 1 ± L ) 4N eff 2k F A Approximation for the Friedel-like oscillations [ N(r;E + i i ) = ( ) 1 ± J ] 1(2k i r) + δn(r;e + A 1 ± L i ) k i r k i A

10 Normalization Averages Fluctuations It is advantageous to split the eigenfunction into its Friedel-like part and fluctuating part A ψ i (r) 2 1 = ( ) [1 ± J 0 (2k 1 ± L F x)] + Aδ ψ i (r) 2 2k F A With a little algebra, one finds quickly that S i = i + i dr J ( 1(2k F x) J 0 (2k F x) = i 1 + 2L ) A k F x πk F A 0 To this first order correction, the average behavior is universal in that it neither depends on the nature of the dynamics nor on whether the boundary conditions are Dirichlet or Neumann.

11 Normalization Averages Fluctuations (a) 0 S i S i 2 cardioid billiard 4 (b) 90 S i S i i stadium billiard i

12 Normalization Averages Fluctuations Interest is in the leading order behavior of the variance Var[S i S j ] = Var[S i ] + Var[S j ] + Covar[S i S j ] as a function of k F L Skipping the gory details, the random plane wave model assumes that Covar[S i S j ] = 0 and the calculations for fully chaotic systems give Var[S i ] = Var[S j ] = k { FL 2ln 2 1 Dirichlet b.c. 4π 3 2ln ln πk F A 2L Neumann b.c. for i = 1000, Neumann gives a result enhanced by approximately a factor 40 as compared to Dirichlet k F L is O(i 1/2 ), i.e. the i scalings are i 1/2 and i 1/2 lni

13 Normalization Averages Fluctuations (a) 3 Var[S i ] 2 cardioid billiard 1 0 (b) 600 Var[S i ] i stadium billiard i

14 Motivating non-local measures Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems Should random plane waves work? Random waves Stadium eigenstate

15 Semiclassical preliminaries Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems Generally speaking, semiclassical theory of chaotic systems necessitates energy smoothing, i.e. def S N = 1 S i N E E 2 <E i<e+ E 2 One can extract individual state properties from the N-scaling because the variance is, in this notation ( Var[ S N ] def = S N S ) 2 = 1 N Var[S i]+ N 1 N Covar i j[s i S j ]

16 Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems In Bogomolny s classic paper (Physica D, 1988), an extremely useful semiclassical approximation for the retarded Green function is given, which is needed here However, one has to be very careful making use of his expression, i.e. one must separate out the extremely short orbits responsible for the Friedel-like oscillations near the boundary keep the first correction terms to a density of states factor His expression then takes the form A Ψ i (r) 2 N = 1 ± J 0(2k F x) 1 πν W Im G osc (r,r,e) E 1 ± J 0(2k F x) ρ osc (E) Aν E W

17 Chaotic quantum billiards Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems From the Bogomolny expression, S i naturally decomposes into 3 components: a short orbit Friedel-like part the fluctuations of interest and spurious terms density of states fluctuations that preserve norm and N, which are needed to cancel the otherwise spurious terms. [ ] S i = S ± S (1) i,osc + S(2) i,osc The δn(r,e) fluctuation terms are not kept as they are of lower order The short orbit term can be taken out of local energy averages A cluster of 4 orbits emerges for each contribution from stationary phase conditions properly applied

18 r (a) Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems r (b) r (c) r (d) y y x The bottom line is that a mean square of an angle function is replaced by its variance and doubled, thus { Var(S i ) = k ( FL (2ln 2 1) π 2 1) 2 2π 3 (2ln 2 1) ( ( ) π 2 1) ln πk F A 2L π 2 x D N

19 (a) 3 Var[S i ] 2 Motivating non-local measures Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems cardioid billiard 1 0 (b) 600 Var[S i ] i stadium billiard i

20 (a) Motivating non-local measures 20 F dos (L) 0 Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems 20 (b) 5.0 F(L) L L

21 Deficiencies of random plane waves Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems 1 Individual eigenstate normalization vs ensemble average normalization directly leads to the replacement of the mean square by the variance 2 Improper dynamical correlations After invoking the Hannay-Ozorio sum rule in the semiclassical theory, leads to factor two difference

22 Bogomolny revisited Chaotic quantum billiards Periodic orbit spectrum Deficiencies of random plane waves Non-chaotic systems 1 For integrable systems, it is possible to give results based on an EBK-quantized tori approach To give an example, consider the circular billiard (note the i-dependence) 2 Var [S i ] = i { π 1 2 ( 1 2 ) 2 π Dirichlet 2 π π1/2 i 1/4 ( 1 π) 2 2 Neumann 2 It is possible using a modification of rectangle eigenstates to derive an expression for the influence of the bouncing ball modes (we rely on Tanner s estimate (J. Phys. A, 1997) of the relative fraction of bouncing ball modes), but the expression is too long and cumbersome to write on a slide... though shown on the variance figure

23 Averages of quantities such as the {S i } are independent of the nature of the dynamics, whereas fluctuations are dependent. It is parallel to the story of level densities and level fluctuations.

24 Averages of quantities such as the {S i } are independent of the nature of the dynamics, whereas fluctuations are dependent. It is parallel to the story of level densities and level fluctuations. There is a large enhancement of the fluctuations with Neumann boundary conditions relative to Dirichlet. It would be interesting to investigate smooth confining potentials.

25 Averages of quantities such as the {S i } are independent of the nature of the dynamics, whereas fluctuations are dependent. It is parallel to the story of level densities and level fluctuations. There is a large enhancement of the fluctuations with Neumann boundary conditions relative to Dirichlet. It would be interesting to investigate smooth confining potentials. corrects two faults of basic random plane wave modeling: i) individual eigenstates are normalized, and ii) dynamical correlations are correctly built into the theory.

26 Averages of quantities such as the {S i } are independent of the nature of the dynamics, whereas fluctuations are dependent. It is parallel to the story of level densities and level fluctuations. There is a large enhancement of the fluctuations with Neumann boundary conditions relative to Dirichlet. It would be interesting to investigate smooth confining potentials. corrects two faults of basic random plane wave modeling: i) individual eigenstates are normalized, and ii) dynamical correlations are correctly built into the theory. As the dynamics tends towards more regular behavior, the fluctuations become enhanced and the residual interaction can have a greater influence on the system s properties.

27 Averages of quantities such as the {S i } are independent of the nature of the dynamics, whereas fluctuations are dependent. It is parallel to the story of level densities and level fluctuations. There is a large enhancement of the fluctuations with Neumann boundary conditions relative to Dirichlet. It would be interesting to investigate smooth confining potentials. corrects two faults of basic random plane wave modeling: i) individual eigenstates are normalized, and ii) dynamical correlations are correctly built into the theory. As the dynamics tends towards more regular behavior, the fluctuations become enhanced and the residual interaction can have a greater influence on the system s properties. Ultra-cold Fermi gasses provide other quantities in the same category as the {S i }, e.g. BCS pairing gap fluctuations [see Garcia-Garcia et al. PRL, 2008 and Olofsson et al. PRL, 2008].

Friedel oscillations in microwave billiards

Friedel oscillations in microwave billiards A. Bäcker, 1 B. Dietz, 2 T. Friedrich, 3 M. Miski-Oglu, 2 A. Richter, 2,4 F. Schäfer, 2 and S. Tomsovic 5 1 Institut für Theoretische Physik, Technische Universität