Interaction Matrix Element Fluctuations

Transcription

1 Interaction Matrix Element Fluctuations in Quantum Dots Lev Kaplan Tulane University and Yoram Alhassid Yale University Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 1/29

2 Outline Ballistic dots in Coulomb blockade regime Conductance peak spacings: need interactions Computing IMEs: Relation to single-particle correlators Random wave model (N ) What happens in actual chaotic dots? Failure of random wave model Failure of leading-order semiclassical theory Can we compute subleading terms in 1/N? Beyond chaos (time permitting) Summary Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 2/29

3 Coulomb Blockade Regime Dot weakly coupled to outside via two leads Decay width Temperature Charging energy Sharp conductance peaks when Fermi energy in leads matches energy needed to move one new electron onto dot (N N + 1) Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 3/29

4 Coulomb Blockade Regime Peaks depend on many-body energies E N and associated wave functions E.g., peak spacings given by E gs N+1 2Egs N + Egs N 1 for T = 0 Statistical properties for N 1? Hartree-Fock approach: E N includes Classical charging energy N 2 e 2 /2C Constant exchange interaction Mean-field single-electron potential (chaotic) Residual two-electron interaction Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 4/29

5 Peak Height Distribution Peak height statistics well explained using constant interaction + chaotic mean field Folk et al, PRL (1996) Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 5/29

6 Peak Spacing Distribution Patel et al, PRL (1998) Peak spacing distribution predicted to be bimodal (e 2 /C + ɛ N+1 ɛ N followed by e 2 /C) in meanfield model (not observed) Two-body interactions essential for understanding spacings Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 6/29

7 Interaction Matrix Elements Diagonal two-body IME v αβ v αβ;αβ Contact interaction model: v αβ = V V d r ψ α( r) 2 ψ β ( r) 2 Interested in fluctuations δvαβ 2, etc. To leading order in g T = kl N (L V ), δv 2 αβ = 2 V 2 V V d r d r C2 ( r, r ) + where C( r, r ) = ψ( r) 2 ψ( r ) 2 ψ( r) 2 ψ( r ) 2 Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 7/29

8 Interaction Matrix Elements Similar expressions for variances δv 2 αα, δv2 αβγδ covariance δv αβ δv αγ (relevant for spectral scrambling) surface charge IME fluctuation δv 2 α Higher moments δv n for n 3 require C( r, r, r ) etc. Aside: IME distributions essential in diverse physical contexts, e.g., mode competition in micron- sized asymmetric dielectric laser resonators (Tureci & Stone) Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 8/29

9 Random wave model (Berry) Typical trajectory in classically ergodic system uniformly explores energy hypersurface Typical single-electron wave function should be composed of random superposition of basis states at fixed energy (e.g., plane waves in hard-wall billiard) Gaussian-distributed ψ( r) Free-space intensity correlation C( r, r ) = 2 β 1 V 2J2 0(k r r ) Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 9/29

10 Random wave model: normalize Normalization in finite volume: C( r, r ) = C( r, r ) 1 d r a C( r, r a ) V V 1 d r a C( r a, r ) V V + 1 d r V 2 a d r b C( r a, r b ) + Satisfies V d r C( r, r ) = 0 δv 2 αβ = 2 3 π V V ( ) 2 [ 2 lnkl + bg β (kl) 2 (Mirlin) + O ( 1 )] (kl) 3 Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 10/29

11 Random wave model: variance δv 2 αβ = 2 3 π ( 2 β ) 2 [ ln kl + bg (kl) 2 + O ( 1 (kl) 3 Leading lnkl/(kl) 2 term depends only on symmetry class, normalization unnecessary b g requires normalized correlator C( r, r ) Shape dependence of b g is weak (< 5%) )] Subleading O(1/(kL) 3 ) corrections are < 10% for systems of experimental interest Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 11/29

12 Subleading effects are small (δv αβ ) error kl kl Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 12/29

13 Weak shape dependence of δv 2 αβ 1.02 (δv αβ ) 2 (r) / (δv αβ ) 2 (1) r Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 13/29

14 Random wave model Within random wave model, other matrix elements differ only by combinatoric factors at leading order δv 2 αβ = 2 3 π ( 2 β ) 2 [ ln kl + bg (kl) 2 δvαα 2 = 2 3 [ lnkl + b π c g β (kl) 2 δvαβγδ 2 = 3 [ ln kl + b g 2 π (kl) 2 + O ( 1 )] (kl) 3 ( )] 1 + O (kl) 3 ( )] 1 + O (kl) 3 Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 14/29

15 Matrix element distributions Naively, should be Gaussian (central limit theorem) Recall δvαβ 2 d r d r C2 ( r, r ) 2 ln kl V V (kl) 2 Similarly δvαβ 3 V V V d r d r d r C 2 ( r, r, r ) where C( r, r, r ) c 3β J 0 (k r r ) J 0 (k r r )J 0 (k r r ) + Thus δvαβ 3 = b 3g c 2 3β 3 (kl) 3 b 3g is geometry-dependent constant c 3β is combinatoric factor Note: no logarithmic divergences Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 15/29

16 Matrix element distributions Skewness γ 1 = δv 3 αβ / [ γ 1 = b 3g c 2 3β Excess kurtosis [ γ 2 = (δv 4αβ 3 δv 2 αβ ] 3/2 ( ) 3 β ( π 3/2 2 3) (ln kl) 3/2 + δv 2 αβ ] 2 ) ( ( ) ) 4 ( ) γ 2 = b 4g c 2 4β + 2 π 2 β 3 / [ δv 2 αβ ] 2 (ln kl) 2 + Very slow convergence of interaction matrix elements to Gaussian statistics even for Gaussian random single-electron wave functions Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 16/29

17 Skewness and excess kurtosis γ i kl Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 17/29

18 What do we have so far? Can use quantum chaos methods to compute universal IME distribution as function of single semiclassical parameter kl Unfortunately, distribution is too narrow to be consistent with low-temperature experimental data on peak spacings Brings into question validity of Hartree-Fock? Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 18/29

19 Actual chaotic systems Example: modified quarter-stadium billiard s r 2 1-s r 1 1 a Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 19/29

20 Variance enhancement over RW (δv αβ ) 2 / (δv αβ,random ) kl Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 20/29

21 Actual chaotic systems: results v αβ variance enhanced by 2 4 over random wave predictions for kl 50 Robust to moderate shape changes No apparent convergence with increasing kl (!) Good: Increased fluctuations consistent with experimental data at low temperatures Good: Support for validity of Hartree Fock Bad: Discrepancy with well-established random wave model Better understanding needed of actual chaotic billiards Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 21/29

22 Actual chaotic systems: results Relation δv 2 αβ = 2 V 2 V still holds V d r d r C2 bill ( r, r ) C2 bill ( r, r ) = intensity correlator for actual billiard (not random waves) Large observable effects on behavior associated with interactions come from subtle correlations within single-particle states How to calculate these correlations? Try semiclassical approach... Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 22/29

23 Semiclassical calculations Correlation C( r, r ) in RW model arises from straight-line path connecting r and r Additional correlation terms from bouncing paths (Hortikar & Srednicki, Urbina & Richter) s r 2 r 1 1-s 1 a Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 23/29

24 Semiclassical calculations Intensity correlator: C sc ( r, r ) = 1 V 2 2 β [ J 2 0(k r r ) + O T clas /T B = correlation time / bounce time ( )] Tclas 1 T B kl δv 2 αβ = 2 3 π ( ) 2 [ 2 ln kl + bg + b sc β (kl) 2 + O ( 1 )] (kl) 3 b sc : formally T clas /T B ; in practice, typically large and overwhelms universal lnkl Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 24/29

25 Semiclassical calculations Semiclassically predicted scaling not observed at all for kl 100 Reason: Formally subleading O(1/(kL) 3 and higher-order terms comparable to leading one Numerical confirmation: quantum maps S Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 25/29 N

26 Short-time calculations Naive semiclassical expressions do not work Nevertheless, we expect (or hope) that IME statistics can be reliably computed using short-time information (few bounces) To have predictive power, statistics must depend only on coarse-scale geometry Confirmed by robustness of results for perturbed modified stadium billiards In maps, v αα (=IPR) may be reliably computed using T t= T α α(t) 2 v αα = v αα, RMT T t= T α α(t) 2 RMT Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 26/29

27 Short-time calculations of v αα 1 λ = 1/8 λ = 1/4 Semiclassical v αα N Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 27/29

28 Summary Observable properties of interacting system computable in terms of single-electron wave function correlations Simple expressions for IME fluctuations in random wave limit Non-Gaussian distribution of IMEs Failure of random wave picture for experimentally relevant system sizes Underestimates v αβ variance by factor of 3 4 Predicts wrong sign for covariance, v αα 3 Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 28/29

29 Summary Dynamical effects essential to obtain agreement with experiment Inadequacy of leading-order semiclassics for computing these effects Hope for robust predictions using short-time dynamics combined with long time RMT Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8, 2008 p. 29/29