Interaction Matrix Element Fluctuations

Size: px

Start display at page:

Download "Interaction Matrix Element Fluctuations"

Gervais Hawkins
6 years ago
Views:

1 Interaction Matrix Element Fluctuations in Quantum Dots Lev Kaplan Tulane University and Yoram Alhassid Yale University Interaction Matrix Element Fluctuations p. 1/29

2 Outline Motivation: ballistic quantum dots in Coulomb blockade regime Conductance peak spacings: need interactions Computing interaction matrix elements: Relation to single-particle correlators Random wave model (N ) What happens in actual chaotic dots? Failure of leading-order semiclassical theory Can we compute subleading terms in 1/N? Summary Interaction Matrix Element Fluctuations p. 2/29

3 Coulomb Blockade Regime Dot weakly coupled to outside via two leads Decay width Temperature Classical charging energy Sharp conductance peaks when Fermi energy in leads matches energy needed to move one new electron onto dot (N N + 1) 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? Interaction Matrix Element Fluctuations p. 3/29

4 Coulomb Blockade Regime 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 Peak height statistics well explained using constant interaction + chaotic mean field Peak spacing distribution predicted to be bimodal (e 2 /C + ǫ N+1 ǫ N followed by e 2 /C) in mean-field model (not observed) Two-body interactions essential for understanding spacings Interaction Matrix Element Fluctuations p. 4/29

5 Peak Height Distribution Folk et al, PRL (1996) Interaction Matrix Element Fluctuations p. 5/29

6 Peak Spacing Distribution Patel et al, PRL (1998) Interaction Matrix Element Fluctuations 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 δv 2 αβ, etc. To leading order in g T = kl N (L V ), δvαβ 2 = 2 V 2 d r d r C2 ( r, r ) + where V V C( r, r ) = ψ( r) 2 ψ( r ) 2 ψ( r) 2 ψ( r ) 2 Interaction Matrix Element Fluctuations 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 p. 8/29

9 Random wave model (Berry) Typical trajectory in classically ergodic system uniformly explores energy hypersurface Analogously, 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 wave function amplitudes ψ( r) Free-space intensity correlation C( r, r ) = 2 β 1 V 2J2 0(k r r ) Interaction Matrix Element Fluctuations 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 ) + V Satisfies V d r C( r, r ) = 0 (Mirlin) δvαβ 2 = 3 ( ) 2 [ ( )] 2 lnkl + bg O π β (kl) 2 (kl) 3 V Interaction Matrix Element Fluctuations 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 p. 11/29

12 δv 2 αβ for random waves in disk (δv αβ ) error kl kl Interaction Matrix Element Fluctuations 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 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 [ 2 lnkl + bg β (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 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) where c 3 3β is a symmetry-dependent combinatoric factor and b 3g is again a geometry-dependent constant Note: no logarithmic divergences Interaction Matrix Element Fluctuations p. 15/29

16 Matrix element distributions Skewness γ 1 = [ δv 3 αβ ] 3/2 δv 2 αβ = b 3g c 2 3β ( ) 3 β ( π 3/2 2 3) (ln kl) 3/2 + Excess kurtosis γ 2 = δv4 αβ 3 [ δv 2 αβ ] 2 [δv 2 αβ ] 2 ( ( ) ) 4 ( ) = b 4g c 2 4β + 2 π 2 β 3 (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 p. 16/29

17 Skewness and excess kurtosis γ i kl Interaction Matrix Element Fluctuations 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 Hartee-Fock? Interaction Matrix Element Fluctuations 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 p. 19/29

20 Variance enhancement over RW 4 (δv αβ ) 2 / (δv αβ,random ) kl Interaction Matrix Element Fluctuations 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 p. 21/29

22 Actual chaotic systems: results Relation δvαβ 2 = 2 V 2 V V d r d r C2 bill ( r, r ) still holds C2 bill ( r, r ) is intensity correlator for actual billiard (not random waves) Subtle correlations within single-particle states may induce large observable effects on behavior associated with interactions How to calculate these correlations? Try semiclassical approach... Interaction Matrix Element Fluctuations p. 22/29

23 Semiclassical calculations Correlation C( r, r ) in random wave model may be thought of as arising from straight-line path connecting r and r Within semiclassical approach, additional correlation terms arise from paths that bounce off the boundary (Hortikar & Srednicki, Urbina & Richter) Intensity correlator: C sc ( r, r ) = 1 V 2 2 β [ J 2 0(k r r ) + O T clas = correlation time, T B = bounce time ( )] Tclas 1 T B kl Interaction Matrix Element Fluctuations p. 23/29

24 Semiclassical calculations δv 2 αβ = 2 3 π ( 2 β ) 2 [ ln kl + bg + b sc (kl) 2 + O ( 1 )] (kl) 3 Formally b sc T clas /T B In practice, b sc is typically large and overwhelms universal ln kl Semiclassically predicted scaling not observed at all for kl 100 Instead scaling (kl) 1.5 seen Reason: Formally subleading O(1/(kL) 3 and higher-order terms comparable to leading one Expansion parameter is 50/kL Numerical confirmation: quantum maps Interaction Matrix Element Fluctuations p. 24/29

25 Variance for quantum maps S N Interaction Matrix Element Fluctuations p. 25/29

26 Short-time calculations Naive semiclassical expressions do not work Nevertheless, we expect (or hope) that matrix element 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 p. 26/29

27 Short-time calculations 1 λ = 1/8 λ = 1/4 Semiclassical v αα N Interaction Matrix Element Fluctuations p. 27/29

28 Summary Observable properties of interacting systems computable in terms of single-electron wave function correlations Simple expressions for IME fluctuations in random wave limit Non-Gaussian distribution Failure of random wave picture for experimentally relevant system sizes Underestimates v αβ variance by factor of 3 4 Also predicts wrong sign for covariance, v αα 3 Dynamical effects essential to obtain agreement with experiment Interaction Matrix Element Fluctuations p. 28/29

29 Summary Inadequacy of leading-order semiclassics for computing these dynamical effects Hope for robust predictions using short-time dynamics combined with random waves on long time scales Interaction Matrix Element Fluctuations p. 29/29

Interaction Matrix Element Fluctuations

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,