Interaction Matrix Element Fluctuations in Quantum Dots Lev Kaplan Tulane University and Yoram Alhassid Yale University Interaction Matrix Element Fluctuations p. 1/37
Outline Motivation: ballistic quantum dots in Coulomb blockade regime Conductance peak spacing: need interactions Interaction matrix elements and single-particle correlators Random wave model (valid for N ) What happens in actual chaotic dots? Can we compute subleading terms in 1/N? Failure of leading-order semiclassical theory Beyond chaos (time permitting) Summary Interaction Matrix Element Fluctuations p. 2/37
Coulomb Blockade Regime Dot weakly coupled to outside via two leads Low temperature Decay width Temperature Classical charging energy Sharp conductance peaks appear when Fermi energy in leads matches energy needed to move one extra electron onto dot 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 Interaction Matrix Element Fluctuations p. 3/37
Peak Height Distribution Folk et al, PRL (1996) Interaction Matrix Element Fluctuations p. 4/37
Coulomb Blockade Regime In Hartree-Fock approach, many-body energies given by 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 + ɛ N+1 ɛ N followed by e 2 /C) in mean-field model (not observed) Two-body interactions essential for understanding spacings Interaction Matrix Element Fluctuations p. 5/37
Peak Spacing Distribution Patel et al, PRL (1998) Interaction Matrix Element Fluctuations p. 6/37
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. 7/37
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. 8/37
Random wave model (Berry) Typical trajectory in classically ergodic system uniformly explores energy hypersurface in phase space 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. 9/37
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 1 2 + O π β (kl) 2 (kl) 3 V Interaction Matrix Element Fluctuations p. 10/37
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 Geometry-dependent coefficient b g requires normalized correlator C( r, r ) Subleading O(1/(kL) 3 ) corrections are < 10% for systems of experimental interest Dependence of b g on shape is weak (< 5%) )] For one-body matrix element variance δv 2 α, log term is absent, and normalization/subtraction changes answer by factor of 10 Interaction Matrix Element Fluctuations p. 11/37
δv 2 αβ for random waves in disk (δv αβ ) 2 2 10-2 1 10-2 error 2 10-3 1 10-3 5 10-4 2 10-4 1 10-4 30 50 70 kl 5 10-3 30 40 50 60 70 kl Interaction Matrix Element Fluctuations p. 12/37
Weak shape dependence of δv 2 αβ 1.02 (δv αβ ) 2 (r) / (δv αβ ) 2 (1) 1 0.98 0.96 0.94 1 2 4 8 16 r Interaction Matrix Element Fluctuations p. 13/37
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. 14/37
Random wave model: ratios However, subleading coefficients b g b g b g cause logarithmically slow convergence of matrix element ratios to asymptotic (combinatoric) values δv 2 αα/δv 2 αβ = 6 + b g b g ln kl + δv 2 αβ /δv2 αβγδ = 4 + b g b g lnkl + (for real β = 1 case) First indication that g T = kl results may not be very helpful in experimentally relevant regime kl 100 Interaction Matrix Element Fluctuations p. 15/37
Variance ratios for RW 3.2 3 variance ratios 2.8 2.6 2.4 2.2 30 40 50 60 70 kl Interaction Matrix Element Fluctuations p. 16/37
Random waves: covariance Important for spectral scrambling (beyond Hartee-Fock-Koopmans, mean field potential changes as electrons are added to dot) For E β near E γ, covariance can be obtained from completeness relation π δv αβ δv αγ 1.85kL δv2 αβ Negative sign appears to contrast with results for diffusive dots Interaction Matrix Element Fluctuations p. 17/37
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. 18/37
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. 19/37
Skewness and excess kurtosis 5 4 3 γ i 2 1 0 30 50 70 100 140 kl Interaction Matrix Element Fluctuations p. 20/37
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. 21/37
Actual chaotic systems Example: modified quarter-stadium billiard s r 2 1-s r 1 1 a Interaction Matrix Element Fluctuations p. 22/37
Actual chaotic systems: results v αβ variance enhanced by factor 2 4 over random wave predictions for kl 50 Robust to moderate shape changes No apparent convergence with increasing kl Good: increased fluctuations are consistent with experimental data at low temperatures Good: support for validity of Hartree Fock picture Bad: Apparent discrepancy with well-established random wave model Good: Better understanding needed of actual chaotic billiards Interaction Matrix Element Fluctuations p. 23/37
Variance enhancement over RW 4 (δv αβ ) 2 / (δv αβ,random ) 2 3.5 3 2.5 2 1.5 1 30 40 50 60 70 kl Interaction Matrix Element Fluctuations p. 24/37
Actual chaotic systems: results Relation δvαβ 2 = 2 V 2 V still holds V d r d r C2 bill ( r, r ) Here C 2 bill ( r, r ) is intensity correlator for actual billiard (not random waves) Thus, Hartree-Fock picture still consistent with experiment, but we must use actual chaotic single-particle states as input 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. 25/37
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) Semiclassical Green s function (Gutzwiller): G( r, r,e) = j D j 1/2 e is j/ iµ j π/2 Interaction Matrix Element Fluctuations p. 26/37
Semiclassical calculations Amplitude correlator: ψ ( r)ψ( r ) = 1 ] [J 0 (k r r ) + h( r, r )(kl) 1/2 V Here h( r, r ) = j 2pLD j πm 2 1/2 cos ( Sj (2µ ) j + 1)π 4 h( r, r ) 1 for bouncing paths with length L Few-bounce paths contribute at same order in kl as straight-line path, but without logarithmic divergence associated with path length L Interaction Matrix Element Fluctuations p. 27/37
Semiclassical calculations Intensity correlator: C sc ( r, r ) = 1 V 2 2 β Finally, δv 2 αβ = 2 V 2 V = 2 3 π where b sc T clas /T B [ J 2 0(k r r ) + O ( )] Tclas 1 T B kl V d r d r C2 bill ( r, r ) ( ) 2 [ 2 ln kl + bg + b sc β (kl) 2 Net effect is to increase numerical coefficient of 1/(kL) 2 term in the matrix element variance ] Interaction Matrix Element Fluctuations p. 28/37
Semiclassical calculations δv 2 αβ = 2 3 π ( 2 β ) 2 [ ln kl + bg + b sc (kl) 2 + O ( 1 )] (kl) 3 In practice, b sc is typically large and overwhelms universal ln kl (even for T clas /T B 1) Semiclassically predicted scaling not observed 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. 29/37
Actual chaotic billiards 5 10-2 2 10-2 (δv αβ ) 2 1 10-2 5 10-3 2 10-3 30 40 50 60 70 kl Interaction Matrix Element Fluctuations p. 30/37
Variance for quantum maps 10-1 10-2 10-3 S 10-4 10-5 10-6 10 100 1000 N Interaction Matrix Element Fluctuations p. 31/37
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. 32/37
Short-time calculations 1 λ = 1/8 λ = 1/4 Semiclassical v αα - 3 0.1 0.01 100 1000 N Interaction Matrix Element Fluctuations p. 33/37
Beyond chaos Universal behavior not expected Predict and see matrix element fluctuations much larger than in chaotic case Implies much wider distribution of conductance peak spacings, consistent with experiments in small dots (Lüscher) Stadium billiard: bouncing ball modes should dominate fluctuations for large kl: v αβ 2 /kl Mixed chaotic/regular phase space: regular states dominate, δv 2 αβ 2 ( f2 f 2 f f 2 ) 2 (kl-independent) Again, formally subleading terms important for experimentally relevant kl Interaction Matrix Element Fluctuations p. 34/37
Beyond chaos: larger variance 20 18 (δv αβ ) 2 / (δv αβ,random ) 2 16 14 12 10 8 6 4 2 30 40 50 60 70 kl Interaction Matrix Element Fluctuations p. 35/37
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 Predicts wrong sign for covariance, v αα 3 Dynamical effects essential to obtain agreement with experiment Interaction Matrix Element Fluctuations p. 36/37
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 Much wider IME distribution for mixed phase space, consistent with small-dot experiments Interaction Matrix Element Fluctuations p. 37/37