Institut de Physique et Chimie des Matériaux de Strasbourg

1. Transmission phase in quantum transport: disorder, chaos and correlation effects Rodolfo A. Jalabert Institut de Physique et Chimie des Matériaux de Strasbourg Philippe Jacquod (Arizona) Rafael A. Molina (Madrid) Peter Schmitteckert (Karlsruhe) Dietmar Weinmann (Strasbourg)

2. CAN WE MEASURE THE SCATTERING PHASE IN A QUANTUM DOT ? Conductance of a quantum dot connected to monochannel leads: Conductance of a two-lead Aharonov-Bohm interferometer: Time reversal: T(Φ)=T(-Φ)  the phase β is locked at 0 or π

3. TWO-LEAD PHASE SENSITIVE MEASUREMENTS The phase is locked at 0 or π trivial ! Levy-Yeyati, Büttiker, PRB’95 Different peaks are in phase mistery !! A. Yacoby et al, PRL’95

4. MULTY-LEAD PHASE SENSITIVE MEASUREMENTS The phase increases continuously by π at each resonance (Friedel sum rule) Different peaks are in phase: π lapses when the transmission vanishes !!! R. Schuster et al, Nature’97

5. CROSSOVER FROM MESOSCOPIC TO UNIVERSAL PHASE Mesoscopic regime: Small dots N < 10, random behavior of phase jumps and lapses M. Avinum-Kalish et al, Nature’05 Universal regime: Large dots N > 14, correlated behavior between phase jumps and lapses

6. quantum dot TRANSPORT THROUGH AN INTERACTING QUANTUM DOT IN THE REGIME OF COULOMB BLOCKADE Charging energy U = e2/C > kBT • Two approaches: • Constant interaction model: ΔVp = U + ΔE(1) • reduced to a one-body problem • 2) Full many-body approach, include electronic correlations

7. PHASE EVOLUTION BETWEEN TWO RESONANCES partial-width amplitude: total-width:

8. PHASE EVOLUTION IN THE COMPLEX PLANE same parity: opposite parity:

9. L ZEROS OF THE TRANSMISSION AND PHASE LAPSES Dn>0 large fluctuations in the partial-width amplitudes, unrestricted off-resonance behavior (UOR): 2 zeros Dn<0

10. PARITY RULE FOR THE TRANSMISSION ZEROS Levy-Yeyati and Büttiker, PRB 2000 • If Dn > 0 there is a zero between the n and the n+1 resonance • If Dn < 0 there is no zero between the n and the n+1 resonance For a disordered quantum dot Experimentally (universal regime) Correlations between wave-functions of different eigenstates ?

11. BERRY CONJECTURE FOR WAVE-FUNCTION CORRELATIONS Random Wave Model for a chaotic billiard: independent random phases uniformly distributed vectors of magnitude k Bessel function for disordered systems

12. UNIVERSAL BEHAVIOR IN THE SEMICLASSICAL LIMIT transmission zero (and phase lapse) between the n and the n+1 resonance Statistical independence of different eigenstates + Berry’s conjecture: Two-dimensional billiard L Probability of not having a phase lapse:

13. L NUMERICAL CALCULATIONS: TRANSMISSION AND PHASES transmission scattering phase accumulated phase plateaus with the same number of resonances and transmission zeros

14. probability of having more resonances than zeroes in a k-interval IS THERE ALWAYS A ZERO BETWEEN TWO RESONANCES ? Nr and Nz grow with the same rate in the semiclassical limit ! number of resonances number of zeroes The probability of observing out-of-phase resonances vanishes as 1/kL

15. VALIDITY OF FRIEDEL SUMMATION RULE number of resonances At finite magnetic field the transmission does not vanish accumulated phase A finite field B lifts the ambiguity in the definition of the accumulated phase

16. ARE CORRELATION NEEDED ? A quantitative description requires correlations to be treated accurately mesoscopic to universal behavior by changing the ratio δ/Γ, provided that U >> Γ

17. STRONGLY CORRELATED SCATTERERS: CONDUCTANCE AND PHASE FROM THE EMBEDDING METHOD Persistent current: ground-state property Scattering phase from the ground state energy of rings with different number of electrons

18. DMRG CALCULATION OF CONDUCTANCE AND PHASE FOR A SIMPLE QUASI-1D SCATTERER Spinless electrons with nearest neighbor interaction U e U=0 -e

19. CAN INTERACTIONS INDUCE UOR BEHAVIOR? Three-level system with large fluctuations among the level couplings: U=0 U=2 UOR

20. INTERACTIONS & CORRELATIONS U=0 N=6 U=2 UOR behavior extra zeros (in pairs) incomplete filling of resonances conductance peaks ≠ resonances

21. INTERACTION EFFECTS IN SMALL QUANTUM DOTS N=8 Disordered quantum dot: random in-site energies non arbitrary couplings U=0 U=2 No new zeros induced by the interactions

22. CONCLUSIONS • Indirect determination of the transmission phase by transport experiments in multilead rings • Phase lapses of π when the transmission vanishes random at low N: mesoscopic regime regular for higher N: universal regime • Universal behavior emerges with probability 1-1/kL • Difference chaotic vs. disordered • The wave-function correlations • are responsible for the universal behavior, not the electronic correlations R.A. Molina, R.A. Jalabert, D. Weinmann, Ph. Jacquod, Phys. Rev. Lett. 2012 R.A. Molina, P. Schmitteckert, D. Weinmann, R.A. Jalabert, Ph. Jacquod, unpublished 2012

23. EMBEDDING METHOD: DETAILS AND IMPLEMENTATION one-particle eigenstates of the ring: transfer matrix of the sample transfer matrix of the lead 1/L expansion of the ground state energy Also, persistent current and scattering phase

24. EMBEDDING METHOD: EFFECTIVE ONE-PARTICLE SCATTERING The interacting region acts as a local non-interacting scatterer