Quantum Impurities out of equilibrium: (Bethe Ansatz for open systems)

1. Dresden, April 2006 Quantum Impurities out of equilibrium:(Bethe Ansatz for open systems) Pankaj Mehta & N.A.

2. Outline

3. Non-equilibrium Dilemmas • Nonequilibriumsystems are relatively poorly understood compared to • their equilibrium counterpart. • No unifying theory such as Boltzman's statistical mechanics • Many of our standard physical ideas and concepts are not applicable • Non-equilibrium systems are all different- it is unclear • what if anything they all have in common. • Interplay between non-equilibrium dynamics and strong correlations

4. Non-equilibrium Dilemmas • Nonequilibrium physics is difficult and compared with equilibrium • physics is poorly understood • No unifying theory such as Bolzman's statistical mechanics • Many of our standard physical ideas and concepts are not applicable • Non-equilibrium systems are all different- it is unclear • what if anything they all have in common. • Interplay of non-equilibrium and strong correlations Study simplest systems: • Non-equilibrium Steady-State • Quantum Impurities

5. Kondo Impurities – Strong Correlations out of Equilibrium Inoshita:Science 24 July 1998: Vol. 281. no. 5376, pp. 526 - 527 • Can control the number of electrons on the dot using gate voltage • For odd number of electrons- quantum dot acts like a quantum impurity • (Kondo, Interacting Resonant Level Model) • Quantum impurity models exhibit new collective behaviors such as the • Kondo effect

6. Quantum Impurities out of Equilibrium Strong Correlations = New Collective Behavior (eg Kondo Effect) No valid perturbation theory Need new degrees of freedom = Nonequilibrium Dynamics No Minimization Principle No Scaling/ RG No simple intuition = Need new conceptual and theoretical tools!

7. Quantum Impurities out of Equilibrium

8. Non-equilibrium: Time-dependent Description

9. The Steady State

10. Non-equilibrium: Time-independent Description

11. Scattering States (QM) • Since we are in a steady-state, can go to a time-independent picture. • Scattering by a localized potential isgiven by theLippman-Schwinger equation:

12. The Scattering state (Many body) A scattering eigenstate is determined by its incoming asymptotics: the baths The wave-function schematically: (the outgoing asymptotics needs to be solved) Must carry out construction for a strongly correlated system.

13. The Scattering State (Many body) To construct the nonequilibrium scattering state, it is useful to unfold the leads so that there are only right-movers: The scattering eigenstate determined by N1 incoming electrons in lead 1, and N2 electrons in lead 2 (determined by m1 and m2 )

14. The Scattering Bethe-Ansatz . .

15. IRL: The Scattering State I .

16. IRL: The Scattering State II .

17. The Scattering State III .

18. Bethe Anstaz basis vs. Fock basis • Energy levels are infinitely degenerate (linear spectrum) • Once again the momentum are not specified - need choose basis • We must choose the momenta of the incoming particles to look like two free Fermi seas S=1 S≠1 S-Matrix Bethe-Ansatz Basis Basis Fock Basis Fermi-sea Momenta Bethe –Ansatz distribution Fermi – Dirac distribution

19. IRL: Current & Dot Occupation

20. IRL: Current vs. Voltage • Exact current as a function of Voltage numerically • Notice the current is non-monotonic in U, with duality between • small and large U • Scaling - out of equilibrium • Can easily generalize to finite temperature

21. IRL: Current vs. Voltage • Exact current as a function of Voltage: • Notice the current is non-monotonic in U, with duality between small and large U • Can easily generalize to finite temperature case GENERAL FRAMEWORK TO CALCULATE STEADY-STATE QUANTITIES EXACTLY!

22. IRL: Current vs. Voltage

23. Kondo: The Current (in progress) Must solve BA equations: In continuum version (Wiener-Hopf):

24. Kondo: The Current (in progress) The Current: Evaluated in the scattering state:

25. Conclusions