Quantum Impurities out of equilibrium Bethe Ansatz for
Quantum Impurities out of equilibrium: (Bethe Ansatz for open systems) Pankaj Mehta & N. A. Dresden, April 2006
Outline
Non-equilibrium Dilemmas Nonequilibrium systems are relatively poorly understood compared 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 unclea what if anything they all have in common. Interplay between non-equilibrium dynamics and strong correlations ●
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 unclea what if anything they all have in common. Interplay of non-equilibrium and strong correlati Study simplest systems: Non-equilibrium Steady-State ● Quantum Impurities ●
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 impur (Kondo, Interacting Resonant Level Model) ●Quantum impurity models exhibit new collective behaviors such as th Kondo effect ●
Quantum Impurities out of Equilibrium Strong Correlations = New Collective Behavior (eg Kondo Effect) Nonequilibrium Dynamics = No valid perturbation theory Need new degrees of freedom = No Minimization Principle No Scaling/ RG No simple intuition Need new conceptual and theoretical tools!
Quantum Impurities out of Equilibrium
Non-equilibrium: Time-dependent Description
The Steady State
Non-equilibrium: Time-independent Description
Scattering States (QM) ● Since we are in a steady-state, can go to a time-independent picture. Scattering by a localized potential is given by the Lippman-Schwinger equation: ●
The Scattering state (Many body) A scattering eigenstate is determined by its incoming asymptotics: the b The wave-function schematically: (the outgoing asymptotics needs to be solved) Must carry out construction for a strongly correlated system.
The Scattering State (Many body) To construct the nonequilibrium scattering state, it is useful to unfold the lea so that there are only right-movers: The scattering eigenstate determined by N 1 incoming electrons in lead 1, and N 2 electrons in lead 2 (determined by m 1 and m 2 )
The Scattering Bethe-Ansatz . .
IRL: The Scattering State I .
IRL: The Scattering State II .
The Scattering State III .
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-Matrix Basis Fermisea Moment a S=1 S≠ 1 Fock Basis Bethe-Ansatz Basis Fermi – Dirac distribution Bethe –Ansatz distribution
IRL: Current & Dot Occupation
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 ●
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 STEADYSTATE QUANTITIES EXACTLY!
IRL: Current vs. Voltage
Kondo: The Current (in progress) Must solve BA equations: In continuum version (Wiener. Hopf):
Kondo: The Current (in progress) The Current: Evaluated in the scattering state:
Conclusions
- Slides: 25