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

Dresden, April 2006

Pankaj Mehta & N.A.

Outline

Non-equilibrium Dilemmas

● Many of our standard physical ideas and concepts are not applicable

● No unifying theory such as Boltzman's statistical mechanics

● Non-equilibrium systems are all different- it is unclear what if anything they all have in common.

● Nonequilibrium systems are relatively poorly understood compared to their equilibrium counterpart.

● Interplay between non-equilibrium dynamics and strong correlations

Non-equilibrium Dilemmas

● Nonequilibrium physics is difficult and compared with equilibrium physics is poorly understood

● Many of our standard physical ideas and concepts are not applicable

● No unifying theory such as Bolzman's statistical mechanics

● Non-equilibrium systems are all different- it is unclear what if anything they all have in common.

● Interplay of non-equilibrium and strong correlations

● Non-equilibrium Steady-State● Quantum Impurities

Study simplest systems:

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

Quantum Impurities out of Equilibrium

Strong Correlations =New Collective Behavior (eg Kondo Effect)

=

No Minimization Principle No Scaling/ RG No simple intuition

Nonequilibrium Dynamics =

No valid perturbation theory Need new degrees of freedom

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)

● Scattering by a localized potential is given by the Lippman-Schwinger equation:

● Since we are in a steady-state, can go to a time-independent picture.

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.

The Scattering State (Many body)

1

To construct the nonequilibrium scattering state, it is useful to unfold the leadsso 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 and )

The Scattering Bethe-Ansatz

.

.

IRL: The Scattering State I

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IRL: The Scattering State II

.

The Scattering State III

.

Bethe Anstaz basis vs. Fock basis

Fock Basis Bethe-Ansatz 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

Fermi – Dirac distribution

Bethe –Ansatz distribution

S=1 S≠1

Basis

Fermi-seaMomenta

S-Matrix

IRL: Current & Dot Occupation 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 STEADY-STATE 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