Transcript Non-equilibrium dynamics of quantum impurities

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:
1
The scattering eigenstate determined by N1 incoming electrons in
lead 1,
and N2 electrons in lead 2 (determined by 1 and 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 (WienerHopf):
Kondo: The Current (in progress)
The Current:
Evaluated in the scattering state:
Conclusions