Quantum Impurities out of equilibrium: Bethe Ansatz for open systems

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Quantum Impurities out of equilibrium(Bethe
Ansatz for open systems)
Pankaj Mehta N.A.
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Outline
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Non-equilibrium Dilemmas

• Nonequilibrium systems 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

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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

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Kondo Impurities Strong Correlations out of
Equilibrium
InoshitaScience 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

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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!
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Quantum Impurities out of Equilibrium
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Non-equilibrium Time-dependent Description
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The Steady State
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Non-equilibrium Time-independent Description
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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

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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.
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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 )
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The Scattering Bethe-Ansatz
.
.
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IRL The Scattering State I
.
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IRL The Scattering State II
.
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The Scattering State III
.
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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

S1
S?1
S-Matrix
Bethe-Ansatz Basis
Basis
Fock Basis
Fermi-sea Momenta
Bethe Ansatz distribution
Fermi Dirac distribution

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IRL Current Dot Occupation
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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

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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!
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IRL Current vs. Voltage
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Kondo The Current (in progress)
Must solve BA equations
In continuum version (Wiener-Hopf)
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Kondo The Current (in progress)
The Current
Evaluated in the scattering state
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Conclusions