Quantum Impurities Out of Equilibrium Natan Andrei With collaborators: P. Mehta - Princeton C. Bolech - Rice A. Jerez - NJIT S.-P Chao - Rutgers G. Palacios - Rutgers Florence, September 2008
Quantum Impurities out-of-Equilibrium • The quantum impurity - experimentally : Goldhaber-Gordon et al, Conenwett et al, Schmid et al • Couple impurity to leads with • Non-equil steady state (NESS) is established: - current’s flow is time independent (after transients) • Measure non-equil current in steady state • The quantum impurity - theoretically : - How to compute Leads = Fermi seas, Non- -equilibrium equilibrium Non
Quantum Impurity Hamiltonian (3d 1d) � � Impurity Hamiltonian (3d): Impurity Hamiltonian (3d): Unfold 3d Hamiltonian 1d field theory: 3- -d d 3 1- -d (R d (R- -movers) movers) Affleck Ludwig 95’ 1 Field Theory of chiral electrons (R-movers): � Impurity Hamiltonian (1d): Low-energy universality
Non-equilibrium: Time-dependent Description Given - - how to set up the non how to set up the non- -equilibrium problem? equilibrium problem? Given Keldysh Keldysh Description of Nonequilibrium requires two elements: , or , ; Equilibrium requires only . For T > 0 : The initial condition at T=0: The initial condition at T=0: For T = 0 :
The Steady State ( open system limit) Non-equilibrium steady states (NESS): when do they occur? ● Leads good thermal baths, infinite volume limit - open system no IR divergences, (B Doyon, NA, PRB ‘05) (order by order in P.T.) Open system limit : ● Dissipation mechanism ● Time-reversal sym. breaking ● Steady-state non- eq. currents I I A steady state ensues A steady state ensues
The Steady State – time independent description The open system open system limit limit : : The a well defined state. a well defined state. Properties: P. Mehta, N.A. PRL 96 , ‘ 06
The Non-equilibrium Steady State - Non-equilibrium T=0 steady state is described by: - Non-equilbrium value: L-S ● For T=0, g.s. of L-S ● Generally, where • For T>0, • For T>0, and: In steady state - “non-thermal” density operator! cf. Hershfield ‘93 - In equilibrium : (Keldysh Boltzmann ) Doyon, N.A. ‘05
Steady-states & Scattering States • Time • dependent ( (Keldysh Keldysh) vs. ) vs. time Scattering) approach ( Scattering) Time- -dependent time- -independent independent approach ( -Keldysh Keldysh approach approach - -Scattering approach Scattering approach - - Scattering approach: non-perturbative Keldysh • The scattering eigenstate describes all aspects of non-equilibrium steady-state physics (NESS): - non-equilibrium currents, - energy dissipation, - entropy production Q : How can an eigenstate describe dissipation, entropy production? A: Scattering eigenstate describes both system and environment (open system)
Entropy production and Dissipation Non-equilibrium currents dissipate heat into environment: ● Scattering state describes system + environment ● Dissipation mechanism: electrons reaching infinity ● Lost high energy electrons generate entropy (entanglement? ) Entropy is produced quasi-statically: - currents ~ 1 - leads ~ L infty
Entropy production and Dissipation ● “Thermodynamic” approach : (discontinuous system - defined w.r.t. quasi-equil , L ~ infty) No accumulation in dot: ● “ Boltzmannian” approach – (distributions) scattering change of distribution: nonequilibrium nonequilibrium equilibrium equilibrium distributions distributions distributions distributions • “ • “ Information Theory Information Theory” ” approach approach – (in the infinite volume limit) : : – (in the infinite volume limit) P. Mehta, N. A. PRL100, ‘ 08 Mixing + Relaxation Mixing + Relaxation mixing relaxation mixing relaxation mixing = mixing = relaxation = relaxation = Kullback- -Leibler Leibler divergence: divergence: Kullback - - amount of work obtained amount of work obtained when relaxes to when relaxes to • Entropy production rate strictly positive, • Entropy production rate strictly positive,
The Scattering Bethe-Ansatz Nonequilibrium described by open-system eigenstates Recent developments: Freq dep-RG TD-DMNRG, TD-DMRG, FRG, Flow-eq . Develop a Bethe Ansatz approach to non-equilibrium : � Traditional � Traditional Bethe Bethe- -Ansatz Ansatz - - inapplicable inapplicable - Periodic boundary conditions Periodic boundary conditions - . - Closed System : Equilibrium, Thermodynamics - Closed System : Equilibrium, Thermodynamics � New technology � New technology Scattering States Scattering States - Asymptotic Boundary conditions on the infinite line Asymptotic Boundary conditions on the infinite line - - Open System : Non- -equilibrium, scattering problems equilibrium, scattering problems - Open System : Non Scattering (Open) Bethe Bethe- -Ansatz Ansatz: : Scattering (Open) 1. Non-equil Interacting Resonance Level model (Non-equil FES) 2. Non-equil Anderson model (Quantum Dot – Non-equil Kondo effect)
The Interacting Resonance Level model out-of-equilibrium ● Non-equil IRL Model: ● Non-equil FES Geim et al 93’ • The 1-d Field Theory Thermodynamic BA Filyov, Wiegman 80’ Diagonalize H via the Open Bethe-Ansatz: - directly on the infinite line (open system) � construct 1-particle eigenstates (with boundary conditions) . � construct N-particle eigenstates out of 1-particle states
IRL: The Scattering State Level width: Phase shift : Single-particle scattering eigenstates - Trans. coeff . Impurity amp. Reflec. amp. Trans. amp. Renormalization prescription Local discontinuity Boundary condition constant - consistent with prescription
IRL: The Scattering State Multi-particle scattering state - N 1 lead-1, N 2 lead-2, with ● eigenstate of for any choice of Bethe momenta . . ● Choose distributions to impose non-eq BC: - incoming particle arrive from free leads at • Distributions must satisfy SBA equation . ( Free baths Fermi-Dirac in Fock basis . Here – free baths in Bethe basis )
The Boundary Conditions II The Boundary Conditions II The boundary conditions become OBA equations for: Bethe chemical potentials determined from minimizing: ● For U=0 distributions reduce to Fermi-Dirac distributions These are OBA eqns for: , (in co-tunneling regime ) - otherwise, eqns more complicated – include complex solutions (Non-equil FES)
Current and Dot Occupation The scattering state is determined in terms of Hybridization width • For U=0, Landauer-Buttiker formulas • For U>0, in the Bethe-basis, expressions look “simple”: - excitations undergo phase shifts only - incorporate interactions and boundary conditions
Current vs. Voltage IRL ● Compute exactly current as a function of Voltage : Non-monotonicity in U - FES : repulsion vs IR Catastrophe ( Borda et al ) Fermi Edge Singularity - Duality : ( Schiller NA) out of equilibrium (Geim et al ’93) (Matveev & Larkin, Levitov, Abanin..) Other approaches: 1. Perturbative RG Borda, Zawadowski 06’ 2. Perturbative expansion Doyon 07’ 3. DMRG Scmitteckert 07’ 4 . Model at self-dual point (U=2) BSG : BA + dressed Landauer : Boulat, Saleur (08’) cf Fendley, Ludwig, Saleur 95’
The Quantum Dot - equilibrium ● Can control number of electrons on dot using gate voltage R L ● For odd number of electrons- U quantum dot acts as quantum impurity ● New collective behaviours , e.g. Kondo effect - formation of narrow peak at the Fermi surface as T 0 van der Wiel et al. Conductance vs Science 2000 Gate voltage Kondo effect - zero bias (equilibrium): filling of odd valleys of Coulomb Blockade at low T T 0
The Quantum Dot - nonquilibrium Conductance vs. bias voltage van der Wiel et al. , Science 2000 • Questions: - The nonequilibrium Kondo Effect - Effects of temperature, - magnetic field - DOS in and out of equilibrium - Decoherence ● Nonequilibrium Anderson model: Equilibrium BA: Wiegmann & Tsvelik, Kawakami & Okiji ‘80-’83 ● Solve: 02 - ‘ 02 Previous attempt - - Konik Konik, Ludwig, , Ludwig, Saleur Saleur ‘ - valid only close to equilibrium valid only close to equilibrium Previous attempt Approach: Landauer Landauer + + dressed dressed BA, BA, Approach: developed by Fendley developed by Fendley Ludwig Ludwig Saleur Saleur ‘ ‘95 95 - Based on - Based on dressed excitations dressed excitations : : holon holon, , spinon spinon. But voltage in leads acts on . But voltage in leads acts on bare electrons bare electrons - Approximation - Approximation : : electron ~ electron ~ spinon spinon + + holon holon. No spinon-antispinons, no holon-antiholons. - Approximation invalid in general - except for for V=0, (cf N.A. ’82)
Anderson Model of the single-level Quantum Dot Anderson Model of the single-level Quantum Dot Anderson model out equilibrium: Open Bethe Ansatz (H=0 T=0) • Similar construction of scattering eigenstates • Bethe momenta - complex strings • Satisfying • Four types of momentum-strings: 11, 12, 21, 22 described by distributions • Distributions determined by SBA-eqn : free leads in Bethe basis • Bethe chemical potentials: determined by physical potentials , minimizing
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