Quantum Impurities In and Out of Equilibrium Natan Andrei HRI 1- Feb 2008
Quantum Impurity Quantum Impurity - a system with a few degrees of freedom interacting with a large (macroscopic) system. Often such systems are strongly correlated: - cannot represent the system as a collection of free particles + weak perturbations Unexpected experimental phenomena Perturbation theory fails New degrees of freedom emerge, collective behavior New theoretical ideas Strong Correlations in equilibrium - Lectures 1,2 Trieste Lectures (1992) N. A. Strong correlations out of equilibrium - Lecture 3 PRL 96, 216802 (2006) P. Mehta, N.A
Quantum Impurity A system with a few degrees of freedom interacting with a large (macroscopic) system Examples: Spin-1/2 impurity embedded in a metal Metal ( Fermi gas of electrons) Quantum dot tiny puddle of electrons coupled to leads - when number of electrons on dot is odd ~ spin -1/2 Both examples exhibit - Kondo effect in and out of equilibrium, resp.
The Kondo Effect -equilibrium Measurements of electric resistance of a metal with dilute concentration of magnetic impurities: Enhanced scattering at low T T 1/5 min c imp De Haas & ven den Berg, 1936
The Kondo Effect - equilibrium Note the upturn at low-t, as opposed to the pure metal behavior: Scattering stronger at low temperatures! Similar behavior in impurity magnetic susceptibility: At high temperatures - free spin susceptibility At low temperatures - spin is screened
The Kondo Hamiltonian 1 To describe the system - Conduction band of electrons (metal) Exchange interaction - electron spin with a local impurity spin: Antiferromagnetic coupling :
The Kondo Hamiltonian 2 Rewrite the Hamiltonian as 1-dim field theory: 1-d problem: keep only s-wave (in) (out) Field Theory if: All scales Bandwidth D universal results - independent of band structure The Kondo Hamiltonian: spherical modes S-waves linearize around Fourier transform (w.r.t.q)
Moment formation The Anderson Impurity 1 Where does the localized spin -1/2 come from? ε d t ε d + U - localized level
Moment formation- The Anderson Impurity 2 Energy scales: Inter-configurational energies ε d and U+ε d Hybridization width Γ = πρv 2 Condition for formation of local moment: Γ << ε d, U + ε d Kondo screening 2Γ 1 ρ J = π ε d Free local moment + U 1 + ε d Charge fluctuations Schrieffer & Wolff 1966 0 T K T LM ε d T Kondo model
The Kondo Effect Study by perturbation theory J. Kondo 64 Perturbation theory breaks down at low temperatures (i.e. in the IR) Resummation?
The Kondo Effect Sum the leading logs of perturbation theory Perturbation theory breaks down at low temperatures (i.e. in the IR) Perturbation theory valid at high temperatures A new low energy scale appears: Strong coupling IR, weak coupling UV How to handle such a theory? The Kondo Problem
The Kondo Problem Many approaches to the Kondo Problem (equilibrium): - Resummation of the perturbation series (fails) - Variational techniques (fail) - Scaling theory (P. W. Anderson) - Renormalization group (K. Wilson) - Fermi Liquid theory of Strong Coupling (P. Nozieres) - Boundary conformal field theory (I. Affleck, A. Ludwig) - Bosonization (A. Luther, I. Peschel, G. Toulouse) - Exact solution - (N. Andrei, P. Wiegman)
The Kondo Problem (equilibrium) What did we learn? - Thermodynamics:
The Kondo Problem Anderson model What did we learn? -The spectral structure (equilibrium) Kondo resonance T K T K πεd ( U + ε d ) exp 2ΓU ε d E F ε d +U A sharp resonance of width T K develops at E F for T<T K Unitary scattering for T=0 and <n>=1 1 π A( ε = 0, T = 0) = sin ( δ ), δ = π Γ 2 2 n
Quantum Impurities out of Equilibrium The quantum dot - experimental The quantum dot - theoretical L Lead U R Lead V g
Quantum Impurities out of Equilibrium H imp = ε d a n a + Un { + t } i daψ ia(0) h.c. n + + i= L, R a Inter-configurational energies ε d and U+ε d 2 2 Hybridization width Γ = ( t + ) Condition for formation of local moment: Γ << ε, U ε d + d πρ L t R
Quantum Impurities out of Equilibrium van der Wiel et al., Science 2000 Conductance vs gate voltage - Kondo enhancement in odd valleys Equilibrium T varies in the range 15-800mK Differential conductance vs bias - Nonequilibrium dynamics Nonequilibrium
Quantum Impurities out of Equilibrium Equilibrium experiments - well understood Kondo resonance increases tunneling DOS, enhances conductance For Γ L =Γ R, unitary limit corresponds to perfect transmission: G = 2e 2 /h Nonequilibrium experiments - need new tools
The Bethe Ansatz Approach Can treat Quantum impurities in and out of equilibrium - Equilibrium Full energy spectrum of the Kondo Hamiltonian - ground state, excitations: spinons, holons.. Complete determination all thermodynamic properties, magnetic susceptibility, specific heat - Nonequilibrium Nonequilibrium expectation values (eg. I(V) curves), Entropy production, time evolution, scattering
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium
Quantum Impurities out of Equilibrium