Interacting impurity out-of- equilibrium: an exact solution Edouard Boulat Université Paris Diderot Collaborators: Hubert Saleur, Peter Schmitteckert
Outline Non-equilibrium in impurity models: • Background • General framework Introduction of the IRL model Analytical approach: TBA Numerical approach: td-DMRG
Background Out-of-equilibrium in quantum impurities • Keldysh approach: perturbative / hard to resum • Dressed TBA (Quantum Hall edge states tunneling) (P.Fendley, A.W.W.Ludwig, H.Saleur 1995) • Map to equilibrium problem (boundary sine Gordon model) (V.Bazhanov, S.Lukyanov, A.B.Zamolodchikov 1999) • Effectively non-interacting system (Toulouse point) (A. Komnik, O. Gogolin 2003) • Scattering Bethe Ansatz (IRLM, Anderson model) (P.Mehta, N.Andrei 2006) • “Impurity conditions” (IRLM) (B.Doyon 2007) 2d INSTANS Summer Conference - Florence 2008
General framework (1) (2) • No interaction: Landauer Büttiker formula T ( E ) I dE f 1 ( E ) f 2 ( E ) scattering approach: Fermi functions Landauer-Büttiker formula for electrons in wires (1) and (2) transmission probability 2d INSTANS Summer Conference - Florence 2008
General framework (1) (2) • No interaction: Landauer Büttiker formula T ( E ) I dE f 1 ( E ) f 2 ( E ) • Interaction: particle production ! + + … 2d INSTANS Summer Conference - Florence 2008
General framework + + … Approach: describe the baths (Hilbert space of the wires) in terms of quasiparticles with the following properties: “equilibrium” (i) they diagonalize the scattering on the impurity, integrability no particle production (diagonal boundary scattering) further (severe) (ii) they survive out of equilibrium. requirement not destroyed by the voltage use the Landauer Büttiker formula for this gas of (interacting) quasiparticles to compute the current. non-Fermi bath 2d INSTANS Summer Conference - Florence 2008
Impurity model: IRLM I nteracting R esonant L evel M odel • Simplest quantum impurity model supporting both interactions and non-equilibrium • Describes strongly polarized electrodes ( spinless ) coupled to nanostructure via: • tunnelling: , 1 2 U • Coulomb repulsion: 1 2 (1) (2) U U Resonance: V G = V /2 I V G V 2d INSTANS Summer Conference - Florence 2008
IRLM (2) Single channel → mapping to 1D (1) (2) Free 1D wire: Free 1D wire: Resonant level: d 1 2 H H 0 H B H V H V V † † † dx 1 2 ( x ) H iv dx ( x ) 1 2 2 0 F a x a - a 1 , 2 † (0) 2 † (0) † † d † d 1 d d † d H B 1 d U : 1 :(0) : 2 :(0) 1 2 1 2 2 d =0 at resonance 2d INSTANS Summer Conference - Florence 2008
Mapping to Kondo Integrable (in equilibrium) (V.Filyov, P.Wiegmann 1978) † d S Mapping to anisotropic Kondo † z 1 d d S model (P.Wiegmann, A.M.Finkel’stein 1980) 2 Kondo temperature T K ↔ Hybridization temperature T B 0 T B Strong Weak T coupling coupling Question: out-of-equilibrium + strong coupling? 2d INSTANS Summer Conference - Florence 2008
Bosonization (I) even/odd basis 1 ( ) e 1 1 2 2 1 ( ) o 2 1 1 2 H ( ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (I) i 4 e a even/odd a basis bosonization 1 ( ) e 1 1 2 2 1 ( ) o 2 1 1 2 H ( ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (I) unitary transformation: cancels interaction along S z i 4 e z a U even/odd i ( ) S ( )( 0 ) U e o e a basis bosonization 1 ( ) e 1 1 2 2 1 ( ) o 2 1 1 2 H ( ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (I) unitary transformation: cancels interaction along S z i 4 e z a U even/odd i ( ) S ( )( 0 ) U e o e a basis bosonization 1 ( ) e 1 1 2 2 ( 1 ) ( 2 U ) U 1 ( ) e o 2 change o 2 1 1 2 8 D H ( ) ( 1 ) ( 2 U ) U of basis o e 2 8 D 2d INSTANS Summer Conference - Florence 2008
Bosonization (I) unitary transformation: cancels interaction along S z i 4 e z a U even/odd i ( ) S ( )( 0 ) U e o e a basis bosonization 1 ( ) e 1 1 2 2 ( 1 ) ( 2 U ) U 1 ( ) e o 2 change o 2 1 1 2 8 D H ( ) ( 1 ) ( 2 U ) U of basis o e 2 8 D • decouples anisotropic Kondo model • scaling dimension H H ( ) H ( ) H 2 1 1 U 1 ( ) ( ) D 1 0 0 B 4 4 4 1 → ( ) T B 8 ( 0 ) i D 1 D H e S h.c. B • duality U 2 U (A.Schiller, N.Andrei 2007) 2d INSTANS Summer Conference - Florence 2008
Voltage operator (1) Simple theory: diagonal boundary scattering H = H + H- trivial scattering Kondo scattering BUT: quasiparticles DESTROYED by the voltage sin H V V 2 D x (1 U ) x D (1 U ) cos 2 sin 2 i / 2 i e H- H + 1 2 → • mixes and • worse: non-local wrt. the Kondo soliton creation operator (exceptions: D =1/2, D =1/4 ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (II) wire 1 wire 2 U(1) SU ( 2 ) 1 † † z relative charge 1 J ( ) iso-spin → 1 1 2 2 2 total charge † mix the wires J 1 2 2d INSTANS Summer Conference - Florence 2008
Bosonization (II) wire 1 wire 2 U(1) SU ( 2 ) 1 † † z relative charge 1 J ( ) iso-spin → 1 1 2 2 2 total charge † mix the wires J 1 2 H ( ) bosonization i 4 e 1 ( 2 ) 1 ( 2 ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (II) wire 1 wire 2 U(1) SU ( 2 ) 1 † † z relative charge 1 J ( ) iso-spin → 1 1 2 2 2 total charge † mix the wires J 1 2 H ( ) iso-spin/charge basis c 2 ( 1 2 ) 1 bosonization 2 ( 1 2 ) 1 i 4 e 1 ( 2 ) 1 ( 2 ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (II) wire 1 wire 2 U(1) SU ( 2 ) 1 † † z relative charge 1 J ( ) iso-spin → 1 1 2 2 2 total charge † mix the wires J 1 2 H ( ) unitary transformation iso-spin/charge basis z U 2 i ( ) S ( 0 ) U c e c 2 ( 1 2 ) 1 bosonization 2 ( 1 2 ) 1 i 4 e 1 ( 2 ) 1 ( 2 ) 2d INSTANS Summer Conference - Florence 2008
Bosonization (II) wire 1 wire 2 U(1) SU ( 2 ) 1 † † z relative charge 1 J ( ) iso-spin → 1 1 2 2 2 total charge † mix the wires J 1 2 H ( ) unitary transformation iso-spin/charge basis z U 2 i ( ) S ( 0 ) U g c e i 2 e c 2 ( 1 2 ) 1 convert to bosonization g e i 2 2 ( 1 2 ) SU(2) 1 variables 1 i 4 e 1 ( 2 ) 1 ( 2 ) 2d INSTANS Summer Conference - Florence 2008
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