1. Non-equilibrium physics Non-equilibrium physics in one dimension Igor Gornyi Москва Сентябрь 2012 Karlsruhe Institute of Technology

2. Nonequilibrium Bosonization Part II Nonequilibrium Bosonization developed by D.Gutman, Y.Gefen, A. Mirlin ’09-10

3. Strongly correlated state (LL) out of equilibrium – ? No energy relaxation in LL (in the absence of inhomogeneities, neglecting non-linearity of spectrum and momentum dependence of interaction) Equilibrium: exact solution via bosonization. Non-equilibrium – ? Fermionic distribution within the bosonization formalism – ?

4. Bosonization

7. Functional bosonization Hubbard-Stratonovich transformation decouples quartic interaction term 1D: gauge transformation with eliminates coupling between fermions and HS-bosons 

8. Averaging over fluctuating bosonic fields Averaging over fluctuating bosonic fields 

10. Tunneling conductance: When the DOS in the tunneling probe is constant, only enters Otherwise, the first term contributes  information on the distribution function inside the wire encoded in Superconducting tip  measurement of both TDOS and distribution function

15. mapping between the Hilbert space of fermions and bosons; construction of the bosonic Hamiltonian representing the original fermionic Hamiltonian in terms of bosonic (particle-hole) excitations, i.e. density fields; expressing fermionic operators in the bosonic language; calculation of observables (Green functions) within the bosonized formalism by averaging with respect to the many body bosonic density matrix

16. Non-interacting electrons: Derivation of non-equilibrium bosonized action Keldysh action: Source term: classical and quantum fields

21. (Dzyaloshinskii-Larkin Theorem)

22. Generating functional as a determinant

23. Single-particle Hamiltonians:

26. Free electrons: Bosonization identity 

27. S is linear in classical component of the density 

34. Disordered Nanowire Drude conductivity at high T: Drude conductivity at high T: White-noise disorder: White-noise disorder: Backscattering amplitude !  – elastic scattering time Renormalization of disorder: Renormalization of disorder: Giamarchi & Schulz

35. “Functional” bosonization Equation of motion for an electron in the fluctuating electric field We use the Hubbard-Stratonovich decoupling scheme Effective action Effective action Green‘s function Green‘s function S eff = + + + … φ(x,t) g0g0g0g0 g0g0g0g0 RPA-terms Non-RPA Single impurity: Grishin, Yurkevich & Lerner

36. Semiclassical Keldysh Green‘s function at x=x‘ Semiclassical Keldysh Green‘s function at x=x‘ Eilenberger equation ( exact for linear spectrum in 1D ! ) Eilenberger equation ( exact for linear spectrum in 1D ! ) We use the ideas of the non- equilibrium superconductivity Kinetic theory of disordered LL Equation of motion for electron in the fluctuating electic field Functional bosonization scheme Functional bosonization scheme Born approximation over impurity scattering Born approximation over impurity scattering ( incoherent limit at T>>T 1 ) ( incoherent limit at T>>T 1 ) Dissipative Keldysh action Dissipative Keldysh action ( 1D ballistic σ-model ) Quantum kinetic equations for electrons and plasmons Quantum kinetic equations for electrons and plasmons D.Bagrets, I.G., D.Polyakov ‘09

38. Kinetic equation for electrons cf. kinetic equations in plasma physics “Poisson” equation Charge density e-e collision integral Motion of e - in the dissipative bosonic environment Motion of e - in the dissipative bosonic environment Full rate of emission Absorption

39. Emission rate (in one-loop) Particle-hole: q=    i     ) / v F Plasmon : q     i       u RPA-like effective e-e interaction: Plasmons exist at      only Poles, if separated, are close to each other.

40. Large energy transfer,    We treat contributions from plasmons and e-h piars separately ! We treat contributions from plasmons and e-h piars separately ! Emission rate of plasmons: Resonant process (u is close to v F !)

41. Collision Kernel Weak interaction limit, α=V q /πv F <<1 Disorder-induced resonant enhancement of inelastic scattering

42. Electron distribution function Hot-electrons with  D = L/v F - dwell time

43. Summary I

44. Summary II