Non-equilibrium physicsNon-equilibrium physics

in one dimensionin one dimension

Igor GornyiIgor Gornyi

Москва Сентябрь 2012Москва Сентябрь 2012

Karlsruhe Institute of Technology

Part IINonequilibrium Nonequilibrium

BosonizationBosonizationdeveloped by D.Gutman, Y.Gefen, A. Mirlin ’09-10

• 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 – ?

Bosonization

Functional bosonization

Hubbard-Stratonovich transformation decouples quartic interaction term

1D: gauge transformation with

eliminates coupling between fermions and HS-bosons

Averaging over fluctuating bosonic fields Averaging over fluctuating bosonic fields

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

• 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

Non-interacting electrons:Derivation of non-equilibrium bosonized action

Keldysh action:

Source term:

classical and quantum fields

(Dzyaloshinskii-Larkin Theorem)

Generating functional as a determinant

Single-particle Hamiltonians:

Free electrons:

Bosonization identity

S is linear in classical component of the density

Disordered NanowireDisordered Nanowire

, Ql GL G

• Drude conductivity at high T:Drude conductivity at high T:

• White-noise disorder:White-noise disorder: 2 2*

1 1( ) ( ) ( ) /(2 )Fb bx x xU vU x

Backscattering amplitude !– elastic scattering time

2 2D Fe v

• Renormalization of disorder:Renormalization of disorder:0

21 1

T

Giamarchi & Schulz

““Functional” bosonizationFunctional” bosonization

*12

ˆ( ) ( , , , ,[ ])''ˆ 1F b bxt z v U U ti i G x tx

eff1

0( , ) ( , , ) e[ ] [ ]xp[ ]G D Gx Stx i Vt i

Equation of motion for an electron in the fluctuating electric field

We use the Hubbard-Stratonovich decoupling schemeWe use the Hubbard-Stratonovich decoupling scheme

• Effective actionEffective action

• Green‘s functionGreen‘s function

SSeff eff = + + + … = + + + …

φφ(x,t(x,t)) gg00

gg00RPARPA-terms-terms

Non-RPANon-RPA

Single impurity: Grishin, Yurkevich & Lerner

• Semiclassical Semiclassical KeldyshKeldysh Green‘s function at Green‘s function at x=x‘x=x‘

1 2 1 2 1 2( , ) ( , ), 0 , 0,, ,( ), Fg it t t t t tx x xGx xGv

1̂g g

• Eilenberger equation Eilenberger equation ( ( exactexact for linear spectrum in 1D ! ) for linear spectrum in 1D ! )

We use the ideas of the We use the ideas of the non-equilibrium non-equilibrium

superconductivitysuperconductivity

Kinetic theory of disordered LLKinetic theory of disordered LL

Equation of motion for electron in the fluctuating electic field

• Functional bosonization schemeFunctional bosonization scheme

• Born approximation over impurity scatteringBorn approximation over impurity scattering ( ( incoherent limit at T>>Tincoherent limit at T>>T11 ) ) • Dissipative Keldysh actionDissipative Keldysh action( 1D ballistic ( 1D ballistic σσ-model )-model )

• Quantum kinetic equations for electrons and plasmonsQuantum kinetic equations for electrons and plasmons

D.Bagrets, I.G., D.Polyakov ‘09

Kinetic equation for electronsKinetic equation for electrons

2

(

1( )

2

1

2(, )(, ,) )

RR R

L

LeR

t

L

x

F

R

L

F

L t

v St

t x

g

d

f

f tx xv

f f

cf. kinetic equations in plasma cf. kinetic equations in plasma physicsphysics

““Poisson” equationPoisson” equation

Charge densityCharge density

e-e collision e-e collision integralintegral

( ) ( ) (1 ) ( ) (1 )ev

St d I f f I f f

• Motion of eMotion of e-- in the dissipative bosonic environment in the dissipative bosonic environment

Full rate of Full rate of emissionemission

AbsorptioAbsorptionn

Emission rate (in one-loop)Emission rate (in one-loop)

( ) , ,( ) Re ( )2 R

qi dI V q D q

,Re ( )RD q

Particle-hole: Particle-hole: q= q= ii ))//vvFF

( ),qV

Plasmon : Plasmon : qqiiuu

RPA-like effective e-e RPA-like effective e-e interaction:interaction:

Plasmons exist at Plasmons exist at onlyonly

Poles, if separated, are close to each other.

Large energy transfer, Large energy transfer,

, ( ) ( )(1 )pI L n

1,2 32 2

1( ) 1 , ( )

2 2F Fv v

L Lu u

We treat contributions from plasmons and e-h We treat contributions from plasmons and e-h piars piars separately ! separately !

Emission rate of Emission rate of plasmons:plasmons:

Resonant process Resonant process (u is close to v(u is close to vFF!)!)

Collision KernelCollision KernelWeak interaction limit,α=Vq/

πvF<<1

Disorder-induced resonant enhancement

of inelastic scattering

Electron distribution functionElectron distribution function

Hot-electrons with Hot-electrons with

D = L/vF - dwell time

3 / 4T eU

Summary I

Summary II