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Non-equilibrium physics Non-equilibrium physics in one dimension Igor Gornyi Москва...
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Non-equilibrium physicsNon-equilibrium physics
in one dimensionin one dimension
Igor GornyiIgor Gornyi
Москва Сентябрь 2012Москва Сентябрь 2012
Karlsruhe Institute of Technology
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Part IINonequilibrium Nonequilibrium
BosonizationBosonizationdeveloped by D.Gutman, Y.Gefen, A. Mirlin ’09-10
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• 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 – ?
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Bosonization
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Functional bosonization
Hubbard-Stratonovich transformation decouples quartic interaction term
1D: gauge transformation with
eliminates coupling between fermions and HS-bosons
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Averaging over fluctuating bosonic fields Averaging over fluctuating bosonic fields
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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
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• 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
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Non-interacting electrons:Derivation of non-equilibrium bosonized action
Keldysh action:
Source term:
classical and quantum fields
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(Dzyaloshinskii-Larkin Theorem)
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Generating functional as a determinant
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Single-particle Hamiltonians:
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Free electrons:
Bosonization identity
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S is linear in classical component of the density
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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
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““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
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• 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
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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
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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.
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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!)!)
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Collision KernelCollision KernelWeak interaction limit,α=Vq/
πvF<<1
Disorder-induced resonant enhancement
of inelastic scattering
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Electron distribution functionElectron distribution function
Hot-electrons with Hot-electrons with
D = L/vF - dwell time
3 / 4T eU
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Summary I
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Summary II