Non-equilibrium physics
in one dimension
Igor Gornyi
Karlsruhe Institute of Technology
Москва
Сентябрь 2012
Part II
Nonequilibrium Bosonization
developed 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 
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 Nanowire
White-noise disorder:
U
(
x
)(
U
x
)


(
x

x
)
/
(
2
v

)
*
b 1 b2

– elastic scattering time
1 2
F
Backscattering amplitude !
• Drude conductivity at high T:
D e2vF2
2
• Renormalization of disorder: 11 Giamarchi
0  
T
l
L
, G G
Q
& Schulz
“Functional” bosonization
We use the Hubbard-Stratonovich decoupling scheme







*
1
ˆ
ˆ
(
i


)

i
v


U

U
G
(
x
,
x
'
,
t
,
t
'
,
[
]
)

1
t
z F
x
b
b

2
Equation of motion for an electron in the fluctuating electric field
• Green‘s function
 



1
GD
(
x
,
t
)

G
(
x
,
t
,
[
]
)
e
x
p
[
i
S
[
]

i
V
e
f
f
0]

• Effective action
φ(x,t)
Seff =
+
RPA-terms
Non-RPA
g0
+
g0
Single impurity: Grishin, Yurkevich & Lerner
+ …
Kinetic theory of disordered LL
D.Bagrets, I.G., D.Polyakov ‘09
• Functional bosonization scheme
• Semiclassical Keldysh Green‘s function at x=x‘
g
(
x
,
t
,
t
)

i
v
G
(
x
,
x

0
,
t
,
t
)

G
(
x
,
x

0
,
t
,
t
)


1
2
F
1
2
1
2
We use the ideas of the nonequilibrium superconductivity
g g  1ˆ
• Eilenberger equation ( exact for linear spectrum in 1D ! )
Equation of motion for electron in the fluctuating electic field
• Born approximation over impurity scattering
( incoherent limit at T>>T1 )
• Dissipative Keldysh action ( 1D ballistic σ-model )
• Quantum kinetic equations for electrons and plasmons
Kinetic equation for electrons

1 R L
R
R




v

f


(
f

f
)

S
t
 t R  F x 


e

2



e-e collision integral

g


R
2
L
“Poisson” equation

1
L

(
t
,
x
)



(
tx
,
)

d

f
(t,x
)
L
L




2

v

F

Charge density
cf. kinetic equations in plasma physics
• Motion of e- in the dissipative bosonic environment












 


S
t
(
)

d
I
(
)
f
(
1

f
)

I
(
)
f
(
1

f
)

e





v
Absorption
Full rate of emission
Emission rate (in one-loop)
RPA-like effective e-e interaction:
V (,q)
Plasmon :
qi 
Particle-hole: q=

u
i
 )/vF
Poles, if separated, are
close to each other.

ReD
,q)
R (
Plasmons exist at
 only





2

id
q





,
I(
)
 Vq
(
,)
R
e(
D
q
)


R
Large energy transfer, 
We treat contributions from plasmons and e-h piars
separately !
Resonant process
(u is close to vF!)
Emission rate of plasmons:








I
()


L
()

(
1

n
)

p
,







1
v
v

F
F
L
(
)


1
,
L
(
)


1
,
2
 3
2
2
2
u
2
u


Collision Kernel
Weak interaction limit, α=Vq/πvF<<1
Disorder-induced resonant
enhancement
of inelastic scattering
Electron distribution function
Hot-electrons with
T 3eU/4
D = L/vF - dwell time
Summary I
Summary II