Nonequilibrium Green’s Function Method in Thermal Transport

Wang Jian-Sheng

• Lecture 1: NEGF basics – a brief history, definition of Green’s functions, properties, interpretation.

• Lecture 2: Calculation machinery – equation of motion method, Feynman diagramatics, Dyson equations, Landauer/Caroli formula.

• Lecture 3: Applications – transmission, transient problem, thermal expansion, phonon life-time, full counting statistics, reduced density matrices and connection to quantum master equation.

References

• J.-S. Wang, J. Wang, and J. T. Lü, “Quantum thermal transport in nanostructures,” Eur. Phys. J. B 62, 381 (2008).

• J.-S. Wang, B. K. Agarwalla, H. Li, and J. Thingna, “Nonequilibrium Green’s function method for quantum thermal transport,” Front. Phys. (2013).

• See also http://staff.science.nus.edu.sg/~phywjs/NEGF/negf.html

Lecture One

History, definitions, properties of NEGF

A Brief History of NEGF

• Schwinger 1961• Kadanoff and Baym 1962• Keldysh 1965• Caroli, Combescot, Nozieres, and Saint-James

1971• Meir and Wingreen 1992

Equilibrium Green’s functions using a harmonic oscillator as an example

• Single mode harmonic oscillator is a very important example to illustrate the concept of Green’s functions as any phononic system (vibrational degrees of freedom in a collection of atoms) at ballistic (linear) level can be thought of as a collection of independent oscillators in eigenmodes. Equilibrium means that system is distributed according to the Gibbs canonical distribution.

Harmonic Oscillator

2 2 2 †

† †

1 ,2 2

1 1 1 ,2 2 2

, [ , ] , [ , ] 12

pH kx u x mm

kH u u a am

u a a x p i a a

Eigenstates, Quantum Mech/Stat Mech

† † † †

1, , 0,1,2,2

1 , 1 1

1Tr ,1

H n E n E n n

a n n n a n n n

aa a a a a aa f

Heisenberg Operator/Equation

† †

( )( ) 1 [ ( ), ]

( ) 1 1 1[ ( ), ] ( ), ( ) ( )2

( ) , ( )

[ ]( )

Ht Hti i

i t i t

O t e OedO t O t H

da t a t H a t a t a tdt i i

a t ae a t a e

O: Schrödinger operatorO(t): Heisenberg operator

Defining >, <, t, Green’s Functions

( ') ( ')

( , ') ( ) ( ') , 1

( ) ( ) ( ) , ( )2

( , ') (1 )2

i t t i t t

ig t t u t u t i

u t a t a t a t ae

ig t t fe f e

( , ') ( ') ( ) ( ', )

( , ') ( ) ( ') ( ') ( , ') ( ' ) ( , ')

( , ') ( ) ( ') ( ' ) ( , ') ( ') ( , ')

1, if 01( ) , if 020, if 0

ig t t u t u t g t t

ig t t Tu t u t t t g t t t t g t t

: time order: anti-time order

Retarded and Advanced Green’s functions

( , ') ( ') [ ( ), ( ')]

sin ( ')( ') ,

( , ') ( ' ) [ ( ), ( ')] ( ', )

( ) ( ) ( ), with ( ) 0 for 0

ig t t t t u t u t

t tt t

ig t t t t u t u t g t t

g t g t t g t t

Fourier Transform

[ ] ( ) , ( ) [ ]2

sin( )[ ] ( )

[ ] [ ] ,

[ ] ( ) (1 ) ( )

r r i t r r i t

r i t t

dg g t e dt g t g e

tg t e dt

g gig f f

Plemelj formula, Kubo-Martin-Schwinger condition

1 1 ( )

[ ] [ ] [ ] ( )

[ ] [ ],

( ) ( )

P i xx i x

g g g f

g t g t i

Valid only in thermal equilibrium

P for Cauchy principle value

Matsubara Green’s Function

( ') ( ')

1( , ') ( ) ( ')

1 (1 )2

where 0 , ' , ( ) ( )

( ) ( )

2[ ] ( ) , , , 1,0,1,2,

[ ] [ ]

iM Mn n

g T u u

fe f e

u u i e ue

ng i g e d n

g g i i

Nonequilibrium Green’s Functions

• By “nonequilibrium”, we mean, either the Hamiltonian is explicitly time-dependent after t0, or the initial density matrix ρ is not a canonical distribution.

• We’ll show how to build nonequilibrium Green’s function from the equilibrium ones through product initial state or through the Dyson equation.

Definitions of General Green’s functions

( , ') ( ) ( ') , ( , ') ( ') ( )jk j k jk k ji iG t t u t u t G t t u t u t

( , ') ( ') ( , ') ( ' ) ( , '),

( , ') ( ' ) ( , ') ( ') ( , ')

G t t t t G t t t t G t t

( , ') ( ') ,

( , ') ( ' )

G t t t t G G

Relations among Green’s functions

( , ') ( ', )

t t r t

t t r a a t

r ajk kj

G G G G

G G G G G G G

G t t G t t

Steady state, Fourier transform

( , ') ( '),

[ ] ( ) ,

[ ] [ ]

G t t G t t

G G t e dt

Equilibrium Green’s Function, Lehmann Representation

| | , ,

( ) , | | 1

( ) Tr ( ) (0)

1| ( ) (0) |

1| | | |

Ht Hti i

jk j k

E E tE i

j kn m

eH n E n Z eZ

u t e u e m m

iG t u t u

i e n u t u nZ

i e n u m m u nZ

Kramers-Kronig Relation

ˆ[ ] ( ) is analytic on the upper half plane of

ˆ[ ] [ ], 0

1 [ '] ( )[ ] 'P , lim'

1 [ '][ ] 'P'

xrr IR

G z e G t dt z

G G z i

G f xG d P dxx x

Eigen-mode Decomposition

( ) 2 ( ) (1) (2) ( )

, , , , , , , , , ,

, ( , , , )

0 00 0

diag0 0

( ) diag ( )

1[ ] diag

j j Nj

t t r a t t r a T

Ku u S u u u

u SQ S S I

G t S g t S

G S S i Ki

Pictures in Quantum Mechanics

• Schrödinger picture: O, (t) =U(t,t0)(t0)

• Heisenberg picture: O(t) = U(t0,t)OU(t,t0) , ρ0, where the evolution operator U satisfies

( , ') ( , '),

( , ') , 't

i H dt

U t ti H U t tt

U t t Te t t

See, e.g., Fetter & Walecka, “Quantum Theory of Many-Particle Systems.”

Calculating correlation

0 0 0 0 0

( '') ''

( ) ( ') Tr ( ) ( ') '

Tr ( ) ( , ) ( , ) ( , ') ( ', )

Tr ( ) ( , ) ( , ') ( ', )

Tr ( ) ,

( , ') ,( , ') ( ', '') ( , '')

i H t dt

A t B t A t B t t t

t U t t AU t t U t t BU t t

t U t t AU t t BU t t

t T e A B

U t t TeU t t U t t U t t

t0 t’ t

Evolution Operator on Contour2

2 1 2 1

3 2 2 1 3 1 3 2 1

11 2 2 1 1 2

( , ) exp ,

( , ) ( , ) ( , ),

( , ) ( , ) ,

( ) ( , ) ( , )

ciU T H d

O U t OU t

Contour-ordered Green’s function

( , ') ( ) ( ')

Tr ( ) C

i H dTC

iG T u u

t T u u e

Contour order: the operators earlier on the contour are to the right. See, e.g., H. Haug & A.-P. Jauho.

Relation to other Green’s function

( , ), or ,

( , ') ( , ') or

G GG G t t G

G G G G

An Interpretation

, 0 0 0

1 1 ,2 2

( ) ( )

Tr ( ) ( ) ( , )

exp ( ) ( , ') ( ') '2

H p p u Ku K K

U t t t U t t

i F G F d d

G is defined with respect to Hamiltonian H and density matrix ρ and assuming Wick’s theorem.

Calculus on the contour

• Integration on (Keldysh) contour

• Differentiation on contour

( ) ( ) ( ) ( )f d f t dt f t dt f t dt

( ) ( )df df td dt

Theta function and delta function• Theta function

• Delta function on contour

where θ(t) and δ(t) are the ordinary theta and Dirac delta functions

1 if is later than ' along the contour( , ')

0 otherwise

( , ') ( ')( , ') ( ' )

( , ') ( , ')( , ') 0( , ') 1

t t t tt t t t

t tt tt t

( , ')( , ') ( , ') ( ')d t t t td

Transformation/Keldysh Rotation'

( , ') ( , ')

1 0 1 11, ,0 1 1 12

r KT T

t t t t

A A A t t

A AA A A

R RR I

A AA R AR R AR

A A A A A A A AA A A A A A A A

Convolution, Langreth Rule

2 3 1 2 2 3 1

( , ) ( , ) ( , )

or , , +

r K r K r K

r r r a a a K r K K a

r r r r r r r r

K K r r K r K a K a a

AB D d d d A B D

C AB C AB C AB

C C A A B BC A B

C A B C A B C A B A B

G g g G G g g G G g

G g g G g G g G

(1 ) (1 )r r a a r aG G g G G G

Lecture two

Equation of motion method, Feynman diagrams, etc

Equation of Motion Method

• The advantage of equation of motion method is that we don’t need to know or pay attention to the distribution (density operator) ρ. The equations can be derived quickly.

• The disadvantage is that we have a hard time justified the initial/boundary condition in solving the equations.

Heisenberg Equation on Contour

2 1 2 1

( , ) exp ,

( ) ( , ) ( , )

( ) [ ( ), ]

ciU T H d

O U t OU t

dOi O Hd

Express contour order using theta function

),'()()'()',()'()(

)'()()',(

uuiuui

Operator A(τ) is the same as A(t) as far as commutation relation or effect on wavefunction is concerned

Iiuu T ])(),([

Equation of motion for contour ordered Green’s function

IuKuTi

uuiuuTi

uuiuui

uuiuuiG

)',()',(

)',())'()((

)',(])'(),([)'()(

),'()()'()',()'()(

)',()'()()',(

),'()()'()',()'()(

),'()()'()',()'()()',(

Equations for Green’s functions

2, , , ,

( , ') ( , ') ( , ')

( , ') ( , ') ( ') , , '

( , ') ( , ') ( ')

( , ') ( , ') 0

r a t r a t

G KG I

G t t KG t t t t It

G t t KG t tt

Solution for Green’s functions

2, , , ,

2 , , , ,

1, , 2

( , ') ( , ') ( ')

using Fourier transform:

[ ] [ ]

[ ] ( ) ( )

[ ] [ ] ( ) , 0

r a t r a t

G t t KG t t t t It

G KG I

G I K c K d K

G G i I K

G f G G G e G

c and d can be fixed by initial/boundary condition.

Feynman diagrammatic method

• The Wick theorem• Cluster decomposition theorem, factor

theorem• Dyson equation• Vertex function• Vacuum diagrams and Green’s function

Handling interaction

Transform to interaction picture, H = H0 + Hn

0 00 0

0 0 00 0 0

( ) ( )

( ) ( ' )

( ) ( )

( ) ( ) ( )

( ) ( ) ( , ) ( , ) ( )

( , ') ( , ')

( ) ( , ) ( , )

( ) ( , ) ( ) (

H Hi t t i t t

I S H I

H Hi t t i t t

H H Hi t t i t t i t t

t e t e U t t S t t t

S t t e U t t e

t e U t t U t t e

A t e A e e U t t A t U t

0( ), )

Hi t tt e

Scattering operator S

0 0 0 0

0 0 0 0( ) ( ) ( ' ) ( ' )

0 0 0 0

( ) ( ') Tr ( ) ( ') '

Tr[ ( , ) ( ) ( , ) ( , ') ( ') ( ', )]Tr[ ( , ) ( ) ( , ') ( ') ( ', )]

H H H Hi t t i t t i t t i t t

A t B t A t B t t t

U t t e A t e U t t U t t e B t e U t tS t t A t S t t B t S t t

0 00 0

( ) ( )

( '') ''

( , ') ( ) ( , '), ( )

( , ') , '

H Hi t t i t tI I Sn n n

i H t dt

i S t t H t S t t H t e H et

S t t Te t t

Contour-ordered Green’s function

)'()(Tr

)'()()',(

τ13n ijk i j k

H T u u u

Perturbative expansion of contour ordered Green’s function

( '') ''

1 1 1 1 2 2

1 2 3 1 2 3

( , ') ( ) ( ')

1( ) ( ') 1 ( ) ( ) ( )2!

1 1( ) ( ') ( ) ( ') ( , , ) ( ) ( ) ( )3! 3

ni H d

jk C j k

C j k n n n

C j k C j k lmn l m n

iG T u u e

i i iT u u H d H d H d

i iT u u T u u T u u u

4 5 6 4 5 6 4 5 6

01 2 3 4 5 6

1 2 5 3 6 4

1 ( , , ) ( ) ( ) ( )3

( , ') ... ( ) ( ') ( ) ( ) ( ) ( ) ( ) ( )

(Wick's theorem)

( ) ( ) ( ) ( ) ( ) ( ) ( )

opq o p qopq

jk C j k l m n o p q

C j l C m p C n q C o

T u u u d d d

G T u u u u u u u u

T u u T u u T u u T u u

General expansion rule

1 2 1 2

( , ')( , , )

( , , , ) ( ) ( ) ( )n n

ijk i j k

j j j n C j j j n

iG T u u u

Single line

3-line vertex

n-double line vertex

Diagrammatic representation of the expansion

21221100 )',(),(),()',()',( ddGGGG n

= + 2iħ + 2iħ

+ 2iħ

Self -energy expansion

Explicit expression for self-energy

' ' ', 0, 0,

'' '' '', ' 0, 0,

'[ ] 2 [ '] [ ']2

'2 '' [0] [ ']2

n jk jlm rsk lr mslmrs

jkl mrs lm rslmrs

di T T G G

Junction system, adiabatic switch-on

• gα for isolated systems when leads and centre are decoupled

• G0 for ballistic system• G for full nonlinear system

t = 0t = −

HL+HC+HR

HL+HC+HR +V

HL+HC+HR +V +Hn

Equilibrium at Tα

Nonequilibrium steady state established

Sudden Switch-on

t = ∞t = −

HL+HC+HR

HL+HC+HR +V +Hn

Green’s function G

Equilibrium at Tα

Nonequilibrium steady state established

Three regions

RCLuuTiG

,,,,)'()()',(

Heisenberg equations of motion in three regions

,1 1 ,2 2

1 1 1[ , ], [ , ],

T LC T RCL C R L C R C n

C CL CRC C C L R C n

H H H H u V u u V u H

H u u u K u

u u H H K u V u V u u Hi i i

u K u V u L R

Force Constant Matrix

1 12 2

CL C CR

LT T T T

L C R C n

K VK V K V

uH p p u u u K u H

p u uu

Relation between g and G0

Equation of motion for GLC

( , ') ( ) ( ') ,

( , ') ( ) ( ')

( , ') ( , '),

( , ') ( , '') ( '', ') '',

( , ') ( , ') ( , ')

TLC C L C

L LCLC CC

LCLC L CC

iG T u u

K G V G

G g V G d

g K g I

Dyson equation for GCC

CCCCCC

TCCCCC

VgVVgV

ddGggG

IdGVgV

dGVgVGK

IGVGVGK

IuuTiG

)',()',()',(

,)',(),(),()',()',(

),',('')',''()'',(

'')',''()'',()',(

)',()',()',()',(

)',()'()()',(

,)'()()',(

212211

Equation of Motion Way (ballistic system)

( , ') ( , ') ( , ')

0 0 0 0, 0 0 , 0

0 0 0 0

( , ') ( , ') '' ( , '') ( '', ')

C CL CR

G KG I

g Dg I

K VK D V D K V V V

G g d g VG

“Partition Function”

( ) ( )

Tr ( ) ( , ) ( , )

Tr ( )

i V H d

Z t U t t U t t

Diagrammatic WayFeynman diagrams for the nonequilibrium transport problem with quartic nonlinearity. (a) Building blocks of the diagrams. The solid line is for gC, wavy line for gL, and dash line for gR; (b) first few diagrams for ln Z; (c) Green’s function G0

CC; (d) Full Green’s function GCC; and (e) re-sumed ln Z where the ballistic result is ln Z0 = (1/2) Tr ln (1 –gCΣ). The number in front of the diagrams represents extra combinatorial factor.

The Langreth theorem

aaarrr

rrrrrr

CBACBACBAD

ddCBAD

dtttBttAdtttBttAttC

BACdtttBttAttC

dtttBttAdBAC

)',(),(),()',(

][][][][][

'')',''()'',('')',''()'',()',(

][][]['')',''()'',()',(

'''')',''()'',('')',''()'',()',(

212211

Dyson equations and solution

GIGGIGGG

GGGGGggG

Energy current

( ', ) ( ', )( , ') ( , ') '

1 Tr [ ]2

1 Tr [ ] [ ] [ ] [ ]2

T LCLL L C

t ar L LCC CC

r aCC L CC L

dHI u V udt

t t t ti G t t G t t dtt t

Landauer/Caroli formula

r aLL CC L CC R L R

r aL L R R

a r r aL R

dHI G G f f ddt

G G G i f f

G G iG G

1D calculation

• In the following we give a complete calculation for a simple 1D chain (the baths and the center are identical) with on-site coupling and nearest neighbor couplings. This example shows the general steps needed for more general junction systems, such as the need to calculate the “surface” Green’s functions.

Ballistic transport in a 1D chain

• Force constants

• Equation of motion64

kkkkkkkk

1 0 1(2 ) , , 1,0,1,2,j j j ju ku k k u ku j

Solution of g

( ) , 0

2 02 0

0 20 0

, 0,1,2, ,

(( ) 2 ) / 0, | | 1

i K g I

k k kk k k k

Kk k k k

i k k k

Lead self energy and transmission

0 00 0 0 00 0 0

1, 4[ ] Tr

0, otherwise

r CL R

j krjk

r aL R

k k kT G G

Heat current and conductance

1lim ,2 1

, 0, 03

T TL R

dI T f f

I f d fT T T e

k T T kh

General recursive algorithm for g

011110

kkkkkk

)|(| while}

Lecture Three

Applications

CARBON NANOTUBE (6,0), FORCE FIELD FROM GAUSSIAN

Transmission

Calculation was performed by J.-T. Lü.

Carbon nanotube, nonlinear effect

The transmissions in a one-unit-cell carbon nanotube junction of (8,0) at 300K. From J-S Wang, J Wang, N Zeng, Phys. Rev. B 74, 033408 (2006).

u4 Nonlinear modelOne degree of freedom (a) and two degrees freedom (b) (1/4) Σ Tijkl ui uj uk ul

nonlinear model. Symbols from quantum master equation, lines from self-consistent NEGF. For parameters used, see Fig.4 caption in J.-S. Wang, et al, Front. Phys. Calculated by Juzar Thingna.

Molecular dynamics with quantum bath2

, , , ,

( ) ( ) ( ') ( ') '

( ) ( ') ( '), , ,

r C r C r r rL R

d u t F t t t u t dtdt

t t i t t L R

See J.-S. Wang, et al, Phys. Rev. Lett. 99, 160601 (2007);Phys. Rev. B 80, 224302 (2009).

Average displacement, thermal expansion

One-point Green’s function

( ) ( )

' '' ''' ( ', '', ''') ( ', '') ( ''', )

( 0) [ 0]

lmn lm njlmn

rj lmn lm nj

iG T u

d d d T G G

G T G t G

dGiM x dT

Thermal expansion

(a) Displacement <u> as a function of position x.

(b) as a function of temperature T. Brenner potential is used. From J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009).

Left edge is fixed.

Graphene Thermal expansion coefficient

The coefficient of thermal expansion v.s. temperature for graphene sheet with periodic boundary condition in y direction and fixed boundary condition at the x=0 edge. is onsite strength. From J.-W. Jiang, J.-S. Wang, and B. Li, Phys. Rev. B 80, 205429 (2009).

Phonon Life-Time

1 , ( )[ ]

Re [ ] 2 ,

Im [ ]

ti tr rq qr

rn q q q q

qrn q q

q q qq

G G t ei

For calculations based on this, see, Y. Xu, J.-S. Wang, W. Duan, B.-L. Gu, and B. Li, Phys. Rev. B 78, 224303 (2008).

Transient problems

, 0( )

( , )( ) Im

T LRL R

Lt t t

H tH t

H u V u t

G t tI t kt

Dyson equation on contour from 0 to t

1 2 1 1 2 2,

( , ') ( ) ( ')

( ) ( ) ( ')

RL R RL L

R RL L LR RL

G d g V g

d d g V g V G

Contour C

TRANSIENT THERMAL CURRENT

The time-dependent current when the missing spring is suddenly connected. (a) current flow out of left lead, (b) out of right lead. Dots are what predicted from Landauer formula. T=300K, k =0.625 eV/(Å2u) with a small onsite k0=0.1k. From E. C. Cuansing and J.-S. Wang, Phys. Rev. B 81, 052302 (2010). See also arXiv:1005.5014.

FINITE LEAD PROBLEM

Top : no onsite (↥ k0=0), (a) NL = NR = 100, (b) NL=NR=50, temperature T=10, 100, 300 K (black, red, blue). TL=1.1T, TR = 0.9T.

Right : with onsite ↦ k0=0.1k and similarly for other parameters. From E. C. Cuansing, H. Li, and J.-S. Wang, Phys. Rev. E 86, 031132 (2012).

Full counting statistics (FCS)• What is the amount of energy (heat) Q transferred in a

given time t ?• This is not a fixed number but given by a probability

distribution P(Q)• Generating function

• All moments of Q can be computed from the derivatives of Z.

• The objective of full counting statistics is to compute Z(ξ).

dQQPeZ Qi )()(

A brief history on full counting statistics

• L. S. Levitov and G. B. Lesovik proposed the concept for electrons in 1993; rederived for noninteracting electron problems by I. Klich, K. Schönhammer, and others

• K. Saito and A. Dhar obtained the first result for phonon transport in 2007

• J.-S. Wang, B. K. Agarwalla, and H. Li, PRB 2011; B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 2012.

Definition of generating function based on two-time measurement

| || |( ) (0, ) ( ,0)

e.g., ( ,0)

L Li H i H t

P P a aH a a aH t U t H U t

Consistent history quantum mechanics (R. Griffiths).

Product initial state

/ 2 / 2

, [ , ] 0, '

Tr (0, ) ( ,0)

( , ') ( , ')

i H i H t

i H i H

i H i H i H

ixH ixHx

e U t e U t

e U t e U t e

U t U t

U t t e U t t e

( , ')

( ) ( )

H t dtixH ixHx

H t dt

ixH ixHx

ixH ixHLC RCL C R

ixH ixHRC LCL C R

RC L T LC C xx I

ixH ixHL L Lx

U t t e Te e

Te t t

H t e H t e

e H H H H H e

H H H H e H e

H H u V u H H

u e u e u x

Schrödinger/Heisenberg/interaction pictures

87tHitHi

ttTettSttStHtttSi

tttStet

tAUtUtA

ttUtHt

iiwtttUt

',)',(),',()()',(

)'(|)',()(|)(|

)](),([)()0(||),0,(),0()(

)',()()',(

||)'(|)',()(|

'')''(

Schrödinger picture

Heisenberg picture

Interaction picture

Compute Z in interaction picture

ALRLCC

ddHHidHiT

)]1ln[(Tr21

)]1)(1det[(ln21)](1[lnTr

][Tr211

')'()(21)(1Tr

)]0,(),0([Tr

Important result

otherwise00if2/)(

0if2/)()(

)',()'('),()',(

)1ln(Tr21ln

ALRLCC

Long-time result, Levitov-Lesovik formula

)/(1),1/(1

)1]([),1]([

)2()1()1(detln4

01lnTr

)1ln(Tr21ln

ffeefefeGGIdt

Arbitrary time, transient result

91 time)long in(

)1ln(Tr21ln

Numerical results, 1D chain

1D chain with a single site as the center. k= 1eV/(uÅ2), k0=0.1k,TL=310K, TC=300K, TR=290K. Red line right lead; black, left lead. B. K. Agarwalla, B. Li, and J.-S. Wang, PRE 85, 051142, 2012.

Cumulants in Finite SystemPlot of the culumants of heat <<Qn>> for n=1,2,3, and 4 for one-atom center connected with two finite lead (one-dimensional chain) as a function of measurement time tM. The black and red line correspond to NL =NR =20 and 30, respectively. The initial temperatures of the left, center, right parts are 310K, 360K, and 290K, respective. We choose k=1 eV/(uÅ2) and k0=0.1 eV/(uÅ2) for all particles. From J.-S. Wang, et al, Front. Phys. 2013.

FCS for coupled leads

LC LRL

CL CRC

RL RCR

K V VK V K V G g gVG

ln ln det ( 1) (1 ) ( 1) (1 ) [ ]4

[ ] , , ,

If center is absent 0, then

[ ] [ ]

i iM L R R L g

r a a rg LR R RL L

r ag I RR R RR L

dZ t I e f f e f f T

T G G i g g L R

T T G G

From H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. E 86, 011141 (2012); H. Li, B. K. Agarwalla, and J.-S. Wang, Phys. Rev. B 86, 165425 (2012). For multiple leads, see B. K. Agarwalla, et al, arXiv:1308.1604

FCS in nonlinear systems

• Can we do nonlinear?

• Yes we can. Formal expression generalizes Meir-Wingreen

• Interaction picture defined on contour

• Some example calculations

Vacuum diagrams, Dyson equation, interaction picture transformation on

contour

( )I1 2 1 2

ln 1 Tr( ) 2 ( )

1( , ) Tr ( ) ( )

, (0 , ) ( ,0 ) /

( ) (0 , ) ( ) ( ,0 )

( ,0 )

x T LC T CRL C C R

i H dI I TC

In M M

i u V u u V u d

Z Gi i

G G G G

iG T u u eZ

Z Z Z V t V t Z

O V O V

Nonlinear system self-consistent results

Single-site (1/4)λu4 model cumulants. k=1 eV/(uÅ2), k0=0.1k, Kc=1.1k, V LC

-1,0=V CR

0,1=-0.25k, TL=660 K, TR=410K.

From H. Li, B. K. Agarwalla, B. Li, and J.-S. Wang, arxiv::1210.2798

( , ') 3 ( , ') ( , ')n i G

Quantum Master Equations

• When we tried to calculation the reduced density matrix from Dyson expansion, we found that the 2nd order (of the system-bath coupling) diagonal terms do not agree with master equation results, and there are also divergences in higher orders. Similarly, the currents also diverge at higher order. We were quite puzzled. The following is our effort to solve this puzzle and found an expression of the high order current which is finite.

Quantum Master Equations0

2 42 4 6

( ) Tr ( , ) ( ) ( , )

Tr ( ) ,

( ) Tr ( ) [ ( ), ( )] ,

( )2! 4!

| |, Tr

O t S t O t S t

iT O t e

d O t T O t O t V t edt

X X V X V O

Unique one-to-one map ρ ↔ ρ0

2 4 42 4 2 2

4 42 3 2 6

2! 4! 2!

[ , ] [ , ]3!

[ , ]2!

( )3! 2!

I T T T T

T LCL C

X V X V X V X V

d i X V V X V Vdt

X V X V V

I VV VV X V VV

V p V u

Group & AcknowledgementsGroup members: front (left to right) Mr. Zhou Hangbo, Prof. Wang Jian-Sheng, Dr./Ms Lan Jinghua, Mr. Wong Songhan; Back: Mr. Li Huanan, Dr. Zhang Lifa, Dr. Bijay Kumar Agarwalla, Mr. Kwong Chang Jian, Dr. Thingna Juzar Yahya. Picture taking 17 April 2012. Others not in the picture: Qiu Hongfei. Thanks also go to Jian Wang, Jingtao Lü, Jin-Wu Jiang, Edardo Cuansing, Meng Lee Leek, Ni Xiaoxi, Baowen Li, Peter Hänggi, Pawel Keblinski.http://staff.science.nus.edu.sg/~phywjs/

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