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Quantum Master Equation Approach to Transport

Wang Jian-Sheng

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NUS, number one in AsiaThis year’s QS university ranking has rated NUS a topmost in Asia.

Department of Physics at NUS is top 32 (QS 2013) world-wide, with renounced research centers such as Graphene Research Center and CQT.

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Outline

• A quick introduction to nonequilibrium Green’s function (NEGF) and some results

• Formulation of quantum master equation to transport (energy, particle, or spin)

• Analytic continuation to

• Application to spin transport

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NEGF

Our review: Wang, Wang, and Lü, Eur. Phys. J. B 62, 381 (2008); Wang, Agarwalla, Li, and Thingna, Front. Phys. (2013), DOI:10.1007/s11467-013-0340-x

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Evolution Operator on Contour

2

1

2 1 2 1

3 2 2 1 3 1 3 2 1

11 2 2 1 1 2

0 0

( , ) exp ,

( , ) ( , ) ( , ),

( , ) ( , ) ,

( ) ( , ) ( , )

c

iU T H d

U U U

U U

O U t OU t

Contour-ordered Green’s function

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( )

0 '

( , ') ( ) ( ')

Tr ( ) C

TC

iH dT

C

iG T u u

t T u u e

t0

τ’

τ

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 functions

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'

( , ), or ,

( , ') ( , ') or

,

,

t

t

t t

G GG G t t G

G G

G G G G

G G G G

t0

τ’

τ

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An Interpretation due to Schwinger

, 0 0 0

1 1,

2 2

( ) ( )

Tr ( ) ( ) ( , )

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

T T T

T

M M

T

C C

H p p u Ku K K

V F u

U t t t U t t

iF G F d d

G is defined with respect to Hamiltonian H and density matrix ρ(t0), and assuming validity of Wick’s theorem.

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Heisenberg Equation on Contour

2

1

2 1 2 1

0 0

( , ) exp ,

( ) ( , ) ( , )

( )[ ( ), ]

c

iU T H d

O U t OU t

dOi O H

d

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Thermal conduction at a junction

Left Lead, TL Right Lead, TR

Junction Partsemi-infinite

Three regions

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RCLuuTi

G

uu

u

u

u

u

u

u

TC

CL

L

L

R

C

L

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

,, 2

1

Junction system, adiabatic switch-on

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• 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

g

G0

G

Equilibrium at Tα

Nonequilibrium steady state established

Sudden Switch-on

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t = ∞t = −

HL+HC+HR

HL+HC+HR +V +Hn

g

Green’s function G

Equilibrium at Tα

Nonequilibrium steady state established

t =t0

Heisenberg equations of motion in three regions

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,

1 1,

2 2

1 1 1[ , ], [ , ],

, ,

T LC T RCL C R L C R C n

T T

C CL CRC C C L R C n

CC

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

Relation between g and G0

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Equation of motion for GLC

2

2

2

2

( , ') ( ) ( ') ,

( , ') ( ) ( ')

( , ') ( , '),

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

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

TLC C L C

TLC C L C

L LCLC CC

LCLC L CC

LL L

iG T u u

iG T u u

K G V G

G g V G d

g K g I

Energy current

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0

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

1Tr [ ]

2

1Tr [ ] [ ] [ ] [ ]

2

T LCLL L C

t ar L LCC CC

t

LCCL

r aCC L CC L

dHI u V u

dt

t t t ti G t t G t t dt

t t

V G d

G G d

Landauer/Caroli formula

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0

1Tr

2

,2

,

( )

r aLL CC L CC R L R

r a

L RL

r aL L R R

a r r aL R

dHI G G f f d

dt

i

I II

G G G i f f

G G iG G

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Self-consistent mean-field NEGF

• Tijkl nonlinear model

1 2

1 2

1 3 4 5 3 4 5 2

3 4 5

2

1 2 1 221

1 1, 1 2

,

( , ) ( , )

( , , ),

(1,2,3,4) (1,2) (3,4)

(1,3) (2,4) (1,4) (2,3)

( ) ( )

Cj j

j j

j j j j j j j jj j j

i i

I K G

T G

iG G G

G G G G

i t j

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u4 Nonlinear model

One degree of freedom (a) and two degrees freedom (b) (1/4) Σ Tiiii ui

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nonlinear model. Symbols are from quantum master equation, lines from self-consistent NEGF. For parameters used, see Fig.4 caption in Wang, et al, Front. Phys 2013.

Calculated by Juzar Thingna.

1

5

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Full Counting Statistics, two-time measurement

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, ,

( )

/ 2 / 2

/ 2 / 2

,

Tr

Tr (0, ) ( ,0)

Tr (0, ) ( ,0)

Tr (0, ) ( ,0)

( , ') ( , ')

L L

L L

L L L

L L

H

L C R

i H i H t

i H i H

i H i H i H

ixH ixHx

e

Z e e

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

Levitov & Lesovik, 1993

Arbitrary time, transient result

21 time)long in(

)(

ln

)(Tr

2

1

)(

ln

)(

ln

)1ln(Tr2

1ln

2

2222

0

0

0

0

ItQ

i

ZQQQ

iG

i

ZQQ

i

ZQ

GZ

M

AL

n

nn

AL

Numerical results, 1D chain

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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.

From Agarwalla, Li, and Wang, PRE 85, 051142, 2012.

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Quantum Master Equation

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Quantum Master Equation

• Advantage of NEGF: any strength of system-bath coupling V; disadvantage: difficult to deal with nonlinear systems.

• QME: advantage - center can be any form of Hamiltonian, in particular, nonlinear systems; disadvantage: weak system-bath coupling, small system.

• Can we improve?

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Dyson Expansion, Divergence

0 0 0

0

0

( )

0

( )

0

2 42 4 6

( ) Tr ( , ) ( ) ( , )

Tr ( ) ,

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

( )2! 4!

| |,

C

C

H B

V d

B c

V d

H B c

T T T

nm

O t S t O t S t

iT O t e

dO t T O t O t V t e

dt

X X V X V O

X n m

0 1Tr ..B cT d

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Unique one-to-one map ρ ↔ ρ0

2

2

2

2 4 42 4 2

0 2

42 3

4

4 42 3 6

2! 4! 2!

[ , ] [ , ]3!

[ , ]2!

( )3! 2!

T

T

T

T T T

X V

T T

T m nmnX V

H X V

T LCL C

X V X V X V

di X V V X V V

dt

E EX V V

I VV VV VV

V p V u

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Order-by-Order Solution to ρ

(0)

(0)

( 2) ( 2) (0)

2(0)

(0) 2 (2) 4

(0)

(2)

3

( )

0

1[ , ]

...

[ , ] 0

1[ , ] [ , ] [ , ]

3!1

[ , ]2!

d

d f

T

T T Td f

f

Tf f

Td

T T Td d d

Td X V

O

X X X

X V Vi

X V V

X V V X V V X V V

X V V

|1 1 | 0 0 ...

0 | 2 2 | 0

0 0 | 3 3 |

... ...

0 |1 2 | |1 3 | ...

| 2 1| 0 | 2 3 |

| 3 1| | 3 2 | 0

... ... ...

d

f

X

X

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DiagrammaticsDiagrams representing the terms for current `V or [X T,V]. Open circle has time t=0, solid dots have dummy times. Arrows indicate ordering and pointing from time -∞ to 0. Note that (4) is cancelled by (c); (7) by (d).

From Wang, Agarwalla, Li, and Thingna, Front. Phys. (2013), DOI: 10.1007/s11467-013-0340-x.

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Analytic Continuation (AC)

• Use the off-diagonal second-order density matrix formula,

as a starting point.

• Let the energy En off the real plane

• Let Em approach En to obtain

• Finally, renormalize ρ

(0)

(2) 1[ , ] ,T m n

f f mn

E EX V V

i

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AC Formula

(0)(2) (0) (0)

, ,

(0)(0)

, , ,

(0) (0)

(0), ,

,

( ) ( ) ( )

( )

( ) ( )

nnnn nj jn nn I jn jj I nj I jn

j n

iinn ij ji I ji

i j i

nj jn nn R jn jj R njj nnn

n nj

V S B

S S V V WE

S S WE

S S V V

E S

,

0

( )

( )( ) ( ) , ( )

( ) ( ) (0)

jn R jnj n

i tR I

S W

WW W iW e C t dt V

C t B t B

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AC: assumptions, and why works?

• Everything else fixed, we assume is an analytic function of the set of energies {Ei}.

• We assume is a function of En only

• Proved to be correct, if system is in equilibrium by comparing with Canonical Perturbation Theory

• Verified numerically to work for a number of models (including a quantum dot and harmonic oscillator center). But still no rigorous proof.

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Comparing AC with DSHDE: discrepancy error for ρ11. Top |AC-DSH|, bottom, difference with a 2nd order time-local Redfield-like quantum master equation solution.

(a) & (b) different temperature bias. See Thingna, Wang, Hänggi, Phys. Rev. E 88, 052127 (2013) for details.

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XXZ spin chain, spin transport

1

1 1 11 1

1

1 1 1

( ) , 0

is conserved in the center

2

N Nx x y y z z z

s i i i i i i ii i

x xL N R

zi

x y y xi i i i i i

H J h J

V B B

j J

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Current & spin chain

• The usual definition j = - dML/dt does not work, as there is no magnetic baths, only thermal baths.

• Tr(ρ(0)j) = 0 exactly so we need to know ρ(2); we use AC.

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Spin current and rectification

(a) Black forward j+, green backward j- currents. Top low temperature (0.5 J), bottom high temperature (5J).

(b) R = |j+ - j-|/|j+ + j-|.

From Thingna and Wang, EPL, 104, 37006 (2013).

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Acknowledgements

Dr. Jose Luis García Palacios

Dr. Juzar Yahya Thingna, University of Augsburg

Prof. Peter Hänggi, University of Augsburg