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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
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
10
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
• 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).