topologic quantum phases
DESCRIPTION
Topologic quantum phases. Pancharatnam phase. The Indian physicist S. Pancharatnam in 1956 introduced the concept of a geometrical phase. Let H(ξ ) be an Hamiltonian which depends from some parameters, represented by ξ ; let |ψ(ξ )> be the ground state. - PowerPoint PPT PresentationTRANSCRIPT
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1
Topologic quantum phasesPancharatnam phase
The Indian physicist S. Pancharatnam in 1956 introduced the concept of a geometrical phase.
Let H(ξ ) be an Hamiltonian which depends from some parameters, represented by ξ ; let |ψ(ξ )> be the ground state.
Compute the phase difference Δϕij between |ψ(ξ i)> and |ψ(ξj)> defined by
This is gauge dependent and cannot have any physical meaning.Now consider 3 points ξ and compute the total phase γ in a closed circuitξ1 → ξ2 → ξ3 → ξ1; remarkably,γ = Δϕ12 + Δϕ23 + Δϕ31
is gauge independent!
Indeed, the phase of any ψ can be changed at will by a gauge transformation, but such arbitrary changes cancel out in computing γ. This clearly holds for any closed circuit with any number of ξ. Therefore γ is entitled to have physical meaning.
There may be observables that are not given by Hermitean operators.
.ijii j i je
22
Consider Evolution of a system when adiabatic theorem holds
(discrete spectrum, no degeneracy, slow changes)
Adiabatic theorem and Berry phase
set of "slowly changing"parameters.
, time-independent sol{ [ ( )]} completeset ofi nstantaneous
utions: [eig
] [ ] [ ] [ ]ensta
( ) [
te
(
s
) )
:
] ( ,
n
n n n
a R
R
t H R a Rt
E R
i
a R
t H R t tt
Assume thatat 0 system is in instantaneous eigenstate,( 0) [ (0)]; then at time t ( ) ( ) is a wave-packet:
( ) ( ) [ ( )] [ ( )] with (0) 1
Then Katoadiabatic theorem ( ) 1 in adiab
n n
n n n m m nm n
n
tt a R t tt c t a R t c a R t c
c t atic limit.
To find the Berry phase, we start from the expansion on instantaneous basis
( ) ( ) [ ( )] [ ( )] with (0) 1 and (t) smalln n n m m n mm n
t c t a R t c a R t c c
3
( ) ( ) ( ) ( )
and plug into .
in the . . . : ( ( ) [ ( )]) ( ) [ ( )] ( ) [
( ) ( )
( )]
( ) )
(
(
)
( )n n n ni t i t i t i tn n n
n n
n n n n
n
n
n R n
n
i Ht
L H S c t a R t c t a R
dRa t
t c t a R tt
c t e c t i
at d
et
t t
t
( ) ( )
0
( ) 1 ( ) , where by definition
( ) ' [ ( ')] dynamical phase, while
( ) geometric phase = Berry phase.
n ni t i tn n
t
n n
n
c t c t e
t dt E R t
t
4
0( ) ' [ ( ')]
( )t
n nii t
n R m
dt E R t
n n n m mm n
mi E i a a
tct eR c a
Negligible because second order (derivative is small, in a
small amplitude)
r.h.s.( ) [ ( )] ( ) ( ) [ ( )]n n n
mn m m m
n
E a R t c t cH t t E a R t
0'[ [ ( ')] ( )]
All together,
0 ( ) [ ] [ ( )]t
n ni dt E R t i t
n R n m m mm n
i R a e c E a R t
Now, scalar multiplication by an removes all other states!
[ ( )] [ ( )]
[ ( )] [ ( )] Berry connection
( ) phase collected at time t
n n R n
n R n
n
iR a R t a R t
a R t a R t
t
55
Professor Sir Michael Berry
6
( ) is a topological phase
and vanishes in simply connected parameter spaces where C can collapse to a point but in a multiply connected spaces yields a quantum number
n n R nC
C i a a dR
0
[ ( )] [ ( )]
The matrix element looks similar to a momentum average, but the gradientis in parameter space. The overall phase change in the transformation is a line integral
n n R n
T
n
iR a R t a R t
i dt
0
. .
This has no physical meaning, it's a gauge, but if C is c
(
losed it bec
) which is gauge inveriant like a magnetic fl
o es
u
m
xn n R n
T
n R n R n
C
n
C
a
i a a dR
a R i a a dR
C
77
Relation of Berry to Pancharatnam phases
( )n n R nC
C i a
B r y
dR
e r
a
Idea: discretize path C assuming regular variation of phase and compute Pancharatnam phase
differences of neighboring ‘sites’.
arg
:
ij
ij
i
ii j i
i j
j
j
Pancharatnam phase de
e
fined by
C
8
1
, 11
Imi ii C C
d8
Pancharatnam phase defined by: ij i ji
i j
e
Pancharatnam for nearby point 1s: i
Limit:
2
thusneglec
Panchar
Wemay conclude
indeed denominator | 1 | 1 ,2
for nearby pointatnam phas
tingsecond ord
s
r,
.e
e
: 1
i
i
i
999
Discrete(Pancharatnam)
1
11
arg ( ) ( )M
i ii
Continuous limit(Berry)
( ) ( )C
i d
Berry’s connection
i
The Pancharatnam formulation is the most useful e.g. in numerics.
Among the Applications:Molecular Aharonov-Bohm effectWannier-Stark ladders in solid state physicsPolarization of solidsPumping
Trajectory C is in parameter space: one needs at least 2 parameters.
Vector Potential Analogy
One naturally writes ( ) · , | . |n n n n R nCC A dR A i a a
introducing a sort of vector potential (which depends on n, however). The gauge invariance arises in the familiar way, that is, if we modify the basis with
( )[ ] [ ], ,i Rn n n n Ra R e a R A A
and the extra term, being a gradient in R space, does not contribute. The Berry phase is real since
| 1 | 0 | | 0| , . 0 | is im aginary
n n R n n R n n n R n
n R n n R n
a a a a a a a aa a c c a a
is rea| | I m |l, |n n R n n n R nA i a a A a a
We prefer to work with a manifestly real and gauge independent integrand; going onwith the electromagnetic analogy, we introduce the field as well, such that
( ) · · .n n nS SC rot A ndS B ndS
10
Im ( | ) Im ( | ) Im[ ( | ) ( | )].i ijk j k ijk j k ijk j kiB a a a a a a a a
The last term vanishes,
and, inserting a compl
( | ) ( | ) ( | ) ( | | )
ete set
Im ( | | ) Im
,
.
ijk j k ijk j ki i
n n n n m m nm n
a a a a a a a a
B a a a a a a
11
| and omittin| g,n n R n n nA i a a B nrot A
To avoid confusion with the electromagnetic field in real space one often speaks about the Berry connection
and the gauge invariant antisymmetric curvature tensor with components
1 23In 3 parameter spaces, , .
Y
d B Y etc
i
Imn n m m nm n
B a a a a
The m,n indices refer to adiabatic eigenstates of H ; the m=n term actually vanishes (vector product of a vector with itself). It is useful to make the Berry conections appearing here more explicit, by taking the gradient of the Schroedinger equation in parameter space:
H R a R E R a [R]
( H R )a R H R a R E R a [R] ( E R )a [R] R n R n n
R n R n n R n R n n
12
Taking the scalar product with an orthogonal am
a H a a H a E a a a ( E ) a
a H a a a E a a
a aa a divergence if degeneracy occurs along C.
E
m R n m R n n m R n m R n n
m R n m m R n n m R n
m R nm R n
m n
E
HE
Formula for the curvature (alias B)
A nontrivial topology of parameter space is associated to the Berry phase, and degeneracies lead to singular lines or surfaces
13
WClassically, the conductance wolud be G=
it should increase without limit for small L.This fails for L < mean free path
L
Ballistic conductor between contacts
W
left electrode right electrode
k
Quantum Transport in nanoscopic devicesBallistic conduction - no resistance. V=RI in not true
If all lengths are small compared to the electron mean free path the transport is ballistic (no scattering, no Ohm law). This occurs in experiments with Carbon Nanotubes (CNT), nanowires, Graphene,…
A graphene nanoribbon field-effect transistor (GNRFET) from Wikipedia
This makes problems a lot easier (if interactions can be neglected). In macroscopic conductors the electron wave functions that can be found by using quantum mechanics for particles moving in an external potential.
14
Number of conducting channels due totransverese degrees of freedom .Electrons available for conduction are those between the Fermi levels
FM k W
Complication: quantum reflection at the contacts
( )k
k
Fermi level right electrode
Fermi level left electrode
Particles lose coherence when travelling a mean free path because of scattering . Dissipative events obliterate the microscopic motion of the electrons . For nanoscopic objects we can do without the theory of dissipation (Caldeira-Leggett (1981). See Altland-Simons- Condensed Matter Field Theory page 130)
If V is the bias, eV= difference of Fermi levels across the junction,How long does it take for an electron to cross the device?
212 12.9Conductance quantum G= per mode resistance=Ge k
h M
2
hopping time given by .
the current is per spin per mode
hophop
hop
ht eVt
e e Vit h
This quantum can be measured! 15
W
left electrode right electrode
k
junction with M conduction modes, i.e. bands of the unbiased hamiltonian at the Fermi level
B.J. Van Wees experiment (prl 1988)A negative gate voltage depletes and narrows down the constriction progressively
Conductance is indeed quantized in units 2e2/h16
1 2 1 2
2
2
1 2
Response Current: I=G V, where:, , electrochemical potentials,
2 Conductance: G= ,
number of modes, quantum transmission probability
2
Line
I=
arV
eM Th
M T
e MTh
Current-Voltage Characteristics J(V) of a junction :
Landauer formula(1957)
17
Phenomenological description of conductance at a junction
1 1FE2 2FE
Rolf LandauerStutgart 1927-New York 1999
18
, 1,
2
2
Extension to finite bias and temperature: Current-Voltage characteristics J(V)
given by J= ( ),
2I(E)= [ ( ) ( ) ( ) ( ) ( )
where ( ) ( ) f=F
( )]L L L R R R
L R
e M E T E f E M E T E f
dEI E
f Eh
f
E
E
ermi function.
Phenomenological description of conductance at a junction
More general formulation, describing the propagation inside a device.
Quantum system
FEFE
leads with Fermi energy EF, Fermi function f(), density of states r
1919
What is the transmission amplitude for electron incoming to eigenstate m and outgoing from eigenstate n?
Quantum system
( ),
Quantum System:eigenstates | , retarded Green's r
mnm g
Quantum System-leads hopping:( ), ( )L R
m mV V
r
r
( ) *
( )
probability to find electron in left wire: ( ). hopping amplitude probability to find final state for electron in right wire: ( ), hopping amplitude amplitide to go from m to
L Ln
R Rn
VV
( ),n : ( )r
mng
*why ? It is the time reversal of L Ln nV V
2020
*
, , , 1,2,
Characteristics:
with ( ) ( )
f=Fermi function.
L R mn mn L Rmn
eJ d ff t t f E f E
This scheme was introduced phenomenologically by Landauer but later confirmed by rigorous quantum mechanical calculations for non-interacting models.
2*
, ,,
Linear response: 2 .mn mnmn
eJ V t t
r
r
r r
( )
( )
( ) ( ),
probability to find electron in left wire: ( )probability to find final state for electron in right wire: ( ) incoming to eigenstate m and outgoing from eigenstate n:
2 ( ) ( )
L
R
L Rmnt V * ( )
, ( )R L rm n mnV g
Multi-terminal extension (Büttiker formula)
,,
voltage between contacts i and j
i i j ij ji j
ij
eJ dE T f E eV f E
V
*
, , , 1,2,
with ( ) ( ) becomesL R mn mn L Rmn
eJ d ff t t f E f E
2222
†hopping 1
hopping
( . .)
C
hain or wire
hopping integr
Hamiltonian
l
:
a
i ii
H t c c h c
t
Microscopic current operator
device
J
, 1 1
† † † †1 1 1 1
,
Continuity equation: 0. Here t
Heisenberg EOM:
ˆ ˆ
[ ( ), (
[ , ] [ ]
(
)]
m m m m
mm hopping m m m m m m m m
m m
J J
dnie e n
dJdivJ divJdx
d Ai
H et c c c c c c c c
A A t H t idt t
dti J
r
† † † †1 1 1 1 1 1) [ ]m m hopping m m m m m m m mJ et c c c c c c c c
2323
† †1 1
Thecurrent operator at site m (Caroli et al.,J.Phys.C(1970))
is physically equivalent.hoppingm m m m m
etJ c c c c
i
†hopping 1
hop
† †,
p ng
1 1 1
i
( . .)
Chain or w
hopping integra
ire Hamilto i
l
n an:
hoppingm m m m m m
i ii
H t c c h c
t
etJ i c c c c
Microscopic current operator
device
J
Partitioned approach (Caroli 1970, Feuchtwang 1976): fictitious unperturbed biased system with left and right parts that obey special boundary conditions: allows to treat electron-electron and phonon interaction by Green’s functions.
device
this is a perturbation (to be treated at all orders = left-right bond
† †1 1 1 1
† †
Time-independent partitioned framework for the calculation of characteristics
ˆ ( ) ( ) ,2
( ) ( ') ( ) , ( ) ( ) ( ')
hopping hoppingm m m m m m mm mm
et et dJ c c c c J J g gi
g t c t c t g t c t c t
24Drawback: separate parts obey strange bc and do not exist.
=pseudo-Hamiltonian connecting left and right
Pseudo-Hamiltonian Approach
25
Simple junction-Static current-voltage characteristics
J
chemical potential 1
2-2 0
1
U=0 (no bias)
no current
Left wire DOS
Right wire DOS
no current
U=2
current
U=1
26
2 2
0
22
2
Half-filled 1d system left wire right wirehopping 1
( ) 1 (1 )
( ) ( )8 2( )( ( ) ( ))
2 4V
gVg geJ V d
V Vg g
1 2 3 4V
0 .1
0 .2
0 .3
0 .4
J
( ) , 0
quantum conductivity
( ) 0, 4 bands mismatch
eUJ V U
e
J V U
Static current-voltage characteristics: example
J