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.
Compute the phase difference Δϕij between |ψ(ξ i)> and |ψ(ξj)> defined
by
i
 i    j  e ij   i    j .
 
 
This is gauge dependent and cannot have any physical meaning.
Now consider 3 points ξ and compute the total phase γ in a closed circuit
ξ1 → 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.
1
1
Adiabatic theorem and Berry phase
Consider Evolution of a system when adiabatic
theorem holds
(discrete spectrum, no degeneracy, slow changes)

i
(t )  H[ R(t )](t ), R  set of "slowly changing" parameters.
t
t ,
time-independent solutions : H[ R]an [ R]  En [ R]an [ R]
 {an [ R(t )]}  complete set of instantaneous eigenstates:
Assume that at t  0 system is in instantaneous eigenstate,
(t  0)  an [ R(0)] ; then at time t (t )   n (t) is a wave-packet:
  n (t )  cn (t )an [ R(t )]   cm am [ R(t )] with cn (0)  1
mn
Then Kato adiabatic theorem 
cn (t )  1 in adiabatic limit.
2
2
cn (t )  1

cn (t )  ein (t )i n (t ) , where by definition
t
 n (t )    dt 'En [ R(t ')]  dynamical phase, while
0
 n (t )  geometric phase = Berry phase.
To find the Berry phase, we start from the expansion on instantaneous basis
 n (t)  cn (t)an [ R(t)]   cm am [ R(t)]
m n
and plug into
i
with cn (0)  1 and cm (t) small

 n  H n .
t


in the L.H .S .:
(cn (t )an [ R(t )])  cn (t ) an [ R(t )]  cn (t ) an [ R (t )]
t
t
cn (t )  ein (t ) i n (t )  cn (t )  i ( n (t )   n (t ))ein ( t ) i n ( t )

dR
an (t ) 
  R an
t
dt
3
i

 i
 i n (t ) 0
 n (t )   En  i n  R  R  an e
t


t
dt ' En [ R ( t ')]
   cm am  cm am 
m n
r.h.s.
H n (t )  En an [ R(t )]cn (t )   cm (t ) Em am [ R(t )]
m n
Negligible because second
order (derivative is small, in a
small amplitude)
All together,
i
t
0  (  n  i R  R )an e 0
dt '[ En [ R ( t ')] i n ( t )]
  [cm  Em ]am [ R(t )]
mn
Now, scalar multiplication by an removes all other states!
 n  iR an [ R(t )]  R an [ R(t )]
an [ R(t )]  R an [ R(t )]  Berry connection
 n (t )  phase collected at time t
4
Professor Sir Michael Berry
5
5
 n  iR an [ R(t )]  R an [ R(t )]
The matrix element looks similar to a momentum average, but the gradient
is in parameter space. The overall phase change in the transformation is a line integral
T
T
 n  i  dt an  R an .R  i 
0
an  R an .dR
0
This has no physical meaning, it's a gauge, but if C is closed it becomes
 n (C )  i 
an  R an dR which is gauge inveriant like a magnetic flux
C
C
 n (C )  i 
an  R an dR is a topological phase
C
and vanishes in simply connected parameter spaces where C can collapse
to a point but in a multiply connected spaces yields a quantum number
6
Relation of Berry to Pancharatnam phases
Berry
C
 n (C )  i 
an  R an dR
C
Pancharatnam phase ij defined by :
 
ij   arg  i    j

 
  i    j
e
 iij
 
  i    j
Idea: discretize path C assuming regular
variation of phase and compute
Pancharatnam phase differences of
neighboring ‘sites’.
7
7
Pancharatnam phase defined by:
Pancharatnam for nearby points:
e
 iij
1  i 

  i    j 
  i    j 
       
       
             
          1        
We may conclude   i       
 2
indeed denominator |1  i | 1 
, thus neglecting second order,
2
Pancharatnam phase for nearby points: 1  i          .
Limit:
   i ,i 1 
i 1

C
d    Im       
C
8
8
Continuous limit
Discrete
(Berry)
(Pancharatnam)
M 1
   arg   (i )  (i 1 )
i 1
   i  ( )  ( ) d
C
Berry’s connection
i       
The Pancharatnam formulation is the most useful e.g. in numerics.
Trajectory C is in parameter space: one needs at least 2 parameters.
Among the Applications:
Molecular Aharonov-Bohm effect
Wannier-Stark ladders in solid state physics
Polarization of solids
Pumping
9
9
Vector Potential Analogy
One naturally writes
 n (C )   An·dR, An  ian | R | an .
C
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
an [ R]  ei ( R) an [ R],
An  An  R  ,

and the extra term, being a gradient in R space, does not contribute. The Berry
phase is real since
 an | an   1 
 R  an | an   0 
 R an | an    an |  R an   0
  an |  R an   c, c.  0   an |  R an  is imaginary
An  ian | R | an  is real,  An   Iman | R | an 
We prefer to work with a manifestly real and gauge independent integrand; going on
with the electromagnetic analogy, we introduce the field as well, such that
 n (C )   rot An· ndS   Bn· ndS.
S
S
10
An  ian | R | an , Bn  rot An and omitting n
Bi   Im   (a | a)i   Im  ijk  j (a |  k a)   Im[ ijk ( j a | k a)   ijk (a |  j  k a)].
The last term vanishes,
  (a | a)i   ijk  j (a |  k a)   ijk ( j a |  k a)  (a |  | a) i
and, inserting a complete set,
Bn   Im (an |  | an )   Im 
mn
an am  am an .
To avoid confusion with the electromagnetic field in real space one
often speaks about the Berry connection
    i      
and the gauge invariant
antisymmetric curvature tensor
with components


Y 
 


 
In 3d parameter spaces, B1  Y23 , etc.
11
Formula for the curvature (alias B)
Bn   Im 
m n
an am  am an
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:
 R H  R  a n  R    R E n  R  a n [R]
 ( R H  R )a n  R   H  R   R a n  R   E n  R   R a n [R] ( R E n  R  )a n [R]
Taking the scalar product with an orthogonal am
a m  R H a n  a m H R a n  E n a m  R a n  a m ( R E n ) a n
a m  R H a n  Em a m  R a n  E n a m  R a n
a m R a n 
a m R H a n
 divergence if degeneracy occurs along C.
Em  E n
A nontrivial topology of parameter space is associated to
the Berry phase, and degeneracies lead to singular lines or
surfaces
12
Quantum Transport in nanoscopic devices
Ballistic conduction - no resistance. V=RI in not true
Ballistic conductor between contacts
W
Classically, the conductance wolud be G=
W
L
it should increase without limit for small L.
This fails for L < mean free path
k
left electrode
right
electrode
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.
13
Number of conducting channels due to
 (k )
transverese degrees of freedom M  k FW .
Fermi level
left electrode
Fermi level right
electrode
Electrons available for conduction are those
between the Fermi levels
k
Complication: quantum reflection at the contacts
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)
14
W
k
left electrode
right
electrode
junction with M conduction modes, i.e. bands
of the unbiased hamiltonian at the Fermi level
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?
thop  hopping time given by eV   
 the current is i 
2e2
 Conductance quantum G=
h
e
thop
e2V

h
h
thop
.
per spin per mode
per mode  resistance=G 1 
12.9
M
k
This quantum can be measured!
15
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/h
16
Current-Voltage Characteristics J(V) of a junction :
Landauer formula(1957)
EF 2   2
EF 1  1
Phenomenological description of conductance at a junction
Linear Response Current:  I=G V, where:
 V  1  2 ,
1 , 2  electrochemical potentials,
Rolf Landauer
Stutgart 1927-New York 1999
2e2
Conductance: G=M
T ,
h
M  number of modes, T  quantum transmission probability

2e 2
 I=
MT  1  2 
h
17
Phenomenological description of conductance at a junction
Extension to finite bias and temperature: Current-Voltage characteristics J(V)
given by J=  dEI ( E ),
2e 2
I(E)=
[ M L ( E )TL ( E ) f L ( E )  M R ( E )TR ( E ) f R ( E )]
h
where f L , R ( E )  f ( E  1,2 ) f=Fermi function.
More general formulation, describing the propagation inside a device.
Quantum system
EF
EF
leads with Fermi energy EF, Fermi function
f(), density of states r
18
Quantum system
Quantum System:
eigenstates | m  , retarded Green's gm( r,)n
Quantum System-leads hopping:
VmL ( ), VmR ( )
What is the transmission amplitude for electron incoming to eigenstate m and
outgoing from eigenstate n?
probability to find electron in left wire: r ( L ) ( ).
hopping amplitude VnL*
probability to find final state for electron in right wire: r ( R) ( ), hopping amplitude VnR
amplitide to go from m to n : gm( r ,)n ( )
why VnL* ? It is the time reversal of VnL
19
19
probability to find electron in left wire: r ( L ) ( )
probability to find final state for electron in right wire: r ( R ) ( )
incoming to eigenstate m and
outgoing from eigenstate n:
tm ,n  2 r ( L) ( ) r ( R) ( )VmR VnL* gm( r ,)n ( )
Characteristics:
e
J   d  f L     f R    tm ,ntm* ,n with f L ,R ( E)  f ( E  1,2 )
m ,n
f=Fermi function.
Linear response: J  2
e2
 V  tm ,ntm* ,n .
m ,n
This scheme was introduced phenomenologically by Landauer but later confirmed
by rigorous quantum mechanical calculations for non-interacting models.
20
20
Multi-terminal extension (Büttiker formula)
J
e
*


d

f


f

t
t
with f L ,R ( E)  f ( E  1,2 ) becomes





R
m ,n m ,n
 L

m ,n
Ji 
e
 dE
i,j


Ti , j  f E  eVij  f j  E 


Vij  voltage between contacts i and j
Microscopic current operator
device
J
Chain or wire Hamiltonian:
H  thopping  (ci†1ci  h.c.)
i
thopping  hopping integral
r
dJ
Continuity equation:
 divJ  0. Here divJ 
 J m ,m 1  J m 1m
t
dx
d
A
Heisenberg EOM: i
A  [ A(t ), H (t )]  i
dt
t
dnˆm
ie
 e[nˆm , H ]  ethopping [ cm† cm 1  cm† 1cm    cm† 1cm  cm† cm 1 ]
dt
 i ( J m,m 1  J m 1m )  ethopping [ cm† cm 1  cm† 1cm    cm† 1cm  cm† cm 1 ]
22
22
Microscopic current operator
device
J
Chain or wire Hamiltonian:
H  thopping  (ci†1ci  h.c.)
i
thopping  hopping integral
J m,m 1  i
ethopping
c
c  cm† cm 1 
†
m 1 m
The current operator at site m (Caroli et al.,J.Phys.C(1970))
Jm 
ethopping
i
c
c  cm† 1cm  is physically equivalent.
†
m 1 m
23
23
Pseudo-Hamiltonian Approach
Time-independent partitioned framework for the calculation of characteristics
et
et
Jˆm  hopping  cm† 1cm  cm† 1cm   J  J m  hopping
i
g  (t )  c † (t ')c(t ) , g  (t )  c(t )c † (t ')



d 
g mm 1 ( )  g  ( ) mm 1 ,

2
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
=pseudo-Hamiltonian connecting left and right
this is a perturbation (to be treated at all orders = left-right bond
Drawback: separate parts obey strange bc and do not exist.
24
Simple junction-Static current-voltage characteristics
J
1
chemical potential   1
-2
U=0 (no bias)
U=1
no current
current
0
2
U=2
Left wire
DOS
Right wire
DOS
no current
25
Static current-voltage characteristics: example
1d system left wire  right wire
Half-filled
J
hopping  1
g( )  1   2  (1   2 )
J(V )  
8 e

g( ) g( 
0
 d
V

2
V
)
2
V 2 V2
( g( )  g(  )) 
2
4
J
0.4
J(V )  
0.3
eU

, U 0
e
0.2
 quantum conductivity
0.1
J(V )  0, U  4 bands mismatch

V
1
2
3
4
26