Pancharatnam phase The Indian physicist S Pancharatnam in 1956 introduced the concept of a geometrical phase Let Hξ be an Hamiltonian which depends from some parameters represented by ID: 799922
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Slide1
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1
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
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 + Δϕ31is 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.
Slide22
2
Consider
Evolution of a system when
adiabatic theorem holds
(discrete spectrum, no degeneracy, slow changes)
Adiabatic theorem and Berry phase
Slide3To find the Berry phase, we start from the expansion on instantaneous basis
3
Slide44
Negligible because second order (derivative is small, in a small amplitude)
Now, scalar multiplication by a
n
removes all other states!
Slide55
5
Professor Sir Michael Berry
Slide66
C
Slide77
7
Relation
of Berry to Pancharatnam
phases
Idea: discretize path C assuming regular variation of phase and compute Pancharatnam phase differences of neighboring ‘sites’.
C
Slide88
8
Limit:
Slide99
9
9
Discrete
(Pancharatnam)
Continuous limit
(Berry)
Berry’s connection
The
Pancharatnam
formulation
is
the
most
useful
e.g.
in
numerics
.
Among
the Applications:
Molecular
Aharonov-Bohm
effect
Wannier-Stark
ladders
in
solid
state
physics
Polarization
of
solids
P
umping
Trajectory
C
is
in
parameter
space
:
one
needs
at
least
2
parameters
.
Slide10Vector Potential Analogy
One naturally writes
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
and the extra term, being a gradient in R space, does not contribute. The Berry phase is real since
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
10
Slide11The last term vanishes,
11
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
Slide12The 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:
12
Taking the scalar product with an orthogonal a
m
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
Slide1313
Ballistic conductor between contacts
W
left electrode
right electrode
Quantum Transport in nanoscopic devices
Ballistic
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.
Slide1414
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)
Slide15If
V
is the bias, eV=
difference of Fermi levels across the junction
,How long does
it take for an electron to cross the device?
This
quantum can be measured!15
W
left electrode
right electrode
junction
with M
conduction
modes
, i.e.
bands
of the
unbiased
hamiltonian
at
the Fermi
level
Slide16B.J. Van Wees experiment (prl 1988)A negative gate voltage depletes and narrows down the constriction progressively
Conductance is indeed quantized in units 2e
2
/h
16
Slide17Current
-Voltage
Characteristics J(V) of a junction
: Landauer
formula(1957)
17Phenomenological
description of conductance at a junction
Rolf Landauer
Stutgart 1927-New York 1999
Slide1818
Phenomenological
description of conductance
at a junction
More general
formulation, describing the propagation inside a device.
Quantum system
J
leads
with Fermi
energy
E
F
, Fermi
function
f(
e
),
density
of
states
r(e)
Slide191919
Quantum
system
J
Slide202020
This scheme was introduced phenomenologically by Landauer but later confirmed by rigorous quantum mechanical calculations for non-interacting models.
Slide21Slide2222
22
Microscopic
current operator
device
J
Slide2323
23
Microscopic
current operator
device
J
Slide24Partitioned 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
24
Drawback: separate parts obey strange bc and do not exist.
=pseudo-Hamiltonian connecting left and right
Pseudo-Hamiltonian Approach
Slide2525
Simple junction-Static current-voltage
characteristics
J
2
-2
0
1
U=0 (no bias)
no current
Left wire
DOS
Right wire
DOS
no current
U=2
current
U=1
Slide2626
Static current-voltage characteristics: example
J