class="bi x0 y0 w1 h1"

Implicit Symmetries in Single-Electron Transport
Konstantin Kikoin
Mikhail Kiselev
Yshai Avishai
Dynamical Symmetries
for Nanostructures
Through Real and Artificial Mol lesecu
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DOI 10.1007/978-3-211-99724-6
SpringerWienNewYork
Sc
Ph.D. Konstantin Kikoin
hool of Physics and Astronomy
Tel Aviv University
69978 Tel Aviv

pose yet another facet of the existing deep and profound relations between quantum
field theory and condensed matter physics.
The concept of symmetry in quantum mechanics has had its golden age in the
middle of the last century. In that period, the beauty, elegance and efficiency of group
theoretical physics has been exposed in numerous remarkable revelations, from clas-
sification of hadron multiplets, isospin in nuclear reactions, the orbital symmetry
in Rydberg atoms, point-groups in crystallography, translational symmetry in solid
state physics, and so on. At the focus of all these studies stands the symmetry group
of the underlying Hamiltonian. Using the powerful formalism of group theory, the
energy spectrum of the physical system possessing the pertinent symmetry could
be extracted within an elegant and time saving formalism. Exploiting the properties
of discrete and infinitesimal rotation and translation operators, general statements
about the basic properties of quantum mechanical systems could be formulated in a
form of theorems (Wigner theorem, Bloch theorem, Goldstone theorem, Adler prin-
ciple, etc). The intimate relation between group theory and quantum mechanics is
therefore well established and has been exposed in numerous excellent handbooks.
A somewhat more subtle aspect featuring group theory and quantum mechanics
emerged and was formulated later on, that is, the concept of dynamical symmetry.
The notion of dynamical symmetry group is distinct from that of the familiar sym-
metry group. To understand this distinction in an heuristic way let us recall that all
generators of the symmetry group of the Hamiltonian
ˆ
H encode certain integrals
of the motion, which commute with
ˆ
H. These operators induce all transformations
which conserve the symmetry of the Hamiltonian, and may have non-diagonal ma-
trix elements only within a given irreducible representation space of
ˆ
H. On the other

The role of dynamical symmetries and their manifestations will be reviewed and
analyzed in several systems such as complex quantum dots (planar, vertical and
self-assembled), molecular complexes adsorbed on metallic surfaces and attached
to quantum wires, cold gases confined in magnetic traps. It will be shown how
these dynamical symmetries are activated by Coulomb and exchange interactions
with itinerant electrons in the macroscopic Fermi or Bose reservoirs (metallic leads
and substrates in various nanodevices). We will then develop the concept within
numerous physical situations, including the Kondo cotunnelling in various environ-
ments. The notion of dynamical symmetry is meaningful also for the systems out of
equilibrium, in presence of electromagnetic field and stochastic noise and in time-
dependent problems like Landau –Zener effect.
Thus, the main goal of this book is to generalize the principles of dynamical
symmetries formulated for the integrable systems to the many-body systems, for
which only the low-energy part of the excitation spectrum is known.
Tel Aviv - Trieste - Beer Sheva, Konstantin Kikoin
October 31, 2011 Mikhail Kiselev
Yshai Avishai
Acknowledgements
We acknowledge fruitful discussions with our colleagues Boris Altshuler, Jan von
Delft, Peter Fulde, Yuri Galperin, Yuval Gefen, Leonid Glazman, Vladimir Gritsev,
David Khmelnitskii, Il’ya Krive, Tetiana Kuzmenko, Stefan Ludwig, Laurens W.
Molenkamp, Florina Onufrieva, Michael Pustilnik, Jean Richert, Robert Shekhter,
Maarten Wegewijs.
ix

Contents
1 INTRODUCTION 1
2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND
MOLECULES 5
2.1 RigidRotator 8

4.3.3 Crossgeometry 156
4.3.4 Parallelgeometry
4.3.5 Multichannel Kondo tunneling 162
4.4 Kondo physics for small rings . . 179
4.4.1 Kondo tunneling and Aharonov – Bohm interference 190
5 DYNAMICAL SYMMETRIES IN MOLECULAR ELECTRONICS . 197
5.1 Kondo effect in molecular environment 197
5.1.1 Chiral symmetry of orbitals and Kondo tunneling . . . 199
5.1.2 Kondo effect in the presence of Thomas-Rashba precession . 201
5.1.3 Scanning tunneling spectroscopy via Kondo impurities . . . . . 206
5.2 Kondo effect in molecular magnets . . . 211
5.3 Phonon assisted tunneling 217
5.3.1 Two-electron tunneling at strong electron-phonon coupling . 227
6 DYNAMICAL SYMMETRIES AND SPECTROSCOPY OF
QUANTUM DOTS 233
6.1 Kondo effect in the presence of electromagnetic field . . . 234
6.2 Excitonicspectroscopyofquantumdots 240
7 DYNAMICAL SYMMETRIES AND NON-EQUILIBRIUM
ELECTRON TRANSPORT 245
7.1 Dynamically induced finite bias anomalies in tunneling spectra . . . . 248
7.2 Dephasing and decoherence in quantum tunneling 258
7.2.1 VectorKeldyshmodelinthetimedomain 276
8 TUNNELING THROUGH MOVING NANOOBJECTS 283
8.1 Conversion of coherent charge input into the Kondo response . 286
8.1.1 Single-electron shuttling 290
8.2 Time-dependentLandau-Zenereffect 292
9 MATHEMATICAL INSTRUMENTATION 309
9.1 SU (2) groupforarbitraryspin 309
9.2 Kinematical constraints for systems with SO(n) and SU(n)
symmetries 311

S
and a macroscopic system B
(”bath” or ”reservoir”). Due to this contact the symmetries of the system S and
the corresponding conservation laws are violated. If the contact between the two
systems is weak enough, the dynamics of interaction may be described in terms of
transitions between the eigenstates of a system S belonging to different irreducible
representations of the group G
S
generated by the operators which obey the algebra
g
S
. If the operators describing transitions between these eigenstates together with
generators of the group G
S
form an enveloping algebra d
S
for the algebra g
S
, one
may say that the system S possesses dynamical symmetry characterized by some
group D
S
. Dynamical symmetry group offers mathematical tool for a unified ap-
proach to quantum objects, which allows one to consider not only the spectrum of
asystemS , but also its response to external perturbation violating the symmetry
G
S
and various complex many-body effects characterizing interaction between the
system S and its environment B.
An initial impact to the study of dynamical symmetries of the above kind was

l = 0,±1 are included in the dynamical group.
In parallel, it was recognized that the well-known fourth dimension hidden in the
Schr
¨
odinger equation for a hydrogen atom and the Runge – Lenz vector related
to this hidden symmetry can also be described in terms of dynamical symmetry: it
was shown that all the discrete levels of an electron in a Coulomb potential form
a multiplet of a conformal group SO(4,2) [274, 294]. When treating the compo-
nents of the Runge – Lenz vector as three more group generators together with the
usual operators of angular moment, one sees that the enveloping o(4) algebra gen-
erates the SO(4) group of 4D rotations [387], which is the real symmetry of the
Schr
¨
odinger equation for an electron in a Coulomb field in accordance with the
early quantum-mechanical solution of this problem [36, 111]. In this case, addi-
tional group operators do not describe transitions within the energy multiplet, and
one may speak about the hidden symmetry of Schr
¨
odinger equation with a Coulomb
potential ∼ 1/r.
The ideas of dynamical symmetry have been applied also to other integrable sys-
tems, in particular to n-dimensional quantum oscillator [40, 145, 173], where the
generators of SO(n,1) group unite all levels of harmonic oscillator into a single
irreducible representation, to non-relativistic electron in quantizing magnetic field,
and to some other problems. Further generalization of the ideas of dynamical sym-
metry includes also the non-stationary states of quantum systems not necessarily
characterized by definite energy.
1 INTRODUCTION 3
The main achievements of the dynamical symmetry approach during the ”Sturm
und Drang” period of its development are summarized in the monograph [275] [pub-

The dynamical symmetries are usually described by the Lie groups SO(n) with
n  4orSU (n) with n  3. Like in integrable systems mentioned above, these
symmetries become a source of specific response of nanoobject S to external fields.
4 1 INTRODUCTION
Dynamical symmetries may be also discerned in time-dependent, non-equilibrium
and stochastic effects.
In this book all facets of dynamical symmetries of nanosystems are discussed
both in terms of strict mathematical definitions and in a context of practical physical
applications in nano- and molecular electronics. Some aspects of dynamical symme-
tries in the physics of complex quantum dots were briefly considered in our reviews
[32, 204, 206]. We start with an exposition of dynamical symmetries in exactly solv-
able models both mentioned above and newly found (Chapter 2), then give a short
description of nanostructures which were practically realized during the last two
decades (Chapter 3). The central part of the book is devoted to studies of dynamical
symmetries in complex quantum dots and molecular complexes (Chapters 4 – 6)
with a special accent on the Kondo-resonance tunneling regime. The latter regime is
a salient example of many-body phenomenon, where the dynamical symmetry plays
a decisive part. Non-equilibrium tunneling through nanoobjects is a special and vast
enough branch of contemporary nanophysics which deserves a special monograph.
In this book we concentrate only on those non-equilibrium effects which are directly
related to dynamical symmetries of quantum dots and molecular complexes (Chap-
ter 7). Special type of temporal phenomena in nanoobjects are adiabatic and nearly
adiabatic effects induced either by classical motion (“shuttling”) of nanoobject or
cyclic variation of the device parameters which result in periodic time-dependent
level crossing (time-dependent Landau – Zener effect). Symmetry related aspect of
these phenomena are discussed in Chapter 8.
It is presumed that the readers of this book possess a basic knowledge of the main
principles of the Group theory and its applications in Quantum mechanics within a
framework of standard textbooks like [92, 130, 132, 151, 327, 428]. We also use
where necessary the method of many-body Green functions. One may address to

mands but due to accidental degeneracy. Such a degeneracy will play important
part in the following chapters of this book. Here we concentrate on two other as-
pects of the symmetry of quantum systems, namely on the dynamical and hidden
symmetries inherent in some integrable quantum objects.
Following the definition used in Ref. [274], we define the dynamical symmetry
group D
S
as a Lie group characterized by the irreducible representations which act
in the whole Hilbert space of eigenstates |l
λ
 of a Schr
¨
odinger equation
ˆ
H|l
λ
 = E
l
|l
λ
 (2.1)
describing quantum system S .Herel is the index of irreducible representation and
λ
enumerates the lines of this representation. Projection operators for an irreducible
K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries
DOI 10.1007/978-3-211-99724-6_2, © 2012 Springer-Verlag/Wien
5
,hrough Real and Artificial Moleculesin Single-Electron Transport T
6 2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES
representation l

These operators are useful for construction of basis functions for irreducible rep-
resentations of G
S
. Group generators obeying algebra g
S
may be represented via
operators (2.2) (see Chapter 9).
To construct an algebra which generates a dynamical group, one should add to
the set (2.2) the operators
X
λμ
(ll

)
= |l
λ
l

μ
| (2.4)
which project the states belonging to different irreducible representations (l = l

) of
the group G
S
one onto another. Unifying the notations |l
λ
 = |
Λ
, one may write

∑
λ
X
λλ
= 1 (2.6)
and the commutation relations for the operators X
κλ
. In general case these relations
may be presented in the following form [170]
[X
κλ
,X
μν
]
∓
= X
κν
δ
λμ
∓X
μλ
δ
κν
(2.7)
“General case” means that the Fock space includes states which may belong to
different charge sectors, where changing the state
λ
for the state
κ
implies changing

groups of the resolvent operator
ˆ
R =(
ˆ
H −E)
−1
or Schr
¨
odinger operator
ˆ
R
−1
.We
will use these operators in a systematic way to construct the irreducible tensor oper-
ators O
(r)
(scalars, r = 0, vectors, r = 1, and tensors r = 2) which transform along
the representation of the dynamical group which characterizes the symmetry prop-
erties of the supermultiplet of the eigenstates of the Schr
¨
odinger equation:
O
(r)
ρ
=
∑
ΛΛ


Λ

we restrict ourself mainly by discrete eigenstates.
In the two following sections we discuss the symmetry properties of two inte-
grable quantum mechanical systems (rigid rotator and hydrogen atom) and show
how the dynamical symmetries D
S
emerge from the apparent symmetry SO(3) of
the Schr
¨
odinger equation.
8 2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES
2.1 Rigid Rotator
A simplified quantum-mechanical description of molecular motion in a framework
of rigid rotator model implies quenching of vibrational excitations, whereas the ro-
tational degrees of freedom are described as rotation of a ”solid body” around some
axis n = n
ξ
,n
η
,n
ζ
which in turn precesses around a fixed z axis in a 3D space. The
Hamiltonian of symmetric rotator is
ˆ
H =
¯
h
2
2I
⊥


1
I

−
1
I
⊥

L
2
ζ
(2.9)
Here the coordinates (
ξ
,
η
,
ζ
) are bound to the rotation axis, (I
⊥
,I

) are two compo-
nent of the moment of inertia. Both types of rotations are quantized but the energy
levels depend only on the quantum number l and the eigenvalues
κ
of the operator
L
ζ
which change in the interval

= I

the levels lose dependence on
κ
and
acquire 2l + 1-fold degeneracy. Additional degeneracy in projection of the angular
momentum on the z-axis of fixed reference frame results in total (2l + 1)
2
-fold de-
generacy of the level E
j
of spherically symmetric rotator. This additional symmetry
is inessential for the level classification, but it is meaningful from the point of view
of the dynamical symmetry of rigid rotator [95, 275, 289]. Indeed, any rotation
Fig. 2.1 Rigid rotator pre-
cessing around the axis z.
x
yy
z
α
ζ
of the coordinates is characterized by three Euler angles in the precessing system
(
ξ
,
η
,
ζ
) and one more angle
α

1
+ 1)= j
2
( j
2
+ 1)=l(l + 1) and their projections J
1
ζ
and
J
2z
have 2l + 1 values. Thus, the total degeneracy of an eigenstate with given l is
(2l + 1)
2
.
The operators L
±
,L
ζ
performing rotations around the axes n
ξ
,n
η
,n
ζ
generate
the o(3) algebra for the subgroup SO(3) (invariance group), whereas the operators
K
±
,K

projection m and the states with other values l

m

of these quantum numbers,
K
(1)
τ
=
∑
lm,l

m

lm|K
(1)
τ
|l

m

X
ll

mm

(2.11)
Then, using the Wigner-Eckart theorem, we represent the coefficients in this expan-
sion as
lm|K

y
of the operator K, work as ladder operators which
connect the states with
Δ
l = ±1 and thus unite all the energy levels of rigid rotator
into an infinite multiplet of the semisimple group SO(4) with generators L,K and
two Casimir operators (9.18) or (9.19).
One may perceive from the above procedure that the choice of dynamical sym-
metry is not a unique procedure. For example, one may change the signature in the
metrics from {+,+,+, +} to {+,+,+, −} and introduce generators
¯
K
j
(9.27) in-
10 2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES
stead of K
j
. These operators represent dynamical symmetry SO(3,1). Usually the
choice of enveloping group is determined by physical reasons (e.g., by the type of
perturbation which actuates the dynamical symmetry). In particular, in case when
this perturbation implies the selection rules
Δ
l = 0,±1,±2 for transitions between
different states, then one has to use the irreducible tensor K
(2)
τ
of the 2nd rank in
the expansion (2.8). Five components of this tensor together with the operators L
j
of the invariance group SO(3) form the set of generators of the dynamical group

Here
Δ
is the 3D Laplacian.
This equation obviously has the spherical symmetry SO(3) generated by opera-
tors of infinitesimal 3D rotations, but the eigenlevels corresponding to discrete states
with E < 0 depend only on the principal quantum number,
E
n
= −
1
2n
2
(2.14)
and not on the orbital momentum l = n −1,n −2, 1,0, thus possessing the n
2
-
fold degeneracy. All peculiarities of the behavior of an electron in a potential ∼ 1/r
stem from the fact that the rotation group SO(3) is only a subgroup of the true
symmetry group of Eq. (2.14) . To reveal this symmetry let us follow the approach
used in Refs. [36, 111] and turn to the momentum representation of the Schr
¨
odinger
equation (2.13)
2.2 Hydrogen atom and Runge-Lenz vector 11
p
2
2
ψ
(p)+
1

2
= E (2.16)
Then we make a conformal mapping of each point (p, p
0
) onto a point on the surface
of the 4D sphere of unit radius with the coordinates (
ξ
1
,
ξ
2
,
ξ
3
,
ξ
4
)
ξ
i
=
2p
0
p
2
0
+ p
2
p
i

ξ
3
,
ξ
4
)=
π
√
8
(p
0
)
5/2
(p
2
0
+ p
2
)
2
ψ
(p) ≡
Φ
(p)
Eq. (2.15) is transformed into
Ψ
(
ξ
1
,

3
,
ξ

4
)d
4
ξ

|
ξ
1
−
ξ

1
|
2
+ |
ξ
2
−
ξ

2
|
2
+ |
ξ
3

operators describing rotations in six 2D planes (
ξ
i
ξ
j
). Details of this construction
may be found in the book [327]. Returning back from
ξ
-space to original variables
{p, p
0
} and changing p
0
for the operator

−
ˆ
H,where
ˆ
H is the Hamiltonian oper-
ator in Eq. (2.13), one eventually finds equations for these generators:
L =
r ×p −p ×r
2
(2.19)
F =

p ×L −L ×p
2
−