dynamical symmetries for nanostructures implicit symmetries in single-electron transport through real and artificial molecules

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

Israel
Ph.D. Mikhail Kiselev
Italy
Strada Costiera 11
34151 Trieste

Ph.D. Yshai Avishai
Ben Gurion University
Israel
84105 Beer Sheva



The Abdus Salam Intl. Center
for Theoretical Physics
2011 94375 2
e-ISBN 978-3-211-99724-6
Dedicated to the memory of Yuval Ne’eman
and John Hubbard, two great physicists
whose ideas are the corner stones of the
theories presented in this book.

Preface
The main goal of this monograph is to demonstrate the relevance of dynamical sym-
metry and its breaking to the rapidly growing field of nanophysics in general, and
nanoelectronics in particular. It is intended to amalgamate seemingly highly abstract
concepts of Group theory with the physics of recently fabricated nanoobjects such
as single electron transistors. In all these systems, dynamical symmetries are shown
to be intimately related with many-body physics, and in particular, the ubiquitous
Kondo effect and other hallmarks of quantum impurity problems. Thereby, we ex-

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

vii
viii Preface
hand, dynamical symmetry of
ˆ
H is realized by transformations implementing tran-
sitions between states belonging to different irreducible representations of the sym-
metry group. One may then say that the generators of dynamical symmetry group of
a quantum mechanical system are in fact the generators of the energy spectrum or
some part of it. Special examples of dynamical symmetries in quantum mechanics
emerge as hidden symmetries, where additional degeneracy exists due to an implicit
symmetry of the interaction. Another example is supersymmetry, where the group
algebra includes both commutation and anticommutation relations.
The starting point in most of our analysis is a generalized Anderson Hamiltonian
which, under certain conditions can be approximated by a generalized spin Hamil-
tonian encoding a myriad of exchange interactions between localized electrons in
nano-objects (such as quantum states in complex quantum dots) and itinerant elec-
trons in the reservoirs made in contact with the localized electrons. These exchange
interactions may be due to spin as well as to orbital degrees of freedom. They lead
to effective exchange Hamiltonians that display a rich pattern of dynamical sym-
metries. Mathematically, these symmetries are exposed as the pertinent exchange
Hamiltonian includes, in addition to the standard spin operators, new sets of vector
operators which form the basis for the representation of irreducible tensor operators
entering the effective Hamiltonian. These operators induce transitions between dif-
ferent spin multiplets and generate dynamical symmetry groups (such as SU (n) and
SO(n)) that are not exposed within the bare Anderson Hamiltonian. Like in quan-
tum field theory, the most dramatic aspects of dynamical symmetry in the present
context is not its relation with the spectrum but, rather, the manner in which it is bro-
ken. An indispensable tool for manipulating the pertinent mathematics required for
identifying the relevant dynamical symmetry groups is the superalgebra of Hubbard
operators, upon which we will heavily rely.

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

2.2 Hydrogen atom and Runge-Lenz vector 10
2.3 Dynamicalsymmetriesforspinsystems 15
2.4 Hubbard atom and Fulde molecule . . . 23
2.4.1 Three-fold way for Hubbard atom . . 29
2.5 Fock–Darwinatom 31
2.6 Dynamicalsymmetryandsupersymmetry 35
2.6.1 Manifestationsofsupersymmetryinatomicmodels 39
2.7 Quasienergy spectrum for periodical time-dependent problems . . . . 45
xi
3 NANOSTRUCTURES AS ARTIFICIAL ATOMS AND
MOLECULES 49
3.1 Introductory remarks. 49
3.2 Planarquantumdots 51
3.3 Verticalquantumdots 61
3.4 Self-assembledquantumdots 66
3.5 Complexquantumdots 70
3.5.1 Double quantum dots . . . 72
3.5.2 Triplequantumdots 80
3.6 Moleculesandmolecularcomplexes 91
3.6.1 Fullerenemoleculesasquantumdots 93
3.6.2 Nanotubesasquantumdots 95
3.6.3 Single electron tunneling through metal organic complexes . 96
3.6.4 Vibrational degrees of freedom in single molecular tunneling 101
xii Contents
4 DYNAMICAL SYMMETRIES IN THE KONDO EFFECT 107
4.1 Kondo mapping and beyond (surplus symmetries) 108
4.2 Kondo effect in quantum dots with even occupation . . . . 126
4.3 Kondo physics for short chains . 136
4.3.1 Serialgeometry 137
4.3.2 Side geometry, Fano – Kondo effect. 147

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

9.2.1 SO(4) group 311
9.2.2 Noncompact groups SO(p,n − p) 313
9.2.3 Groups of conformal transformations 314
9.2.4 From SU (2) to SU (n) 315
9.3 Bosonizationandfermionizationforarbitraryspins 320
160
Contents xiii
9.3.1 Schwinger boson representation for the SU(2) group 321
9.3.2 Holstein – Primakoff boson representation for the SU (2)
group 322
9.3.3 Dyson – Maleev representation for the SU(2) group 323
9.3.4 Pomeranchuk – Abrikosov spin fermion representation for
the SU (2) group 323
9.3.5 Spin-fermion representations for the SO(n) groups . . 325
9.3.6 Popov–Fedotovsemi-fermionrepresentation 326
9.3.7 Majoranafermionization 327
9.3.8 Mixedfermion-bosonrepresentations 327
10 CONCLUSIONS AND PROSPECTS 329
Index 335
References 341

Chapter 1
INTRODUCTION
In modern theoretical physics the word-combination ”dynamical symmetry” is fre-
quently used in the context of various mechanisms of dynamical symmetry break-
ing of vacuum expectation value (Higgs – Anderson mechanism in particle physics,
Anderson – Nambu mechanism in superconductivity, etc, see [101] for basic refer-
ences). This book is devoted to analysis of dynamical symmetries that arise when
a group theoretical approach is used in a description of a contact between a few
electron nanosystem S with definite symmetry G

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

given in a context of classification of elementary particle multiplets. The first rep-
resentative example of dynamical symmetry was an attempt to construct hadron
multiplets and transitions between states within this miltiplet by means of gener-
ators of the group SU(3) [126, 127, 296, 447]. This group of unitary matrices of
1
K. Kikoin et al., Dynamical Symmetries for Nanostructures: Implicit Symmetries
DOI 10.1007/978-3-211-99724-6_1, © 2012 Springer-Verlag/Wien
,hrough Real and Artificial Moleculesin Single-Electron Transport T
2 1 INTRODUCTION
the 3-rd order describes the states of 3-level system and all transition between these
states. Three states were identified with three quarks labeled by u,d,s ”colors”. The
concept of dynamical symmetry (the “eightfold way”), as an approximate symmetry
generated by some operator algebra which describes transitions between the states
belonging to various irreducible representations of the group G
S
as a result of dy-
namics was formulated quite distinctly in Ref. [81] (see also [39]). Following this
paradigm, the dynamical symmetry group D
S
of a quantum mechanical system may
be defined as a finite-dimensional Lie group whose irreducible representations act
in a Hilbert space of all states of a subsystem S in a given energy interval E which
characterize the scale of interaction of this subsystem with its environment B.
A short time later the energy spectrum of an integrable quantum mechanical sys-
tem , namely the rigid rotator, also was described in terms of dynamical symmetry
given by SO(4) group of 4-dimensional orthogonal matrices [46, 289]. The conven-
tional symmetry group of rigid rotator is the usual SO(3) group of 3D rotations,
and additional dimension arises when the ”supermultiplets” with different orbital
moments l and ”selection rules”
Δ

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-

lished in Russian]. Various facets of the Coulomb problem for the hydrogen atom
treated in terms of hidden and dynamical symmetry approach are discussed in two
more books [95, 201]. In the latter book the supersymmetry of hydrogen atom which
is closely related to existence of the Runge – Lenz vector is also discussed.
During the last decade of past century novel approximately symmetric few-body
quantum objects became available for theoretical analysis due to rapid progress of
nanotechnology and nanophysics. These nano-objects are quantum dots with count-
able number of electrons and controllable spin states incorporated in electric cir-
cuits, where metallic electrodes play part of a reservoir B for a quantum dot S
[238, 359]. Another class of quantum objects with similar properties are molecular
complexes which form bridges between metallic electrodes or between the metallic
substrate and the tip of tunnel microscope [70, 295].
It was recognized [203] that the concept of dynamical symmetry is highly useful
for the study of many-body effects which accompany tunneling through quantum
dots and molecular bridges. In case of strong Coulomb blockade which suppresses
charge fluctuations in a quantum dot, the spin state of a dot with given number N
of electrons is usually well defined. Then electron tunneling through the dot which
may be detected as a single electron tunnel current between the source and drain
electrodes, breaks the spin symmetry of this dot. This symmetry violation as well
as the many-body effects which accompany electron tunneling through quantum
dots may be quite elegantly described within a framework of dynamical symmetry
approach.
Unlike the integrable systems with dynamical symmetries described in the mono-
graphs [95, 201, 275], the problems of complex quantum dots and molecules in con-
tact with boson or fermion bath as a rule cannot be solved exactly. Moreover, the
type of dynamical symmetry strongly depends on the characteristic energy scale E
of the coupling between the nanoobject S and the bath B. Besides, this symmetry
may be changed with decreasing temperature and varying control parameters, thus
resulting in quantum criticality phenomena, which may be easily detected as varia-
tions of current-voltage characteristics in single-electron tunneling experiments.

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

the monograph [106] as an introductory course to this field. However we considered
expedient to collect in the Mathematical Annex (Chapter 9) all relevant informa-
tion about the characteristic properties of those Lie groups which are responsible
for dynamical symmetries in nanosystems and to present other useful mathemati-
cal information related to the physical problems discussed in this book. In Chapter
10, which terminates the book, the implementation of dynamical symmetry ideas
in nanophysics is summarized and possible future development of this approach is
discussed.
Chapter 2
HIDDEN AND DYNAMICAL SYMMETRIES
OF ATOMS AND MOLECULES
We concentrate in this book on the symmetry properties of nanoobjects (quantum
dots, rings and short chains of quantum dots, molecular complexes) in a weak tun-
neling and/or capacitive contact with reservoirs (metallic electrodes attached to
quantum dots, metallic substrates or edges of nanowires for molecular complexes
deposited on these surfaces and points, etc). Before turning to these artificially en-
gineered devices, we will review in brief the origin of dynamical symmetry in ”nat-
ural” quantum objects, i.e. in some integrable quantum systems with well defined
energy spectrum and quantum numbers. Conventionally the symmetry of such sys-
tems is considered in terms of the symmetry group G
S
of Schr
¨
odinger equation.
This description is based on the fundamental Wigner theorem [428] which states
that the eigenfunctions which belong to a given energy level E are transformed
along the same irreducible representation of the group G
S
.
Sometimes two or more energy levels coincide not because of symmetry de-

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

X
λμ
(l)
= |l
λ
l
μ
| (2.2)
play central part in the procedure of construction of irreducible representations of
a group of Schr
¨
odinger equation G
S
. The basic property of these operators is given
by the equation
X
λμ
(l)
|l

ν
 =
δ
ll

δ
μν
|l
λ
. (2.3)

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

the commutation relation
[X
ΛΛ

,
ˆ
H]=(E
Λ

−E
Λ
)
ˆ
H (2.5)
The right hand side of this relation turns into zero provided the states
Λ
and
Λ

belong to the same irreducible representation of the group G
S
.
If one succeeds in constructing a closed algebra d
S
from the set of operators
(2.2),(2.4) then it is possible to say that the system described by the Hamiltonian
(2.1) possesses the dynamical symmetry D
S
. This algebra is conditioned by the
norm

∑
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

the number of fermions N
λ
→ N
κ
in a many-particle system. If both N
λ
−N
κ
and
N
ν
−N
μ
are odd numbers(Fermi-type operators), the plus sign should be chosen
2 HIDDEN AND DYNAMICAL SYMMETRIES OF ATOMS AND MOLECULES 7
in Eq. (2.7). If at least one of these difference is zero or even number (Bose-type
operators), one should take the minus sign.
The operators X
κλ
were exploited by J. Hubbard as a convenient tool for de-
scription of elementary excitations in strongly correlated electron systems (SCES).
His seminal model of interacting electron motion in a narrow band, known now as a
Hubbard model [169, 170, 171] was the first microscopic model of SCES for which
the conventional perturbative approach based on Landau Fermi liquid hypothesis
turned out to fail (see detailed discussion in Ref. [157]. Now the realm of SCES is
really vast, and the most of artificial nanostructures belong to this realm. In particu-
lar, complex quantum dots under strong Coulomb blockade are typical examples of
short Hubbard chains or rings (see Chapter 4).
The Hubbard operators (2.4) obeying the commutation relation (2.5) is a con-
venient tool for construction of the algebras generating the dynamical symmetry

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

|O
(r)
ρ
|
Λ

X
ΛΛ

. (2.8)
Here the index
ρ
stands for components of irreducible tensor operator of the rank
r. On the one hand, it is clear that the operators X
ΛΛ

are able to generate all the
eigenstates of the Hamiltonian
ˆ
H from any given initial state
Λ

. On the other hand,
the components of the operators O
(r)
form a closed algebra, which characterizes
the dynamical symmetry group provided the Hamiltonian
ˆ
H possesses such sym-
metry. Having in mind future applications to geometrically confined nanoobjects,

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
⊥


L
2
ξ
+ L
2
η

+
¯
h
2
2I

L
2
ζ
=
¯
h
2
2I
⊥
L
2
+
¯
h
2
2


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

κ
= −l, + l
E
j
κ
=
¯
h
2
2I
⊥
l(l + 1)+
¯
h
2
2

1
I

−
1
I
⊥

κ
2
. (2.10)
In case of fully symmetric rotator with I
⊥

= 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

between the axes
ζ
and z (Fig. (2.1). Corresponding
spherical coordinates are (r,
ϕ
,
ϑ
,
α
). The basis functions for description of such a
motion are the hyperspherical harmonics Y
nlm
(
α
,
ϑ
,
ϕ
). On the other hand, these
2.1 Rigid Rotator 9
harmonics form the basis for the irreducible representations of the group SO(4) of
rotations on a 4D sphere (see Section 9.2.1). The six generators L,K of this group
obey o(4) algebra defined in Eq. (9.14). One may then turn to linear combinations
J
(1,2)
=(L ±K)/2 (9.16). These two vectors describe the two types of rotations
mentioned above. Due to the kinematic constraint (9.19) these operators have the
same eigenvalues j
1
( j

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

ζ
, which depend on all rotation angles (
ϕ
,
ϑ
,
α
) (9.21) generate the dynamical
algebra o(4), and thus define the dynamical symmetry group SO(4) of a rigid rota-
tor which unites all the energy levels of rigid rotator in an infinite “supermiltiplet”
[289].
To show this, let us consider the vector K as an irreducible tensor of the 1-st rank
and express its components K
1
τ
via projection operators (2.4) following the pattern
(2.8). Here
τ
= 0,±1 stands for K
ζ
,K
±
, respectively. In this case, one should use the
operators X
(ll

)
mm

describing transitions between states with given momentum l and its

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

(1)
τ
|l

m

 =(−1)
l−m

l 1 l

−m
τ
m


lK
(1)
l

 (2.12)
(cf. Eq. 2.8) The factors in the r.h.s. of this equation are the Wigner 3 j-symbol and
the reduced matrix element of the vector K. Explicit form of these matrix elements
may be found in [275, 289, 291].
It follows from the triangle rule for 3 j-coefficients in Eq. (2.12) that the transver-
sal components K
±
= K
x
±iK

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

SU (3) (see Sections 2.3 and 9.2.4).
2.2 Hydrogen atom and Runge-Lenz vector
The quantum mechanical problem of electronic spectrum of hydrogen atom inher-
ited its peculiar properties from its classical analog, i.e. from the mechanical prob-
lem of rotation of one celestial body in the gravitational field ∼ 1/r of another body
(Kepler problem). It was realized three centuries ago that the orbital rotation of ce-
lestial body in such a field is characterized by a specific constant of motion which
is known now as a Laplace-Runge-Lenz vector (although two latter physicists only
used this vector in their own tutorial and scientific texts). This vector arose anew in
the analysis of the Schr
¨
odinger equation for an electron wavefunction
ψ
(r) in the
potential field created by a proton. This equation in atomic units (e = 1,
¯
h = 1,m = 1)
has the form:

−
Δ
2
−
1
r

ψ
(r)=E
ψ
(r). (2.13)

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
π

ψ
(q)dq
(p −q)
2
= −
p
2
0
2
ψ
(p) (2.15)
where
ψ
(p)=
1
(2
π
)
3/2

ψ
(r)e
−i(p·r)
dr, −
p
2
0

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

(i = 1,2,3);
ξ
4
=
p
2
0
− p
2
p
2
0
+ p
2
;
∑
i=1−4
ξ
2
i
= 1 (2.17)
Then by means of substitution
Ψ
(
ξ
1
,
ξ
2
,

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
,

ξ
2
,
ξ
3
,
ξ
4
)= (2.18)
1
2
π
2
p
0

R
4
Ψ
(
ξ

1
,
ξ

2
,
ξ


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

−
ξ

3
|
2
+ |
ξ
4
−
ξ

4
|
2
The latter equation is invariant relative to rotations in a 4D space. Thus, we see that
the real symmetry of an electron in a Coulomb field is SO(4). One may construct
the infinitesimal rotation operators in the space {
ξ
1
,
ξ
2
,
ξ
3
,
ξ
4
}. There are six such

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
−

r
r


1
−2
ˆ
H
The vector of orbital momentum L contains three generators of the group SO(3),
which in this case is only a subgroup of the true symmetry group SO(4) [111, 324].
Three more generators of the latter group are given by the components of the vector
F. The Runge – Lenz vector mentioned above is in fact
A = F

−2
ˆ
H. (2.20)