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DYNAMICAL SYMMETRIES IN NANOPHYSICS

K. Kikoin (Tel-Aviv University)

PLAN

I.

• Group theory in quantum mechanics. Basic definitions .• Symmetry of Hamiltonian. • Hidden symmetry of Hamiltonian• Dynamical symmetries of multiplet

II.• Nanoobjects: single-electron tunneling through quantum dots and molecules• Dynamical symmetries of few-electron systems• Kondo effect and Kondo-regime in single electron tunneling• Kondo exotics

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OMNIPRESENT GROUP THEORY

1. Closeness: Product of two elements AB=C in the set belongs to the set2. Associativity: (AB)C = A(BC)3. Existence of a unite element: EA = AE = A4. Existence of inverse elements: AA-1 = A-1A = E for any A

Set of elements E,A,B,C,… N plus multiplication rule

The set without (4) forms semigroup

E J K L M N

J K E N L M

K E J M N L

L M N E J K

M N L K E J

N L M J K E

E J K L M N

EJKLMN

Tutorial example: Multiplication table:

Isomorphism: = Group D3 of discrete rotations = Permutation group P123

1

23

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GROUP THEORY IN QUANTUM MECHANICS

Noether theorem: Any differentiable symmetry of the action has a corresponding conservation law. Homogeneity in time – energy conservation Homogeneity in space: – momentum conservation Isotropy in space: – angular momentum conservation

Ingredients:

δr

The wave functions belonging to a given eigen energy E transform along a representation of the group G of the Schroedinger equation

Wigner theorem:

Hψ({x}) = Eψ({x}) HRψ({x}) = ERψ({x}).

Infinitesimal operators as generators of Lie algebras

These operators perform infinitesimal translations and rotations in space time and form the basis if irreducible representations of continuous Lie groups. Example: rotations on a 3D sphere

Laplacian Δ is invariant under transformations in Euclidean space. In case of rotation group SO(3) we deal with invariant under rotations on the sphere.

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These operator form closed Lie algebra o(3)

J2 is the Casimir operator, which commutes with unit operator. In general case several Casimir operators may be constructed from group generators. Usually Casimir operators explicitly enter the Hamiltonian. In particular, one may use J2 instead of Δ .

In some special cases Casimir operators do not enter the Hamiltonian. This is the sign of

hidden symmetry

Example: electron in a Coulomb potential ~ 1/r .

It was noticed (W. Pauli, 1927, V.A. Fock, 1935) that the Schroedinger equation with this potential is in fact 4-dimensional (in {p,ε}-space) . As a result three more generators may be introduced which form a Runge-Lenz vector (L=iħI)

Two last equation define two Casimir operators for a group of 4D rotations SO(4) in this specific potential. But the operator N does not enter the Hamiltonian!

Column 1

Column 2

Column 3

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Let us add one more dimension to our Euclidean 3D world. Time t may play part of his additional coordinate. To describe rotations on 4D sphere one needs three more infinitesimal operators

SO(4) group of rotations on 4D sphere – example of semisimple group.

These 6 operators form closed so(4) algebra

Two Casimir operators are

A linear transformation converts (*) into another basis

with commutation relations

(*)

Triads with commutation relations of this type form ideals of the o(4) algebra. Algebra possessingnon-Abelian ideal is qualified as a semisimple algebra.

Two Casimir operators in these terms are

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Now we are prepared to discussion of dynamical symmetries

Representation theory in quantum mechanics implies symmetry operations on matrices. Each symmetry group is realized in some basis forming the irreducible representations. In case of rotation group SO(3) these are spherical functions Ylm ( φ,θ). Wigner theorem in these terms means that the matrix representation of Schroedinger equation has a block-diagonal form

s

p

d

l=0

l=1

l=2

… …

l – operators have nonzero matrix elements only within a given block. Each block corresponds to the eigen energy El

s

p

d

…

…Question:what about off-diagonal blocks ?

Δl=1 Δl=2 …

To find the answer let us return to the physical realizationof SO(4) symmetry, namely to rigid quantum rotator

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αz’ z

Angular coordinates contain three angles: (φ,θ,α). The latter angle describes precessionof rotator axis z’ around cartesian axis z in 3D space. This angle play the same part asthe time t in definition of the operator M . Thus, effective dimension of phase space for the states of rigid rotation is D=4.

Basis of representations of SO(4) group in angular coordinates is given by hyperspherical functions

Ladder operators Mi act on the spherical functions with given n in the following way:

(cl and al are some constants). Thus M-operators unlike L-operators raise (lower) orbital index l , and the set of operators L, M involves both diagonal blocks and off-diagonal blocks with Δl = 1 in the Hamiltonian matrix.

One says that the group SO(4) realizes the dynamical symmetry of rigid rotator.

Generators of dynamical symmetry group describes transitions between the levels of the supermultiplet, i.e., the levels belonging to different irreducible representations and different eigenstates of the Hamiltonian.

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s1

0

-1R

SO(4) symmetry in spin systems: singlet-triplet multiplet formed by two electrons

s

T

S=0

S=1

…

Spin 1 operators S generate o(3) algebra and generators Sz, S+, S- describe rotations on a 3D spherein spin space. Adding vector R (allowing singlet-triplet transitions) we extend the rotation group fromSO(3) to SO(4).

4 states form spinor of 4th rank

These operators were invented by J. Hubbard

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Commutation relations are the same as for operators L and M in quantum rotator problem.

Casimir operators:

Electron spins are not independent in SO(4) group: they are constrained by the second Casimir operator.

When the dynamical symmetry becomes actual?

In stationary state spin is conserved as well as angular momentum, and we are satisfied with conventional (Wigner’s) symmetry of the Hamiltonian.

Dynamical symmetry is activated in interaction with external fields (electromagnetic field etc) and in interaction with environment.

The latter possibility is realized in nanophysics.

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S= 2 X 10 +X 0−1 ; S z =X 11−X−1−1

R=2 X 1S− X S−1 ; R z=− X 0S+X S0

Commutation relations

Kinematics: three Casimir operators

s1

0

-1

Singlet GS T-exciton

One more physical example: SO(5) algebra for Wannier excitons for three vectors S,R,M and scalar A

S-Exciton

( ) ( )

( )

1 1 0 02 ;E E z E E

SE ES

X X X X

i X

M

X

M

A

+ -= - = - +

= -

R

A M

[Si,S

k] = ie

ijkS

k, [S

i,R

k] = ie

ijkS

k, [R

i,R

k] = ie

ijkS

k,

[Si,M

k] = ie

ijkM

k, [M

i,M

k] = ie

ijkS

k, [R

i,M

k] = iAΔ

ik

S2 + R2 + M2 + A2 = 4

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This construction may be generalized for higher groups SO(n)

Group generators are combined in vectors (V) and scalars (A)

Multiplet consists of singlets (S) and triplets (T)

In some physical problems group generators form tensors (e.g. spin systems with uniaxial magnetic anistropy). In the latter case the dynamical symmetry SU(3) is realized by one vector and one tensor.

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A little more about SU(n) groups in a context of dynamical symmetries

Mathematically SU(n) is a group of unitary matrices of n-th rank. In nanophysics one frequentlydeals with the groups SU(3) and SU(4).

3x3SU(3) group describes all interlevel transitions in a three-level system.

Its generators are so called Gell-Mann matrices which are the generalization of su(2) Pauli matrices σ. The GM matrices may be combined in three triads:

Only eight of these matrices may be usedas linearly independent generators of SU(3) group.

σ+ σ- σ3

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The simplest physical realization of SU(4) symmetry group is a two-well potential with one electron inside.

Electron is characterized by spin with two projections (up and down)and its position in the wells (left and right). Both quantum numbers arerealized by means of Pauli matrices σ and τ (spin and pseudospin, resp.).

15 generators of SU(4) group are constructed as

These generators describe all positions of electron in the potential relief and all possible transitions between the quantum states. If one is interested only in spin conserving transitions between two wells, but the spin flips in a given well are allowed, the matrix (*) reduces to

(*)

In this case the dynamical symmetry is reduced:

Now we are well equipped for investigation of dynamical symmetries in quantum dots and related nano-objects.

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Planar quantum dots

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Coulomb blockade due to noticeable capacitive energy of QD.

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odd oddeven even

Coulomb windows diagram: tunnel conductance G as a function of eV, Vg

Single electron tunneling

G = dI/dV

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σ’ σ

(a) (b)

σ σ’

(c) (d)

In a Coulomb window direct tunneling is suppressed by Coulomb blockade and only cotunneling processes are possible. These processes are accompanied by creation of electron- hole pairseither in the same lead or in different leads. The latter process is responsible for charge transport.The electron spin is not conserved in cotunneling processes.

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N=1

Kondo Hamiltonian H = J (S s)

S=1/2

E+U

E

Zero energy spin reversal peak ~δ(0)develops into resonance due to dynamicalscreening by electron-hole pairsfrom continuum (Kondo cloud)

TK ~ D0 exp(-1/2νJ(0))

CONVENTIONAL KONDO COTUNNELING: theory

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a b

How does Kondo effect in quantum dots looks experimentally:

G = dI/dV – tunneling conductance

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KONDO SCREENINGUndersreened screened overscreened (two-channel)

N

N-1 N-1

N N

N-1

N

N-1

. . .Kondo singlet +free electrons (FL)

reduced spin+free electrons

Column 1

Column 2

Column 3

1/T + const const - (T/TK)2

non-Fermi liquid (NFL)

T a

Magnetic response:

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S

T

Generalized Kondo Hamiltonian H= J1 (S s) + J2 (R s)

S=1 plus singletΔTS ~ T K ΔTS

Kondo cotunnelling through DQD: N=2

Two electrons always form spin Singlet and spin Triplet

Usually ground state is S,but the S/T level crossingis possible under certain conditions.Unconventional features of Kondo tunneling may be observedwhen

S

1

0 T

-1R

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General view on emergence of dynamical symmetries

δE

CONT INUUM

Flow RG?

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Dynamical symmetry alters (or arises) in the course of contraction of the energy scale δE. Let us look how does this mechanism work in case of Kondo effect. Even conventional Kondo effect (N=1) may be described in terms of dynamical symmetries. Basic Hamiltonian is that of the Anderson model describing cotunneling through a quantum dot.

2D0

εd

εd +U

0

0D0 D

(a) (b)

E

The Renormalization Group procedure is applied to this Hamiltonian.In the course of energy scale reduction the dot level εd shifts ~ ln (D0 -D).When the renormalized level crosses the boundary D the first stage of RG procedure terminates. We are now in a regime of localized spin described by the Kondo Hamiltonian H = J (S s) with J ~ W2/ εd , and in the second stage of scaling procedure the coupling constant J is renormalized. The scaling equation reads with

Its solution is

J(T) turns to infinity (reaches the stable fixed point) at T=TK

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The same procedure in terms of dynamical symmetries:

Spin Charge

0

0, 2

1

0

E

N= 0 1 2

Symmetric Hubbard parabola

Kinematic scheme of possible spin and charge transitions

All these transitions in 4-level system are described by generators of SU(4) group. This is the basic symmetry of Hubbard atom with varying N =0,1,2

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E

N= 0 1 2

Asymmetric Hubbard parabola

If E(2) – E(1) >> E(0) – E(1), the doubly occupied stateis first to be integrated out in the process if RG procedureand we remain with SU(3) symmetry

Spin Charge

1

0

0

Transition from Anderson model to exchange model means elimination of empty state andeventually we come to the scheme containing only spin states and describing SU(2) Kondo effect.

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This construction is only rephrasing of the well known procedure which means not too much from practical point of view, but dynamical symmetries are deeply involved in Kondo effect at even occupation .

H= J1 (S s) + J2 (R s)

ΔTS ~ T K ΔTS Within the energy interval δE ~ the exchange (cotunneling) Hamiltonian is

and both parameters J1 and J2 are subject to RG renormalization. The system of scaling flow equations

0 - ΔTS

Solution of this system gives for TK as afunction of exchange gap

Maximum TK is reached at ΔTS = 0, where the singlet and triplet states are degenerate. The sign and the magnitude of this gap may be controlled by the gate voltage vg , so this effect is experimentally checkable.

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Magnetic field induced (!) Kondo effect

R=2 X 1S , R z =X 11− X SS

H Kondo =J R⋅s

-1

0

S

1B

ET

ES

It was the first example of Kondo effect due to dynamical symmetry of DQD D. Kobden et al, Nature, 408, 342 (2000)

M. Pustilnik, Y. Avishai & KK, PRL, 84, 1756 (2000)

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Intermezzo: other types of quantum dots.

Vertical dots with parabolic confinement potential

“Fock – Darwin” atoms with cylindrical symmetry

●● ●● ●●

●● ●●

●●

l = -2 -1 0 +1 +2

2

1

0

n

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“Periodic table” for vertical dots

Transition between the oscillator levels are describedby SO(2,1) two-dimensional Lorenz group.

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Fullerene quantum dots Endofullerene

Nanotube quantum dots

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Double and triple quantum dots Ring-like dots

Self-assembled semiconductor quantum dots

Excitonic atom

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(a) (b) (c) (d)

V

Ws

dW

l r

Basic configurations of double quantum dots

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W

(e) (f) (g) (h) (i)

V

(a) (b) (c) (d)

… triple quantum dots

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(a) (b) (c) (d)

…vertical quantum dots

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Kondo effect and Aharonov – Bohm interference in “which path” geometry

W

V

Ф

Two paths have different chirality when turning the magnetic flux from the left and from the right. As a result two partial waves meet in the dot 3 with phase difference

The interference effect ~ 1 – may partially or completele suppress Kondo tunneling

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Exotic Kondo effect where spin and charge variables change their role.

εc

εi + Qic

εi

εc + Qc

W

s

d

Qic

Triple quantum dot in parallel configuration. Central dot is smaller than side dots, and the Coulombblockade is stronger.

rlc

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Two electrons, singlet ground state with S = 0

Spin reversal is impossible in a singlet state, but position of the level have changed (pseudospin reversal).

This is possible realization of two-channel Kondo effect, where spin enumerates channels andposition plays part of spin!

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CONCLUDING REMARKS

In artificial nanoobjects various types of dynamical symmetries may be realized and used in practical (?) applications

Y. Avishai (Israel)Y. Oreg (Israel)M. Kiselev (Italy)T. Kuzmenko (Israel)R. Shekhter (Sweden)M. Wegewijs (Germany)