222

Laurea in Scienza dei Materiali A.A. 2015-2016ELEMENTI DI FISICA TEORICA (EFT) (7 crediti ) aula 29Laurea Magistrale in Fisica

Teoria dei Solidi (TS) (6 crediti) aula 29

Prof. Michele Cini Tel. 4596

michele.cini@roma2.infn.itRicevimento Studenti (stanza 9 corridoio C1) Lunedi e Mercoledi 14-16

9-10

10-11

11-12

Lunedi Martedi Mercoledi Giovedi Venerdi

Ts

EFT EFTEFT

EFT

EFT

http://people.roma2.infn.it/~cini/

Ts Ts Ts

Ts

files delle lezioni:

invito a mandare un mail a: cini@roma2.infn.it per presa contatto

Programma di massima del corso

Teoria della simmetria, Jahn-

Teller, Space group, Gruppi

doppi e Gruppi magnetici

Ore 14

Seconda quantizzazione, teorema

adiabatico, funzioni di Green, metodi

diagrammatici,

applicazioni: risonanze, screening, Kubo,

Kondo, spettroscopie elettroniche

Ore 21

q

E

0

Programma di massima del corso

Trasporto quantico,

quantum pumping Ore 3

Fase di Berry, polarizzazione dei

solidi Ore 2

Programma di massima del corso

Effetto hall quantistico

intero e frazionario,Grafene

Effetti di bassa

dimensionalita’ e topologici:

Cariche frazionarie, anyons,

applicazioni a grafene e

nanotubi, isolanti topologici

Ore 8

Totale Ore 48

hole has moved

spinonholon

666

Mai piu di un’ora al giorno

Non occorre prendere appunti !

Libro Springer-Verlag (disponibile in biblioteca)

PowerPoint se la frequenza in aula e’ soddisfacente se no,no!

Teoria dei Solidi

esame solo orale con prima domanda a piacere

7

Leonhard Euler (April 15, 1707 – September 7, 1783)

Group Representations for Physicists

Groups are central to Theoretical Physics, particularly for Quantum Mechanics,

from atomic to condensed-matter and to particle theory, not only as

mathematical aids to solve problems, but above all as conceptual tools. They

were introduced by Lagrange and Euler dealing with permutations , Ruffini,

Abel and Galois dealing with the theory of algebraic equations.

Évariste Galois(Bourg-la-Reine, 25 ottobre 1811– Parigi, 31 maggio 1832)

Joseph-Louis Lagrange(Giuseppe Lodovico Lagrangia) (Torino, 25 gennaio 1736 – Parigi, 10 aprile 1813)

88

Abstract Groups

A Group G is a set with a binary operation or multiplicationbetween any two elements satisfying:

-1 -1 -1

1) G is closed, i.e. a G, b G ab G.

2) The product is associative : a(bc) = (ab)c.

3) e G ( identity): ea = ae = a, a G.

4) a G, a : a a =aa = e.

It is not necessary that G be commutative and generally ab ba.

Commutative Groups are called Abelian. Quantum Mechanical operators do

not generally commute, and we are also interested in non-Abelian Groups

Abstract: no matter what the elements are, we are interested in their operations.

Usually we shall consider symmetry operations, permutations, Lorentz

transformations …

10

i ij jj

ij

not Abelian, in general :

GL n General Linear Group in n dimensions

Matrix

.

GL(n) is the set of linear operations x' = a x ,

where A = {a } is such that Det

group

A

s

0.

Order of Group NG =number of elements.

Many Groups of interest have a finite order NG, like: point Groups like the Group

C3v of symmetry operations of an equilateral triangle, the Group S(N) of

permutations of N objects.

Important infinite order Groups may be discrete or continuous

(Lie Groups have the structure of a differentiable manifold).

Integers with the + operation (Abelian), identity e=0

Real numbers with the + operation (Abelian), identity e=0

Real numbers excluding 0 with the * operation (Abelian), identity e=1

11

SL(n)= Special Linear Group in n dimensions

Or Unimodular Group

i ij j ijSL(n)=the set of linear operations x' = a x ,where A = {a }

is nXn matrix such that DetA = 1.

n

j

Let A and B denote two Groups with all the elements different,

that is, a A a not in B (except the identity, of course).

We also assume that all the elements of A commute with those of B.

This

is what happens if the two Groups have nothing to do with each other,

for instance one could do permutations of 7 objects

and the other spin rotations. In such cases it is often useful to define

a d C = A×B, which is a Group whose

elements

irect prod

are ab =

uct

ba.

Direct product

1212

Rotation Group O(n) of the Orthogonal transformations, or of the orthogonal matrices AT=A-1

Columns and rows are real orthogonal vectors

Examples of infinite Groups of physical interest

Orthogonal Groups

ATA= A AT=I Det(A) =1 or -1

abstract view: transformation and matrices are same Group

cos( ) sin( ) 2 dim

sin( ) cos( )In

1313

† 1Special Unitary Group SU(n): nxn Unitary matrices ( )

with det(A)=1.

1 1 belongs to SU(2). This acts on spin 1/2.

1 12

A A

i

Space Group (Translations and rotations leaving a solid invariant,

not Abelian)

Translations of a Bravais lattice (Abelian)

Lorentz Group: transformations (x,y,z,t) (x’,y’,z’,t’) that preserve the interval.

Examples of infinite Groups of physical interest

Examples of infinite Groups of physical interest

0 = , where I =identity nxn.

0

Symplectic matrices are 2n x2n matrices M such that:

M . One can prove that Det(M)=1.

Symplectic matrices are a Group Sp(2n,R)

where R means that entr

n

n

n

T

ILet

I

M

ies are real.

1 1

1 1

In classical mechanics the transformation from a set q ... , ,... to

a new set Q ... , ,...

preserving the form of Hamilton's equations are Sp(2n,R).

They are canonical transformations.

n n

n n

q p p

Q P P

15

U(1) gauge group of electromagnetism,QED

SU(2) group of rotations of spin 1/2

U(1)xSU(2) gauge group of electro-weak theory (Salam)

SU(3) quantum chromodynamics, quarks

Groups of Particle Physics

16

U(1)xSU(2)xSU(3) gauge group of the Standard Model

Groups of Particle Physics

17

2 4

( ), =1,..4

( ) 10 | | | |

L V

V

Unstable maximum of V at =0 with U(1) symmetry

Infinite minima at = 5 : symmetry is broken (changing the state changes)ie

Ferromagnetic materials:symmetry above Curie temperature broken below

Solids break the rotational symmetry of fundamental laws

Electroweak theory: Higgs field=order parameter breaks electroweak symmetry

at the electroweak temperature

Superconducting order parameter breaks U(1) as well

Spontaneous

Symmetry Breaking

complex variable

1818

The operations of the 32 point Groups are rotations (proper and improper) and

reflections. In the Schönflies notation, which is frequently used

in molecular Physics, proper rotations by an angle 2π/ n are denoted Cn and

reflections by σ; improper rotations Sn are products of Cn and σ (S for Spiegel=mirror).

Point Symmetry in Molecules and Solids

Low symmetry: C1 has only E

example CFClBrI

http://www.chem.uiuc.edu/weborganic/chiral/mirror/flatFClBrI.htm

Cs has E sh O=N-Cl reflection in molecular plane

is the only symmetry

The reflection plane is orthogonal or parallel to the rotation axis. The molecular axis

is one of those with highest n. A symmetry plane can be vertical (i.e. contain the

molecular axis) or horizontal (i.e. orthogonal to it), and the reflections are σv

or σh accordingly.

1919

High symmetry: Td tetrahedron CH4

Oh octahedron SF6

2020

Otherwise: choose axis of maximum order. It will be the vertical

axis. Most often rotations are proper, and the axis is Cn,.

Cn

Sn

If the rotation is improper the axis is Sn.and the Group is also Sn.

3 3Staggered conformation of CH CH

2121

When the axis is Cn the Group is is Cn if there are no more

symmetry elements.

Cn

Cn

sv

Cn

sh

Otherwise:

If it is improper the Group is Sn

When the axis is Cn and there are no C2 axes orthogonal to

molecular axis the Group is also Cn; then we look for more

elements:

vertical reflections, horizontal reflections

When the axis is Cn and there are C2 axes orthogonal to molecular

axis the Group is Dn then we look for more elements

22

Cn becomes Cnh if there are horizontal

planes

CHBr CHBr is C2h

Cn becomes Cnv for vertical planes H2O is

C2v

22

Cn Groups with symmetry planes

2323

24

If there are C2 axes orthogonal to molecular axis,

The Group is Dn

C2

Dn

If in addition there are horizontal planes,

Dnh,

Benzene D6h

If there are vertical planes, Dnd,

Allene CH2 CH CH2 is D2D

( both CH2 planes are mirror ) 24

26

2

:

( )

( )v

h

linear

C vertical plane HCl

D horizontal plane CO

HCN (prussic acid)

Linear molecules

http://www.phys.ncl.ac.uk/staff/njpg/symmetry/Molecules_pov.html

http://www.uniovi.es/qcg/d-MolSym/26

2727

2828

29

Icosahedron Ih group

The images contained in this page have been created and are copyrighted © by

V. Luaña (2005). Permission is hereby granted for their use and reproduction for

any kind of educational purpose, provided that their origin is properly attributed.29

30

C60

buckminsterfullerene

The images contained in this page have been created and are copyrighted © by

V. Luaña (2005). Permission is hereby granted for their use and reproduction for

any kind of educational purpose, provided that their origin is properly attributed.30

31

32

33

34

Symmetry in quantum mechanics

Symmetry operators (space or spin rotations, reflections, ...)

preserve normalisation are unitary

1 † is unitaryR R R

†, 1.R RG R RR

REQUIRES:

.a pi

aT e

.

†

a pi

a aT e T

Examples: TRANSLATIONS:

.Li

R e

.

†

Li

R e R

ROTATIONS:

3535

REFLECTIONS:

: , , , ,Z x y z x y zs †

1 0 0 1 0 0

: 0 1 0 0 1 0

0 0 1 0 0 1

Zs

ASSOCIATED MATRIX:

2† † * † † † * †

Indeed, , onsider eigenvalue equation

v v v v v v= v v 1

R G c

R R R R

One can writ it Ree , w hie

All eigenvalues of unitary matrices have modulus

unity

Symmetry operators are unitary they can be diagonalized

36

*

be (1-body or many-body) eigenstate of R:

Multiply the second by :

,and the . .

i

i

i i

Let

R e

R e

R R e c c is R e

0or

36

Unitary operators

Different eigenvalues imply orthogonal eigenvectos

** † 1

but

iR R R R e

37

Matrix Representation of symmetry operators

Evidently, D S S

Let { } orthonormal basis, S D S

Let , therefore .R G RS G

.RS R D S D R D S

Since , it must also be true that RSR SS D RG

D R D S D RS

37

Consider for example the action of a symmetry S on a one-partice basis

(the argument runs in a similar way in the many-body case)

38

38

1

is represented by matrix

on (1-body or many-body) basis s

Symmetry means , 0

Symmetry Group mea

et .

Then , , 0 trivially implies :

ˆ , 0.

The matri

ns , , 0.

x

S D S S

S G S H

H H

S H SHS H S H

G S G S

S H

H

of H commutes with the matrices of the symmetries.

Let H = Quantum Hamiltonian to diagonalize:

If G is Abelian diagonalize all S simultaneously, and get

all symmetry labels.

Each set of labels is an independent subspace.

Consider again the one-particle example (the argument again runs in a

similar way in the many-body case)

: crystal translationEx Grampl upe oT GG N

. .i iip t t

iT e e

it primitive translation vectors of Bravais lattice

unitary translation operators for 1-electron states

39

Born- Von Karman Boundary Conditions: the lattice is a 3-Torus

Using supercell of size N with pbc :

1 Abelian cyclic finite Group.

We can diagonalize all lattice translations at once :

the eigenvalue equation reads:

N

i T

i

T G

T

,ikx

k kx e u x ka

with (lattice periodic)k k iu x u x t

.i

i iT x x t e x

Solve by means of Bloch’s theorem:

40

Born- Von Karman Boundary Conditions

4141

( )1.iik N t

e

The solution is , where the reciprocal lattice vectors areNk G

iG.tG reciprocal lattice vector: e 1 for any lattice translation

i

i i

N iN

i

T x x t e x

T x e x x

The allowed k values are obtained from the Born- Von Karman BC:

ka

1 1 2 2 3 3

2 31

1 2 3

,

2 and cyclic.( )

G m b m b m b

a ab

a a a

4242

.G ensures lattice periodic plane wave: 1iG te t

2

( ) .2

k k k

p kV x u x u x

m

iG.tG reciprocal lattice vector: e 1 for any lattice translation

This elementary example shows some features of the Group theory methods.

By introducing a symmetry-related quantum number k and writing wavefunctions on a Bloch basis, we reduce to a much easier subproblem: to find

the cell periodic solutions of