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Laurea in Scienza dei Materiali A.A. 2013-2014 ELEMENTI DI FISICA TEORICA (EFT) (7 crediti ) aula 29 Laurea Magistrale in Fisica

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

Prof. Michele Cini Tel. 4596 michele.cini@roma2.infn.it Ricevimento Studenti (stanza 9 corridoio C1) Lunedi e Mercoledi 14-16

9-10

10-11

11-12

Lunedi Martedi Mercoledi Giovedi Venerdi

Ts

EFT EFT EFT

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

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Mai piu di un’ora al giorno

Non occorre prendere appunti !

Libro Springer-Verlag (disponibile in biblioteca)

PowerPoint ogni settimana aggiornato sul web

Teoria dei Solidi

esame solo orale con prima domanda a piacere

Possibilita’ di trattare sul programma per un 20%

Programma di massima del corso

Teoria della simmetria Ore 14

Seconda quantizzazione, teorema

adiabatico, funzioni di Green, metodi

diagrammatici, applicazioni

Ore 24

Effetto hall quantistico

Effetti di bassa dimensionalita’ e

topologici: Cariche frazionarie,

anyons, applicazioni a grafene e

nanotubi

Ore 6

Fase di Berry, trasporto, pumping

polarizzazione dei solidi Ore 4

Totale Ore 48

4

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)

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Abstract Groups

A Group G is a set with an operation or multiplication between 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 mainly interested in non-Abelian Groups

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

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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 tyhe 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

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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.

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

abstract view: transformation and matrices are same Group

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

1 1with det(A)=1, like .

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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 evident physical interest

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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

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

Groups of Particle Physics

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2 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

Convective cells in liquids…..

Spontaneous

Symmetry Breaking

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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.

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High symmetry: Td tetrahedron CH4

Oh octahedron SF6

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Otherwise: choose axis of maximum order, say Cn; it will be the

vertical axis. If rotations are proper, and there are no C2 axes

orthogonal to molecular axis the Group is also Cn;

vertical reflections, horizontal reflections

Cn

Cn

sv

Cn

sh

Otherwise:

If it is improper the Group is Sn

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Cn becomes Cnh if there are horizontal

planes

CHBr CHBr is C2h

Cn becomes Cnv for vertical planes H2O is

C2v

If there are C2 axes orthogonal to molecular axis, Dn

C2

Dn

If there are horizontal planes, Dnh,

Benzene D6h

If there are vertical planes, Dnd,

Allene CH2 CH CH2 is D2D

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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/ 17

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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. 20

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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. 21

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Symmetry in quantum mechanics

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

are unitary they can be diagonalized

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:

Nuovo orario: sala riunioni 2 vicino al magazzino

Lu-mer 11-12 mar 13,30-14,30 secondo orario ven festa

gio 10-11 in 29

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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

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* † 1

1-body or many-body eigenstate

,and the . .

but

i

i

i i

i

R e

R e

R R e c c is e

R R R e

0or

Different eigenvalues orthogonal eigenvectors

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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

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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 no need for Group theory: diagonalize all

S simultaneously, and get all symmetry labels.

Each set of labels is an independent subspace.

: crystal translationEx Grampl upe oT GG N

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

. .i iip t t

iT e e

it primitive translation vectors of Bravais lattice

unitary translation operators for 1-electron states

,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:

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

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

the cell periodic solutions of

2

( ) .2

k k k

p kV x u x u x

m

( )pbc+uniqueness of wave 1.function iik N t

e

The solution is , whereNk G

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

i

i iT x x t e x .ikx

k kx e u x

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