Post on 15-Feb-2019
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