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

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
Lunedi
9-10
EFT
10-11
Ts
11-12
Martedi
EFT
Mercoledi
Giovedi
Venerdi
EFT
EFT
EFT
Ts
Ts
Ts
Ts
Prof. Michele Cini Tel. 4596
[email protected]
Ricevimento Studenti (stanza 9 corridoio C1) Lunedi e Mercoledi 14-16
invito a mandare un mail a: [email protected] per presa contatto
files delle lezioni:
http://people.roma2.infn.it/~cini/
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Teoria dei Solidi
Mai piu di un’ora al giorno
Non occorre prendere appunti !
Libro Springer-Verlag (disponibile in biblioteca)
PowerPoint ogni settimana aggiornato sul web
esame solo orale con prima domanda a piacere
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Possibilita’ di trattare sul programma per un 20%
Programma di massima del corso
Teoria della simmetria
Seconda quantizzazione, teorema
adiabatico, funzioni di Green, metodi
diagrammatici, applicazioni
Fase di Berry, trasporto, pumping
polarizzazione dei solidi
Effetto hall quantistico
Effetti di bassa dimensionalita’ e
topologici: Cariche frazionarie,
anyons, applicazioni a grafene e
nanotubi
Totale
Ore 14
Ore 24
Ore 4
Ore 6
Ore 48
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)
Leonhard Euler (April 15,
1707 – September 7, 1783)
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Abstract Groups
A Group G is a set with an operation or multiplication between any two elements
satisfying:
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 -1 : a -1a =aa -1 = e.
Abstract: no matter what the elements are, we are interested in their operations
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
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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
Matrix groups
 not Abelian, in general  :
GL  n   General Linear Group in n dimensions.
GL(n) is the set of linear operations x'i =  a ij x j ,
j
where A = {a ij } is such that DetA  0.
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SL(n)= Special Linear Group in n dimensions
Or Unimodular Group
n
SL(n)=the set of linear operations x'i =  a ijx j ,where A = {a ij }
j
is nXn matrix such that DetA = 1.
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 direct product C = A×B, which is a Group whose
elements are ab = ba.
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Examples of infinite Groups of evident physical interest
Rotation Group O(n) of the Orthogonal transformations, or of the
orthogonal matrices AT=A-1
abstract view: transformation and matrices are same Group
Special Unitary Group SU(n): nxn Unitary matrices (A†  A1 )
with det(A)=1, like
i 1 1 

.
2 1 1
Translations of a Bravais lattice (Abelian)
Lorentz Group: transformations (x,y,z,t) (x’,y’,z’,t’) that preserve the interval.
Space Group (Translations and rotations leaving a solid invariant,
not Abelian)
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Groups of Particle Physics
U(1)
gauge group of electromagnetism,QED
SU(2)
U(1)xSU(2)
SU(3)
group of rotations of spin 1/2
gauge group of electro-weak theory (Salam)
quantum chromodynamics, quarks
U(1)xSU(2)xSU(3)
gauge group of the Standard Model
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Spontaneous
Symmetry Breaking
L       V ( )
V ( )  10 |  |2  |  |4
Unstable maximum of V at  =0 with U(1) symmetry
Infinite minima at  = 5ei : symmetry is broken (changing  the state changes)
Solids break the rotational symmetry of fundamental laws
Ferromagnetic materials:symmetry above Curie temperature broken below
Superconducting order parameter breaks U(1) as well
Electroweak theory: Higgs field=order parameter breaks electroweak symmetry
at the electroweak temperature
Convective cells in liquids…..
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Point Symmetry in Molecules and Solids
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).
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.
Low symmetry: C1 has only E
example
CFClBrI
Cs has E sh O=N-Cl reflection in molecular plane
is the only symmetry
http://www.chem.uiuc.edu/weborganic/chiral/mirror/flatFClBrI.htm
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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
sv
Cn
Cn
sh
Otherwise:
If it is improper the Group is Sn
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Cn becomes Cnh if there are horizontal
planes
C2
CHBr CHBr is C2h
Cn becomes Cnv for vertical planes H2O is
C2v
Dn
If there are C2 axes orthogonal to molecular axis, 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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Linear molecules
HCN (prussic acid)
linear :
Cv
vertical plane ( HCl)
Dh
horizontal plane (CO2 )
http://www.phys.ncl.ac.uk/staff/njpg/symmetry/Molecules_pov.html
http://www.uniovi.es/qcg/d-MolSym/
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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
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V. Luaña (2005). Permission is hereby granted for their use and reproduction for
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any kind of educational purpose, provided that their origin is properly attributed.
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Symmetry in quantum mechanics
Symmetry operators (space or spin rotations, reflections, ...)
are unitary  they can be diagonalized
R  G,
REQUIRES:
R R   R† R      1.
R 1  R †

R is unitary
Examples:
TRANSLATIONS:
Ta  e
i
a. p
Ta  e
†
i
a. p
 T a
ROTATIONS:
R  e
i
 .L
R †  e
i
 .L
 R
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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
REFLECTIONS:
sZ :
ASSOCIATED MATRIX:
 x, y, z    x, y,  z 
1 0 0  1 0 0 
s Z :  0 1 0    0 1 0 
 0 0 1  0 0 1

 

†
All eigenvalues of unitary matrices have modulus
unity
Indeed, R  G,
consider eigenvalue equation
Rv   v  v† R†   *v†  v† R† Rv= *v† v 
 1
2
One can write   ei , with Re   
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Different eigenvalues  orthogonal eigenvectors

1-body or many-body eigenstate
 R   ei 

i
R


e



R   R   ei   , and the c.c. is e  i  
but
*
R
  R †    R 1   e  i  
  
or
   0
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Matrix Representation of symmetry operators
Let {  }  orthonormal basis, S      D  S 

Evidently,
D  S     S 
Let R  G , therefore RS  G.
RS    R   D  S      D  R  D  S  .


Since RS  G, it must also be true that RS      D  RS 



D  R  D  S    D  RS 

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Let H = Quantum Hamiltonian to diagonalize:
S  Symmetry means H  SHS 1  H
G  Symmetry Group means
S is represented by matrix
  S , H   0
S  G,
 S , H   0.
D  S     S  
on (1-body or many-body) basis set
 S , H   0 trivially implies :
Hˆ  
    S , H   
 .
Then S  G,
H    
 0.
The matrix of H commutes with the matrices of the symmetries.
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.
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Example : GT  crystal translation Group
ti
NG  
primitive translation vectors of Bravais lattice
Ti  eip.ti  e.ti
unitary translation operators for 1-electron states
Using supercell of size N with pbc :
Ti N  1  GT
Abelian cyclic finite Group.
We can diagonalize all lattice translations Ti at once :
the eigenvalue equation reads:
Ti  x     x  ti   ei  x  .
Solve by means of Bloch’s theorem:
 k  x   eikxuk  x  ,   ka
with uk  x   uk  x  ti  (lattice periodic)
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Ti  x     x  ti   e   x 
i
 k  x   eikxuk  x  .
pbc+uniqueness of wave function  eik ( N ti )  1.
The solution is Nk  G, where
G reciprocal lattice vector: eiG.t  1 for any lattice translation
G ensures lattice periodic plane wave: eiG .t  1 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
  p  k 2

 V ( x )  uk  x    k uk  x  .

 2m

This elementary example shows some features of the Group theory methods.
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