Introduction to Group Theory

Home , Abelian group, Cyclic permutation, Group theory, Mathematical structure, Projection (mathematics), Triangle Group

With Applications to Quantum Mechanics and Solid State Physics

Roland Winkler [email protected]

August 2011

(Lecture notes version: November 3, 2015)

Please, let me know if you ﬁnd misprints, errors or inaccuracies in these notes. Thank you.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 General Literature

I J. F. Cornwell, Group Theory in Physics (Academic, 1987) general introduction; discrete and continuous groups

I W. Ludwig and C. Falter, Symmetries in Physics (Springer, Berlin, 1988). general introduction; discrete and continuous groups

I W.-K. Tung, Group Theory in Physics (World Scientiﬁc, 1985). general introduction; main focus on continuous groups

I L. M. Falicov, Group Theory and Its Physical Applications (University of Chicago Press, Chicago, 1966). small paperback; compact introduction

I E. P. Wigner, Group Theory (Academic, 1959). classical textbook by the master

I Landau and Lifshitz, Quantum Mechanics, Ch. XII (Pergamon, 1977) brief introduction into the main aspects of group theory in physics

I R. McWeeny, Symmetry (Dover, 2002) elementary, self-contained introduction

I and many others Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Specialized Literature

I G. L. Bir und G. E. Pikus, Symmetry and Strain-Induced Eﬀects in Semiconductors (Wiley, New York, 1974) thorough discussion of group theory and its applications in solid state physics by two pioneers

I C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon, 1972) comprehensive discussion of group theory in solid state physics

I G. F. Koster et al., Properties of the Thirty-Two Point Groups (MIT Press, 1963) small, but very helpful reference book tabulating the properties of the 32 crystallographic point groups (character tables, Clebsch-Gordan coeﬃcients, compatibility relations, etc.)

I A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1960) comprehensive discussion of the (group) theory of angular momentum in quantum mechanics

I and many others

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 These notes are dedicated to

Prof. Dr. h.c. Ulrich R¨ossler from whom I learned group theory

R.W.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Introduction and Overview Deﬁnition: Group A set G = {a, b, c,...} is called a group, if there exists a group multiplication connecting the elements in G in the following way (1) a, b ∈ G : c = a b ∈ G (closure) (2) a, b, c ∈ G :(ab)c = a(bc) (associativity) (3) ∃ e ∈ G : a e = a ∀a ∈ G (identity / neutral element) (4) ∀a ∈ G ∃ b ∈ G : a b = e, i.e., b ≡ a−1 (inverse element) Corollaries (a) e−1 = e (b) a−1a = a a−1 = e ∀a ∈ G (left inverse = right inverse) (c) e a = a e = a ∀a ∈ G (left neutral = right neutral) (d) ∀a, b ∈ G : c = a b ⇔ c−1 = b−1a−1 Commutative (Abelian) Group (5) ∀a, b ∈ G : a b = b a (commutatitivity) Order of a Group = number of group elements Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Examples

I integer numbers Z with addition (Abelian group, inﬁnite order)

I rational numbers Q\{0} with multiplication (Abelian group, inﬁnite order)

I complex numbers {exp(2πi m/n): m = 1,..., n} with multiplication (Abelian group, ﬁnite order, example of cyclic group)

I invertible (= nonsingular) n × n matrices with matrix multiplication (nonabelian group, inﬁnite order, later important for representation theory!)

I permutations of n objects: Pn (nonabelian group, n! group elements)

I symmetry operations (rotations, reﬂections, etc.) of equilateral triangle ≡ P3 ≡ permutations of numbered corners of triangle – more later! n I (continuous) translations in R : (continuous) translation group ≡ vector addition in Rn

I symmetry operations of a sphere only rotations: SO(3) = special orthogonal group in R3 = real orthogonal 3 × 3 matrices

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics

Group theory is the natural language to describe symmetries of a physical system

I symmetries correspond to conserved quantities

I symmetries allow us to classify quantum mechanical states • representation theory • degeneracies / level splittings

I evaluation of matrix elements ⇒ Wigner-Eckart theorem e.g., selection rules: dipole matrix elements for optical transitions

I Hamiltonian Hˆ must be invariant under the symmetries of a quantum system ⇒ construct Hˆ via symmetry arguments

I ...

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Classical Mechanics

I Lagrange function L(q, q˙ ), d ∂L ∂L I Lagrange equations = i = 1,..., N dt ∂q˙ i ∂qi ∂L ∂L I If for one j : = 0 ⇒ pj ≡ is a conserved quantity ∂qj ∂q˙ j

Examples

I qj linear coordinate I qj angular coordinate • translational invariance • rotational invariance • linear momentum pj = const. • angular momentum pj = const. • translation group • rotation group

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (1) Evaluation of matrix elements

I Consider particle in potential V (x) = V (−x) even

I two possiblities for eigenfunctions ψ(x)

ψe (x) even: ψe (x) = ψe (−x)

ψo (x) odd: ψo (x) = −ψo (−x) R ∗ I overlapp ψi (x) ψj (x) dx = δij i, j ∈ {e, o} R ∗ I expectation value hi|x|ii = ψi (x) x ψi (x) dx = 0 well-known explanation

I product of two even / two odd functions is even

I product of one even and one odd function is odd

I integral over an odd function vanishes

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (1) Evaluation of matrix elements (cont’d)

Group theory provides systematic generalization of these statements

I representation theory ≡ classiﬁcation of how functions and operators transform under symmetry operations

I Wigner-Eckart theorem ≡ statements on matrix elements if we know how the functions and operators transform under the symmetries of a system

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (2) Degeneracies of Energy Eigenvalues ˆ ˆ I Schr¨odingerequation Hψ = Eψ or i~∂t ψ = Hψ ˆ ˆ ˆ ˆ ˆ I Let O with i~∂t O = [O, H] = 0 ⇒ O is conserved quantity ˆ ˆ ⇒ eigenvalue equations Hψ = Eψ and Oψ = λOˆ ψ can be solved simultaneously ˆ ⇒ eigenvalue λOˆ of O is good quantum number for ψ Example: H atom 2 ∂2 2 ∂ Lˆ2 e2 Hˆ = ~ + + − ⇒ group SO(3) I 2m ∂r 2 r ∂r 2mr 2 r 2 2 ⇒ [Lˆ , Hˆ ] = [Lˆz , Hˆ ] = [Lˆ , Lˆz ] = 0 2 ⇒ eigenstates ψnlm(r): index l ↔ Lˆ , m ↔ Lˆz

I really another example for representation theory

I degeneracy for 0 ≤ l ≤ n − 1: dynamical symmetry (unique for H atom) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (3) Solid State Physics in particular: crystalline solids, periodic assembly of atoms ⇒ discrete translation invariance

(i) Electrons in periodic potential V (r)

I V (r + R) = V (r) ∀ R ∈ {lattice vectors}

⇒ translation operator TˆR : TˆR f (r) = f (r + R)

[TˆR, Hˆ ] = 0

ik·r ⇒ Bloch theorem ψk(r) = e uk(r) with uk(r + R) = uk(r) ⇒ wave vector k is quantum number for the discrete translation invariance, k ∈ ﬁrst Brillouin zone

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (3) Solid State Physics

(ii) Phonons rotation by 90o I Consider square lattice

I frequencies of modes are equal

I degeneracies for particular propagation directions

(iii) Theory of Invariants

I How can we construct models for the dynamics of electrons or phonons that are compatible with given crystal symmetries?

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (4) Nuclear and Particle Physics

Physics at small length scales: strong interaction

Proton mp = 938.28 MeV rest mass of nucleons almost equal

Neutron mn = 939.57 MeV ∼ degeneracy

I Symmetry: isospin ˆI with [ˆI , Hˆstrong] = 0

1 1 1 1 I SU(2): proton | 2 2 i, neutron | 2 − 2 i

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Mathematical Excursion: Groups Basic Concepts

Group Axioms: see above

Deﬁnition: Subgroup Let G be a group. A subset U ⊆ G that is itself a group with the same multiplication as G is called a subgroup of G.

Group Multiplication Table: compilation of all products of group elements ⇒ complete information on mathematical structure of a (ﬁnite) group

P3 e a b c d f Example: permutation group P3 e e a b c d f 1 2 3 1 2 3 1 2 3 e = a = b = a a b e f c d 1 2 3 2 3 1 3 1 2 b b e a d f c 1 2 3 1 2 3 1 2 3 c c d f e a b c = d = f = 1 3 2 3 2 1 2 1 3 d d f c b e a f f c d a b e

I {e}, {e, a, b}, {e, c}, {e, d}, {e, f }, G are subgroups of G

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 P3 e a b c d f Conclusions from e e a b c d f Group Multiplication Table a a b e f c d b b e a d f c I Symmetry w.r.t. main diagonal ⇒ group is Abelian c c d f e a b d d f c b e a n I order n of g ∈ G: smallest n > 0 with g = e f f c d a b e

2 n I {g, g ,..., g = e} with g ∈ G is Abelian subgroup (a cyclic group)

I in every row / column every element appears exactly once because:

Rearrangement Lemma: for any ﬁxed g 0 ∈ G, we have G = {g 0g : g ∈ G} = {gg 0 : g ∈ G} i.e., the latter sets consist of the elements in G rearranged in order. 0 0 0 proof: g1 6= g2 ⇔ g g1 6= g g2 ∀g1, g2, g ∈ G

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Goal: Classify elements in a group (1) Conjugate Elements and Classes

−1 I Let a ∈ G. Then b ∈ G is called conjugate to a if ∃ x ∈ G with b = x ax . Conjugation b ∼ a is equivalence relation: • a ∼ a reﬂexive • b ∼ a ⇔ a ∼ b symmetric • a ∼ co a = xcx −1 ⇒ c = x −1ax ⇒ a ∼ b transitive b ∼ c b = ycy −1 = (xy −1)−1a(xy −1)

I For ﬁxed a, the set of all conjugate elements Example: P3 C = {x ax −1 : x ∈ G} is called a class. x e a b c d f e e a b c d f • identity e is its own class x e x −1 = e ∀x ∈ G a e a b d f c • Abelian groups: each element is its own class b e a b f c d x ax −1 = ax x −1 = a ∀a, x ∈ G c e b a c f d d e b a f d c • Each b ∈ G belongs to one and only one class f e b a d c f ⇒ decompose G into classes ⇒ classes {e}, {a, b}, • in broad terms: “similar” elements form a class {c, d, f }

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Goal: Classify elements in a group (2) Subgroups and Cosets I Let U ⊂ G be a subgroup of G and x ∈ G. The set x U ≡ {x u : u ∈ U} (the set Ux) is called the left coset(right coset) of U.

I In general, cosets are not groups. If x ∈/ U, the coset x U lacks the identity element: suppose ∃u ∈ U with xu = e ∈ x U ⇒ x −1 = u ∈ U ⇒ x = u−1 ∈ U 0 0 0 I If x ∈ x U, then x U = x U any x ∈ x U can be used to deﬁne coset x U

I If U contains s elements, then each coset also contains s elements (due to rearrangement lemma).

I Two left (right) cosets for a subgroup U are either equal or disjoint (due to rearrangement lemma).

I Thus: decompose G into cosets G = U ∪ x U ∪ y U ∪ ... x, y,.../∈ U Thus Theorem 1: Let h order of G h I ⇒ ∈ N Let s order of U ⊂ G s I Corollary: The order of a ﬁnite group is an integer multiple of the orders of its subgroups.

I Corollary: If h prime number ⇒ {e}, G are the only subgroups ⇒ G is isomorphic to cyclic group Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Goal: Classify elements in a group (3) Invariant Subgroups and Factor Groups connection: classes and cosets

I A subgroup U ⊂ G containing only complete classes of G is called invariant subgroup (aka normal subgroup).

I Let U be an invariant subgroup of G and x ∈ G ⇔ x Ux −1 = U ⇔ x U = Ux (left coset = right coset)

I Multiplication of cosets of an invariant subgroup U ⊂ G: x, y ∈ G :(x U)(y U) = xy U = z U where z = xy

well-deﬁned: (x U)(y U) = x (U y) U = xy UU = z UU = z U

I An invariant subgroup U ⊂ G and the distinct cosets x U form a group, called factor group F = G/U • group multiplication: see above • U is identity element of factor group • x −1 U is inverse for x U

I Every factor group F = G/U is homomorphic to G (see below).

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Permutation Group P3

e a b c d f invariant subgroup U = {e, a, b} e e a b c d f a a b e f c d ⇒ one coset cU = dU = f U = {c, d, f } b b e a d f c factor group P /U = {U, cU} c c d f e a b 3 d d f c b e a U cU f f c d a b e U U cU cU cU U

I We can think of factor groups G/U as coarse-grained versions of G.

I Often, factor groups G/U are a helpful intermediate step when working out the structure of more complicated groups G.

I Thus: invariant subgroups are “more useful” subgroups than other subgroups.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Mappings of Groups 0 0 I Let G and G be two groups. A mapping φ : G → G assigns to each g ∈ G an element g 0 = φ(g) ∈ G0, with every g 0 ∈ G0 being the image of at least one g ∈ G.

I If φ(g1) φ(g2) = φ(g1 g2) ∀g1, g2 ∈ G, then φ is a homomorphic mapping of G on G0.

• A homomorphic mapping is consistent with the group structures • A homomorphic mapping G → G0 is always n-to-one (n ≥ 1): The preimage of the unit element of G0 is an invariant subgroup U of G. G0 is isomorphic to the factor group G/U.

I If the mapping φ is one-to-one, then it is an isomomorphic mapping of G on G0. • Short-hand: G isomorphic to G0 ⇒ G ' G0 • Isomorphic groups have the same group structure.

I Examples: 0 • trivial homomorphism G = P3 and G = {e}

• isomorphism between permutation group P3 and symmetry group C3v of

equilateral triangle Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Products of Groups

I Given two groups G1 = {ai } and G2 = {bk }, their outer direct product is the group G1 × G2 with elements (ai , bk ) and multiplication

(ai , bk ) · (aj , bl ) = (ai aj , bk bl ) ∈ G1 × G2

• Check that the group axioms are satisﬁed for G1 × G2.

• Order of Gn is hn (n = 1, 2) ⇒ order of G1 × G2 is h1h2

• If G = G1 × G2, then both G1 and G2 are invariant subgroups of G. Then we have isomorphisms G2 'G/G1 and G1 'G/G2. • Application: built more complex groups out of simpler groups

I If G1 = G2 = G = {ai }, the elements

(ai , ai ) ∈ G ⊗ G deﬁne a group G˜ ≡ G ⊗ G called the inner product of G. • The inner product G ⊗ G is isomorphic to G (⇒ same order as G) • Compare: product representations (discussed below)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Matrix Representations of a Group

Motivation C e i e = identity i I Consider symmetry group Ci = {e, i} i = inversion e e i i i e I two “types” of basis functions: even and odd

I more abstract: reducible and irreducible representations matrix representation (based on 1 × 1 and 2 × 2 matrices) Γ = {D = 1, D = 1}  1 e i  Γ = {D = 1, D = −1} consistent with group 2 e i multiplication table 1 0 1 0  Γ3 = De = 0 1 , Di = 0 −1 where Γ1 : even function fe (x) = fe (−x) irreducible representations Γ2 : odd functions fo (x) = −fo (−x)

Γ3 : reducible representation: decompose any f (x) into even and odd parts 1 fe (x) = f (x) + f (−x) f (x) = f (x)+f (x) with 2 e o 1 fo (x) = 2 f (x) − f (−x) How to generalize these ideas for arbitrary groups? Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Matrix Representations of a Group I Let group G = {gi : i = 1,..., h}

I Associate with each gi ∈ G a nonsingular square matrix D(gi ). If the resulting set {D(gi ): i = 1,..., h} is homomorphic to G it is called a matrix representation of G.

• gi gj = gk ⇒ D(gi ) D(gj ) = D(gk ) • D(e) = 1 (identity matrix) −1 −1 • D(gi ) = D (gi )

I dimension of representation = dimension of representation matrices

Example (1): G = C∞ = rotations around a ﬁxed axis (angle φ) I C∞ is isomorphic to group of orthogonal 2 × 2 matrices SO(2) cos φ − sin φ D (φ) = ⇒ two-dimensional (2D) representation 2 sin φ cos φ

I C∞ is homomorphic to group {D1(φ) = 1} ⇒ trivial 1D representation 1 0 higher-dimensional I C∞ is isomorphic to group ⇒ 0 D2(φ) representation

I Generally: given matrix representations of dimensions n1 and n2, we can construct (n1 + n2) dimensional representations Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Matrix Representations of a Group (cont’d)

Example (2): Symmetry group C3v of equilateral triangle (isomorphic to permutation group P3)

y ◦ ◦ ◦ ◦ y x = 120 = 240 = 240 = 120 ↔ − identity rotation φ rotation φ reflection y roto- reflection φ roto- reflection φ 2 P3 e a b = a c = ec d = ac f = bc 2 Koster E C3 C3 σv σv σv

Γ1 (1) (1) (1) (1) (1) (1)

Γ2 (1) (1) (1) (−1) (−1) (−1) √ √ √ √ 1 3 1 3 1 3 1 3 1 0 − 2 − 2 − 2 2 1 0 − 2 − 2 − 2 2 Γ √ √ √ √ 3 0 1 3 1 3 1 0 −1 3 1 3 1 2 − 2 − 2 − 2 − 2 2 2 2

P3 e a b c d f I mapping G → {D(gi )} homomorphic, e e a b c d f but in general not isomorphic (not faithful) multipli- a a b e f c d I consistent with group multiplication table cation b b e a d f c table c c d f e a b I Goal: characterize matrix representations of G d d f c b e a I Will see: G fully characterized by its “distinct” f f c d a b e matrix representations (only three for G = C3v !) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Goal: Identify and Classify Representations

I Theorem 2: If U is an invariant subgroup of G, then every representation of the factor group F = G/U is likewise a representation of G. Proof: G is homomorphic to F, which is homomorphic to the representations of F. Thus: To identify the representations of G it helps to identify the representations of F.

I Deﬁnition: Equivalent Representations

Let {D(gi )} be a matrix representation for G with dimension n. Let X be a n-dimensional nonsingular matrix. 0 −1 The set {D (gi ) = X D(gi ) X } forms a matrix representation called equivalent to {D(gi )}. 0 Convince yourself: {D (gi )} is, indeed, another matrix representation.

Matrix representations are most convenient if matrices {D} are unitary. Thus

I Theorem 3: Every matrix representation {D(gi )} is equivalent to a unitary 0 0 † 0 −1 representation {D (gi )} where D (gi ) = D (gi )

I In the following, it is always assumed that matrix representations are unitary.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Theorem 3 (cf. Falicov)

Challenge: Matrix X has to be choosen such that it makes all matrices 0 D (gi ) unitary simultaneously.

I Let {D(gi ) ≡ Di : i = 1,..., h} be a matrix representation for G (dimension h). h P † I Deﬁne H = Di Di (Hermitean) i=1 I Thus H can be diagonalized by means of a unitary matrix U. −1 P −1 −1 P −1 −1 −1 d ≡ U HU = U Di Di U = U Di U U Di U i i | {z } | {z } † † =D˜i ˜ P ˜ ˜ =Di = Di Di with dµν = dµδµν diagonal i I Diagonal entries dµ are positive: PP ˜ ˜† P ˜ ˜∗ P ˜ 2 dµ = (Di )µλ(Di )λµ = (Di )µλ(Di )µλ = |(Di )µλ| > 0 i λ iλ iλ ±1/2 I Take diagonal matrix d˜± with elements (d˜±)µν ≡ dµ δµν 1 ˜ ˜ ˜ P ˜ ˜† ˜ I Thus = d− d d− = d− Di Di d− (identity matrix) i Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Theorem 3 (cont’d)

0 ˜ ˜ ˜ ˜ −1 ˜ I Assertion: Di = d− Di d+ = d−U Di U d+ are unitary matrices equivalent to Di

−1 • equivalent by construction: X = d˜−U

• unitarity: =1 z }| { 0 0 † ˜ ˜ ˜ ˜ P ˜ ˜† ˜ ˜ ˜† ˜ Di Di = d−Di d+ (d− Dk Dk d−) d+Di d− k ˜ P ˜ ˜ ˜† ˜† ˜ = d− Di Dk Dk Di d− k | {z } | {z } ˜ ˜† = Dj = Dj (rearrangement lemma) | {z } = d = 1 qed

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Reducible and Irreducible Representations (RRs and IRs)

I If for a given representation {D(gi ): i = 1,..., h}, an equivalent 0 representation {D (gi ): i = 1,..., h} can be found that is block diagonal 0 0 D1(gi ) 0 D (gi ) = 0 ∀gi ∈ G 0 D2(gi )

then {D(gi ): i = 1,..., h} is called reducible, otherwise irreducible.

I Crucial: the same block diagonal form is obtained for all representation matrices D(gi ) simultaneously. 0 0 I Block-diagonal matrices do not mix, i.e., if D (g1) and D (g2) are 0 0 0 block diagonal, then D (g3) = D (g1) D (g2) is likewise block diagonal. ⇒ Decomposition of RRs into IRs decomposes the problem into the smallest subproblems possible.

I Goal of Representation Theory Identify and characterize the IRs of a group.

I We will show The number of inequivalent IRs equals the number of classes. Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Schur’s First Lemma

Schur’s First Lemma: Suppose a matrix M commutes with all matrices D(gi ) of an irreducible representation of G

D(gi ) M = M D(gi ) ∀gi ∈ G (♠) then M is a multiple of the identity matrix M = c1, c ∈ C.

Corollaries

I If (♠) holds with M 6= c1, c ∈ C, then {D(gi )} is reducible.

I All IRs of Abelian groups are one-dimensional

Proof: Take gj ∈ G arbitrary, but ﬁxed. G Abelian ⇒ D(gi ) D(gj ) = D(gj ) D(gi ) ∀gi ∈ G Lemma ⇒ D(gj ) = cj 1 with cj ∈ C, i.e., {D(gj ) = cj } is an IR.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Schur’s First Lemma (cf. Bir & Pikus)

† † † † I Take Hermitean conjugate of (♠): M D (gi ) = D (gi ) M † −1 † † Multiply with D (gi ) = D (gi ): D(gi ) M = M D(gi )

† I Thus: (♠) holds for M and M , and also the Hermitean matrices 0 1 † 00 i † M = 2 (M + M ) M = 2 (M − M )

0 00 I It exists a unitary matrix U that diagonalizes M (similar for M ) −1 0 d = U M U with dµν = dµδµν

0 0 0 −1 I Thus (♠) implies D (gi ) d = d D (gi ), where D (gi ) = U D(gi ) U 0 more explicitly: Dµν (gi )(dµ − dν ) = 0 ∀i, µ, ν

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Schur’s First Lemma (cont’d)

Two possibilities:

I All dµ are equal, i.e, d = c1. So M0 = UdU−1 and M00 are likewise proportional to 1, and so is M = M0 − iM00.

I Some dµ are diﬀerent:

Say {dκ : κ = 1,..., r} are diﬀerent from {dλ : λ = r + 1,..., h}. ∀κ = 1,..., r; Thus: D0 (g ) = 0 κλ i ∀λ = r + 1,..., h 0 Thus {D (gi ): i = 1,..., h} is block-diagonal, contrary to the assumption that {D(gi )} is irreducible qed

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Schur’s Second Lemma

Schur’s Second Lemma: Suppose we have two IRs {D1(gi ), dimension n1} and {D2(gi ), dimension n2}, as well as a n1 × n2 matrix M such that

D1(gi ) M = M D2(gi ) ∀gi ∈ G (♣)

(1) If {D1(gi )} and {D2(gi )} are inequivalent, then M = 0.

(2) If M 6= 0 then {D1(gi )} and {D2(gi )} are equivalent.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Schur’s Second Lemma (cf. Bir & Pikus)

† −1 −1 I Take Hermitean conjugate of (♣); use D (gi ) = D (gi ) = D(gi ), so † −1 −1 † M D1(gi ) = D2(gi )M −1 −1 I Multiply by M on the left; Eq. (♣) implies M D2(gi ) = D1(gi ) M, so † −1 −1 † −1 MM D1(gi ) = D1(gi )MM ∀gi ∈ G

† I Schur’s ﬁrst lemma implies that MM is square matrix with MM† = c1 with c ∈ C (*)

I Case a: n1 = n2 • If c 6= 0 then det M 6= 0 because of (*), i.e., M is invertible.

So (♣) implies −1 M D1(gi ) M = D2(gi ) ∀gi ∈ G

thus {D1(gi )} and {D2(gi )} are equivalent.

• If c = 0 then MM† = 0, i.e., P † P ∗ P 2 Mµν Mνµ = Mµν Mµν = |Mµν | = 0 ∀µ ν ν ν so that M = 0.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Schur’s Second Lemma (cont’d)

I Case b: n1 6= n2 (n1 < n2 to be speciﬁc)

• Fill up M with n2 − n1 rows to get matrix M˜ with det M˜ = 0.

• However M˜ M˜ † = MM†, so that det(MM†) = det(M˜ M˜ †) = (det M˜ ) (det M˜ †) = 0

• So c = 0, i.e., MM† = 0, and as before M = 0. qed

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Orthogonality Relations for IRs Notation:

I Irreducible Representations (IR): ΓI = {DI (gi ): gi ∈ G} I nI = dimensionality of IR ΓI I h = order of group G

Theorem 4: Orthogonality Relations for Irreducible Representations (1) two inequivalent IRs ΓI 6= ΓJ h 0 0 P ∗ ∀ µ , ν = 1,..., nI DI (gi )µ0ν0 DJ (gi )µν = 0 i=1 ∀ µ, ν = 1,..., nJ

(2) representation matrices of one IR ΓI h nI P ∗ 0 0 DI (gi )µ0ν0 DI (gi )µν = δµ0µ δν0ν ∀ µ , ν , µ, ν = 1,..., nI h i=1 Remarks I [DI (gi )µν : i = 1,..., h] form vectors in a h-dim. vector space p I vectors are normalized to h/nI (because ΓI assumed to be unitary)

I vectors for different I , µν are orthogonal P 2 P 2 I in total, we have I nI such vectors; therefore I nI ≤ h Corollary: For finite groups the number of inequivalent IRs is finite. Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Theorem 4: Orthogonality Relations for IRs

(1) two inequivalent IRs ΓI 6= ΓJ I Take arbitrary nJ × nI matrix X 6= 0 (i.e., at least one Xµν 6= 0) P −1 I Let M ≡ DJ (gi ) X DI (gi ) i =M =1 z }| { z }| { P −1 −1 ⇒D J (gk ) M = DJ (gk ) DJ (gi ) X DI (gi ) DI (gk ) DI (gk ) i | {z } | {z } P −1 = DJ (gk gi ) X DI (gk gi ) DI (gk ) i |{z} |{z} =gj =gj P −1 = DJ (gj ) X DI (gj ) DI (gk ) j | {z } = M DI (gk ) ⇒ (Schur’s Second Lemma) 0 0 = Mµµ0 ∀µ, µ P P −1 in particular correct for = DJ (gi )µκ Xκλ DI (g )λµ0 i X = δ δ 0 i κ,λ κλ νκ λν P −1 = DJ (gi )µν DI (gi )ν0µ0 i P ∗ = DI (gi )µ0ν0 DJ (gi )µν qed i Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Theorem 4: Orthogonality Relations for IRs (cont’d)

(2) representation matrices of one IR ΓI First steps similar to case (1): P −1 I Let M ≡ DI (gi ) X DI (gi ) with nI × nI matrix X 6= 0 i ⇒D I (gk ) M = M DI (gk ) ⇒ (Schur’s First Lemma): M = c 1, c ∈ C

P P −1 I Thus c δµµ0 = DI (gi )µκ Xκλ DI (gi )λµ0 choose Xκλ = δνκ δλν0 i κ,λ P −1 = DI (gi )µν DI (gi )ν0µ0 = Mµµ0 i 1 P 1 P P −1 h I c = Mµµ = DI (gi )µν DI (gi )ν0µ = δνν0 qed nI µ nI i µ nI | {z } −1 DI (gi gi = e)ν0ν = δνν0

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Goal: Characterize diﬀerent irreducible representations of a group Characters

I The traces of the representation matrices are called characters P χ(gi ) ≡ tr D(gi ) = i D(gi )µµ

I Equivalent IRs are related via a similarity transformation 0 −1 D (gi ) = X D(gi )X with X nonsingular 0 This transformation leaves the trace invariant: tr D (gi ) = tr D(gi ) ⇒ Equivalent representations have the same characters.

I Theorem 5: If gi , gj ∈ G belong to the same class Ck of G, then for every representation ΓI of G we have χI (gi ) = χI (gj )

Proof: −1 • gi , gj ∈ C ⇒ ∃ x ∈ G with gi = x gj x −1 • Thus DI (gi ) = DI (x) DI (gj ) DI (x )

• −1 (trace invariant under χI (gi ) = tr DI (x) DI (gj ) DI (x ) cyclic permutation) −1 = tr DI (x ) DI (x) DI (gj ) = χI (gk ) | {z } =1 Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Characters (cont’d) Notation I χI (Ck ) denotes the character of group elements in class Ck

I The array [χI (Ck )] with I = 1,..., N (N = number of IRs) k = 1,..., N˜ (N˜ = number of classes) is called character table.

Remark: For Abelian groups the character table is the table of the 1 × 1 representation matrices

Theorem 6: Orthogonality relations for characters

Let {DI (gi )} and {DJ (gi )} be two IRs of G. Let hk be the number of elements in class Ck and N˜ the number of classes. Then N˜ P hk ∗ Proof: Use orthogonality χI (Ck ) χJ (Ck ) = δIJ ∀ I , J = 1,..., N relation for IRs k=1 h

I Interpretation: rows [χI (Ck ): k = 1,... N˜ ] of character table are like N orthonormal vectors in a N˜ -dimensional vector space ⇒ N ≤ N˜ .

I If two IRs ΓI and ΓJ have the same characters, this is necessary and suﬃcient for Γ and Γ to be equivalent. I J Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Symmetry group C3v of equilateral triangle

(isomorphic to permutation group P3)

Γ1 (1) (1) (1) (1) (1) (1)

Γ2 (1) (1) (1) (−1) (−1) (−1) √ √ √ √ 1 3 1 3 1 3 1 3 1 0 − 2 − 2 − 2 2 1 0 − 2 − 2 − 2 2 Γ √ √ √ √ 3 0 1 3 1 3 1 0 −1 3 1 3 1 2 − 2 − 2 − 2 − 2 2 2 2

P3 e a b c d f Character table e e a b c d f P3 e a, b c, d, f multipli- a a b e f c d cation b b e a d f c C3v E 2C3 3σv table c c d f e a b Γ1 1 1 1 d d f c b e a Γ2 1 1 −1 f f c d a b e Γ3 2 −1 0

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Interpretation: Character Tables

I A character table is the uniquely deﬁned signature of a group and its IRs ΓI [independent of, e.g., phase conventions for representation matrices DI (gi ) that are quite arbitrary].

I Isomorphic groups have the same character tables.

I Yet: the labeling of IRs ΓI is a matter of convention. – Customary: • Γ1 = identity representation: all characters are 1 • IRs are often numbered such that low-dimensional IRs come ﬁrst; higher-dimensional IRs come later

• If G contains the inversion, a superscript ± is added to ΓI indicating ± the behavior of ΓI under inversion (even or odd) • other labeling schemes are inspired by compatibility relations (more later)

I Different authors use different conventions to label IRs. To compare such notations we need to compare the uniquely defined characters for each class of an IR. (See, e.g., Table 2.7 in Yu and Cardona: Fundamentals of Semiconductors; here we follow Koster et al.)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Decomposing Reducible Representations (RRs) Into Irreducible Representations (IRs)

Given an arbitrary RR {D(gi )} the representation matrices {D(gi )} can be brought into block-diagonal form by a suitable unitary transformation

 D1(gi )  ) . 0  ..  a1 times    D1(gi )  0  .  . D(gi ) → D (gi ) =  ..  .   .  DN (gi )  )  .   ..  aN times 0 DN (gi )

Theorem 7: Let aI be the multiplicity, with which the IR ΓI ≡ {DI (gi )} is contained in the representation {D(gi )}. Then N P (1) χ(gi ) = aI χI (gi ) I =1 h N˜ 1 P ∗ P hk ∗ (2) aI = χI (gi ) χ(gi ) = χI (Ck ) χ(Ck ) h i=1 k=1 h

We say: {D(gi )} contains the IR ΓI aI times.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof: Theorem 7 (1) due to invariance of trace under similarity transformations N h X 1 X ∗ (2) we have aJ χJ (gi ) = χ(gi ) χ (gi ) × h I J=1 i=1 N h h X 1 X 1 X ⇒ a χ∗(g ) χ (g ) = χ∗(g ) χ(g ) qed J h I i J i h I i i J=1 i=1 i=1 | {z } =δIJ

Applications of Theorem 7:

I Corollary: The representation {D(gi )} is irreducible if and only if h X 2 |χ(gi )| = h i=1 1 for one I Proof: Use Theorem 7 with a = I 0 otherwise

I Decomposition of Product Representations (see later)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Where Are We?

We have discussed the orthogonality relations for

I irreducible representations

I characters These can be complemented by matching completeness relations. Proving those is a bit more cumbersome. It requires the introduction of the regular representation.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 The Regular Representation

Finding the IRs of a group can be tricky. Yet for ﬁnite groups we can derive the regular representation which contains all IRs of the group.

I Interpret group elements gν as basis vectors {|gν i : ν = 1,... h} for a h-dim. representation

⇒ Regular representation: νth column vector of DR (gi ) gives image |gµi = gi |gν i ≡ |gi gν i of basis vector |gν i 1 if g g −1 = g ⇒ D (g ) = µ ν i R i µν 0 otherwise

I Strategy: −1 −1 −1 g1 g2 g3 ... • Re-arrange the group multiplication table g1 e ... as shown on the right g2 e . • For each gi ∈ G we have DR (gi )µν = 1, g3 . e if the entry (µ, ν) in the re-arranged . . e group multiplication table equals gi , otherwise DR (gi )µν = 0. Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Properties of the Regular Representation {DR (gi )}

(1) {DR (gi )} is, indeed, a representation for the group G

(2) It is a faithful representation, i.e., {DR (gi )} is isomorphic to G = {gi }. h if g = e (3) χ (g ) = i R i 0 otherwise Proof:

(1) Matrices {DR (gi )} are nonsingular, as every row / every column contains “1” exactly once.

Show: if gi gj = gk , then DR (gi )DR (gj ) = DR (gk ) Take i, j, µ, ν arbitrary, but ﬁxed −1 −1 DR (gi )µλ = 1 only for gµ gλ = gi ⇔ gλ = gi gµ −1 DR (gj )λν = 1 only for gλ gν = gj ⇔ gλ = gj gν P −1 ⇔ DR (gi )µλ DR (gj )λν = 1 only for gi gµ = gj gν λ −1 ⇔ gµ gν = gi gj = gk [deﬁnition of DR (gk )µν ]

(2) immediate consequence of deﬁnition of DR (gi ) 1 if g = g g −1 = e (3) D (g ) = i µ µ R i µµ 0 otherwise h if g = e ⇒ χ (g ) = PD (g ) = i R i R i µµ 0 otherwise µ Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Regular Representation for P3

g e a b c d f g −1 e b a c d f e e a b c d f e e b a c d f a a b e f c d a a e b f c d b b e a d f c ⇒ b b a e d f c c c d f e a b c c f d e a b d d f c b e a d d c f b e a f f c d a b e f f d c a b e

Thus  1 0 0   0 0 1   0 1 0   0 1 0   1 0 0   0 0 1         0 0 1   0 1 0   1 0 0  e =   a =   b =    1 0 0   0 1 0   0 0 1   0 1 0   0 0 1   1 0 0  0 0 1 1 0 0 0 1 0

 1 0 0   0 1 0   0 0 1   0 1 0   0 0 1   1 0 0         0 0 1   1 0 0   0 1 0  c =   d =   f =    1 0 0   0 0 1   0 1 0   0 1 0   1 0 0   0 0 1  0 0 1 0 1 0 1 0 0

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Completeness of Irreducible Representations

Lemma: The regular representation contains every IR nI times, where nI = dimensionality of IR ΓI . P Proof: Use Theorem 7: χR (gi ) = aI χI (gi ) where i 1 P ∗ 1 ∗ aI = h χI (gi ) χR (gi ) = h χI (e) χR (e) = nI i | {z } | {z } =nI =h Corollary (Burnside’s Theorem): For a group G of order h, the dimensionalities nI of the IRs ΓI obey P P 2 P 2 Proof: h = χR (e) = I aI χI (e) = I nI nI = h I serious constraint for dimensionalities of IRs Theorem 8: The representation matrices DI (gi ) of a group G of order h obey the completeness relation X X nI D∗(g ) D (g ) = δ ∀ i, j = 1,..., h (*) h I i µν I j µν ij I µ,ν Proof:

I Theorem 4: Interpret [DI (gi )µν : i = 1,..., h] as orthonormal row vectors of a matrix M ⇒ M is square matrix: unitary I Corollary: M has h columns ⇒ column vectors also orthonormal = completeness (*) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Completeness Relation for Characters

Theorem 9: Completeness Relation for Characters If χI (Ck ) is the character for class Ck and irreducible representation I , then h k X ∗ 0 χ (C )χ (C 0 ) = δ 0 ∀ k, k = 1,..., N˜ h I k I k kk I

 χ1(Ck )   .  ˜ I Interpretation: columns  .  of character table [k = 1,..., N] χN (Ck ) are like N˜ orthonormal vectors in a N-dimensional vector space

˜ (from com- ) I Thus N ≤ N. pleteness) Number N of irreducible representations ˜ (from ortho- = Number N˜ of classes I Also N ≤ N gonality)

I Character table • square table • rows and column form orthogonal vectors

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Theorem 9: Completeness Relation for Characters

Lemma: Let {DI (gi )} be an nI -dimensional IR of G. Let Ck be a class of G with hk elements. Then X hk DI (gi ) = χI (Ck ) 1 nI i∈Ck The sum over all representation matrices in a class of an IR is proportional to the identity matrix. Proof of Lemma:

I For arbitrary gj ∈ G P −1 P −1 P DI (gj ) DI (gi ) DI (gj ) = DI (gj ) DI (gi ) DI (gj ) = DI (gi 0 ) x 0 i∈Ck i∈Ck | {z }  i ∈Ck 0  =DI (g 0 ) with i ∈Ck  i  because gj maps gi 6= gi onto g 0 6= g 0 1 2 i1 i2 P ⇒ (Schur’s First Lemma): DI (gi ) = ck 1 i∈Ck

1 P hk I ck = tr DI (gi ) = χI (Ck ) qed nI i∈Ck nI

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Proof of Theorem 9: Completeness Relation for Characters

I Use Theorem 8 (Completeness Relations for Irreducible Representations) N P P nI ∗ P P D (gi )µν DI (gj )µν = δij h I I =1 µ,ν i∈Ck j∈Ck0

P nI P P ∗ P ⇒ DI (gi ) DI (gj ) = hk δkk0 I h µ,ν i∈C j∈C 0 k µν k µν | {z } | {z } hk ∗ hk0 χI (Ck ) δµν χI (Ck0 ) δµν (Lemma) nI nI | {z } hk hk0 ∗ P 2 χI (Ck ) χI (Ck0 ) δµν qed nI µ,ν | {z } =nI

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Summary: Orthogonality and Completeness Relations

Theorem 4: Orthogonality Relations for Irreducible Representations n h I , J = 1,..., N I X ∗ 0 0 D (g ) 0 0 D (g ) = δ δ 0 δ 0 µ , ν = 1,..., n h I i µ ν J i µν IJ µµ νν I i=1 µ, ν = 1,..., nJ

Theorem 8: Completeness Relations for Irreducible Representations N X X nI D∗(g ) D (g ) = δ ∀ i, j = 1,..., h h I i µν I j µν ij I =1 µ,ν

Theorem 6: Orthogonality Relations for Characters

N˜ X hk χ∗(C ) χ (C ) = δ ∀ I , J = 1,..., N h I k J k IJ k=1

Theorem 9: Completeness Relation for Characters h N k X ∗ 0 χ (C )χ (C 0 ) = δ 0 ∀ k, k = 1,..., N˜ h I k I k kk I =1 Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Unreducible h hhhh More on Irreduciblehhhh Problems

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Quantum Mechanics

Topics:

I Behavior of quantum mechanical states and operators under symmetry operations

I Relation between irreducible representations and invariant subspaces of the Hilbert space

I Connection between eigenvalue spectrum of quantum mechanical operators and irreducible representations

I Selection rules: symmetry-induced vanishing of matrix elements and Wigner-Eckart theorem

Note: Operator formalism of QM convenient to discuss group theory. Yet: many results also applicable in other areas of physics.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symmetry Operations in Quantum Mechanics (QM)

I Let G = {gi } be a group of symmetry operations of a qm system e.g., translations, rotations, permutation of particles

I Translated into the language of group theory: In the Hilbert space of the qm system we have a group of unitary 0 0 operators G = {Pˆ(gi )} such that G is isomorphic to G.

Examples

I translations Ta ˆ → unitary operator P(Ta) = exp (ipˆ · a/~) (pˆ = momentum) ˆ 1 2 P(Ta) ψ(r) = 1 + ∇ · a + 2 (∇ · a) + ... ψ(r) = ψ(r + a)

I rotations Rφ ˆ (L = angular momentum → unitary operator Pˆ(n, φ) = exp iLˆ · nφ/~ φ = angle of rotation n = axis of rotation)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Transformation of QM States

I Let {|νi} be an orthonormal basis

I Let Pˆ(gi ) be the symmetry operator for the symmetry transformation gi with symmetry group G = {gi }.

P I Then Pˆ(gi ) |νi = |µi hµ| Pˆ(gi ) |νi ↑ µ | {z } 1 P D(gi )µν = µ |µihµ| matrix of a unitary P I So Pˆ(gi ) |νi = D(gi )µν |µi where D(gi )µν = representation of G µ because Pˆ(gi ) unitary

I Note: bras and kets transform according to complex conjugate representations ˆ † P ∗ hν|P(gi ) = hµ| D(gi )µν µ (consistent with I Let gi , gj ∈ G with gi gj = gk ∈ G. Then matrix multiplication) D(gi )κµ D(gi )µν z }| { z }| {  P ˆ ˆ P  |κihκ| P(gi ) |µi hµ| P(gi ) |νi = D(gi )κµD(gi )µν |κi κµ κµ Pˆ(gi ) Pˆ(gj ) |νi = P  Pˆ(gk ) |νi = D(gk )κν |κi κ Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Transformation of (Wave) Functions ψ(r) ˆ 0 ˆ 0 −1 ˆ † ˆ −1 I P(gi )|ri = |r =gi ri ⇔ hr|P(gi ) = hr =gi r| b/c P(g) = P(g )

I Let ψ(r) ≡ hr|ψi ˆ ˆ −1 ⇒ P(gi ) ψ(r) ≡ hr|P(gi )|ψi = ψ(gi r) ≡ ψi (r)

I In general, the functions V = {ψi (r): i = 1,..., h} are linear dependent

⇒ Choose instead linear independent functions ψν (r) ≡ hr|νi spanning V

⇒ Expand images Pˆ(gi ) ψν (r) in terms of {ψν (r)}: P P Pˆ(gi ) ψ(r) = hr|Pˆ(gi )|ψi = hr|µihµ|Pˆ(gi )|νi = D(gi )µν ψµ(r) µ µ

ˆ −1 P ⇒ P(gi ) ψν (r) = ψν (gi r) = D(gi )µν ψµ(r) µ

I Thus: every function ψ(r) induces a matrix representation Γ = {D(gi )}

I Also: every representation Γ = {D(gi )} is completely characterized by a (nonunique) set of basis functions {ψν (r)} transforming according to Γ.

I Dirac bra-ket notation convenient for formulating group theory of functions. Yet: results applicable in many areas of physics beyond QM. Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Important Representations in Physics (usually reducible)

(1) Representations for polar and axial (cartesian) vectors

I generally: two types of point group symmetry operations

• proper rotations gpr = (n, θ) about axis n, angle θ

D[gpr = (n, θ)] = Rodrigues’ rotation formula  2  nx (1 − cos θ) + cos θ nx ny (1 − cos θ) − nz sin θ nx nz (1 − cos θ) + ny sin θ 2  ny nx (1 − cos θ) + nz sin θ ny (1 − cos θ) + cos θ ny nz (1 − cos θ) − nx sin θ  2 nz nx (1 − cos θ) − ny sin θ nz ny (1 − cos θ) + nx sin θ nz (1 − cos θ) + cos θ

I det D(gpr) = +1

I χ(gpr) = tr D(gpr) = 1 + 2 cos θ independent of n

• improper rotations gim ≡ i gpr = gpri where i = inversion

• inversion i: Dpol(i) = −13×3 I polar vectors • improper rotations gim = i gpr: • proper rotations gpr: I Dpol(gim) = −Dpol(gpr) I det Dpol(gpr) = +1 I det Dpol(gim) = −1 I tr Dpol(gpr) = 1 + 2 cos θ I tr Dpol(gim) = −(1 + 2 cos θ)

• Γpol = {Dpol(g)} ⊆ O(3) always a faithful representation (i.e., isomorphic to G) • examples: position r, linear momentum p, electric ﬁeld E Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Important Representations in Physics (1) Representations for polar and axial (cartesian) vectors (cont’d)

I axial vectors • proper rotations gpr:

I Dax(gpr) = Dpol(gpr)

I det Dax(gpr) = +1

I tr Dax(gpr) = 1 + 2 cos θ

• inversion i: Dax(i) = +13×3

• improper rotations gim = i gpr:

I Dax(gim) = Dax(gpr) = −Dpol(gpr)

I det Dax(gim) = +1

I tr Dax(gim) = 1 + 2 cos θ

• Γax = {Dax(g)} ⊆ SO(3) • examples: angular momentum L, magnetic ﬁeld B

I systems with discrete symmetry group G = {gi : i = 1,..., h}: We have a “universal recipe” to Γpol = {Dpol(gi ): i = 1,..., h} construct the 3 × 3 matrices Dpol(g) and Dax(g) for each group element Γax = {Dax(gi ): i = 1,..., h} gpr = (n, θ) and gim = i (n, θ) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Important Representations in Physics (cont’d)

(2) Equivalence Representations Γeq I Consider symmetric object (symmetry group G) • vertices, edges, and faces of platonic solids are equivalent by symmetry • atoms / atomic orbitals |µi in a molecule may be equivalent by symmetry

I Equivalence representation Γeq describes mapping of equivalent objects P I Generally: Pˆ(g) |µi = Deq(g)νµ |νi ν

I Example: orbitals of equivalent H atoms in NH3 molecule (group C3v )

Equivalent to: permutations of corners of triangle (group P3)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Symmetry group C3v of equilateral triangle

(isomorphic to permutation group P3)

z ◦ ◦ ◦ ◦

y y 2 3 = 120 = 120 = 240 = 240 ↔ − 1 x identity rotation φ rotation φ reflection y roto- reflection φ roto- reflection φ 2 P3 e a b = a c = ec d = ac f = bc 2 Koster E C3 C3 σv σv σv

Γ1 (1) (1) (1) (1) (1) (1)

Γ2 (1) (1) (1) (−1) (−1) (−1) √ √ √ √ 1 3 1 3 1 3 1 3 1 0 − 2 − 2 − 2 2 1 0 − 2 − 2 − 2 2 Γ √ √ √ √ 3 0 1 3 1 3 1 0 −1 3 1 3 1 2 − 2 − 2 − 2 − 2 2 2 2 √ √ 3 1 3 1 n, θ (0, 0, 1), 0 (0, 0, 1), 2π/3 (0, 0, 1), 4π/3 (0, 1, 0), π (− 2 , − 2 , 0), π (− 2 , 2 , 0), π √ √ √ √ 1 3 ! 1 3 ! 1 3 ! 1 3 ! 1 − − − 1 − − − Γpol = √2 2 √2 2 √2 2 √2 2 −1 1 3 − 1 − 3 − 1 − 3 1 3 1 Γ1 + Γ3 1 2 2 2 2 1 2 2 2 2 1 1 1 1 √ √ √ √ 1 3 ! 1 3 ! 1 3 ! 1 3 ! 1 − − − −1 − Γax = √2 2 √2 2 √2 2 2√ 2 1 3 3 1 3 − 1 − 3 − 1 − 1 − − 1 Γ2 + Γ3 1 2 2 2 2 −1 2 2 2 2 1 1 −1 −1

Γeq = 1 1 1 1 1 1 1 1 1 1 1 1 Γ1 + Γ3 1 1 1 1 1 1 Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Transformation of QM States (cont’d)

I in general: representation {D(gi )} of states {|νi} is reducible

0 −1 I We have D (gi ) = U D(gi ) U

0 P −1 I More explicitly: D (gi )µ0ν0 = Uµ0µ D(gi )µν Uνν0 µν | {z } hµ| Pˆ(gi ) |νi P −1 ˆ = hµ|Uµ0µ P(gi ) Uνν0 |νi µν 0 0 = hµ | Pˆ(gi ) |ν i 0 P with |ν i = Uνν0 |νi ν

I Thus: block diagonalization 0 −1 {D(gi )} → {D (gi ) = U D(gi ) U} corresponds to change of basis 0 P {|νi} → {|ν i = Uνν0 |νi} ν

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Basis Functions for Irreducible Representations

I matrices {DI (gi )} are fully characterized by basis functions I {ψν (r): ν = 1,... nI } transforming according to IR ΓI ˆ I −1 P I P(gi ) ψν (r) = ψν (gi r) = DI (gi )µν ψµ(r) µ

I convenient if we need to spell out phase conventions for {DI (gi )} (→ Koster) I identify IRs for (components of) polar and axial vectors

Example: Symmetry group C3v z ◦ ◦ ◦ ◦

y y = 240 = 120 = 240 = 120 ↔ − x identity rotation φ rotation φ reflection y roto- reflection φ roto- reflection φ basis functions x 2 + y 2; z; Γ1 (1) (1) (1) (1) (1) (1) 2 2 Lx + Ly Γ2 (1) (1) (1) (−1) (−1) (−1) Lz √ √ √ √ 1 3 1 3 1 3 1 3 1 0 − 2 − 2 − 2 2 1 0 2 2 2 − 2 Γ √ √ √ √ x, y 3 0 1 3 1 3 1 0 −1 3 1 3 1 2 − 2 − 2 − 2 2 − 2 − 2 − 2 √ √ √ √ 1 3 1 3 1 3 1 3 1 0 − 2 − 2 − 2 2 −1 0 − 2 − 2 − 2 2 √ √ √ √ Lx , Ly 0 1 3 1 3 1 0 1 3 1 3 1 2 − 2 − 2 − 2 − 2 2 2 2 r = (x, y, z) = polar vector, L = (Lx , Ly , Lz ) = axial vector Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Relevance of Irreducible Representations Invariant Subspaces Definition:

I Let G = {gi } be a group of symmetry transformations. Let H = {|µi} be a Hilbert space with states |µi. A subspace S ⊂ H is called invariant subspace (with respect to G) if

Pˆ(gi ) |µi ∈ S ∀ gi ∈ G, ∀ |µi ∈ S I If an invariant subspace can be decomposed into smaller invariant subspaces, it is called reducible, otherwise it is called irreducible.

Theorem 10: An invariant subspace S is irreducible if and only if the states in S transform according to an irreducible representation.

Proof:

I Suppose {D(gi )} is reducible. 0 −1 I ∃ unitary transformation U with {D (gi ) = U D(gi ) U} block diagonal 0 0 P I For {D (gi )} we have the basis {|µ i = µ Uµµ0 |µi} 0 0 I The block diagonal form of {D (gi )} implies that {|µ i is reducible

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Invariant Subspaces (cont’d)

Corollary: Every Hilbert space H can be decomposed into irreducible invariant subspaces SI transforming according to the IR ΓI Remark: Given a Hilbert space H we can generally have multiple α (possibly orthogonal) irreducible invariant subspaces SI α SI = |I ν αi : ν = 1,..., nI transforming according to the same IR ΓI P Pˆ(gi ) |I ν αi = DI (gi )µν |I µαi µ Theorem 11: (1) States transforming according to diﬀerent IRs are orthogonal (2) For states |I µαi and |I ν βi transforming according to the same IR ΓI we have hI µα | I ν βi = δµν hI α||I βi where the reduced matrix element hI α||I βi is independent of µ, ν.

Remark: This theorem lets us anticipate the Wigner-Eckart theorem Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Invariant Subspaces (cont’d)

Proof of Theorem 11 ˆ 1 ˆ † ˆ 1 Pˆ † ˆ I Use unitarity of P(gj ): = P(gj ) P(gj ) = h P(gi ) P(gi ) i I Then 1 P ˆ † ˆ hI µα |J ν βi = h hI µα| P(gi ) P(gi ) |J ν βi i | {z ∗ } | {z } P hI µ0 α|D (g ) P D (g ) 0 |J ν0 βi µ0 I i µ0µ ν0 J i ν ν P 0 0 1 P ∗ = hI µ α |J ν βi h i DI (gi )µ0µ DJ (gi )ν0ν µ0ν0 | {z } (1/nI ) δIJ δµν δµ0ν0 1 P 0 0 = δIJ δµν hI µ α |I µ βi nI µ0 | {z } ≡hI α||I βi qed

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Discussion Theorem 11

I ΓJ × ΓI contains the identity representation Γ1 if and only if the IR ΓJ is the ∗ ∗ complex conjugate of ΓI , i.e., ΓJ = ΓI ⇔ DJ (g) = DI (g) ∀g.

I If the ket |J µαi transforms according to the IR ΓJ , the bra hJ µα| ∗ transforms according to the complex conjugate representation ΓJ . I Thus: hJ µα|I ν βi= 6 0 equivalent to • bra and ket transform according to complex conjugate representations • hJ µα|I ν βi contains the identity representation

I Indeed, common theme of representation theory applied to physics: Terms are only nonzero if they transform according to a representation that contains the identity representation.

I Variant of Theorem 11 (Bir & Pikus): Z If fI (x) transforms according to some IR ΓI , then fI (x) dx 6= 0

only if ΓI is the identity representation.

I Applications • Wigner-Eckart Theorem • Nonzero elements of material tensors • Our universe would be zero “by symmetry” if the apparently trivial identity

representation did not exist. Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Decomposition into Irreducible Invariant Subspaces

I Goal: Decompose general state |ψi ∈ H into components from irreducible invariant subspaces SI

ˆ I nI P ∗ ˆ I Generalized projection operator Πµµ0 := h DI (gi )µµ0 P(gi ) i ˆ I I Theorem 12: (i) Πµµ0 |Jναi = δIJ δµ0ν |I µαi ˆ I ˆ J ˆ J (ii) Πµµ0 Πνν0 = δIJ δµ0ν Πµν0 P ˆ I 1 (iii) Πµµ = Proof: I µ I n ∗ n ∗ 0 ˆ I P ˆ P I P 0 (i) Πµµ0 |Jναi = h DI (gi )µµ0 P(gi ) |Jναi = h i DI (gi )µµ0 DJ (gi )ν ν |Jν αi i | {z } ν0 | {z } P 0 δ δ δ ν0 DJ (gi )ν0ν |Jν αi IJ µν0 µ0ν ˆ I ˆ J nI P nJ P ∗ ∗ ˆ ˆ (ii) Πµµ0 Πνν0 = h h DI (gi )µµ0 DJ (gj )νν0 P(gi ) P(gj ) subst. gi gj = gk i j nI P nJ P ∗ −1 ∗ ˆ = h h DI (gi )µµ0 DJ (gi gk )νν0 P(gk ) i k nJ P nI P ∗ −1 ∗ ∗ ˆ = h h DI (gi )µµ0 DJ (gi )νλ DJ (gk )λν0 P(gk ) kλ i | {z } DJ (gi )λν | {z } [or use (i)] δIJ δµλδµ0ν P ˆ I P 1 P P ∗ ˆ P ˆ ˆ 1 (iii) Πµµ = h µDI (gi )µµ0 nI P(gi ) = δgi e P(gi ) = P(e) ≡ I µ i I | {z } |{z} i χ∗(g ) χ (e) I i I Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Decomposition into Invariant Subspaces (cont’d)

ˆ I Pˆ I nI P ∗ ˆ I Let Π ≡ Πµµ = h χI (gi ) P(gi ): µ i I P • Πˆ |ψi = cI να |I ναi να I • Πˆ projects |ψi on the invariant subspace SI (IR ΓI )

I For functions ψ(r) ≡ hr|ψi:

ˆ I nI P ∗ −1 we need not know Πµµ0 ψ(r) = h DI (gi )µµ0 ψ(gi r) the expansion (*) i

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Irreducible Invariant Subspaces (cont’d) Example: C e i e = identity i I Group Ci = {e, i} i = inversion e e i i i e

Ci e i

I character table Γ1 1 1 Γ2 1 −1

I Pˆ(e) ψ(x) = ψ(x), Pˆ(i) ψ(x) = ψ(−x)

ˆ I nI P ∗ ˆ I Projection operator Π = h χI (gi ) P(gi ) with nI = 1, h = 2 i

ˆ 1 1 ˆ ˆ ˆ 1 1 I Π = 2 [P(e) + P(i)] ⇒ Π ψ(x) = 2 [ψ(x) + ψ(−x)] even part ˆ 2 1 ˆ ˆ ˆ 2 1 Π = 2 [P(e) − P(i)] ⇒ Π ψ(x) = 2 [ψ(x) − ψ(−x)] odd part

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Product Representations

I Let {|I µi : µ = 1,... nI } and {|Jνi : ν = 1,... nJ } denote basis functions for invariant subspaces SI and SJ (need not be irreducible) Consider the product functions

{|I µi |Jνi : µ = 1,..., nI ; ν = 1,..., nJ }. How do these functions transform under G?

I Deﬁnition: Let DI (g) and DJ (g) be representation matrices for g ∈ G.

The direct product (Kronecker product) DI (g) ⊗ DJ (g) denotes the matrix whose elements in row (µν) and column (µ0ν0) are given by 0 µ, µ = 1,..., nI [DI (g) ⊗ DJ (g)]µν,µ0ν0 = DI (g)µµ0 DJ (g)νν0 0 ν, ν = 1,..., nJ x11 x12 y11 y12 I Example: Let DI (g) = and DJ (g) = x21 x22 y21 y22   x11y11 x11y12 x12y11 x12y12 x11 DJ (g) x12 DJ (g)  x11y21 x11y22 x12y21 x12y22  DI (g) ⊗ DJ (g) = =   x21 DJ (g) x22 DJ (g)  x21y11 x21y12 x22y11 x22y12  x21y21 x21y22 x22y21 x22y22

I Details of the arrangement in the following not relevant Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Product Representations (cont’d)

I Dimension of product matrix

dim [DI (g) ⊗ DJ (g)] = dim DI (g) dim DJ (g)

I Let ΓI = {DI (gi )} and ΓJ = {DJ (gi )} be representations of G. Then

ΓI × ΓJ ≡ {DI (g) ⊗ DJ (g)} is a representation of G called product representation.

I ΓI × ΓJ is, indeed, a representation:

Let DI (gi ) DI (gj ) = DI (gk ) and DJ (gi ) DJ (gj ) = DJ (gk ) ⇒ [D (g ) ⊗ D (g )] [D (g ) ⊗ D (g )] I i J i I j J j µν,µ0ν0 P = DI (gi )µκ DJ (gi )νλ DI (gj )κµ0 DJ (gj )λν0 κλ → DJ (gk ) 0 → DI (gk )µµ0 νν

= [DI (gk ) ⊗ DJ (gk )]µν,µ0ν0

I The characters of the product representation are χI ×J (gi ) = χI (gi ) χJ (gi )

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Decomposing Product Representations

I Let ΓI = {DI (gi )} and ΓJ = {DJ (gi )} be irreducible representations of G

The product representation ΓI × ΓJ = {DI ×J (gi )} is generally reducible

I According to Theorem 7, we have N˜ h Γ × Γ = P aIJ Γ where aIJ = P k χ∗ (C ) χ (C ) I J K K K h K k I ×J k K k=1 | {z } =χI (Ck ) χJ (Ck )

I The multiplication table for the ΓI × ΓJ Γ1 Γ2 ... irreducible representations ΓI of G Γ1 lists P aIJ Γ Γ k K K .2 .

I Example: Permutation group P3

χ(C) e a, b c, d, f ΓI × ΓJ Γ1 Γ2 Γ3 Γ1 1 1 1 Γ1 Γ1 Γ2 Γ3 Γ2 1 1 −1 Γ2 Γ1 Γ3 Γ3 2 −1 0 Γ3 Γ1 + Γ2 + Γ3

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 (Anti-) Symmetrized Product Representations

Let {|σµi} and {|τν i} be two sets of basis functions for the same n-dim. (again: need not be representation Γ = {D(g)} with characters {χ(g)}. irreducible)

(1) “Simple” Product: (discussed previously)

2 I Character tr[D(g) ⊗ D(g)] = χ (g)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 (Anti-) Symmetrized Product Representations (cont’d)

(2) Symmetrized Product: µ = 1,..., n o s 1 total: 1 n(n + 1) I |ψµν i = 2 (|σµi|τν i + |σν i|τµi), ν = 1, . . . µ 2 n n ˆ s 1 P P I P(g)|ψµν i = 2 Dµ0µDν0ν (|σµi|τν i + |σν i|τµi) µ0=1 ν0=1 n µ0−1 P P s s = (Dµ0µDν0ν + Dµ0ν Dν0µ)|ψµ0ν0 i + Dµ0µDµ0ν |ψµ0µ0 i µ0=1 ν0=1 n µ0 P P (s) s ≡ [D(g) ⊗ D(g)]µ0ν0,µν |ψµ0ν0 i µ0=1 ν0=1 n µ−1 (s) P P I tr[D(g) ⊗ D(g)] = (DµµDνν + Dµν Dνµ) + DµµDµµ µ=1 ν=1 n n 1 P P = 2 [Dµµ(g)Dνν (g) + Dµν (g)Dνµ(g)] µ=1 ν=1 n n 1 P P 2 = 2 Dµµ(g) Dνν (g) + Dµµ(g ) µ=1 ν=1 1 2 2 = 2 [χ(g) + χ(g )]

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 (Anti-) Symmetrized Product Representations (cont’d)

(3) Antisymmetrized Product: µ = 1,..., n o a 1 total: 1 n(n − 1) I |ψµν i = 2 (|σµi|τν i − |σν i|τµi), ν = 1, . . . µ − 1 2 n n ˆ a 1 P P I P(g)|ψµν i = 2 Dµ0µDν0ν (|σµi|τν i − |σν i|τµi) µ0=1 ν0=1 n µ0−1 P P a = (Dµ0µDν0ν − Dµ0ν Dν0µ)|ψµ0ν0 i µ0=1 ν0=1 n µ0−1 P P (a) a ≡ [D(g) ⊗ D(g)]µ0ν0,µν |ψµ0ν0 i µ0=1 ν0=1 n µ−1 (a) P P I tr[D(g) ⊗ D(g)] = (DµµDνν − Dµν Dνµ) µ=1 ν=1 n n 1 P P = 2 [Dµµ(g)Dνν (g) − Dµν (g)Dνµ(g)] µ=1 ν=1 n n 1 P P 2 = 2 Dµµ(g) Dνν (g) − Dµµ(g ) µ=1 ν=1 1 2 2 = 2 [χ(g) − χ(g )]

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Intermezzo: Material Tensors to be added ...

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Discussion

I Representation – Vector Space The matrices {D(gi )} of an n-dimensional (reducible or irreducible) representation describe a linear mapping of a vector space V onto itself. D(gi ) 0 0 P u = (u1,..., un) ∈ V : u −−−→ u ∈ V with uµ = D(gi )µν uν ν I Irreducible Representation (IR) – Invariant Subspace The decomposition of a reducible representation into IRs ΓI corresponds to a decomposition of the vector space V into invariant subspaces SI such that DI (gi ) SI −−−−→ SI ∀gi ∈ G (i.e., no mixing)

This decomposition of V lets us break down a big physical problem into smaller, more tractable problems

I Product Representation – Product Space A product representation ΓI × ΓJ describes a linear mapping of the product space SI × SJ onto itself DI ×J (gi ) SI × SJ −−−−−→ SI × SJ ∀gi ∈ G

P IJ I The block diagonalization ΓI × ΓJ = K aK ΓK corresponds to a decomposition of SI × SJ into invariant subspaces SK Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Discussion (cont’d) Clebsch-Gordan Coeﬃcients (CGC)

P IJ I The block diagonalization ΓI × ΓJ = K aK ΓK corresponds to a decomposition of SI × SJ into invariant subspaces SK ⇒ Change of Basis: unitary transformation IJ aK ( PP ` SI × SJ −→ SK K `=1 I J K` old basis {eµeν } −→ new basis {eκ }

K` P IJ K ` I J index ` not needed Thus eκ = eµ eν if often aIJ ≤ 1 µν µ ν κ K IJ K ` where µ ν κ = Clebsch-Gordan coeﬃcients (CGC)

Clebsch-Gordan coeﬃcients describe the unitary transformation for the ` decomposition of the product space SI × SJ into invariant subspaces SK

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Clebsch-Gordan Coeﬃcients (cont’d) Remarks

I CGC are independent of the group elements gi

I CGC are tabulated for all important groups (e.g., Koster, Edmonds)

I Note: Tabulated CGC refer to a particular deﬁnition (phase convention) I for the basis vectors {eµ} and representation matrices {DI (gi )}

I Clebsch-Gordan coeﬃcients C describe a unitary basis transformation C† C = C C† = 1

I Thus Theorem 13: Orthogonality and completeness of CGC ∗ X IJ K ` IJ K 0 `0 µ ν κ µ ν κ0 = δKK 0 δκκ0 δ``0 µν ∗ X IJ K ` IJ K ` µ ν κ µ0 ν0 κ = δµµ0 δνν0 K`κ

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Clebsch-Gordan Coeﬃcients (cont’d)

† Clebsch-Gordan coeﬃcients (1) I x J = C C block-diagonalize the representation matrices (unitary transformation) (2) = C† C

More explicitly:

Theorem 14: Reduction of Product Representation ΓI × ΓJ ∗ X X IJ K ` IJ K ` (1) DI (gi )µµ0 DJ (gi )νν0 = µ ν κ DK (gi )κκ0 µ0 ν0 κ0 K` κκ0

(2) DK (gi )κκ0 δKK 0 δ``0 ∗ X X IJ K ` IJ K 0 `0 = µ ν κ DI (gi )µµ0 DJ (gi )νν0 µ0 ν0 κ0 µµ0 νν0

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Evaluating Clebsch-Gordan Coeﬃcients

I A group G is called simply reducible if its product representations IJ ΓI × ΓJ contain the IRs ΓK only with multiplicities aK = 0 or 1. I For simply reducible groups( ⇒ no index `) according to Theorem 14 (1): nK X ∗ D (g ) 0 D (g ) 0 D (g ) 0 h I i µµ J i νν K i κ˜κ˜ i 0 0 ∗ X X IJ K IJ K nK X ∗ = D 0 (g ) 0 D (g ) 0 µ ν κ µ0 ν0 κ0 h K i κκ K i κ˜κ˜ K 0 κκ0 i ∗ IJ K IJ K | {z } = δ 0 δ δ 0 0 (Theorem 4) = µ ν κ˜ µ0 ν0 κ˜0 K K κκ˜ κ˜ κ 0 0 0 I Choose triple µ = µ = µ0, ν = ν = ν0, andκ ˜ =κ ˜ = κ0 such that LHS 6= 0 r IJ K nK X ∗ ⇒ = DI (gi )µ µ DJ (gi )ν ν D (gi )κ κ > 0 µ0 ν0 κ0 h 0 0 0 0 K 0 0 i Given the representation matrices {DI (g)}, the CGCs are unique for each triple I , J, K up to an overall phase that we choose such that IJ K > 0 µ0 ν0 κ0 1 nK X ⇒ IJ K = D (g ) D (g ) D∗ (g ) µ ν κ IJ K h I i µµ0 J i νν0 K i κκ0 i µ0 ν0 κ0 ∀µ, ν, κ IJ I If aK > 1: CGCs not unique ⇒ trickier! Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: CGC for group P3 ' C3v

IJ This group is simply reducible, aK ≤ 1, so we may drop the index `.

Here: For Γ3 use the representation matrices {D3(g)} corresponding to the basis functions x, y.

1 1 1 1 2 2 2 2 1 = = = 1 1 1 1 1 1 1 1 1 1

1 3 3 1 0 2 3 3 0 1 = = 0 1 −1 0 1 µ ν µν 1 µ ν µν √ √ 3 3 1 1/ 2 0 3 3 2 0 1/ 2 = √ = √ 0 1/ 2 −1/ 2 0 µ ν 1 µν µ ν 1 µν √ √ 3 3 3 1/ 2 0 3 3 3 0 −1/ 2 = √ = √ 0 −1/ 2 −1/ 2 0 µ ν 1 µν µ ν 2 µν

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Comparison: Rotation Group

I Angular momentum j = 0, 1/2, 1, 3/2,... corresponds to the irreducible representations of the rotation group

I For each j, these IRs are (2j + 1)-dimensional, i.e., the z component of angular momentum labels the basis states for the IR Γj .

I Γj=0 is the identity representation of the rotation group

I The product representation Γj1 × Γj2 corresponds to the addition of angular momenta j1 and j2;

Γj1 × Γj2 = Γ|j1−j2| + ... + Γj1+j2 Here all multiplicities aj1j2 are one. j3

I In our lecture, Clebsch-Gordan coeﬃcients have the same meaning as in the context of the rotation group: They describe the unitary transformation from the reducible product space to irreducible invariant subspaces. This unitary transformation depends only on (the representation matrices of) the IRs of the symmetry group of the problem so that the CGC can be tabulated. Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symmetry of Observables

I Consider Hermitian operator (observable) Oˆ.

Let G = {gi } be a group of symmetry transformations with {Pˆ(gi )} the group of unitary operators isomorphic to G. • For arbitrary |φi we have |ψi = O|ˆ φi. 0 0 • Application of gi gives |ψ i = Pˆ(gi )|ψi and |φ i = Pˆ(gi )|φi. 0 0 0 0 −1 • Thus |ψ i = Oˆ |φ i requires Oˆ = Pˆ(gi ) Oˆ Pˆ(gi )

0 −1 If Oˆ = Pˆ(gi ) Oˆ Pˆ(gi ) = Oˆ ⇔ [Pˆ(gi ), Oˆ] = 0 ∀gi ∈ G we call G the symmetry group of Oˆ which leaves Oˆ invariant. Of course, we want the largest G possible.

I Lemma: If |ni is an eigenstate of Oˆ, i.e., Oˆ |ni = λn |ni, and [Pˆ(gi ), Oˆ] = 0, then Pˆ(gi ) |ni is likewise an eigenstate of Oˆ for the same eigenvalue λn.

As always Pˆ(gi ) |ni need not be orthogonal to |ni.

Proof: Oˆ [Pˆ(gi ) |ni] = Pˆ(gi ) Oˆ |ni = λn [Pˆ(gi ) |ni]

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symmetry of Observables (cont’d)

I Theorem 15: Let G = {Pˆ(gi )} be the symmetry group of the observable Oˆ. Then the eigenstates of a d-fold degenerate eigenvalue λn of Oˆ form a d-dimensional invariant subspace Sn. The proof follows immediately from the preceding lemma.

I Most often: Sn is irreducible • central property of nature for applying group theory to physics problems • unless noted otherwise, always assumed in the following • Identify d-fold degeneracy of λn with d-dimensional IR of G.

I Under which cirumstances can Sn be reducible? 0 • G does not include all symmetries realized in the system, i.e., G $ G 0 (“hidden symmetry”). Then Sn is an irreducible invariant subspace of G . Examples: hydrogen atom, m-dimensional harmonic oscillator (m > 1). • A variant of the preceding case: The extra degeneracy is caused by the antiunitary time reversal symmetry (more later). • The degeneracy cannot be explained by symmetry: rare! (Usually such “accidental degeneracies” correspond to singular points

in the parameter space of a system.) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symmetry of Observables (cont’d) Remarks:

I IRs of G give the degeneracies that may occur in the spectrum of observable Oˆ.

I Usually, all IRs of G are realized in the spectrum of observable Oˆ (reasonable if eigenfunctions of Oˆ form complete set)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Application 1: Symmetry-Adapted Basis Let Oˆ = Hˆ = Hamiltonian

I Classify the eigenvalues and eigenstates of Hˆ according to the IRs ΓI of the symmetry group G of Hˆ . α: distinguish diﬀerent Notation: Hˆ |I µ, αi = EI α |I µ, αi µ = 1,..., nI levels transforming according to same ΓI

If ΓI is nI -dimensional, then eigenvalues EI α are nI -fold degenerate. Note: In general, the “quantum number” I cannot be associated directly with an observable.

I For given EI α, it suﬃces to calculate one eigenstate |I µ0, αi. Then

{|I µ, αi : µ = 1,..., nI } = {Pˆ(gi ) |I µ0, αi : gi ∈ G} (i.e., both sets span the same subspace of H)

J = 1,..., N; I Expand eigenstates |I µ, αi n o |Jν, βi : ν = 1,..., nJ ; in a symmetry-adapted basis β = 1, 2,... |I µ, αi = P hJν, β|I µ, αi |Jν, βi = PhI α||I βi |I µ, βi Jν,β | {z } β =δIJ δµν hI α||I βi see Theorem 11 ⇒ partial diagonalization of Hˆ independent of speciﬁc details

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Application 2: Eﬀect of Perturbations

(0) I Let Hˆ = Hˆ0 + Hˆ1, Hˆ0 = unperturbed Hamiltonian: Hˆ0|ni = En |ni

Hˆ1 = perturbation ˆ 0 2 (0) ˆ P |hn|H1|n i| I Perturbation expansion En = En + hn|H1|ni + + ... 0 (0) (0) n 6=n En − En0 ˆ 0 (0) ˆ 0 ⇒ need matrix elements hn|H|n i = En δnn0 + hn|H1|n i

I Let G0 = symmetry group of Hˆ0 o usually G $ G0 G = symmetry group of Hˆ 0 I The unperturbed eigenkets {|ni} transform according to IRs ΓI of G0 0 I {ΓI } are also representations of G, yet then reducible 0 I Every IR ΓI of G0 breaks down into (usually multiple) IRs {ΓJ } of G 0 P ΓI = aJ ΓJ (see Theorem 7) J ⇒ compatibility relations for irreducible representations 0 0 0 0 0 0 I Theorem 16: hn = Jµα|Hˆ |n = J µ α i = δJJ0 δµµ0 hJα||Hˆ ||J α i ˆ ˆ † ˆ ˆ 1 P ˆ † ˆ ˆ Proof: Similar to Theorem 11 with H = P(gj ) H P(gj ) = h i P(gi ) H P(gi ) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Compatibility Relations for C3v 'P3

I Character table C3v 'P3 C3v E 2C3 3σv

P3 e a, b c, d, f

Γ1 1 1 1 Γ2 1 1 −1 Γ3 2 −1 0

2 −1 I C3v 'P3 has two subgroups C3 = {E, C3, C3 = C3 }' G1 = {e, a, b} Cs = {E, σv }' G2 = {e, c} = {e, d} = {e, f }

I Both subgroups are Abelian, so they have only 1-dim. IRs 2 2 C3 EC3 C3 C3 EC3 C3 Cs E σi Cs E σi 2 E EC3 C3 Γ1 1 1 1 E E σi Γ1 1 1 2 ∗ C3 C3 C3 E Γ2 1 ω ω σi σi E Γ2 1 −1 2 2 ∗ C3 C3 EC3 Γ3 1 ω ω 2πi/3 C 3 C 3v C s ω≡e Γ Γ 1 2 Γ 2 2−fold degen. Γ Γ 2 I compatibility relations 3 Γ Γ 2 C P Γ Γ Γ Γ 2 nondegen. 3v 3 1 2 3 1 Γ C3 G1 Γ1 Γ1 Γ2 + Γ3 Γ 2 nondegen. Γ 1 1 Cs G2 Γ1 Γ2 Γ1 + Γ2 Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Discussion: Compatibility Relations

Compatibility relations and Theorem 16 tell us how a degenerate level 0 transforming according to the IR ΓI of G0 splits into multiple levels transforming according to certain IRs {ΓJ } of G when the perturbation Hˆ1 reduces the symmetry from G0 to G $ G0. Thus qualitative statements:

I Which degenerate levels split because of Hˆ1?

I Which degeneracies remain unaﬀected by Hˆ1?

I These statements do not require any perturbation theory in the conventional sense. (For every pair G0 and G, they can be tabulated once and forever!)

I These statements do not require some kind of “smallness” of Hˆ1.

I But no statement whether (or how much) a level will be raised or lowered by Hˆ1.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 (Ir)Reducible Operators

I Up to now: symmetry group of operator Oˆ requires −1 Pˆ(gi ) Oˆ Pˆ(gi ) = Oˆ ∀gi ∈ G

I More general: A set of operators {Qˆν : ν = 1,..., n} with

n −1 P ∀ ν = 1,..., n Pˆ(gi ) Qˆν Pˆ(gi ) = D(gi )µν Qˆµ µ=1 ∀ gi ∈ G

is called reducible (irreducible), if Γ = {D(gi ): gi ∈ G} is a reducible (irreducible) representation of G. −1 Often a shorthand notation is used: gi Qˆν ≡ Pˆ(gi ) Qˆν Pˆ(gi )

I We say: The operators {Qˆν } transform according to Γ.

I Note: In general, the eigenstates of {Qˆν } will not transform according to Γ.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 (Ir)Reducible Operators (cont’d) Examples:

I Γ1 = “identity representation”; D(gi ) = 1 ∀gi ∈ G; nI = 1 −1 ⇒ Pˆ(gi ) Qˆ Pˆ(gi ) = Qˆ ∀gi ∈ G We say: Qˆ is a scalar operator or invariant.

I most important scalar operator: the Hamiltonian Hˆ i.e., Hˆ always transforms according to Γ1 The symmetry group of Hˆ is the largest symmetry group that leaves Hˆ invariant.

I position operatorx ˆν ν = 1, 2, 3 (polar vectors) momentum operatorp ˆν = −i~ ∂xν

⇒ {xˆν } and {pˆν } transform according to 3-dim. representation Γpol (possibly reducible!)

I composite operators (= tensor operators) P e.g., angular momentum ˆlν = ελµν xˆλpˆµ ν = 1, 2, 3 (axial vector) λ,µ Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Tensor Operators

ˆ I ˆI I Let Q ≡ {Qµ : µ = 1,..., nI } transform according to ΓI = {DI (gi )} ˆ J ˆJ Q ≡ {Qν : ν = 1,..., nJ } transform according to ΓJ = {DJ (gi )}

Then QˆI QˆJ : µ = 1,..., nI transforms according to the product I µ ν ν = 1,..., nJ representation ΓI × ΓJ

I ΓI × ΓJ is, in general, reducible ˆI ˆJ ⇒ The set of tensor operators Qµ Qν is likewise reducible

I A unitary transformation brings ΓI × ΓJ = {DI (gi ) ⊗ DJ (gi )} into block-diagonal form ˆI ˆJ ⇒ The same transformation decomposes Qµ Qν into irreducible tensor operators (use CGC)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Where Are We?

We have discussed

I the transformational properties of states

I the transformational properties of operators

Now:

I the transformational properties of matrix elements ⇒ Wigner-Eckart Theorem

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Wigner-Eckart Theorem

Let {|I µ, αi : µ = 1,..., nI } transform according to ΓI = {DI (gi )} 0 0 0 0 {|I µ , α i : µ = 1,..., nI 0 } transform according to ΓI 0 = {DI 0 (gi )} ˆ J ˆJ Q = {Qν : ν = 1,..., nJ } transform according to ΓJ = {DJ (gi )}

X 0 Then hI 0µ0, α0 | QˆJ | I µ, αi = JI I ` hI 0α0 || QˆJ || I αi ν ν µ µ0 ` ` 0 0 ˆJ where the reduced matrix element hI α || Q || I αi` is independent of µ, µ0 and ν.

Proof:

J µ = 1,..., nI Qˆ |I µ, αi : transforms according to ΓI × ΓJ I ν ν = 1,..., nJ ˆJ P JI K ` I Thus CGC expansion Qν |I µ, αi = |Kκ, `i Kκ,` ν µ κ 0 0 0 ˆJ P JI K ` 0 0 0 I hI µ , α | Qν | I µ, αi = ν µ κ hI µ , α | Kκ, `i Theorem 11 Kκ,` | {z } 0 0 ˆJ ≡ δI 0K δµ0κ hI α || Q || I αi`

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Discussion: Wigner-Eckart Theorem

I Matrix elements factorize into two terms • the reduced matrix element independent of µ, µ0 and ν 0 • CGC indexing the elements µ, µ and ν of ΓI ,ΓI 0 and ΓJ . (CGC are tabulated, independent of Qˆ J )

I Thus: reduced matrix element = “physics” Clebsch-Gordan coeﬃcients = “geometry”

0 I Matrix elements for diﬀerent values of µ, µ and ν have a ﬁxed ratio independent of Qˆ J

∗ Equivalent to: If ΓI 0 × ΓJ × ΓI does not I If ΓI 0 is not contained in ΓI × ΓJ contain the identity representation 0 JI I ` 0 ⇒ ν µ µ0 = 0 ∀ ν, µ, µ

0 0 0 ˆJ 0 ⇒h I µ , α | Qν | I µ, αi = 0 ∀ ν, µ, µ Many important selection rules are some variation of this result.

I Theorems 11 and 16 are special cases of the WE theorem for Qˆ1 = 1 and Qˆ1 = Hˆ (yet we proved the WE theorem via Theorem 11)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Discussion: Wigner-Eckart Theorem (cont’d)

Application: Perturbation theory

I Compatibility relations and Theorem 16 describe splitting of degenerate levels using the symmetry group G of perturbed problem

I Alternative approach Splitting of levels using the symmetry group G0 of unperturbed problem (i.e., no need to know group G of perturbed problem) J • Let Qˆ be tensor operator transforming according to IR ΓJ of G0 ˆ J ˆ J J ˆJ ˆ • Often: perturbation H1 = F · Q = Fν Qν i.e., H1 is proportional to only νth component of tensor operator Qˆ J ˆ J ˆJ J Q projected on component Qν via suitable orientation of ﬁeld F ˆ ˆ ˆ ˆJ • Symmetry group of H = H0 + H1 is subgroup G ⊂ G0 which leaves Qν invariant. • WE Theorem: 0 ˆ 0 J ˆJ 0 0 0 0 P JI I ` ˆJ 0 0 hn|H1|n i = Fν hn=I µα|Qν |n =I µ α i = 0 hI α || Q || I α i` (∗) ` ν µ µ • Changing the orientation of F J changes only the CGCs in (∗) ˆJ 0 0 The reduced matrix elements hI α || Q || I α i` are “universal” Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Optical Selection Rules

Example: Optical transitions for a system with symmetry group C3v (e.g., NH3 molecule)

I Optical matrix elements hiI |e · ˆr|fJ i (dipole approximation)

where |iI i = initial state (with IR ΓI ); |fJ i = ﬁnal state (IR ΓJ ) e = (ex , ey , ez ) = polarization vector ˆr = (ˆx, yˆ, zˆ) = dipole operator (≡ position operator)

I xˆ, yˆ transform according to Γ3 zˆ transforms according to Γ1

I e.g., light xy polarized: hi1|ex xˆ + ey yˆ|f3i • transition allowed because Γ3 × Γ3 = Γ1 + Γ2 + Γ3 • in total 4 diﬀerent matrix elements, but only one reduced matrix element

I z polarized: hi1|ez zˆ|f3i • transition forbidden because Γ1 × Γ3 = Γ3

I These results are independent of any microscopic models for the NH3 molecule! Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Goal: Spin 1/2 Systems and Double Groups Rotations and Euler Angles

I So far: transformation of functions and operators dependent on position

I Now: systems with spin degree of freedom ⇒ wave functions are two-component Pauli spinors ψ↑(r) Ψ(r) = ψ↑(r) |↑i + ψ↓(r) |↓i ≡ ψ↓(r) I How do Pauli spinors transform under symmetry operations?

I Parameterize rotations via Euler angles α, β, γ

z z z z z’ z z y’ z’ y’ z’ y’’ y’ y’ y’ y β y β y x’’’ x’’ γ x’’ x’ x’ x’ α α α x x x axis z, angle α axis y’, angle β axis z’, angle γ

I Thus general rotation R(α, β, γ) = Rz0 (γ) Ry 0 (β) Rz (α)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Rotations and Euler Angles (cont’d)

I General rotation R(α, β, γ) = Rz0 (γ) Ry 0 (β) Rz (α) 0 0 I Difficulty: axes y and z refer to rotated body axes (not fixed in space) −1 I Use Rz0 (γ)= Ry 0 (β) Rz (γ) Ry 0 (β) preceding rotations −1 are temporarily undone Ry 0 (β) = Rz (α) Ry (β) Rz (α) −1 I Thus R(α, β, γ) = Ry 0 (β) Rz (γ) Ry 0 (β) Ry 0 (β) Rz (α) | {z } | {z } −1 =1 rotations about Rz (α) Ry (β) Rz (α) z axis commute rotations about I Thus R(α, β, γ) = Rz (α) Ry (β) Rz (γ) space-fixed axes!

3 I More explicitly: rotations of vectors r = (x, y, z) ∈ R  cos α − sin α 0   cos β 0 − sin β  Rz (α) = sin α cos α 0  , Ry (β) = 0 1 0  etc. 0 0 1 sin β 0 cos β

I SO(3) = set of all rotation matrices R(α, β, γ) = set of all orthogonal 3 × 3 matrices R with det R = +1.

I R(2π, 0, 0) = R(0, 2π, 0) = R(0, 0, 2π) = 1 ≡ e

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Rotations: Spin 1/2 Systems

I Rotation matrices for spin-1/2 spinors (axis n) i 1 Rn(φ) = exp − 2 σ · n φ = cos(φ/2) − in · σ sin(φ/2)

I R(α, β, γ) = Rz (α) Ry (β) Rz (γ) e−i(α+γ)/2 cos(β/2) −e−i(α−γ)/2 sin(β/2) = e−i(α−γ)/2 sin(β/2) ei(α+γ)/2 cos(β/2) transformation matrix for spin 1/2 states

I SU(2) = set of all matrices R(α, β, γ) = set of all unitary 2 × 2 matrices R with det R = +1. 1 rotation by 2π I R(2π, 0, 0) = R(0, 2π, 0) = R(0, 0, 2π) = − ≡ e¯ is not identity 1 rotation by 4π I R(4π, 0, 0) = R(0, 4π, 0) = R(0, 0, 4π) = = e is identity

I Every SO(3) matrix R(α, β, γ) corresponds to two SU(2) matrices R(α, β, γ) and R(α + 2π, β, γ) = R(α, β + 2π, γ) = R(α, β, γ + 2π) =e ¯ R(α, β, γ) = R(α, β, γ)e ¯ ⇒ SU(2) is called double group for SO(3) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Double Groups

I Deﬁnition: Double Group Let the group of spatial symmetry transformations of a system be

G = {gi = R(αi , βi , γi ): i = 1,..., h} ⊂ SO(3) Then the corresponding double group is

Gd = {gi = R(αi , βi , γi ): i = 1,..., h} ∪ {gi = R(αi + 2π, βi , γi ): i = 1,..., h} ⊂ SU(2)

I Thus with every element gi ∈ G we associate two elements gi and g¯i ≡ e¯ gi = gi e¯ ∈ Gd

I If the order of G is h, then the order of Gd is 2h.

I Note: G is not a subgroup of Gd because the elements of G are not a closed subset of Gd . Example: Let g = rotation by π • in G: g 2 = e the same group element g is thus 2 • in Gd : g =e ¯ interpreted diﬀerently in G and Gd

I Yet: {e, e¯} is invariant subgroup of Gd and the factor group Gd /{e, e¯} is isomorphic to G.

⇒ The IRs of G are also IRs of Gd Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Double Group C3v

C3v E E¯ 2C3 2C¯3 3σv 3¯σv

Γ1 1 1 1 1 1 1 Γ2 1 1 1 1 −1 −1 Γ3 2 2 −1 −1 0 0

Γ4 2 −2 1 −1 0 0 Γ5 1 −1 −1 1 i −i Γ6 1 −1 −1 1 −i i

I For Γ1,Γ2, and Γ3 the “barred” group elements have the same characters as the “unbarred” elements. Here the double group gives us the same IRs as the “single group”

I For other groups a class may contain both “barred” and “unbarred” group elements. ⇒ the number of classes and IRs in the double group need not be twice the number of classes and IRs of the “single group”

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Time Reversal (Reversal of Motion)

I Time reversal operator θˆ : t → −t

−1 I Action of θˆ: θˆˆr θˆ = ˆr independent of t θˆpˆ θˆ−1 = −pˆ   θˆLˆ θˆ−1 = −Lˆ linear in t  θˆSˆ θˆ−1 = −Sˆ

ˆ ˆ I Consider time evolution: U(δt) = 1 − iH δt/~ ⇒ Uˆ(δt) θˆ|ψi = θˆUˆ(−δt) |ψi ⇔ −iHˆ θˆ|ψi = θˆiHˆ |ψi but need also [θ,ˆ Hˆ ] = 0

The explicit form of θˆ depends on the representation −1 ∗ I position representation: θˆˆr θˆ = ˆr ⇒ θˆ ψ(r) = ψ (r) −1 ∗ I momentum representation: θˆpˆ θˆ = −pˆ ⇒ θˆ ψ(p) = ψ (−p)

I spin 1/2 systems: 2 • θˆ = iσy K ⇒ θˆ = −1 • all eigenstates |ni of Hˆ are at least two-fold degenerate (Kramers degeneracy)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Time Reversal and Group Theory

I Consider a system with Hamiltonian Hˆ .

I Let G = {gi } be the symmetry group of spatial symmetries of Hˆ

[Pˆ(gi ), Hˆ ] = 0 ∀gi ∈ G

I Let {|I νi : ν = 1,..., nI } be an nI -fold degenerate eigenspace of Hˆ which transforms according to IR ΓI = {DI (gi )}

Hˆ |I νi = EI |I νi ∀ν P Pˆ(gi ) |I νi = DI (gi )µν |I µi µ

I Let Hˆ be time-reversal invariant: [Hˆ , θˆ] = 0 ⇒ θˆ is additional symmetry operator (beyond {Pˆ(gi )}) with

[θ,ˆ Pˆ(gi )] = 0 ˆ ˆ ˆ ˆ ˆP P ∗ ˆ I P(gi ) θ |I νi = θ P(gi )|I νi = θ DI (gi )µν |I µi = DI (gi )µν θ |I µi µ µ

I Thus: time-reversed states {θˆ|I νi} transform according to ∗ ∗ complex conjugate IR ΓI = {DI (gi )} Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Time Reversal and Group Theory (cont’d)

I time-reversed states {θˆ|I νi} transform according to ∗ ∗ complex conjugate IR ΓI = {DI (gi )} Three possiblities (known as “cases a, b, and c”) (a) {|I νi} and {θˆ|I νi} are linear dependent

(b) {|I νi} and {θˆ|I νi} are linear independent ∗ ∗ The IRs ΓI and ΓI are distinct, i.e., χI (gi ) 6= χI (gi ) (c) {|I νi} and {θˆ|I νi} are linear independent ∗ ∗ ΓI = ΓI , i.e., χI (gi ) = χI (gi ) ∀ gi Discussion I Case (a): time reversal is additional constraint for {|I νi} e.g., nI = 1 ⇒ |νi reell

I Cases (b) and (c): time reversal results in additional degeneracies

I Our definition of cases (a)–(c) follows Bir & Pikus. Often (e.g., Koster) a different classification is used which agrees with Bir & Pikus for spinless systems. But cases (a) and (c) are reversed for spin-1/2 systems.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Time Reversal and Group Theory (cont’d)

I When do we have case (a), (b), or (c)?

Criterion by Frobenius & Schur  η case (a) 1  P χ (g 2) = 0 case (b) h I i i  −η case (c)

+1 systems with integer spin where η = −1 systems with half-integer spin

Proof: Tricky! (See, e.g., Bir & Pikus)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Cyclic Group C3

2 3 I C3 is Abelian group with 3 elements: C3 = {q, q , q ≡ e}

I Multiplication table Character table C e q q2 2 time 3 C3 e q q reversal 2 e e q q Γ 1 1 1 a 2 1 q q q e Γ 1 ω ω∗ b 2 2 2 q q e q ∗ Γ3 1 ω ω b ω≡e2πi/3

I IR Γ1: no additional degeneracies because of time reversal

I IRs Γ2 and Γ3: these complex IRs need to be combined ⇒ two-fold degeneracy because of time reversal symmetry (though here no spin!)

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Solid State Physics

First: Some terminology

I Lattice: periodic array of atoms (or groups of atoms)

I Bravais lattice: 3 n = (n1, n2, n3) ∈ Z Rn = nx ax + ny ay + nz az ai linearly independent

Every lattice site Rn is occupied with one atom Example: 2D honeycomb lattice is not a Bravais lattice

I Lattice with basis:

• Every lattice site Rn is occupied with z atoms

• Position of atoms relative to Rn: τ i , i = 1,..., q

• These q atoms with relative positions τ i form a basis. • Example: two neighboring atoms in 2D honeycomb lattice

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symmetry Operations of Lattice

I Translation t (not necessarily by lattice vectors Rn)

I Rotation, inversion → 3 × 3 matrices α

I Combinations of translation, rotation, and inversion ⇒ general transformation for position vector r ∈ R3: r0 = αr + t ≡ {α|t} r

I Notation {α|t} includes also • Mirror reflection = rotation by π about axis perpendicular to mirror plane followed by inversion • Glide reflection = translation followed by reflection • Screw axis = translation followed by rotation Symmetry operations {α|t} form a group 0 0 0 0 0 0 0 0 I Multiplication {α |t }{α|t} r = α r + t = α αr + α t + t | {z } 0 0 0 r0=αr+t = {α α|α t + t } r −1 −1 −1 I Inverse Element {α|t} = {−α | − α t} because {α|t}−1 {α|t} = {α−1α|α−1t − α−1t} = {1|0}

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Classiﬁcation Symmetry Groups of Crystals to be added ...

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symmetry Groups of Crystals Translation Group

Translation group = set of operations {1|Rn}

I {1|Rn0 }{1|Rn} = {1|Rn0 + 1 Rn} = {1|Rn0+n} ⇒ Abelian group

I associativity (trivial)

I identity element {1|0} = {1|R0}

−1 I inverse element {1|Rn} = {1| − Rn}

Translation group Abelian ⇒ only one-dimensional IRs

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Irreducible Representations of Translation Group

(for clarity in one spatial dimension)

I Consider translations {1|a} ˆ ˆ I Translation operator Ta = T{1|a} is unitary operator ⇒ eigenvalues have modulus 1 −iφ I eigenvalue equation Tˆa |φi = e |φi −π < φ ≤ π −inφ more generally Tˆna |φi = e |φi n ∈ Z −inφ I ⇒ representations D({1|Rna}) = e − π < φ ≤ π Physical Interpretation of φ hr−a| z }| { −iφ Consider hr | Tˆa | φi ⇒ hr − a|φi = e hr|φi | {z } e−iφ|φi Thus: Bloch Theorem (for φ = ka)

I The wave vector k (or φ = ka) labels the IRs of the translation group I The wave functions transforming according to the IR Γk are ikr Bloch functions hr|φi = e uk (r) with ik(r−a) ikr −ika e uk (r − a) = e uk (r)e or uk (r − a) = uk (r) Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Irreducible Representations of Space Groups to be added ...

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Luttinger (1956) Theory of Invariants Bir & Pikus Idea:

I Hamiltonian must be invariant under all symmetry transformations of the system

I Example: free particle 2 ˆ pˆ H X 4 H = Ekin = + c1Hpˆ + c2Xˆr ·Xpˆ + c4 pˆ + ... 2m |{z} | {z } | {z } |{z} not inversion breaks time scalar scalar symmetric reversal

I Crystalline solids:

Ekin = E(k) = kinetic energy of Bloch electron with crystal momentum p = ~k ⇒ dispersion E(k) must reﬂect crystal symmetry

2 3 E(k) = a0 + a1 k +a2 k + a3 k + ... |{z} | {z } only in crystals without inversion symmetry

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Theory of Invariants (cont’d)

More generally: E

I Bands E(k) at expansion point k0 n-fold degenerate conduction band valence band I Bands split for k 6= k 0 HH

I Example: GaAs (k0 = 0) LH

SO I Band structure E(k) for small k via diagonalization k of n × n matrix Hamiltonian H(k).

I Goal: Set up matrix Hamiltonian H(k) by exploiting the symmetry at expansion point k0

I Incorporate also perturbations such as • spin-orbit coupling (spin S) • electric ﬁeld E, magnetic ﬁeld B • strain ε • etc.

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Invariance Condition

I Consider n × n matrix Hamiltonian H(K)

I K = K (k, S, E, B, ε,...) = general tensor operator where k = wave vector E = electric field ε = strain field S = spin B = magnetic field etc.

I Basis functions {ψν (r): ν = 1,..., n} transform according to representation Γψ = {Dψ(gi )} of group G. (Γψ does not have to be IR)

I Symmetry transformation gi ∈ G applied to tensor K −1 K → gi K ≡ Pˆ(gi ) K Pˆ(gi )

⇒ H(K) → H(gi K)

−1 I Equivalent to inverse transformation gi applied to ψν (r): P −1 ψν (r) → ψν (gi r) = Dψ(gi )µν ψµ(r) µ −1 ⇒ H(K) → Dψ(gi ) H(K) Dψ(gi )

−1 I Thus Dψ(gi ) H(gi K) Dψ(gi ) = H(K) ∀ gi ∈ G really n2 equations! Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Invariance Condition (cont’d) Remarks

I If −k0 is in the same star as the expansion point k0, additional constraints arise from time reversal symmetry.

in particular: −k0 = k0 = 0

I If Γψ is reducible, the invariance condition can be applied to each “irreducible block” of H(K) (see below).

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Invariant Expansion

Expand H(K) in terms of irreducible tensor operators and basis matrices

J I Decompose tensors K into irreducible tensors K transforming according to IR ΓJ of G J J P J Kν → gi Kν ≡ DJ (gi )µν Kµ µ 2 2 I n linearly independent basis matrices {Xq : q = 1,..., n } transforming as −1 P Xq → gi Xq ≡ Dψ(gi ) Xq Dψ(gi ) = DX (gi )pq Xp p with “expansion coeﬃcents” DX (gi )pq ∗ I Representation ΓX ' Γψ × Γψ is usually reducible.

We have IR Γψ for the ket basis functions of H and ∗ IR Γψ (i.e., the complex conjugate IR) for the bras 2 I ∗ ⇒ from {Xq : q = 1,..., n } form linear combinations Xν ∗ ∗ transforming according to the IRs ΓI occuring in Γψ × Γψ I ∗ I ∗ P ∗ I ∗ Xµ → gi Xµ = DI (gi )µν Xµ ν

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Invariant Expansion (cont’d)

I Consider most general expansion ∗ P P IJ I J IJ H(K) = bµν Xµ Kν bµν = expansion coeﬃcients IJ µν

I Transformations gi ∈ G: I ∗ P ∗ I ∗ J P J Xµ → DI (gi )µ0µ Xµ0 , Kν → DJ (gi )ν0ν Kν0 µ0 ν0

I Use invariance condition (must hold ∀ gi ∈ G)

P IJ I ∗ J P IJ P 1 P ∗ I ∗ J bµν Xµ Kν = bµν h DI (gi )µ0µ DJ (gi )ν0ν Xµ0 Kν0 µν µν µ0ν0 i | {z } δIJ δµ0ν0 δµν P II P I ∗ I = δIJ bµµ Xµ0 Kµ0 µ µ0 | {z } ≡aI

P P I ∗ I I Then H(K) = aI Xν Kν I ν

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Irreducible Tensor Operators

Construction of irreducible tensor operators K = K(k, S, E, B, ε)

I Components ki , Si , Ei , Bi , εij transform according to some IRs ΓI of G. ⇒ “elementary” irreducible tensor operators KI

I Construct higher-order irreducible tensor operators with CGC: If we have multiplicities aIJ > 1 K P IJ K ` I J K Kκ = µ ν κ Kµ Kν we get diﬀerent tensor operators µν for each value `

I I Irreducible tensor operators K are “universally valid” for any matrix Hamiltonian transforming according to G

I Yet: if for a particular matrix Hamiltonian H(K) with basis functions {ψν } transforming according to Γψ an IR ΓI does not appear in ∗ I Γψ × Γψ, then the tensor operators K may not appear in H(K).

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Basis Matrices

I In general, the basis functions {ψν (r): ν = 1,..., n} include E several irreducible representations ΓJ Γc ⇒ Decompose H(K) into nJ × nJ0 blocks HJJ0 (K), such that ∗ • rows transform according to IR ΓJ (dimension nJ ) • columns transform according to IR ΓJ0 (dimension nJ0 ) Γv   Hcc Hcv Hcv 0 Γv’ † 0 k H(K) =  Hcv Hvv Hvv  † † Hcv 0 Hvv 0 Hv 0v 0

I ∗ ∗ ∗ I Choose basis matrices Xν transforming according to the IRs ΓI in ΓJ × ΓJ0 I ∗ −1 I ∗ P ∗ I ∗ Xν → DJ (gi ) Xν DJ0 (gi ) = DI (gi )µν Xµ µ

0 ∗ I ∗ IJ J ` I More explicitly: (Xν )λµ = ν µ λ

I 0 which reﬂects the transformation rules for matrix elements hJλ|Kν |J µi

P P I ∗ I ⇒ For each block we get HJJ0 (K) = aI Xν Kν I ν Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Time Reversal

I time reversal θˆ connects expansion point k0 and −k0 ˆ I often: θ ψk0λ(r) and ψ−k0λ(r) linearly dependent

ˆ P 0 0 θ ψk0λ(r) = Tλλ ψ−k0λ (r) λ0 I thus additional condition ζ = +1 : K even under θ T −1 H(ζK) T = H∗(K) = Ht (K) ζ = −1 : K even under θ applicable in particular for k0 = −k0 = 0

I k0 6= −k0: often k0 and −k0 also connected by spatial symmetries R −1 −1 H−k0 (K) = D(R) Hk0 (R K) D (R). k y

−1 −1 ∗ t K’Γ K ⇒ T H (R K)T = H (ζK) = H (ζK) k x k0 k0 k0

graphene: k0 = K

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Example: Graphene

strictly speaking I Electron states at K point: point group C3v D3h = C3v + inversion

It must be Γ3 because I “Dirac cone”: IR Γ3 of C3v this is the only 2D IR of C3v

∗ ∗ I We have Γ3 × Γ3 = Γ1 + Γ2 + Γ3 (with Γ3 = Γ3) 1 1 2 3 3 ⇒ basis matrices X1 = ; X1 = σy ; X1 = σz , X2 = −σx

I Irreducible tensor operators K up to second order in k: 2 2 2 2 Γ1 : kx + ky Γ3 : kx , ky ; ky − kx , 2kx ky Γ2:[kx , ky ] ∝ Bz

I Hamiltonian here: basis functions |xi and |yi 2 2 1 2 2 H(K) = a31(kx σz − ky σx ) + a11(kx + ky ) + a32[(ky − kx )σz − 2kx ky σx ]

I More common: basis functions |x − iyi and |x + iyi 1 1 2 3 3 ⇒ basis matrices X1 = ; X1 = σz ; X1 = σx , X2 = σy 2 2 1 2 2 H(K) = a31(kx σx + ky σy ) + a11(kx + ky ) + a32[(ky − kx )σx + 2kx ky σy ]

I additional constraints for H(K) from time reversal symmetry Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Graphene: Basis Matrices (D3h)

Symmetrized matrices for the invariant expansion of the blocks Hαβ for the point group D3h.

Block Representations Symmetrized matrices H Γ∗ × Γ Γ : 1  55 5 5 1  = Γ1 + Γ2 + Γ6 Γ2 : σz no spin  Γ6 : σx , σy ∗ 1 H77 Γ7 × Γ7 Γ1 :   = Γ1 + Γ2 + Γ5 Γ2 : σz   Γ : σ , −σ  5 x y   H Γ∗ × Γ Γ : 1  99 9 9 1  = Γ1 + Γ2 + Γ3 + Γ4 Γ2 : σz with spin  Γ3 : σx   Γ4 : σy   ∗ 1  H79 Γ7 × Γ9 Γ5 : , −iσz   = Γ5 + Γ6 Γ6 : σx , σy 

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Graphene: Irreducible Tensor Operators (D3h) K Terms printed in bold give rise to invariants in H55(K) allowed by 1 time-reversal invariance. Notation: {A, B} ≡ 2 (AB + BA).

2 2 2 2 Γ1 1; kx + ky; {kx, 3ky − kx}; kx Ex + ky Ey ; xx + yy; (yy − xx)kx + 2xyky;(yy − xx )Ex + 2xy Ey ; sxBx + syBy; szBz; (sxky − sykx)Ez; sz(kxEy − kyEx); 2 2 Γ2 {ky , 3kx − ky }; Bz; kxEy − kyEx; (xx − yy )ky + 2xy ky ; (xx + yy)Bz; (xx − yy)Ey + 2xyEx; sz; sx By − sy Bx ;(sx kx + sy ky )Ez ; sz(xx + yy); Γ3 Bx kx + By ky ; Ex Bx + Ey By ; Ez Bz ;(yy − xx )Bx + 2xy By ; sx kx + sy ky ; sx Ex + sy Ey ; sz Ez ; sx (yy − xx ) + 2sy xy Γ4 Bx ky − By kx ; Ez ; Ex By − Ey Bx ;(xx − yy )By + 2xy Bx ; (xx + yy )Ez ; sx ky − sy kx ; sx Ey − sy Ex ; sy (xx − yy ) + 2sx xy Γ5 Bx , By ; By ky − Bx kx , Bx ky + By kx ; ky Ez , −kx Ez ; Ey By − Ex Bx , Ey Bx + Ex By ;(xx + yy )(Bx , By ); (xx − yy )Bx + 2xy By , (yy − xx )By + 2xy Bx ; 2xy Ez , (xx − yy )Ez ; sx , sy ; sy ky − sx kx , sx ky + sy kx ; sy Bz , −sx Bz ; sz By , −sz Bx ; sy Ey − sx Ex , sx Ey + sy Ex ;(sx , sy )(xx + yy ); sx (xx − yy ) − 2sy xy , sy (yy − xx ) − 2sx xy Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Graphene: Irreducible Tensor Operators (cont’d)

Γ6 kx, ky; {ky + kx, ky − kx}, 2{kx, ky}; 2 2 2 2 {kx, kx + ky}, {ky, kx + ky}; Bz ky , −Bz kx ; Ex , Ey ; ky Ey − kx Ex , kx Ey + ky Ex ; EyBz, −ExBz; EzBy, −EzBx; yy − xx, 2xy; (xx + yy)(kx, ky); (xx − yy)kx + 2xyky, (yy − xx)ky + 2xykx; 2xy Bz , (xx − yy )Bz ; (xx − yy )Ex + xy Ey , (yy − xx )Ey + xy Ex ; (xx + yy )(Ex , Ey ); sz ky , −sz kx ; syBy − sxBx, sxBy + syBx; szEy, −szEx; syEz, −sxEz; sz(kxEy + kyEx), sz(kxEx − kyEy); (sxky + sykx)Ez, (sxkx − syky)Ez; 2sz xy , sz (xx − yy );

Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Winkler and Z¨ulicke, Graphene: Full Hamiltonian PRB 82, 245313

H(K) = a61(kx σx + ky σy ) “Dirac term” 2 2 1 2 2 nonlinear + anisotropic + a12(kx + ky ) + a62[(ky − kx )σx + 2kx ky σy ] corrections + a22(kx Ey − ky Ex )σz orbital Rashba term +a21Bz σz orbital Zeeman term + a14(xx + yy )1 + a66[(yy − xx )σx + 2xy σy ] strain-induced terms +a15[(yy − xx )kx + 2xy ky ]1 isotropic velocity +a67(xx + yy )(kx σx + ky σy ) renormalization a68{[(xx − yy )kx + 2xy ky ]σx anisotropic velocity +[(yy − xx )ky + 2xy kx ]σy } renormalization +a23(xx + yy )Bz σz strain - orbital Zeeman +a24[(xx − yy )Ey + 2xy Ex ]σz strain - orbital Rashba + a21 sz σz intrinsic SO coupling + a61 sz (Ey σx − Ex σy ) + a62 Ez (sy σx − sx σy ) Rashba SO coupling +a63 sz [(kx Ey + ky Ex )σx + (kx Ex − ky Ey )σy ] +a64 Ez [(sx ky + sy kx )σx + (sx kx − sy ky )σy ] +a11 (sx ky − sy kx )Ez + a12 sz (kx Ey − ky Ex ) + a26(xx + yy )sz σz strain-mediated SO coupling Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Symbols G group U subgroup a, b, c,... group elements i, j, k,... indeces labeling group elements e unit element (= identity element) of a group h order of a group (= number of group elements) Ck classes of a group hk number of group elements in class Ck N˜ number of classes D(gi ) matrix representation for group element gi Γ = {D(gi )} (irreducible) representation I , J, K,... indeces labeling irreducible representations N number of irreducible representations µ, ν, λ, . . . indeces labeling the elements of representation matrices D(gi ) nI dimensionality of irreducible representation ΓI χI (gi ) character of representation matrix for group element gi IJ aK multiplicity with which ΓK is contained in ΓI × ΓJ H Hilbert space, multiband Hamiltonian SI invariant subspace (IR ΓI ) α α, β we may have multiple irreducible invariant subspaces SI for one IR ΓI Pˆ(gi ) unitary operator that realizes the symmetry element gi in the Hilbert space

IR irreducible representation Roland Winkler, NIU, Argonne, and NCTU 2011−2015