Representation theory & the Hubbard model

Transcription

1 Representation theory & the Hubbard model Simon Mayer March 17, 2015

2 Outline 1. The Hubbard model 2. Representation theory of the symmetric group S n 3. Representation theory of the special unitary group SU(n) 4. Ordering of energy levels

3 The Hubbard model General Hamiltonian H = ) t ij (c iσ c jσ + h.c. i,j,σ }{{} hopping term Simplifying assumptions + i i, j Λ, σ {1,..., N} U i (n i µ) 2 }{{} on-site interaction t ij = t for nearest neighbours, U i = U for all sites

4 Strong coupling limit Strong on-site interaction, weak (relative to on-site) hopping term: U t 1 Second order perturbation theory H eff = J i,j S i S j, J = 2t2 U (generalized) Heisenberg antiferromagnet, S i is spin operator at site i: usual spin operator for N = 2, generalized spin operator for N > 2

5 Representation theory of the symmetric group S n S n = group of permutations of {1,..., n} notation for σ S n σ = ( ) 1 2 n σ(1) σ(2) σ(n) or in cycle notation ( ) (15423), ( ) (132)(45) Conjugacy class of a S n Cl(a) = {b S n : g S n with b = gag 1 } determined by cycle structure (ν 1,..., ν n ), index marks length of cycle

6 Representation theory of the symmetric group S n Constraint ν 1 + 2ν nν n = n Switch variables λ 1 = ν ν n, λ 2 = ν ν n,..., λ n = ν n Then λ 1 λ n and λ λ n = n λ = (λ 1,..., λ n ) is a partition of n, represented by Young diagram Example: λ = (3, 2, 1, 1) represented by 1-1 correspondence between conjugacy class and Young diagram

7 Representation theory of the symmetric group S n Result: # of Young diagrams = # of conjugacy classes = # of inequivalent irreducible representations (irreps) of S n Attach to each distinct Young diagram λ a distinct irrep F λ of S n Young tabloid corresponding to a Young diagram is a decomposition of {1,..., n} into a union of disjoint sets with # of elements given by λ i. E.g. tabloid for λ = (3, 2, 1, 1) is {t} = {2, 3, 5}{1, 7}{4}{6} M λ : set of all tabloids corresponding to λ, #M λ = n! λ 1! λ n!.

8 Representation theory of the symmetric group S n Let S n act on M λ, get a representation of S n on F(M λ ), the space of functions on M λ Examples: λ = (n), λ = (n 1, 1), λ = (n 2, 2) Goal: To each λ corresponds unique new irrep F λ of F(M λ ); the space F(M λ ) decomposes into direct sum of irreps isomorphic to certain of the F µ with µ λ (with multiplicity) together with the one unique new rep F λ Each Young diagram determines an irrep of S n.

9 Representation theory of the symmetric group S n Young tableau t corresponding to λ: assignment of the numbers 1,..., n to each of the boxes of λ; order matters! Every Young tableau gives rise to a Young tabloid Example: Young diagram λ = (3, 2, 1, 1), Young tableau t = , Young tabloid {t} = {2, 3, 5}{1, 7}{4}{6} C t = subgroup of S n permuting the numbers in the columns of t among themselves; e.g. for t as above, C t = S {3,1,4,6} S {5,7}

10 Representation theory of the symmetric group S n e t = π C t sgn(π)δ π{t} for δ the unit function on F(M λ ) Then define F λ = span(e t ) where t ranges over all tableaux corresponding to λ Useful: Hook formula dim F λ = b λ n! hook length(b) Example: Construct e t for S 3 with the Young diagram λ = (2, 1) =

11 Representation theory of the special unitary group SU(n) Strategy: Establish connection between irreps of the symmetric group and irreps of GL(n) Since SU(n) SL(n, C) GL(n, C), we will get rep of SU(n) Show that they are irreducible and all of the irreps of SU(n)

12 Representation theory of the special unitary group SU(n) Look at V V for V a finite dim. vector space S 2 acts on V V by (12)x y = y x, has two 1-dim. irreps: trivial rep and sgn rep Have decomposition V V = S 2 (V ) Λ 2 (V ) into symmetric and antisymmetric tensors S 2 (V ) is direct sum of 1 2 n(n + 1) copies of trivial rep of S 2 Λ 2 (V ) is direct sum of 1 2 n(n 1) copies of sgn rep of S 2 A linear transformation on V, A(x y) = Ax Ay, commutes with the action of S 2 : A(12)(x y) = A(y x) = Ay Ax = (12)Ax Ay = (12)A(x y) Get rep of GL(V ) on V V that commutes with S 2 (and is irreducible on the subspaces S 2 and Λ 2 )

13 Representation theory of the special unitary group SU(n) Now decompose T r V = V V (r factors) into irreducibles There exist distinct irreps (ρ i, U i ) of GL(V ), associated with a different rep (σ i, F i ) of S r. Have decomposition T r V = (U 1 F 1 ) (U p F p ) f i = dim F i, s i = dim U i, then T r V decomposes under GL(V ) into a direct sum of f i copies of U i and under S r into a direct sum of s i copies of F i

14 Representation theory of the special unitary group SU(n) Take Young diagram λ, let dim V = n. Then define the entries of the dimension table by d ij = n + j i, where i labels the rows and j labels the columns of λ. We have b λ hook length(b) dim U λ = b λ d(b) Example: λ = (4, 3, 1) d = n n + 1 n + 2 n + 3 n 1 n n + 1 n 2, hook length = So for n = 3, we have dim U λ = = 15

15 Representation theory of the special unitary group SU(n) λ a Young diagram, F λ corresponding irrep of S r. T r V has component W λ = U λ F λ, t a tableau of type λ Define E t = σ C t,π R t sgn(σ)σπ Then E t (T r V ) = U λ e t = {φ(e t ) : φ Hom Sr (F λ, T r V )}, so E t (T r V ) gives a copy of the irrep U λ ; in particular E t (v i1 v ir ) span the space E t (T r V ) when v 1,..., v r is a basis of V and the indices i j range from 1 to n Get basis of E t (T r V ) by arranging the v ij so that ij are non-decreasing along the rows strictly increasing on the columns of λ

16 Representation theory of the special unitary group SU(n) Goal: the U λ are all the finite-dim. irreps of SL(V ), take V = C n subgroups of SL(n, C) {( 1... )} N + =, N = 0 1 {( δ1 ) } 0 n H =... : δ i = 1 0 δ n i=1 {( )} weight vector: simultaneous eigenvector for all elements of H there exists function µ on H s.t. ρ(δ)v = µ(δ)v for all δ H µ(δ) = δ m 1 1 δn mn, where the weight m = (m 1,..., m n ) is only defined up to adding a constant s(1,..., 1),

17 Representation theory of the special unitary group SU(n) One can prove: Every finite-dim. rep of SL(d, C) has a maximal weight vector (i.e. a simultaneous eigenvector of B + = H N + ), it is determined (up to equivalence) by the corresponding highest weight m Passing to the Lie algebra of SL(d, C), one shows that a maximal weight vector must obey m 1 m 2 m n (set m n = 0 by subtracting m n (1,..., 1)) This is just a Young diagram with (at most) n 1 rows! In particular, we have constructed all irreps of SU(n)

18 Ordering of energy levels Lieb/Mattis (1962), bipartite lattice with Heisenberg Hamiltonian (N = 2 in our model); the ground state of H belongs at most to total spin s := S A S B and (denoting E(S) the lowest energy eigenvalue with total spin S) E(S + 1) > E(S) for all S s; E(S) > E(s) for S > s and a special type of lattice Proof utilizes M subspaces and the Perron-Frobenius theorem

19 Ordering of energy levels Generalization to our case: space of states H = Λ i=1 CN, decompose space as H = λ HN λ, λ are Young diagrams with at most N 1 rows (since H is invariant under SU(N)) Denote E(λ) relative ground state energy in the space H N λ Equivalent of LM theorem (Hakobyan, 2010 for the chain): If λ λ, then E(λ ) > E(λ) and the relative ground state energy levels are non-degenerate (inside H N λ )

20 Ordering of energy levels Perron-Frobenius: If a connected hermitian matrix has no positive off-diagonal elements then its ground state is non-degenerate and has positive components Special case of chain: construct non-positive basis N {x 1 },..., {x N } = (c x1 σ,σc x2 σ,σ c xnσ σ,σ ) 0 σ=1 Basic states in the same subspace Hλ N are connected by the kinetic term in the Hamiltonian. Obtain uniqueness of relative ground state Ω n1...n N = #{x σ }=n σ ω {x 1 } {x N } {x 1 },..., {x N } with strictly positive components from PF theorem