Bethe equations for generalized Hubbard models
Victor Fomin
Laboratoire d’Annecy-le-Vieux de Physique Theorique´ (LAPTH), France CNRS, Universite´ de Savoie
16 april 2010
work with L.Frappat and E.Ragoucy arXiv:1002.1806v1 [hep-th]
Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 1 / 23 Plan of the talk
1 Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Integrability and R-matrix formalism Coordinate Bethe Ansatz
2 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability Coordinate Bethe Ansatz and results
Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 2 / 23 only 1 band nearest-neighbour electrons interact small range of Coulomb interaction = forbid double occupancy
Starting from the model: electrons in the static lattice with the Coulomb repulsion
N ! X ~p2 X H = i + V (~x ) + V (~x − ~x ) 2m I i C i j i=1 1≤i Using the approximations: Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model I Model named after John Hubbard describes electronic correlations in narrow energy band metals Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 3 / 23 only 1 band nearest-neighbour electrons interact small range of Coulomb interaction = forbid double occupancy Using the approximations: Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model I Model named after John Hubbard describes electronic correlations in narrow energy band metals Starting from the model: electrons in the static lattice with the Coulomb repulsion N ! X ~p2 X H = i + V (~x ) + V (~x − ~x ) 2m I i C i j i=1 1≤i Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 3 / 23 only 1 band nearest-neighbour electrons interact small range of Coulomb interaction = forbid double occupancy Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model I Model named after John Hubbard describes electronic correlations in narrow energy band metals Starting from the model: electrons in the static lattice with the Coulomb repulsion N ! X ~p2 X H = i + V (~x ) + V (~x − ~x ) 2m I i C i j i=1 1≤i Using the approximations: Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 3 / 23 small range of Coulomb interaction = forbid double occupancy Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model I Model named after John Hubbard describes electronic correlations in narrow energy band metals Starting from the model: electrons in the static lattice with the Coulomb repulsion N ! X ~p2 X H = i + V (~x ) + V (~x − ~x ) 2m I i C i j i=1 1≤i Using the approximations: only 1 band nearest-neighbour electrons interact Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 3 / 23 Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model I Model named after John Hubbard describes electronic correlations in narrow energy band metals Starting from the model: electrons in the static lattice with the Coulomb repulsion N ! X ~p2 X H = i + V (~x ) + V (~x − ~x ) 2m I i C i j i=1 1≤i Using the approximations: only 1 band nearest-neighbour electrons interact small range of Coulomb interaction = forbid double occupancy Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 3 / 23 Mott metal-insulator transition relevant to high-Tc super-conductivity in 2D,3D → approximations, but in 1D periodic Hubbard model is integrable! Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model II Second-quantized Hamiltonian in the basis of Wannier states is X X † X HHub = −t ci,acj,a + U ni,↑ni,↓ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 4 / 23 Mott metal-insulator transition relevant to high-Tc super-conductivity in 2D,3D → approximations, but in 1D periodic Hubbard model is integrable! Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model II Second-quantized Hamiltonian in the basis of Wannier states is X X † X HHub = −t ci,acj,a + U ni,↑ni,↓ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 4 / 23 relevant to high-Tc super-conductivity in 2D,3D → approximations, but in 1D periodic Hubbard model is integrable! Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model II Second-quantized Hamiltonian in the basis of Wannier states is X X † X HHub = −t ci,acj,a + U ni,↑ni,↓ Mott metal-insulator transition Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 4 / 23 in 2D,3D → approximations, but in 1D periodic Hubbard model is integrable! Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model II Second-quantized Hamiltonian in the basis of Wannier states is X X † X HHub = −t ci,acj,a + U ni,↑ni,↓ Mott metal-insulator transition relevant to high-Tc super-conductivity Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 4 / 23 Introduction to one-dimensional Hubbard model On the origin of the Hubbard model Origin of the Hubbard model II Second-quantized Hamiltonian in the basis of Wannier states is X X † X HHub = −t ci,acj,a + U ni,↑ni,↓ Mott metal-insulator transition relevant to high-Tc super-conductivity in 2D,3D → approximations, but in 1D periodic Hubbard model is integrable! Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 4 / 23 If ∃ matrix R12(u1, u2) ∈ End(V1) ⊗ End(V2) which satisfies some properties and the Yang-Baxter equation: R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2) QL R12(u1, u2) ⇒ T0(u) = i=1 R0i (0, u) - monodronomy matrix with ”0” - aux.space and {1, 2...L} - physical spaces, ⇒ τ(u) = Tr0(T0(u)) - transfer matrix such that [τ(u), τ(v)] = 0 d Hamiltonian H is in τ(u): H = du (ln τ(0)) - spin chain Hamiltonian on L sites. PL i Charges Qi are also in τ(u): τ(u) = i=0 u Qi R-matrix formalism allows to construct Qi : Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism R-matrix formalism In general ( basics from lectures of Faddeev ) Integrability ∝ ∃ independent Qi such that [Qi , H] = 0 and [Qi , Qj ] = 0 for i, j = 1, 2, ... Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 5 / 23 If ∃ matrix R12(u1, u2) ∈ End(V1) ⊗ End(V2) which satisfies some properties and the Yang-Baxter equation: R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2) QL R12(u1, u2) ⇒ T0(u) = i=1 R0i (0, u) - monodronomy matrix with ”0” - aux.space and {1, 2...L} - physical spaces, ⇒ τ(u) = Tr0(T0(u)) - transfer matrix such that [τ(u), τ(v)] = 0 d Hamiltonian H is in τ(u): H = du (ln τ(0)) - spin chain Hamiltonian on L sites. PL i Charges Qi are also in τ(u): τ(u) = i=0 u Qi Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism R-matrix formalism In general ( basics from lectures of Faddeev ) Integrability ∝ ∃ independent Qi such that [Qi , H] = 0 and [Qi , Qj ] = 0 for i, j = 1, 2, ... R-matrix formalism allows to construct Qi : Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 5 / 23 QL R12(u1, u2) ⇒ T0(u) = i=1 R0i (0, u) - monodronomy matrix with ”0” - aux.space and {1, 2...L} - physical spaces, ⇒ τ(u) = Tr0(T0(u)) - transfer matrix such that [τ(u), τ(v)] = 0 d Hamiltonian H is in τ(u): H = du (ln τ(0)) - spin chain Hamiltonian on L sites. PL i Charges Qi are also in τ(u): τ(u) = i=0 u Qi Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism R-matrix formalism In general ( basics from lectures of Faddeev ) Integrability ∝ ∃ independent Qi such that [Qi , H] = 0 and [Qi , Qj ] = 0 for i, j = 1, 2, ... R-matrix formalism allows to construct Qi : If ∃ matrix R12(u1, u2) ∈ End(V1) ⊗ End(V2) which satisfies some properties and the Yang-Baxter equation: R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 5 / 23 d Hamiltonian H is in τ(u): H = du (ln τ(0)) - spin chain Hamiltonian on L sites. PL i Charges Qi are also in τ(u): τ(u) = i=0 u Qi Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism R-matrix formalism In general ( basics from lectures of Faddeev ) Integrability ∝ ∃ independent Qi such that [Qi , H] = 0 and [Qi , Qj ] = 0 for i, j = 1, 2, ... R-matrix formalism allows to construct Qi : If ∃ matrix R12(u1, u2) ∈ End(V1) ⊗ End(V2) which satisfies some properties and the Yang-Baxter equation: R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2) QL R12(u1, u2) ⇒ T0(u) = i=1 R0i (0, u) - monodronomy matrix with ”0” - aux.space and {1, 2...L} - physical spaces, ⇒ τ(u) = Tr0(T0(u)) - transfer matrix such that [τ(u), τ(v)] = 0 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 5 / 23 PL i Charges Qi are also in τ(u): τ(u) = i=0 u Qi Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism R-matrix formalism In general ( basics from lectures of Faddeev ) Integrability ∝ ∃ independent Qi such that [Qi , H] = 0 and [Qi , Qj ] = 0 for i, j = 1, 2, ... R-matrix formalism allows to construct Qi : If ∃ matrix R12(u1, u2) ∈ End(V1) ⊗ End(V2) which satisfies some properties and the Yang-Baxter equation: R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2) QL R12(u1, u2) ⇒ T0(u) = i=1 R0i (0, u) - monodronomy matrix with ”0” - aux.space and {1, 2...L} - physical spaces, ⇒ τ(u) = Tr0(T0(u)) - transfer matrix such that [τ(u), τ(v)] = 0 d Hamiltonian H is in τ(u): H = du (ln τ(0)) - spin chain Hamiltonian on L sites. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 5 / 23 Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism R-matrix formalism In general ( basics from lectures of Faddeev ) Integrability ∝ ∃ independent Qi such that [Qi , H] = 0 and [Qi , Qj ] = 0 for i, j = 1, 2, ... R-matrix formalism allows to construct Qi : If ∃ matrix R12(u1, u2) ∈ End(V1) ⊗ End(V2) which satisfies some properties and the Yang-Baxter equation: R12(u1, u2)R13(u1, u3)R23(u2, u3) = R23(u2, u3)R13(u1, u3)R12(u1, u2) QL R12(u1, u2) ⇒ T0(u) = i=1 R0i (0, u) - monodronomy matrix with ”0” - aux.space and {1, 2...L} - physical spaces, ⇒ τ(u) = Tr0(T0(u)) - transfer matrix such that [τ(u), τ(v)] = 0 d Hamiltonian H is in τ(u): H = du (ln τ(0)) - spin chain Hamiltonian on L sites. PL i Charges Qi are also in τ(u): τ(u) = i=0 u Qi Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 5 / 23 4 × 4 matrix associated with gl(1|1) algebra σ with R12(λ) - free fermion model’s R-matrix (XX spin chain) and f - coupling function → Hubbard model constant U , 0 λ12 = λ1 − λ2, λ12 = λ1 + λ2 satisfies graded Yang-Baxter equation (Olmedilla, Wadati, Akutsu): ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ R12 (λ1, λ2)R13 (λ1, λ3)R23 (λ2, λ3) = R23 (λ2, λ3)R13 (λ1, λ3)R12 (λ1, λ2) Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism Hubbard model’s R-matrix The Hubbard model’s R-matrix (Shastry): 16 × 16 matrix: ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1, ↑↓ from R-matrix R12(λ1, 0) → Hubbard Hamiltonian Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 6 / 23 4 × 4 matrix associated with gl(1|1) algebra and f - coupling function → Hubbard model constant U , 0 λ12 = λ1 − λ2, λ12 = λ1 + λ2 satisfies graded Yang-Baxter equation (Olmedilla, Wadati, Akutsu): ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ R12 (λ1, λ2)R13 (λ1, λ3)R23 (λ2, λ3) = R23 (λ2, λ3)R13 (λ1, λ3)R12 (λ1, λ2) Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism Hubbard model’s R-matrix The Hubbard model’s R-matrix (Shastry): 16 × 16 matrix: ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1, σ with R12(λ) - free fermion model’s R-matrix (XX spin chain) ↑↓ from R-matrix R12(λ1, 0) → Hubbard Hamiltonian Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 6 / 23 and f - coupling function → Hubbard model constant U , 0 λ12 = λ1 − λ2, λ12 = λ1 + λ2 satisfies graded Yang-Baxter equation (Olmedilla, Wadati, Akutsu): ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ R12 (λ1, λ2)R13 (λ1, λ3)R23 (λ2, λ3) = R23 (λ2, λ3)R13 (λ1, λ3)R12 (λ1, λ2) Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism Hubbard model’s R-matrix The Hubbard model’s R-matrix (Shastry): 16 × 16 matrix: ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1, σ with R12(λ) - free fermion model’s R-matrix (XX spin chain) 4 × 4 matrix associated with gl(1|1) algebra ↑↓ from R-matrix R12(λ1, 0) → Hubbard Hamiltonian Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 6 / 23 satisfies graded Yang-Baxter equation (Olmedilla, Wadati, Akutsu): ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ R12 (λ1, λ2)R13 (λ1, λ3)R23 (λ2, λ3) = R23 (λ2, λ3)R13 (λ1, λ3)R12 (λ1, λ2) Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism Hubbard model’s R-matrix The Hubbard model’s R-matrix (Shastry): 16 × 16 matrix: ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1, σ with R12(λ) - free fermion model’s R-matrix (XX spin chain) 4 × 4 matrix associated with gl(1|1) algebra and f - coupling function → Hubbard model constant U , 0 λ12 = λ1 − λ2, λ12 = λ1 + λ2 ↑↓ from R-matrix R12(λ1, 0) → Hubbard Hamiltonian Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 6 / 23 Introduction to one-dimensional Hubbard model Integrability and R-matrix formalism Hubbard model’s R-matrix The Hubbard model’s R-matrix (Shastry): 16 × 16 matrix: ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1, σ with R12(λ) - free fermion model’s R-matrix (XX spin chain) 4 × 4 matrix associated with gl(1|1) algebra and f - coupling function → Hubbard model constant U , 0 λ12 = λ1 − λ2, λ12 = λ1 + λ2 satisfies graded Yang-Baxter equation (Olmedilla, Wadati, Akutsu): ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ R12 (λ1, λ2)R13 (λ1, λ3)R23 (λ2, λ3) = R23 (λ2, λ3)R13 (λ1, λ3)R12 (λ1, λ2) ↑↓ from R-matrix R12(λ1, 0) → Hubbard Hamiltonian Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 6 / 23 II) N-particle eigenfunction P ~ QN † φN,~σ = ~x∈[1,L] Ψ~σ(x) i=1 cxi ,σi φ0 where ~σ = (σ1, ..., σN ) and ~x = (x1, ..., xN ) Ψ~σ(~x) -? ⇒ Coordinate Bethe Ansatz! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach I Direct diagonalization of the periodic Hamiltonian : L X X † † X HHub = ci,aci+1,a + ci+1,aci,a + U ni,↑ni,↓, L + 1 ≡ 1 i=1 a=↑,↓ i PL Symmetry: [HHub, i=1 ni,σ] =0 - number of particles is conserved N↑, N↓ are quantum numbers of eigenfunction I) we choose pseudo-vacuum: φ0 such that ci,σφ0 = 0, the energy E = 0 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 7 / 23 Ψ~σ(~x) -? ⇒ Coordinate Bethe Ansatz! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach I Direct diagonalization of the periodic Hamiltonian : L X X † † X HHub = ci,aci+1,a + ci+1,aci,a + U ni,↑ni,↓, L + 1 ≡ 1 i=1 a=↑,↓ i PL Symmetry: [HHub, i=1 ni,σ] =0 - number of particles is conserved N↑, N↓ are quantum numbers of eigenfunction I) we choose pseudo-vacuum: φ0 such that ci,σφ0 = 0, the energy E = 0 II) N-particle eigenfunction P ~ QN † φN,~σ = ~x∈[1,L] Ψ~σ(x) i=1 cxi ,σi φ0 where ~σ = (σ1, ..., σN ) and ~x = (x1, ..., xN ) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 7 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach I Direct diagonalization of the periodic Hamiltonian : L X X † † X HHub = ci,aci+1,a + ci+1,aci,a + U ni,↑ni,↓, L + 1 ≡ 1 i=1 a=↑,↓ i PL Symmetry: [HHub, i=1 ni,σ] =0 - number of particles is conserved N↑, N↓ are quantum numbers of eigenfunction I) we choose pseudo-vacuum: φ0 such that ci,σφ0 = 0, the energy E = 0 II) N-particle eigenfunction P ~ QN † φN,~σ = ~x∈[1,L] Ψ~σ(x) i=1 cxi ,σi φ0 where ~σ = (σ1, ..., σN ) and ~x = (x1, ..., xN ) Ψ~σ(~x) -? ⇒ Coordinate Bethe Ansatz! Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 7 / 23 Pauli exclusion principle Ψ~σ(~x) = 0 for two particles on the same site with the same spin ikx Ψσ(x) = e Energy E = 2 cos k, Bethe root k is to be determined Periodic boundary condition ⇒ eikL = 1 - trivial Bethe equation, n k = 2π L with n = −L/2...L/2 - usual solid state physics result! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach II Schrodinger equation The Hamiltonian applied on N-particle eigenfunction (HHubφN,~σ = EφN,~σ): N X X Ψ~σ(~x + ~ek ) + Ψ~σ(~x − ~ek ) + U δ(xk = xl )δ(σk 6= σl )Ψ~σ(~x) = EΨ~σ(~x) k=1 k We have conditions on Ψ~σ(~x): periodic boundary conditions: Ψ~σ(~x + ~ek L) = Ψ~σ(~x) One-particle solution: k ↓ notation: ~ek - vector (0, ...0, 1, 0, ...0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 8 / 23 ikx Ψσ(x) = e Energy E = 2 cos k, Bethe root k is to be determined Periodic boundary condition ⇒ eikL = 1 - trivial Bethe equation, n k = 2π L with n = −L/2...L/2 - usual solid state physics result! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach II Schrodinger equation The Hamiltonian applied on N-particle eigenfunction (HHubφN,~σ = EφN,~σ): N X X Ψ~σ(~x + ~ek ) + Ψ~σ(~x − ~ek ) + U δ(xk = xl )δ(σk 6= σl )Ψ~σ(~x) = EΨ~σ(~x) k=1 k We have conditions on Ψ~σ(~x): periodic boundary conditions: Ψ~σ(~x + ~ek L) = Ψ~σ(~x) Pauli exclusion principle Ψ~σ(~x) = 0 for two particles on the same site with the same spin One-particle solution: k ↓ notation: ~ek - vector (0, ...0, 1, 0, ...0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 8 / 23 Energy E = 2 cos k, Bethe root k is to be determined Periodic boundary condition ⇒ eikL = 1 - trivial Bethe equation, n k = 2π L with n = −L/2...L/2 - usual solid state physics result! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach II Schrodinger equation The Hamiltonian applied on N-particle eigenfunction (HHubφN,~σ = EφN,~σ): N X X Ψ~σ(~x + ~ek ) + Ψ~σ(~x − ~ek ) + U δ(xk = xl )δ(σk 6= σl )Ψ~σ(~x) = EΨ~σ(~x) k=1 k We have conditions on Ψ~σ(~x): periodic boundary conditions: Ψ~σ(~x + ~ek L) = Ψ~σ(~x) Pauli exclusion principle Ψ~σ(~x) = 0 for two particles on the same site with the same spin One-particle solution: ikx Ψσ(x) = e k ↓ notation: ~ek - vector (0, ...0, 1, 0, ...0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 8 / 23 Periodic boundary condition ⇒ eikL = 1 - trivial Bethe equation, n k = 2π L with n = −L/2...L/2 - usual solid state physics result! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach II Schrodinger equation The Hamiltonian applied on N-particle eigenfunction (HHubφN,~σ = EφN,~σ): N X X Ψ~σ(~x + ~ek ) + Ψ~σ(~x − ~ek ) + U δ(xk = xl )δ(σk 6= σl )Ψ~σ(~x) = EΨ~σ(~x) k=1 k We have conditions on Ψ~σ(~x): periodic boundary conditions: Ψ~σ(~x + ~ek L) = Ψ~σ(~x) Pauli exclusion principle Ψ~σ(~x) = 0 for two particles on the same site with the same spin One-particle solution: ikx Ψσ(x) = e Energy E = 2 cos k, Bethe root k is to be determined k ↓ notation: ~ek - vector (0, ...0, 1, 0, ...0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 8 / 23 n k = 2π L with n = −L/2...L/2 - usual solid state physics result! Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach II Schrodinger equation The Hamiltonian applied on N-particle eigenfunction (HHubφN,~σ = EφN,~σ): N X X Ψ~σ(~x + ~ek ) + Ψ~σ(~x − ~ek ) + U δ(xk = xl )δ(σk 6= σl )Ψ~σ(~x) = EΨ~σ(~x) k=1 k We have conditions on Ψ~σ(~x): periodic boundary conditions: Ψ~σ(~x + ~ek L) = Ψ~σ(~x) Pauli exclusion principle Ψ~σ(~x) = 0 for two particles on the same site with the same spin One-particle solution: ikx Ψσ(x) = e Energy E = 2 cos k, Bethe root k is to be determined Periodic boundary condition ⇒ eikL = 1 - trivial Bethe equation, k ↓ notation: ~ek - vector (0, ...0, 1, 0, ...0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 8 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach II Schrodinger equation The Hamiltonian applied on N-particle eigenfunction (HHubφN,~σ = EφN,~σ): N X X Ψ~σ(~x + ~ek ) + Ψ~σ(~x − ~ek ) + U δ(xk = xl )δ(σk 6= σl )Ψ~σ(~x) = EΨ~σ(~x) k=1 k We have conditions on Ψ~σ(~x): periodic boundary conditions: Ψ~σ(~x + ~ek L) = Ψ~σ(~x) Pauli exclusion principle Ψ~σ(~x) = 0 for two particles on the same site with the same spin One-particle solution: ikx Ψσ(x) = e Energy E = 2 cos k, Bethe root k is to be determined Periodic boundary condition ⇒ eikL = 1 - trivial Bethe equation, n k = 2π L with n = −L/2...L/2 - usual solid state physics result! k ↓ notation: ~ek - vector (0, ...0, 1, 0, ...0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 8 / 23 P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 for N particles → N! possible dispositions. 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! for N particles → N! possible dispositions. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! for N particles → N! possible dispositions. 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! for N particles → N! possible dispositions. 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! for N particles → N! possible dispositions. 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! for N particles → N! possible dispositions. 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach III Bethe hypothesis N-particles solution: in 1D particles can be ordered! From x1 < x2 to x1 > x2 - interaction! for N particles → N! possible dispositions. 1 2 3 e.g. N = 3,Q = in the sector DQ : xQ(1) ≤ ... ≤ xQ(N) with Q ∈ SN , Q(1) Q(2) Q(3) Bethe Ansatz: for k1 < k2 < ... < kN and ~x ∈ DQ P (Q) P sg[P] −1 i kP(i)xi Ψ (~x) = (−1) A~σ(PQ, P )e i , ~σ P∈SN example, for N = 2: (a) Identical particles (σ1 = σ2), only 1 sector (x1 < x2): ( ) Ψ id ( , ) = ( , ) i(k1x1+k2x2) − (Π , ) i(k2x1+k1x2) ~σ x1 x2 A~σ id id e A~σ 12 id e (b) Non-identical particles, σ1 6= σ2, 2 sectors (Q = id, x1 < x2 and Q = Π12, x2 < x1): ( i(k x +k x ) i(k x +k x ) ( ) Q = id, A (id, id)e 1 1 2 2 − A (Π , Π )e 2 1 1 2 Ψ Q ( , ) = ~σ ~σ 12 12 ~σ x1 x2 i(k x +k x ) i(k x +k x ) Q = Π12, A~σ (Π12, id)e 1 1 2 2 − A~σ (id, Π12)e 2 1 1 2 PN −1 Energy E = 2 i=1 cos ki , Bethe roots k’s and A~σ(PQ, P ) are to be determined Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 9 / 23 for N = 2: a) Identical particles or σ1 = σ2, only 1 sector: A~σ(id, id) and A~σ(Π12, id), b) Non-identical particles, σ1 6= σ2, 2 sectors: A~σ(id, id), A~σ(Π12, Π12), A~σ(id, Π12) and A~σ(Π12, id) A↑,↑(Π12, id) 1 0 0 0 A↑,↑(id, id) A↑,↓(Π12, id) 0 t12 r12 0 A↑,↓(id, id) Thus, = , A↑,↓(Π12, Π12) 0 r12 t12 0 A↑,↓(id, Π12) A↓,↓(Π12, id) 0 0 0 1 A↓,↓(id, id) 2i(sin ka−sin kb) −u where tab = , rab = 2i(sin ka−sin kb)−u 2i(sin ka−sin kb)−u Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe hypothesis N-particles solution: Schrodinger equation + Pauli exclusion principle → relations between −1 A~σ(PQ, P ), ”S-matrix” Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 10 / 23 A↑,↑(Π12, id) 1 0 0 0 A↑,↑(id, id) A↑,↓(Π12, id) 0 t12 r12 0 A↑,↓(id, id) Thus, = , A↑,↓(Π12, Π12) 0 r12 t12 0 A↑,↓(id, Π12) A↓,↓(Π12, id) 0 0 0 1 A↓,↓(id, id) 2i(sin ka−sin kb) −u where tab = , rab = 2i(sin ka−sin kb)−u 2i(sin ka−sin kb)−u Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe hypothesis N-particles solution: Schrodinger equation + Pauli exclusion principle → relations between −1 A~σ(PQ, P ), ”S-matrix” for N = 2: a) Identical particles or σ1 = σ2, only 1 sector: A~σ(id, id) and A~σ(Π12, id), b) Non-identical particles, σ1 6= σ2, 2 sectors: A~σ(id, id), A~σ(Π12, Π12), A~σ(id, Π12) and A~σ(Π12, id) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 10 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe hypothesis N-particles solution: Schrodinger equation + Pauli exclusion principle → relations between −1 A~σ(PQ, P ), ”S-matrix” for N = 2: a) Identical particles or σ1 = σ2, only 1 sector: A~σ(id, id) and A~σ(Π12, id), b) Non-identical particles, σ1 6= σ2, 2 sectors: A~σ(id, id), A~σ(Π12, Π12), A~σ(id, Π12) and A~σ(Π12, id) A↑,↑(Π12, id) 1 0 0 0 A↑,↑(id, id) A↑,↓(Π12, id) 0 t12 r12 0 A↑,↓(id, id) Thus, = , A↑,↓(Π12, Π12) 0 r12 t12 0 A↑,↓(id, Π12) A↓,↓(Π12, id) 0 0 0 1 A↓,↓(id, id) 2i(sin ka−sin kb) −u where tab = , rab = 2i(sin ka−sin kb)−u 2i(sin ka−sin kb)−u Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 10 / 23 0 0 0 All coefficients A~σ(P , Q ) with all possible spin ~σ and sectors Q are in the vector Aˆ(P0) Sab(sin ka, sin kb) acts non-trivially on a and b particles. Sab(sin ka, sin kb) satisfies the Yang-Baxter equation. Periodic boundary conditions ⇒ ”Auxiliary problem” ˆ ikj L ˆ Sj+1j ...SNj S1j ...Sj−1j A(id) = e A(id) Physical chain → ”small” chain Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? ˆ ˆ for any N: A(ΠabP) = Sab(sin ka, sin kb)A(P) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 11 / 23 Sab(sin ka, sin kb) acts non-trivially on a and b particles. Sab(sin ka, sin kb) satisfies the Yang-Baxter equation. Periodic boundary conditions ⇒ ”Auxiliary problem” ˆ ikj L ˆ Sj+1j ...SNj S1j ...Sj−1j A(id) = e A(id) Physical chain → ”small” chain Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? ˆ ˆ for any N: A(ΠabP) = Sab(sin ka, sin kb)A(P) 0 0 0 All coefficients A~σ(P , Q ) with all possible spin ~σ and sectors Q are in the vector Aˆ(P0) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 11 / 23 Sab(sin ka, sin kb) satisfies the Yang-Baxter equation. Periodic boundary conditions ⇒ ”Auxiliary problem” ˆ ikj L ˆ Sj+1j ...SNj S1j ...Sj−1j A(id) = e A(id) Physical chain → ”small” chain Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? ˆ ˆ for any N: A(ΠabP) = Sab(sin ka, sin kb)A(P) 0 0 0 All coefficients A~σ(P , Q ) with all possible spin ~σ and sectors Q are in the vector Aˆ(P0) Sab(sin ka, sin kb) acts non-trivially on a and b particles. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 11 / 23 Periodic boundary conditions ⇒ ”Auxiliary problem” ˆ ikj L ˆ Sj+1j ...SNj S1j ...Sj−1j A(id) = e A(id) Physical chain → ”small” chain Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? ˆ ˆ for any N: A(ΠabP) = Sab(sin ka, sin kb)A(P) 0 0 0 All coefficients A~σ(P , Q ) with all possible spin ~σ and sectors Q are in the vector Aˆ(P0) Sab(sin ka, sin kb) acts non-trivially on a and b particles. Sab(sin ka, sin kb) satisfies the Yang-Baxter equation. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 11 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach IV Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? ˆ ˆ for any N: A(ΠabP) = Sab(sin ka, sin kb)A(P) 0 0 0 All coefficients A~σ(P , Q ) with all possible spin ~σ and sectors Q are in the vector Aˆ(P0) Sab(sin ka, sin kb) acts non-trivially on a and b particles. Sab(sin ka, sin kb) satisfies the Yang-Baxter equation. Periodic boundary conditions ⇒ ”Auxiliary problem” ˆ ikj L ˆ Sj+1j ...SNj S1j ...Sj−1j A(id) = e A(id) Physical chain → ”small” chain Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 11 / 23 sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 we choose pseudo-vacuum, for example spin down particles we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Periodic boundary conditions ⇒ ”Lieb and Wu equations” Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Periodic boundary conditions ⇒ ”Lieb and Wu equations” Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” we choose pseudo-vacuum, for example spin down particles Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Periodic boundary conditions ⇒ ”Lieb and Wu equations” Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” we choose pseudo-vacuum, for example spin down particles we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 Periodic boundary conditions ⇒ ”Lieb and Wu equations” Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” we choose pseudo-vacuum, for example spin down particles we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” we choose pseudo-vacuum, for example spin down particles we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Periodic boundary conditions ⇒ ”Lieb and Wu equations” Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” we choose pseudo-vacuum, for example spin down particles we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Periodic boundary conditions ⇒ ”Lieb and Wu equations” sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz Lieb and Wu approach V Bethe equations −1 N-particles solution: What are k’s and A~σ(PQ, P ) ? We can use again Coordinate Bethe Ansatz approach to ”Auxiliary problem” we choose pseudo-vacuum, for example spin down particles we find the one-particle solution fy (a) of new Hamiltonian. a - new Bethe root, y - coordinate on the small chain. (1) P QM Bethe hypothesis: Ψ (~y) = B(π) fy (aπ(i)) with y1 < ... < yM M π∈SM i=1 i and a1 < ... < aM . Periodic boundary conditions ⇒ ”Lieb and Wu equations” sin k −a +iU/4 eikj L = QM j α , for j = 1, ..., N α=1 sin kj −aα−iU/4 sin k −a +iU/4 QN j α = − QM aβ −aα+iU/2 , for α = 1, ..., M j=1 sin kj −aα−iU/4 β=1 aβ −aα−iU/2 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 12 / 23 Beisert, nlin/0610017v2: Shastry’s R-matrix ↔ AdS/CFT S-matrix with centrally extended psu(2|2) symmetry. Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz AdS/CFT application In papers of Rej, Serban, Staudacher ’05 and Beisert ’06, 1D Hubbard model in appears in N = 4 SYM. Rej, Serban, Staudacher, hep-th/0512077v2: Hubbard Hamiltonian at half-filling ↔ 3 loop dilatation operator of planar N = 4 SYM in su(2) sector. or Lieb and Wu equations at half-filing ↔ BDS Bethe equations However, BDS Bethe equations → BES Bethe equations with transcendental phase appeared. hep-th/0610251v2 One of the motivations for studies... Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 13 / 23 Introduction to one-dimensional Hubbard model Coordinate Bethe Ansatz AdS/CFT application In papers of Rej, Serban, Staudacher ’05 and Beisert ’06, 1D Hubbard model in appears in N = 4 SYM. Rej, Serban, Staudacher, hep-th/0512077v2: Hubbard Hamiltonian at half-filling ↔ 3 loop dilatation operator of planar N = 4 SYM in su(2) sector. or Lieb and Wu equations at half-filing ↔ BDS Bethe equations Beisert, nlin/0610017v2: Shastry’s R-matrix ↔ AdS/CFT S-matrix with centrally extended psu(2|2) symmetry. However, BDS Bethe equations → BES Bethe equations with transcendental phase appeared. hep-th/0610251v2 One of the motivations for studies... Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 13 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models ( also extensible on superalgebras ) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 14 / 23 Generalized XX spin chain model: R12(λ) depend on choice of π and π¯ R12(λ) = Σ12 P12 + Σ12 sin λ + (P12 − Σ12P12) cos λ 1 Matrix Σ12 constructed using π and π¯: Σab = 2 (Iab − CaCb) C-matrix: C = E 11 − E 22 ∈ C2 × C2 XX spin chain model: R12(λ) depends in similar way on C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism I (G. Feverati, L. Frappat and E. Ragoucy arXiv:0903.0190 [math-ph]) We introduce n × n matrix projectors: π and π¯ such that π +π ¯ = Id, insert in basic ingredients in Hubbard R-matrix construction: C-matrix: C = π − π¯ ∈ Cn × Cn comparing with Hubbard model’s ones: n projectors π,π¯ → [ 2 ] inequivalent models. Maassarani’s generalizations of XX R-matrix and Shastry’s R-matrix correspond the particular choice of projectors. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 15 / 23 C-matrix: C = E 11 − E 22 ∈ C2 × C2 XX spin chain model: R12(λ) depends in similar way on C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism I (G. Feverati, L. Frappat and E. Ragoucy arXiv:0903.0190 [math-ph]) We introduce n × n matrix projectors: π and π¯ such that π +π ¯ = Id, insert in basic ingredients in Hubbard R-matrix construction: C-matrix: C = π − π¯ ∈ Cn × Cn Generalized XX spin chain model: R12(λ) depend on choice of π and π¯ R12(λ) = Σ12 P12 + Σ12 sin λ + (P12 − Σ12P12) cos λ 1 Matrix Σ12 constructed using π and π¯: Σab = 2 (Iab − CaCb) comparing with Hubbard model’s ones: n projectors π,π¯ → [ 2 ] inequivalent models. Maassarani’s generalizations of XX R-matrix and Shastry’s R-matrix correspond the particular choice of projectors. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 15 / 23 XX spin chain model: R12(λ) depends in similar way on C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism I (G. Feverati, L. Frappat and E. Ragoucy arXiv:0903.0190 [math-ph]) We introduce n × n matrix projectors: π and π¯ such that π +π ¯ = Id, insert in basic ingredients in Hubbard R-matrix construction: C-matrix: C = π − π¯ ∈ Cn × Cn Generalized XX spin chain model: R12(λ) depend on choice of π and π¯ R12(λ) = Σ12 P12 + Σ12 sin λ + (P12 − Σ12P12) cos λ 1 Matrix Σ12 constructed using π and π¯: Σab = 2 (Iab − CaCb) comparing with Hubbard model’s ones: C-matrix: C = E 11 − E 22 ∈ C2 × C2 n projectors π,π¯ → [ 2 ] inequivalent models. Maassarani’s generalizations of XX R-matrix and Shastry’s R-matrix correspond the particular choice of projectors. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 15 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism I (G. Feverati, L. Frappat and E. Ragoucy arXiv:0903.0190 [math-ph]) We introduce n × n matrix projectors: π and π¯ such that π +π ¯ = Id, insert in basic ingredients in Hubbard R-matrix construction: C-matrix: C = π − π¯ ∈ Cn × Cn Generalized XX spin chain model: R12(λ) depend on choice of π and π¯ R12(λ) = Σ12 P12 + Σ12 sin λ + (P12 − Σ12P12) cos λ 1 Matrix Σ12 constructed using π and π¯: Σab = 2 (Iab − CaCb) comparing with Hubbard model’s ones: C-matrix: C = E 11 − E 22 ∈ C2 × C2 XX spin chain model: R12(λ) depends in similar way on C n projectors π,π¯ → [ 2 ] inequivalent models. Maassarani’s generalizations of XX R-matrix and Shastry’s R-matrix correspond the particular choice of projectors. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 15 / 23 ↑ with R12(λ) - generalized gl(n) XX model’s R-matrix ↓ with R12(λ) - generalized gl(m) XX model’s R-matrix the same coupling function f (λ1, λ2, U) when n and m equal 2 11 22 2 2 projectors π,π¯ are such that Cσ = Eσ − Eσ ∈ C × C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix associated with gl↑(n) ⊕ gl↓(m): ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1 In limit one get ”bosonic” Hubbard model: (n−1)(m−1)+1 projectors π,π¯ → [ 2 ] non-trivial inequivalent models. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 16 / 23 ↓ with R12(λ) - generalized gl(m) XX model’s R-matrix the same coupling function f (λ1, λ2, U) when n and m equal 2 11 22 2 2 projectors π,π¯ are such that Cσ = Eσ − Eσ ∈ C × C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix associated with gl↑(n) ⊕ gl↓(m): ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1 ↑ with R12(λ) - generalized gl(n) XX model’s R-matrix In limit one get ”bosonic” Hubbard model: (n−1)(m−1)+1 projectors π,π¯ → [ 2 ] non-trivial inequivalent models. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 16 / 23 the same coupling function f (λ1, λ2, U) when n and m equal 2 11 22 2 2 projectors π,π¯ are such that Cσ = Eσ − Eσ ∈ C × C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix associated with gl↑(n) ⊕ gl↓(m): ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1 ↑ with R12(λ) - generalized gl(n) XX model’s R-matrix ↓ with R12(λ) - generalized gl(m) XX model’s R-matrix In limit one get ”bosonic” Hubbard model: (n−1)(m−1)+1 projectors π,π¯ → [ 2 ] non-trivial inequivalent models. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 16 / 23 when n and m equal 2 11 22 2 2 projectors π,π¯ are such that Cσ = Eσ − Eσ ∈ C × C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix associated with gl↑(n) ⊕ gl↓(m): ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1 ↑ with R12(λ) - generalized gl(n) XX model’s R-matrix ↓ with R12(λ) - generalized gl(m) XX model’s R-matrix the same coupling function f (λ1, λ2, U) In limit one get ”bosonic” Hubbard model: (n−1)(m−1)+1 projectors π,π¯ → [ 2 ] non-trivial inequivalent models. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 16 / 23 11 22 2 2 projectors π,π¯ are such that Cσ = Eσ − Eσ ∈ C × C Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix associated with gl↑(n) ⊕ gl↓(m): ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1 ↑ with R12(λ) - generalized gl(n) XX model’s R-matrix ↓ with R12(λ) - generalized gl(m) XX model’s R-matrix the same coupling function f (λ1, λ2, U) In limit one get ”bosonic” Hubbard model: when n and m equal 2 (n−1)(m−1)+1 projectors π,π¯ → [ 2 ] non-trivial inequivalent models. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 16 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix associated with gl↑(n) ⊕ gl↓(m): ↑↓ ↑ ↓ ↑ 0 ↓ 0 R12 (λ1, λ2) = R12(λ12) R12(λ12) + f (λ1, λ2, U)R12(λ12) C↑1 R12(λ12) C↓1 ↑ with R12(λ) - generalized gl(n) XX model’s R-matrix ↓ with R12(λ) - generalized gl(m) XX model’s R-matrix the same coupling function f (λ1, λ2, U) In limit one get ”bosonic” Hubbard model: when n and m equal 2 11 22 2 2 projectors π,π¯ are such that Cσ = Eσ − Eσ ∈ C × C (n−1)(m−1)+1 projectors π,π¯ → [ 2 ] non-trivial inequivalent models. Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 16 / 23 π¯-particles = electron-like particles with spin ↑, ↓ and additional degree of freedom. π-particles = heavy particles with spin ↑, ↓ and additional degree of freedom which only interact with π¯-particles. generates Hubbard-like integrable Hamiltonians depending on the choice of Cσ: Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix : satisfies the Yang-Baxter Equation for example, kinetic term: X † X † † T = (bi,σ,π¯ bi+1,σ,π¯ + h.c.) + (bi,σ,π¯ bi,σ,πbi+1,σ,πbi+1,σ,π¯ + h.c.) i,σ,π¯ i,σ,π,π¯ and Hubbard model’s one: X † THub = (bi,σbi+1,σ + h.c.) i,σ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 17 / 23 π¯-particles = electron-like particles with spin ↑, ↓ and additional degree of freedom. π-particles = heavy particles with spin ↑, ↓ and additional degree of freedom which only interact with π¯-particles. Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix : satisfies the Yang-Baxter Equation generates Hubbard-like integrable Hamiltonians depending on the choice of Cσ: for example, kinetic term: X † X † † T = (bi,σ,π¯ bi+1,σ,π¯ + h.c.) + (bi,σ,π¯ bi,σ,πbi+1,σ,πbi+1,σ,π¯ + h.c.) i,σ,π¯ i,σ,π,π¯ and Hubbard model’s one: X † THub = (bi,σbi+1,σ + h.c.) i,σ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 17 / 23 π-particles = heavy particles with spin ↑, ↓ and additional degree of freedom which only interact with π¯-particles. Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix : satisfies the Yang-Baxter Equation generates Hubbard-like integrable Hamiltonians depending on the choice of Cσ: π¯-particles = electron-like particles with spin ↑, ↓ and additional degree of freedom. for example, kinetic term: X † X † † T = (bi,σ,π¯ bi+1,σ,π¯ + h.c.) + (bi,σ,π¯ bi,σ,πbi+1,σ,πbi+1,σ,π¯ + h.c.) i,σ,π¯ i,σ,π,π¯ and Hubbard model’s one: X † THub = (bi,σbi+1,σ + h.c.) i,σ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 17 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models generalized R-matrix and integrability R-matrix formalism II The generalized Hubbard-like model’s R-matrix : satisfies the Yang-Baxter Equation generates Hubbard-like integrable Hamiltonians depending on the choice of Cσ: π¯-particles = electron-like particles with spin ↑, ↓ and additional degree of freedom. π-particles = heavy particles with spin ↑, ↓ and additional degree of freedom which only interact with π¯-particles. for example, kinetic term: X † X † † T = (bi,σ,π¯ bi+1,σ,π¯ + h.c.) + (bi,σ,π¯ bi,σ,πbi+1,σ,πbi+1,σ,π¯ + h.c.) i,σ,π¯ i,σ,π,π¯ and Hubbard model’s one: X † THub = (bi,σbi+1,σ + h.c.) i,σ Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 17 / 23 and Bethe equations : N +N N 1 3 λ − sin k − iU 4 ikj L Y l j Y e = bi , for j ∈ [1, N] λ − sin k + iU l=1 l j i=1 N +N 2 4 = , ( ) < ( ), ∈ [ , ], bi 1 arg bi arg bi+1 with i 1 N4 N N N +N N λ − sin k − iU 2πi P 3 n¯ 4 1 3 Y l j N +N i=1 i Y Y λl − λm − 2iU = e 1 3 bi , λ − sin k + iU λ − λ + 2iU j=1 l j i=1 m=1 l m m6=l with 1 ≤ n¯ < ... < n¯ ≤ N + N , for l ∈ [1, N + N ] 1 N3 1 3 1 3 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz I example of gl(1|2) ⊕ gl(1|2) model 11 22 33 Projectors: πσ = Eσ and π¯σ = Eσ + Eσ , only electron-like particles. The periodic Hamiltonian: † † † † H = HHub(c, c , U) + HHub(d, d , U) + Vint (c, c , d, d , U) N1 N2 N3 N4 z }| { z }| { z }| { z }| { N particles excited state: (c ↑, ..., c ↑, c ↓, ..., c ↓, d ↑, ..., d ↑ d ↓, ..., d ↓) PN with energy: E = U L − 2N − 2 l=1 cos kl with N = N1 + N2 + N3 + N4 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 18 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz I example of gl(1|2) ⊕ gl(1|2) model 11 22 33 Projectors: πσ = Eσ and π¯σ = Eσ + Eσ , only electron-like particles. The periodic Hamiltonian: † † † † H = HHub(c, c , U) + HHub(d, d , U) + Vint (c, c , d, d , U) N1 N2 N3 N4 z }| { z }| { z }| { z }| { N particles excited state: (c ↑, ..., c ↑, c ↓, ..., c ↓, d ↑, ..., d ↑ d ↓, ..., d ↓) PN with energy: E = U L − 2N − 2 l=1 cos kl with N = N1 + N2 + N3 + N4 and Bethe equations : N +N N 1 3 λ − sin k − iU 4 ikj L Y l j Y e = bi , for j ∈ [1, N] λ − sin k + iU l=1 l j i=1 N +N 2 4 = , ( ) < ( ), ∈ [ , ], bi 1 arg bi arg bi+1 with i 1 N4 N N N +N N λ − sin k − iU 2πi P 3 n¯ 4 1 3 Y l j N +N i=1 i Y Y λl − λm − 2iU = e 1 3 bi , λ − sin k + iU λ − λ + 2iU j=1 l j i=1 m=1 l m m6=l with 1 ≤ n¯ < ... < n¯ ≤ N + N , for l ∈ [1, N + N ] 1 N3 1 3 1 3 Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 18 / 23 p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 1 types of π¯ ↓-particles n + m − 3 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 2 types of π¯ ↓-particles Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz II General case: gl(n) ⊕ gl(m) 1d Hubbard-like model P{p,q} ii P{n,m} ii Projectors: π{↑,↓} = i=1 E{↑,↓} and π¯{↑,↓} = i={p,q}+1 E{↑,↓}. The periodic Hamiltonian: L X H = (ΣP)↑ x,x+1 + (ΣP)↓ x,x+1 + u C↑x C↓x , L + 1 ≡ 1 x=1 pseudo−vacuum N↑2 N↑n pseudo−vacuum N↓2 N↓m z }| { z }| { z }| { z }| { z }| { z }| { particle content: ( 1 ↑, ..., 1 ↑ , 2 ↑, ..., 2 ↑, ..., n ↑, ..., n ↑, 1 ↓, ..., 1 ↓ , 2 ↓, ..., 2 ↓, ..., m ↓, ..., m ↓) n + m − 2 types particles Periodic boundary conditions → Auxiliary problem Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 19 / 23 q types of π ↓-particles and m − q − 1 types of π¯ ↓-particles n + m − 3 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 2 types of π¯ ↓-particles Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz II General case: gl(n) ⊕ gl(m) 1d Hubbard-like model P{p,q} ii P{n,m} ii Projectors: π{↑,↓} = i=1 E{↑,↓} and π¯{↑,↓} = i={p,q}+1 E{↑,↓}. The periodic Hamiltonian: L X H = (ΣP)↑ x,x+1 + (ΣP)↓ x,x+1 + u C↑x C↓x , L + 1 ≡ 1 x=1 pseudo−vacuum N↑2 N↑n pseudo−vacuum N↓2 N↓m z }| { z }| { z }| { z }| { z }| { z }| { particle content: ( 1 ↑, ..., 1 ↑ , 2 ↑, ..., 2 ↑, ..., n ↑, ..., n ↑, 1 ↓, ..., 1 ↓ , 2 ↓, ..., 2 ↓, ..., m ↓, ..., m ↓) n + m − 2 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles Periodic boundary conditions → Auxiliary problem Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 19 / 23 n + m − 3 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 2 types of π¯ ↓-particles Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz II General case: gl(n) ⊕ gl(m) 1d Hubbard-like model P{p,q} ii P{n,m} ii Projectors: π{↑,↓} = i=1 E{↑,↓} and π¯{↑,↓} = i={p,q}+1 E{↑,↓}. The periodic Hamiltonian: L X H = (ΣP)↑ x,x+1 + (ΣP)↓ x,x+1 + u C↑x C↓x , L + 1 ≡ 1 x=1 pseudo−vacuum N↑2 N↑n pseudo−vacuum N↓2 N↓m z }| { z }| { z }| { z }| { z }| { z }| { particle content: ( 1 ↑, ..., 1 ↑ , 2 ↑, ..., 2 ↑, ..., n ↑, ..., n ↑, 1 ↓, ..., 1 ↓ , 2 ↓, ..., 2 ↓, ..., m ↓, ..., m ↓) n + m − 2 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 1 types of π¯ ↓-particles Periodic boundary conditions → Auxiliary problem Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 19 / 23 p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 2 types of π¯ ↓-particles Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz II General case: gl(n) ⊕ gl(m) 1d Hubbard-like model P{p,q} ii P{n,m} ii Projectors: π{↑,↓} = i=1 E{↑,↓} and π¯{↑,↓} = i={p,q}+1 E{↑,↓}. The periodic Hamiltonian: L X H = (ΣP)↑ x,x+1 + (ΣP)↓ x,x+1 + u C↑x C↓x , L + 1 ≡ 1 x=1 pseudo−vacuum N↑2 N↑n pseudo−vacuum N↓2 N↓m z }| { z }| { z }| { z }| { z }| { z }| { particle content: ( 1 ↑, ..., 1 ↑ , 2 ↑, ..., 2 ↑, ..., n ↑, ..., n ↑, 1 ↓, ..., 1 ↓ , 2 ↓, ..., 2 ↓, ..., m ↓, ..., m ↓) n + m − 2 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 1 types of π¯ ↓-particles Periodic boundary conditions → Auxiliary problem n + m − 3 types particles Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 19 / 23 q types of π ↓-particles and m − q − 2 types of π¯ ↓-particles Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz II General case: gl(n) ⊕ gl(m) 1d Hubbard-like model P{p,q} ii P{n,m} ii Projectors: π{↑,↓} = i=1 E{↑,↓} and π¯{↑,↓} = i={p,q}+1 E{↑,↓}. The periodic Hamiltonian: L X H = (ΣP)↑ x,x+1 + (ΣP)↓ x,x+1 + u C↑x C↓x , L + 1 ≡ 1 x=1 pseudo−vacuum N↑2 N↑n pseudo−vacuum N↓2 N↓m z }| { z }| { z }| { z }| { z }| { z }| { particle content: ( 1 ↑, ..., 1 ↑ , 2 ↑, ..., 2 ↑, ..., n ↑, ..., n ↑, 1 ↓, ..., 1 ↓ , 2 ↓, ..., 2 ↓, ..., m ↓, ..., m ↓) n + m − 2 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 1 types of π¯ ↓-particles Periodic boundary conditions → Auxiliary problem n + m − 3 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 19 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz II General case: gl(n) ⊕ gl(m) 1d Hubbard-like model P{p,q} ii P{n,m} ii Projectors: π{↑,↓} = i=1 E{↑,↓} and π¯{↑,↓} = i={p,q}+1 E{↑,↓}. The periodic Hamiltonian: L X H = (ΣP)↑ x,x+1 + (ΣP)↓ x,x+1 + u C↑x C↓x , L + 1 ≡ 1 x=1 pseudo−vacuum N↑2 N↑n pseudo−vacuum N↓2 N↓m z }| { z }| { z }| { z }| { z }| { z }| { particle content: ( 1 ↑, ..., 1 ↑ , 2 ↑, ..., 2 ↑, ..., n ↑, ..., n ↑, 1 ↓, ..., 1 ↓ , 2 ↓, ..., 2 ↓, ..., m ↓, ..., m ↓) n + m − 2 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 1 types of π¯ ↓-particles Periodic boundary conditions → Auxiliary problem n + m − 3 types particles p types of π ↑-particles and n − p − 1 types of π¯ ↑-particles q types of π ↓-particles and m − q − 2 types of π¯ ↓-particles Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 19 / 23 no mixing between π¯ and π Bethe roots k’s. non-trivial Bethe root’s ”quantification” only for π¯ Bethe roots k’s: sin k −a +iU/4 eikj L = e2πiΦ QM j α α=1 sin kj −aα−iU/4 sin k −a +iU/4 Q j α = −e2πiΨ Q aβ −aα+iU/2 j∈π¯↑,π¯↓ sin kj −aα−iU/4 β∈π¯↑ aβ −aα−iU/2 Phases Φ and Ψ depend on trivially ”quantized” π Bethe root k’s and other level Bethe roots from nested CBA Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz III General case: gl(n) ⊕ gl(m) 1d Hubbard-like model Results for general case: P only electron-like particles contribute in energy: E = 2 cos(kl ) l∈π¯↑,π¯↓ (V.F., L. Frappat and E. Ragoucy arXiv:1002.1806[hep-th]) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 20 / 23 non-trivial Bethe root’s ”quantification” only for π¯ Bethe roots k’s: sin k −a +iU/4 eikj L = e2πiΦ QM j α α=1 sin kj −aα−iU/4 sin k −a +iU/4 Q j α = −e2πiΨ Q aβ −aα+iU/2 j∈π¯↑,π¯↓ sin kj −aα−iU/4 β∈π¯↑ aβ −aα−iU/2 Phases Φ and Ψ depend on trivially ”quantized” π Bethe root k’s and other level Bethe roots from nested CBA Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz III General case: gl(n) ⊕ gl(m) 1d Hubbard-like model Results for general case: P only electron-like particles contribute in energy: E = 2 cos(kl ) l∈π¯↑,π¯↓ no mixing between π¯ and π Bethe roots k’s. (V.F., L. Frappat and E. Ragoucy arXiv:1002.1806[hep-th]) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 20 / 23 Phases Φ and Ψ depend on trivially ”quantized” π Bethe root k’s and other level Bethe roots from nested CBA Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz III General case: gl(n) ⊕ gl(m) 1d Hubbard-like model Results for general case: P only electron-like particles contribute in energy: E = 2 cos(kl ) l∈π¯↑,π¯↓ no mixing between π¯ and π Bethe roots k’s. non-trivial Bethe root’s ”quantification” only for π¯ Bethe roots k’s: sin k −a +iU/4 eikj L = e2πiΦ QM j α α=1 sin kj −aα−iU/4 sin k −a +iU/4 Q j α = −e2πiΨ Q aβ −aα+iU/2 j∈π¯↑,π¯↓ sin kj −aα−iU/4 β∈π¯↑ aβ −aα−iU/2 (V.F., L. Frappat and E. Ragoucy arXiv:1002.1806[hep-th]) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 20 / 23 Generalized gl(n) ⊕ gl(m) 1d Hubbard-like models Coordinate Bethe Ansatz and results Coordinate Bethe Ansatz III General case: gl(n) ⊕ gl(m) 1d Hubbard-like model Results for general case: P only electron-like particles contribute in energy: E = 2 cos(kl ) l∈π¯↑,π¯↓ no mixing between π¯ and π Bethe roots k’s. non-trivial Bethe root’s ”quantification” only for π¯ Bethe roots k’s: sin k −a +iU/4 eikj L = e2πiΦ QM j α α=1 sin kj −aα−iU/4 sin k −a +iU/4 Q j α = −e2πiΨ Q aβ −aα+iU/2 j∈π¯↑,π¯↓ sin kj −aα−iU/4 β∈π¯↑ aβ −aα−iU/2 Phases Φ and Ψ depend on trivially ”quantized” π Bethe root k’s and other level Bethe roots from nested CBA (V.F., L. Frappat and E. Ragoucy arXiv:1002.1806[hep-th]) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 20 / 23 new integrable Hamiltonians thermodynamic limit and string hypothesis ? application to N = 4SYM theory ? application to condensed matter physics ? Conclusions Conclusions and open questions ? generalized Hubbard-like models: extensions of Shastry’s R-matrix ? open questions: Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 21 / 23 thermodynamic limit and string hypothesis ? application to N = 4SYM theory ? application to condensed matter physics ? Conclusions Conclusions and open questions ? generalized Hubbard-like models: extensions of Shastry’s R-matrix new integrable Hamiltonians ? open questions: Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 21 / 23 application to N = 4SYM theory ? application to condensed matter physics ? Conclusions Conclusions and open questions ? generalized Hubbard-like models: extensions of Shastry’s R-matrix new integrable Hamiltonians ? open questions: thermodynamic limit and string hypothesis ? Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 21 / 23 application to condensed matter physics ? Conclusions Conclusions and open questions ? generalized Hubbard-like models: extensions of Shastry’s R-matrix new integrable Hamiltonians ? open questions: thermodynamic limit and string hypothesis ? application to N = 4SYM theory ? Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 21 / 23 Conclusions Conclusions and open questions ? generalized Hubbard-like models: extensions of Shastry’s R-matrix new integrable Hamiltonians ? open questions: thermodynamic limit and string hypothesis ? application to N = 4SYM theory ? application to condensed matter physics ? Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 21 / 23 Conclusions Outlook Thank you Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 22 / 23 Conclusions Appendix L X X ( , †, , †, ) = − ( † + † )( d d − d − d ) Vint c c d d U ckσ ck+1σ ck+1σ ckσ nk,σ nk+1,σ nk,σ nk+1,σ i=1 σ=↑,↓ + ( † + † )( c c − c − c ) dkσ dk+1σ dk+1σ dkσ nk,σ nk+1,σ nk,σ nk+1,σ L U X + (1 − 2nc )(1 − 2nd ) + (1 − 2nd )(1 − 2nc ) 4 k↑ k↑ k↑ k↑ k=1 c d c d c d c d − 2(1 − nk↑ − nk↑)(1 + 2nk↓nk↓) − 2(1 + 2nk↑nk↑)(1 − nk↓ − nk↓) c d c d + (1 + 2nk↑nk↑)(1 + 2nk↓nk↓) Victor Fomin (LAPTH) ICFT10, University of Kent 16/04/2010 23 / 23