Properties of the Bethe Ansatz equations for Richardson-Gaudin models

Transcription

1 Properties of the Bethe Ansatz equations for Richardson-Gaudin models Inna Luyaneno, Phillip Isaac, Jon Lins Centre for Mathematical Physics, School of Mathematics and Physics, The University of Queensland

2 Some History Quantum Inverse Scattering Method (QISM) for twisted periodic boundary conditions [Faddeev, Kulish, Slyanin, Tahtajan 1979]. BCS model of superconductivity [Bardeen, Cooper and Schrieffer 1957]. Solved for the rational case [Richardson 1963]. Integrals of motion for the Richardson model [Cambiaggio et al. 1997]. Eigenvalues for the Richardson model [Sierra 000]. Generalized to the trigonometric case using Gaudin s method [Amico et al. 001], [Duelsy et al. 001]. Reformulated through the quasi-classical limit of QISM [Zhou et al. 00], [von Delft and Poghossian 00]. Some extensions: [Ovchinniov 003], [Dunning and Lins 004], [Ibañez et al. 009], [Srypny 009], [Duelsy et al. 010, 011], [Lins and Marquette 013].

3 Some History Boundary QISM (BQISM) for open-boundary conditions, for the case of XXZ spin chain [Slyanin 1988]. periodic boundary vs What is the effect of the boundary for Richardson-Gaudin models? Gaudin magnet with boundary [Hiami 1995]. Quasi-classical limit of the BQISM [Di Lorenzo et al. 00]. Generalized Gaudin systems [Srypny 006, 007, 010]. Trigonometric Gaudin model [Cirilio António et al. 013].

4 Outline In the quasi-classical limit (QC) QC trig. BQISM variable change QC trig. QISM variable change QC rat. BQISM QC rat. QISM

5 R-matrix R-matrix is an operator R(u) End(V V ) (V = C, u C) satisfying the Yang-Baxter equation (YBE) in End(V V V ): R 1 (u v)r 13 (u)r 3 (v) = R 3 (v)r 13 (u)r 1 (u v) Rational solution (η C, P(u v) = v u, u, v V ) u + η R rat (u) = 1 1 (ui I + ηp) = u + η u + η 0 u η 0 0 η u u + η. Trigonometric solution sinh(u + η) R trig 1 (u) = 0 sinh u sinh η 0 sinh(u + η) 0 sinh η sinh u sinh(u + η) Rational limit sinh(νx) lim = x : Trigonometric Rational ν 0 ν

6 QISM [Faddeev et al. 1979] Monodromy matrix End ( V a V L) (where V a = C is the auxiliary space, V L = V V... V is the quantum space, L N) L times ( ) ( ) e ηγ 0 A(u) B(u) T a (u) := 0 e ηγ R al (u ε L )...R a (u ε )R a1 (u ε 1 ) = C(u) D(u) Transfer matrix t(u) := tr a (T a (u)) = A(u) + D(u) End(V L ): [t(u), t(v)] = 0 u, v C Then t(u) generates a set of mutually commuting operators {C j }: t(u) = C j u j. j= Tae any function of {C j } as the Hamiltonian. Then {C j } are mutually commuting integrals of motion.

7 Algebraic Bethe Ansatz Start with a reference state Ω V L : B(u)Ω = 0, A(u)Ω = a(u)ω, D(u)Ω = d(u)ω, C(u)Ω 0. Then (for the trigonometric R-matrix) Φ(v 1,..., v N ) = C(v 1 )...C(v N )Ω is an eigenstate of t(u) with the eigenvalue Λ(u, v 1,..., v N ) = a(u) N =1 sinh(u v + η) sinh(u v ) + d(u) N =1 if Φ 0 and v s satisfy the Bethe Ansatz equations (BAE) sinh(u v η), sinh(u v ) a(v ) N d(v ) = sinh(v v i η) sinh(v v i + η), = 1,..., N For the rational R-matrix: sinh(x) x.

8 Algebraic Bethe Ansatz Start with a reference state Ω V L : B(u)Ω = 0, A(u)Ω = a(u)ω, D(u)Ω = d(u)ω, C(u)Ω 0. Then (for the trigonometric R-matrix) Φ(v 1,..., v N ) = C(v 1 )...C(v N )Ω is an eigenstate of t(u) with the eigenvalue Λ(u, v 1,..., v N ) = a(u) N =1 sinh(u v + η) sinh(u v ) + d(u) N =1 if Φ 0 and v s satisfy the Bethe Ansatz equations (BAE) sinh(u v η), sinh(u v ) a(v ) N d(v ) = sinh(v v i η) sinh(v v i + η), = 1,..., N For the rational R-matrix: sinh(x) x.

9 Quasi-classical limit (QC) ( ) L 0 Derive the expressions for a(u) and d(u) for Ω = : 1 a(u) = e ηγ L Then the BAE: e ηγ sinh(u ε l η/) L, d(u) = e ηγ sinh(u ε l ) L sinh(u ε l + η/). sinh(u ε l ) sinh(v ε l η/) N sinh(v ε l + η/) = sinh(v v i η) sinh(v v i + η). Tae first non-zero term as η 0: QISM η 0 QC QISM

10 Quasi-classical limit (QC) ( ) L 0 Derive the expressions for a(u) and d(u) for Ω = : 1 a(u) = e ηγ L Then the BAE: e ηγ sinh(u ε l η/) L, d(u) = e ηγ sinh(u ε l ) L sinh(u ε l + η/). sinh(u ε l ) sinh(v ε l η/) N sinh(v ε l + η/) = sinh(v v i η) sinh(v v i + η). Tae first non-zero term as η 0: QISM η 0 QC QISM

11 QC trig. QISM [Amico et al. 001], [Duelsy et al. 001] L N γ + coth(v ε l ) = coth(v v i ) ( ) QC rat. QISM [Richardson 1963] γ + L 1 v ε l = N v v i ( ) QC trig. QISM ( ) QC rat. QISM ( ) Change of variables v i = ln y i, ε l = ln z l in ( ) gives L y (N L/ + γ 1) + y z l = N y y y i ( )

12 QC trig. BQISM variable change QC trig. QISM ( ) variable change QC rat. BQISM QC rat. QISM ( )

13 Reflection equations [Cheredni 1984] Start with the trigonometric R-matrix. In addition to the YBE we require that it satisfies the reflection equations for some K ± End(V ), referred to as the reflection matrices: { R1 (u v)k 1 (u)r 1(u + v)k (v) = K (v)r 1(u + v)k 1 (u)r 1(u v), R 1 (v u)k + 1 (u)r 1(u + v)k + (v) = K + (v)r 1(u + v)k + 1 (u)r 1(v u), where R(u) R( u η). Easy to chec that ( K sinh(ξ (u) = ) + u) 0 0 sinh(ξ, u) ( K + sinh(ξ (u) = + ) + u + η) 0 0 sinh(ξ + u η) satisfy these equations for any ξ ± C.

14 BQISM [Slyanin 1988] Define the double row monodromy matrix End ( V a V L) : T a (u) := R al (u ε L )...R a1 (u ε 1 )Ka (u) ( ) Ra1 1 ( u ε 1)...R 1 A(u) B(u) al ( u ε L) =. C(u) D(u) The transfer matrix satisfies t(u) := tr a ( K + a (u)t a (u) ) [t(u), t(v)] = 0 u, v C Thus, it is a generating function for the integrals of motion!

15 Algebraic Bethe Ansatz Introduce ã(u) = (u)a(u) ηd(u). Start with a reference state Ω and loo for other eigenstates in the form Φ(v 1,..., v N ) = C(v 1 )...C(v N )Ω. Eigenvalues: Λ(u, v 1,..., v N ) = ã(u) sinh(ξ+ + u + η/) sinh u + d(u) sinh(u + η) sinh(ξ+ u + η/) sinh u BAE: N =1 N =1 sinh(u v + η) sinh(u + v + η) + sinh(u v ) sinh(u + v ) sinh(u v η) sinh(u + v η), sinh(u v ) sinh(u + v ) ã(v ) sinh(ξ + + v + η/) N d(v ) sinh(v η) sinh(ξ + v + η/) = sinh(v v i η) sinh(v + v i η) sinh(v v i + η) sinh(v + v i + η)

16 Quasi-classical limit (QC) If we substitute η = 0 the BAE will tae the following form: sinh(ξ + v ) sinh(ξ + + v ) sinh(ξ v ) sinh(ξ + v ) = 1. Choose ξ = ξ (η), ξ + = ξ + (η), so that this holds as η 0. Tae QC trig. BQISM ξ + = ηα, ξ = ηβ L (α + β + 1) coth v + (coth(v ε l ) + coth(v + ε l )) = N = (coth(v v i ) + coth(v + v i )) ( )

17 QC rat. BQISM L v (α + β + 1) + v ε l = N v v v i ( ) Same expression as ( ) QC trig. QISM with the change of variables! QC trig. BQISM QC rat. BQISM ( ) QC trig. QISM ( ) Change of variables v = ln y, ε l = ln z l in ( ) gives (α + β + 1) y + 1 L y 1 + ( y y z l = N ( y y y i + y y i 1 ) 1 + y = z l 1 ) ( )

18 QC trig. BQISM ( ) variable change QC trig. QISM ( ) variable change ( ) ( ) QC rat. BQISM ( ) QC rat. QISM ( )

19 Introduce an additional parameter ρ: v v + ρ, ε l ε l + ρ. QC rat. BQISM ( ) ρ QC rat. QISM ( ) L v (α + β + 1) + + ρv + ρ /4 N v ε l + ρ(v ε l ) = v + ρv + ρ /4 v v i + ρ(v v i ). Rescale α + β = ρ(α + β )/4 and consider ρ : (α + β ) + QC trig. BQISM ( ) (α+β+1) σ y + 1 L ( y σ y 1+ y z l and consider σ : (α + β + 1) + L 1 v ε l = N v v i. ρ QC trig. QISM ( ) Set ρ/ = ln σ: + L ) 1 σ 4 y = z l 1 y y z l = N N ( ) y y y i + σ 4 y y i 1 y y y i.

20 Thus, we have QC trig. BQISM ( ) QC trig. QISM ( ) ( ) ( ) QC rat. BQISM ( ) QC rat. QISM ( )

21 But in fact QC trig. BQISM ( ) variable change QC trig. QISM ( ) variable change ( ) ( ) QC rat. BQISM ( ) QC rat. QISM ( )

22 Variable change Substitute v y y 1, ε l z l z 1 l (α+β+1) y + y 1 y y 1 Using we obtain ( ): + L into ( ): y y y + y zl z = l y y y + y zl z = y y l z l (α+β+1) y + 1 L ( y y 1+ y z l QC rat. BQISM ( ) + ) 1 y = z l 1 + N 1 y z l 1 N y y y + y yi y i ( ) y y y i + y y i. 1 variable change QC trig. BQISM ( ).

23 Than you for your attention! QC trig. BQISM ( ) variable change QC trig. QISM ( ) variable change ( ) ( ) QC rat. BQISM ( ) QC rat. QISM ( )