Trigonometric SOS model with DWBC and spin chains with non-diagonal boundaries

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1 Trigonometric SOS model with DWBC and spin chains with non-diagonal boundaries N. Kitanine IMB, Université de Bourgogne. In collaboration with: G. Filali RAQIS 21, Annecy June 15 - June 19, 21 Typeset by FoilTEX RAQIS1, Annecy, 21

2 The XXZ spin-1/2 Heisenberg chain 1. Periodic chain. H bulk = M P m=1 `σx m σx m+1 + σy m σy m+1 + (σz m σz m+1 1) h 2 MP σ z m m=1 - anisotropy h- external magnetic field. Periodic boundary conditions: σ M+1 = σ Open chain H = M 1 X m=1 `σx m σx m+1 + σy m σy m+1 + (σz m σz m+1 1) h σ z 1 + C (e iφ σ e iφ σ 1 ) h + σ z M + C +(e iφ+ σ + M + e iφ+ σ M ) h ±, C ±, φ ± - boundary paprameters. Typeset by FoilTEX RAQIS1, Annecy, 21 1

3 Bethe Ansatz 1. Periodic case: coordinate Bethe ansatz: H. Bethe 1931, R. Orbach 1958 algebraic Bethe ansatz L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan Open chain, diagonal boundary terms: coordinate Bethe ansatz: F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter and G.R.W. Quispel algebraic Bethe ansatz E.K. Sklyanin Open chain, non-diagonal boundary terms Different forms of Bethe ansatz: J. Cao, H.-Q. Lin, K.-J. Shi, Y. Wang, 23, R. I. Nepomechie 23 Algebraic BA, Vertex-IRF correspondence: W.-L. Yang, Y.-Z. Zhang 27 Typeset by FoilTEX RAQIS1, Annecy, 21 2

4 Algebraic Bethe Ansatz, periodic case L.D. Faddeev, E.K. Sklyanin, L.A. Takhtajan (1979): 1. Yang-Baxter equation: R 12 (λ 12 ) R 13 (λ 13 ) R 23 (λ 23 ) = R 23 (λ 23 ) R 13 (λ 13 ) R 12 (λ 12 ). Trigonometric solution: R(λ) = Properties: sinh(λ + η) sinh λ sinh η sinh η sinh λ sinh(λ + η) 1 C A, = cosh η. R() = sinh η P Unitarity R(λ)R( λ) = f 1 (λ) I, Typeset by FoilTEX RAQIS1, Annecy, 21 3

5 Crossing symmetry σ y 1 Rt 1(λ η) σ y 1 R(λ) = f 2(λ) I. 2. Monodromy matrix. T a (λ) = R am (λ ξ M )... R a1 (λ ξ 1 ) = ξ j are arbitrary inhomogeneity parameters. Yang-Baxter algebra: generators A, B, C, D A(λ) «B(λ) C(λ) D(λ) commutation relations given by the R-matrix of the model Transfer matrix: t(λ) = tr a T a (λ) = A(λ) + D(λ) Commuting charges: [t(λ), t(µ)] = periodic Hamiltonian: H bulk = c λ R ab (λ µ) T a (λ)t b (µ) = T b (µ)t a (λ) R ab (λ µ) log t(λ) hs z + const. λ= [a] Conserved quantities: [H bulk, t(λ)] = Typeset by FoilTEX RAQIS1, Annecy, 21 4

6 Cherednik 1984 Reflection equation R 12 (λ µ) K 1 (λ) R 12 (λ + µ) K 2 (µ) = K 2 (µ) R 12 (λ + µ) K 1 (λ) R 12 (λ µ). General 2 2 solution (Ghoshal Zamolodchikov 1994): K cosh( + 2ξ )e λ e λ cosh e τ «sinh 2λ (λ) = f(λ) e τ sinh 2λ cosh( + 2ξ )e λ cosh e λ Second boundary: K + (λ) = K ( λ η), ξ ξ + We could replace +, τ τ + and get commuting charges. However to construct the eigenstates we use a constrain and keep 4 independent boundary parameters (instead of 6), τ, ξ ± Algebraic Bethe Ansatz, Sklyanin 1988 U (λ) = T(λ) K (λ) σ y T t ( λ) σ y = A (λ) C (λ) «B (λ), D (λ) Typeset by FoilTEX RAQIS1, Annecy, 21 5

7 1. Reflection algebra R 12 (λ µ) (U ) 1 (λ) R 12 (λ + µ η)(u ) 2 (µ) =(U ) 2 (µ) R 12 (λ + µ η) (U ) 1 (λ) R 12 (λ µ) 2. Transfer matrix: T (λ) = tr {K + (λ) U (λ)}. 3. Hamiltonian (homogeneous limit): d H = C 1 dλ T (λ) + constant. λ=η/2 [T (λ), T (µ)] = No reference state!, such that C (λ) =, λ Typeset by FoilTEX RAQIS1, Annecy, 21 6

8 Vertex-IRF transformation Cao et al: 8-vertex like scheme. Gauge transformation to diagonalize the boundary matrices. non diagonal K(λ) K(λ) diagonal, six vertex R(λ) R(λ, ) SOS, dynamical R-matrix S(λ; ) = exp (λ++τ) exp (λ +τ) «1 1 S(ξ, ) 1 S(λ, ησ z 1 ) R 1 (λ ξ, ) = R 1 (λ ξ)s(λ, ) S(ξ, ησ ) 1 K-matrix S 1 (λ, )K (λ, )S( λ, ) = K (λ, ) sinh( λ+ξ ) sinh(+λ+ξ ) sinh( λ+ξ ) sinh(λ+ξ ) 1 A Typeset by FoilTEX RAQIS1, Annecy, 21 7

9 Dynamical Yang-Baxter equation R(λ, ) = 1 sinh(λ + η) sinh λ sinh( η) sinh η sinh( λ) sinh sinh sinh η sinh(+λ) sinh λ sinh(+η) sinh sinh B A sinh(λ + η) satisfies the dynamical Yang Baxter equation (DYBE) R 12 (λ 1 λ 2, ησ z 3 )R 13(λ 1 λ 3, )R 23 (λ 2 λ 3, ησ z 1 ) =R 23 (λ 2 λ 3, )R 13 (λ 1 λ 3, ησ z 2 )R 12(λ 1 λ 2, ) Typeset by FoilTEX RAQIS1, Annecy, 21 8

10 Dynamical monodromy matrix T(λ, ) = R 1 (λ ξ 1, η Dynamical Yang-Baxter algebra NX σ z i )...R N(λ ξ N, ) = i=2 A(λ, ) «B(λ, ) C(λ, ) D(λ, ) R 12 (λ 1 λ 2, η NX σ z i )T 1(λ 1, )T 2 (λ 2, ησ z 1 ) i=1 =T 2 (λ 2, )T 1 (λ 1, ησ z 2 )R 12(λ 1 λ 2, ) Reflection equation R 12 (λ 1 λ 2, )K 1 (λ 1 )R 21 (λ 1 + λ 2, )K 2 (λ 2 ) =K 2 (λ 2 )R 12 (λ 1 + λ 2, )K 1 (λ 1, )R 21 (λ 1 λ 2, ) Typeset by FoilTEX RAQIS1, Annecy, 21 9

11 Dynamical reflection algebra Double row monodromy matrix U(λ, ) A(λ, ) «B(λ, ) C(λ, ) D(λ, ) = T(λ, )K(λ; ) bt(λ, ), bt(λ, ) =R N (λ + ξ N, )...R 1 (λ + ξ 1, η NX σ z i ) i=2 Reflection Algebra R 12 (λ 1 λ 2, η NX σ z i )U 1(λ 1, )R 21 (λ 1 + λ 2, η i=1 NX σ z i )U 2(λ 2, ) i=1 = U 2 (λ 2,)R 12 (λ 1 + λ 2, η NX NX σ z i )U 1(λ 1, )R 21 (λ 1 λ 2, η σ z i ) i=1 i=1 Typeset by FoilTEX RAQIS1, Annecy, 21 1

12 Algebraic Bethe Ansatz After the Vertex-IRF transformation there are two reference states:, (all the spins up, all the spins down): eigenstates of A(λ, ), D(λ, ) and C(λ, ) =, B(λ, ) =. Then we can construct states: Ψ 1 SOS ({λ}) = B(λ 1, )... B(λ M2, ), Ψ 2 SOS ({µ}) = C(µ 1, )... C(µ M2, ), For this boundary constraint: number of operators is always M/2 necessary to diagonalize K + (λ). More precisely, if {λ} is a solution of the corresponding Bethe equations tr K + (µ)u(µ, ) Ψ 1 SOS ({λ}) = Λ 1(µ, ({λ}) Ψ 1 SOS ({λ}) with a diagonal boundary matrix: K + (λ) sinh( η)sinh(+λ+η+ξ + ) sinh sinh( λ η+ξ + ) sinh(+η)sinh(λ+η+ξ + ) sinh sinh( λ η+ξ + ) 1 A Typeset by FoilTEX RAQIS1, Annecy, 21 11

13 Back to XXZ Eigenstates for the XXZ transfer matrix Ψ 1,2 XXZ ({λ}) = S 1...M() Ψ 1,2 SOS ({λ}) S 1...M () = S M (ξ M, )...S 1 (ξ 1, η If Ψ 1 SOS ({λ}) is a SOS eigenstate, then MX σ i ) i=2 T (µ) Ψ 1 XXZ ({λ}) = Λ 1(µ, ({λ}) Ψ 1 XXZ ({λ}) M/2 H XXZ Ψ 1 XXZ ({λ}) = X j=1 ǫ 1 (λ j ) Ψ 1 XXZ ({λ}) Typeset by FoilTEX RAQIS1, Annecy, 21 12

14 Partition function To compute the scalar products and then correlation functions a necessary step is to obtain the partition function of the corresponding two-dimensional model with domain wall boundary conditions (DWBC). Z M (λ 1,..., λ M ; ξ 1,..., ξ M ; ) = B(λ 1, )... B(λ M, ) This quantity can be interpreted as a partition function of a SOS model with the statistical weights given by the entries of R(λ, ): η 2η + η + 2η η η + η + η + η + η η η + η η Typeset by FoilTEX RAQIS1, Annecy, 21 13

15 with one reflecting end η + η and DWBC -η -(N-1)η -Nη -(N-1)η -(N-2)η -(N-3)η +(N-3)η +(N-2)η +(N-1)η +η +(N-1)η +Nη Typeset by FoilTEX RAQIS1, Annecy, 21 14

16 Six vertex and SOS partition functions The DWBC were first introduced for the six vertex model on a square lattice in the framework of the computation of the correlation functions of the periodic XXZ chain in 1982 (Izergin, Korepin) Korepin 1982: properties defining the partition function of the six vertex model with DWBC Izergin 1987: Determinant representation for the partition function of the six vertex model with DWBC (permits to compute correlation functions for the periodic XXZ chain) Tsuchiya 1998: Modification of Izergin s determinant for a six vertex model with reflecting end (permits to compute correlation functions for the open XXZ chain with diagonal boundaries). Rosengren 29: Representation for the partition function of the SOS model on a square lattice. Not a single determinant! Typeset by FoilTEX RAQIS1, Annecy, 21 15

17 Properties of the partition function 1. For each parameter λ i the normalized partition function! NX Z N ({λ}, {ξ}, ) = exp (2N + 2) λ i i=1 sinh( + ξ + λ i )sinh( + λ i )Z N ({λ}, {ξ}, ) is a polynomial of degree at most 2N + 2 in e 2λ i. 2. Z 1 (λ, ξ, ) can be easily computed as there are only two configurations possible. η η η + η + η + η 3. Z N ({λ}, {ξ}, ) is symmetric in λ j as [B(λ, ), B(µ, )] =. 4. Z N ({λ}, {ξ}, ) is symmetric in ξ k (simple consequence of the DYBE). Typeset by FoilTEX RAQIS1, Annecy, 21 16

18 5. Recursion relations: Setting λ 1 = ξ 1 we fix the configuration in the lower right corner (N 2)η... + Nη + (N 1)η + (N 2)η + (N 1)η + Nη fixes configuration of two lines and one column Z N ({λ}, {ξ}, ) is reduced to Z N 1 (λ 2,... λ N, ξ 2... ξ N, ) Setting λ N = ξ 1 we fix the configuration in the upper right corner (N 2)η (N 1)η Nη... Nη (N 1)η (N 2)η fixes configuration of two lines and one column Z N ({λ}, {ξ}, ) is reduced to Z N 1 (λ 1,... λ N 1, ξ 2... ξ N, ) Typeset by FoilTEX RAQIS1, Annecy, 21 17

19 Note: these are general conditions for the partition function, but here we have 2N conditions and polynomials of degree 2N + 2. We need more conditions. 6. Crossing symmetry: R-matrix: σ y 1 : Rt 1 12 ( λ η, + ησ z 1 ) : sinh( ησ z σy 2 ) 1 sinh = R 21 (λ, ) Normal ordering: σ z 1 in the argument of the R matrix (which does not commute with it) is always on the right of all other operators involved in the definition of R. It implies for the B operators (and hence for the partition function) B( λ η, ) = ( 1) N+1 sinh(λ + ζ) sinh(2(λ + η)) sinh(λ + ζ + ) B(λ, ) sinh(2λ)sinh(λ ζ + η)sinh(λ ζ + η) Now the polynomial of degree 2N + 2 is defined in 4N points Partition function is defined uniquely by these conditions. Typeset by FoilTEX RAQIS1, Annecy, 21 18

20 Determinant representation The unique function satisfying properties is the following determinant Y N sinh( + η(n 2i)) Z N,2N ({λ}, {ξ}, {}) = ( 1) N det M ij sinh( + η(n 2i + 1)) NQ i,j=1 Q 1 i<j N i=1 «N i+1 sinh(λ i + ξ j )sinh(λ i ξ j )sinh(λ i + ξ j + η) sinh(λ i ξ j + η) sinh(ξ j + ξ i ) sinh(ξ j ξ i ) sinh(λ j λ i ) sinh(λ j + λ i + η) M i,j = sinh( + ζ + ξ j) sinh( + ζ + λ i ) sinh(ζ ξ j) sinh(ζ + λ i ) sinh(2λ i )sinh η sinh(λ i ξ j + η) sinh(λ i + ξ j + η)sinh(λ i ξ j )sinh(λ i + ξ j ) Very similar to the six vertex case. Typeset by FoilTEX RAQIS1, Annecy, 21 19

21 Next steps 1. Scalar products (Dual reflection algebra + F-basis) 2. Boundary inverse problem? 3. Correlation functions 4. Some results for = 1 2, three colors model. 5. Weaker constraints for the boundary terms non-equilibrium systems (ASEP)? Typeset by FoilTEX RAQIS1, Annecy, 21 2