The XYZ spin chain/8-vertex model with quasi-periodic boundary conditions Exact solution by Separation of Variables

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1 The XYZ spin chain/8-vertex model with quasi-periodic boundary conditions Exact solution by Separation of Variables Véronique TERRAS CNRS & Université Paris Sud, France Workshop: Beyond integrability. The mathematics and physics of integrability and its breaking in low-dimensional strongly correlated quantum phenomena July 13-17, 2015 CRM Montreal In collaboration with G. Niccoli.

2 8-vertex model 2-d square lattice model link ɛ j = ± vertex Boltzmann weight R 8V (z 1 z 2) ɛ 1,ɛ 2 ɛ 1,ɛ 2 = z 2 ɛ 2 ɛ 1 ɛ 1 ɛ 2 z 1 a 8V b 8V c 8V d 8V

3 8-vertex model 2-d square lattice model link ɛ j = ± vertex Boltzmann weight R 8V (z 1 z 2) ɛ 1,ɛ 2 ɛ 1,ɛ 2 = z 2 ɛ 2 ɛ 1 ɛ 1 ɛ 2 R 8V (z) = 0 a 8V (z) 0 0 d 8V 1 (z) B 0 b 8V (z) c 8V (z) 0 0 c 8V (z) b 8V (z) 0 A d 8V (z) 0 0 a 8V (z) z 1 z : spectral parameter p = e iπω : elliptic parameter a 8V (z) = ρ θ 4(z 2ω) θ 4 (η 2ω) θ 4 (z η 2ω) θ 4 (0 2ω), b8v (z) = ρ θ 1(z 2ω) θ 4 (η 2ω) θ 1 (z η 2ω) θ 4 (0 2ω), c 8V (z) = ρ θ 4(z 2ω) θ 1 (η 2ω) θ 1 (z η 2ω) θ 4 (0 2ω), d8v (z) = ρ θ 1(z 2ω) θ 1 (η 2ω) θ 4 (z η 2ω) θ 4 (0 2ω).

4 8-vertex model 2-d square lattice model link ɛ j = ± vertex Boltzmann weight R 8V 12 (z 1 z 2) ɛ 1,ɛ 2 ɛ 1,ɛ 2 = z 2 ɛ 2 ɛ 1 ɛ 1 ɛ 2 R 8V 12 (z) = 0 a 8V (z) 0 0 d 8V 1 (z) B A 0 b 8V (z) c 8V (z) 0 0 c 8V (z) b 8V (z) 0 d 8V (z) 0 0 a 8V (z) z 1 End(V 1 V 2) V i C 2 satisfying the Quantum Yang-Baxter Equation (QYBE) on V 1 V 2 V 3, V i C 2 : R 8V 12 (z 1 z 2) R 8V 13 (z 1) R 8V 23 (z 2) = R 8V 23 (z 2) R 8V 13 (z 1) R 8V 12 (z 1 z 2)

5 Commuting transfer matrices for the 8-vertex model Monodromy Matrix: (on V 0 V N, V N = V 1 V 2... V N, V i C 2 ) satisfying M (8V) 0 (λ) = R (8V) 0N (λ ξ N) R (8V) 02 (λ ξ 2) R (8V) 01 (λ ξ 1) «A (8V) (λ) B (8V) (λ) = C (8V) (λ) D (8V) (λ) R (8V) 00 (λ 1 λ 2) M (8V) 0 (λ 1) M (8V) 0 (λ 2) = M (8V) 0 (λ 2) M (8V) 0 (λ 1) R (8V) 00 (λ 1 λ 2) commutation relations for A (8V), B (8V), C (8V), D (8V) [0] Transfer Matrix: T (8V) (λ) = tr 0 M(8V) 0 (λ) [T (8V) (u), T (8V) (v)] = 0

6 Commuting transfer matrices for the 8-vertex model Monodromy Matrix: (on V 0 V N, V N = V 1 V 2... V N, V i C 2 ) satisfying M (8V) 0 (λ) = R (8V) 0N (λ ξ N) R (8V) 02 (λ ξ 2) R (8V) 01 (λ ξ 1) «A (8V) (λ) B (8V) (λ) = C (8V) (λ) D (8V) (λ) R (8V) 00 (λ 1 λ 2) M (8V) 0 (λ 1) M (8V) 0 (λ 2) = M (8V) 0 (λ 2) M (8V) 0 (λ 1) R (8V) 00 (λ 1 λ 2) commutation relations for A (8V), B (8V), C (8V), D (8V) [0] Transfer Matrix: (periodic boundary conditions) T (8V) (λ) = tr 0 M(8V) 0 (λ) [T (8V) (u), T (8V) (v)] = 0

7 Commuting transfer matrices for the 8-vertex model Monodromy Matrix: (on V 0 V N, V N = V 1 V 2... V N, V i C 2 ) satisfying M (8V) 0 (λ) = R (8V) 0N (λ ξ N) R (8V) 02 (λ ξ 2) R (8V) 01 (λ ξ 1) «A (8V) (λ) B (8V) (λ) = C (8V) (λ) D (8V) (λ) R (8V) 00 (λ 1 λ 2) M (8V) 0 (λ 1) M (8V) 0 (λ 2) = M (8V) 0 (λ 2) M (8V) 0 (λ 1) R (8V) 00 (λ 1 λ 2) Transfer Matrix: T (8V) (λ) = tr 0 M(8V) 0 (λ) [T (8V) (u), T (8V) (v)] = 0 Remark. [R (8V) (λ), σ α σ α ] = 0 for α = x, y, z for any fixed α, T α (8V) (λ) = tr 0 σα 0 M (8V) 0 (λ) defines a one-parameter family of commuting quasi-periodic transfer matrices log T (8V) α (λ) = H XYZ = 1 NX λ 2 λ=0 ξ n=0 with σ β N1 = σα 1 σ β 1 σα 1 n=1 [0] n o J xσnσ x n1j x y σnσ y y n1 Jzσz nσn1 z 1 2 J0, Goal: find the (complete set of) eigenvalues and eigenstates of T α (8V) (λ)

8 Commuting transfer matrices for the 8-vertex model Goal: However: Monodromy Matrix: (on V 0 V N, V N = V 1 V 2... V N, V i C 2 ) satisfying M (8V) 0 (λ) = R (8V) 0N (λ ξ N) R (8V) 02 (λ ξ 2) R (8V) 01 (λ ξ 1) «A (8V) (λ) B (8V) (λ) = C (8V) (λ) D (8V) (λ) R (8V) 00 (λ 1 λ 2) M (8V) 0 (λ 1) M (8V) 0 (λ 2) = M (8V) 0 (λ 2) M (8V) 0 (λ 1) R (8V) 00 (λ 1 λ 2) Transfer Matrix: T (8V) (λ) = tr 0 M(8V) 0 (λ) [T (8V) (u), T (8V) (v)] = 0 T α (8V) (λ) = tr 0 σα 0 M (8V) 0 (λ) [T α (8V) [0] (u), T (8V) α (v)] = 0 find the (complete set of) eigenvalues and eigenstates of T α (8V) (λ) no simple reference state [X (8V) (λ), X (8V) (µ)] 0 for X = A, B, C, D not directly solvable by Bethe ansatz nor by separation of variables Baxter s solution (Ann.Phys.73) map onto an IRF model (SOS model)

9 (8V)SOS model (dynamical 6-vertex model) 2-d square lattice model vertex local height t j t j t k = ±η (adjacent) face Boltzmann weight R(λ i ξ j t) ɛ i,ɛ j = ɛ i,ɛ j ξ j t t ηɛ i t ηɛ j t η(ɛ i ɛ j ) λ i = t η(ɛ i ɛ j) t t t t t t t t 2 t t t t 1 b(u t) c(u t) t t t t t t t t 2 t t t t 1 b(u t) c(u t)

10 (8V)SOS model (dynamical 6-vertex model) 2-d square lattice model vertex local height t j t j t k = ±η (adjacent) face Boltzmann weight R(λ t) = B0 b(λ t) c(λ t) 0 0 c(λ t) b(λ t) 0A R(λ i ξ j t) ɛ i,ɛ j = ɛ i,ɛ j ξ j t t ηɛ i t ηɛ j t η(ɛ i ɛ j ) λ i = t η(ɛ i ɛ j) λ : spectral parameter t t 0 ηz : dynamical parameter b(λ t) = θ(t η) θ(λ) θ(t) θ(λ η) c(λ t) = θ(λ t) θ(η) θ(t) θ(λ η) θ(u) = θ 1 (u ω) satisfying the Dynamical Quantum Yang-Baxter Equation: R 12(λ 1 λ 2 t ησ3) z R 13(λ 1 λ 3 t) R 23(λ 2 λ 3 t ησ1) z = R 23(λ 2 λ 3 t) R 13(λ 1 λ 3 t ησ2) z R 12(λ 1 λ 2 t) Spin conservation, solvable by Bethe Ansatz

11 Baxter s Vertex-IRF Transformation It is equivalent to the following Dynamical Gauge Equivalence: R (8V) 12 (λ 12) S 1(λ 1 t) S 2(λ 2 t ησ1) z = S 2(λ 2 t) S 1(λ 1 t ησ2) z R (SOS) 12 (λ 12 t) «θ2( λ t 2ω) θ 2(λ t 2ω) with S(λ t) = λ θ 3( λ t 2ω) θ 3(λ t 2ω) 12 λ 1 λ 2 relation between monodromy matrices: where M (8V) 0 (λ) S 0(λ t) S q(t ησ z 0) = S q(t) S 0(λ t ηs z ) M (SOS) 0 (λ t) M (8V) 0 (λ) = R (8V) 0N (λ ξ N) R (8V) 02 (λ ξ 2) R (8V) 01 (λ ξ 1) N 1 X M (SOS) 0 (λ t) = R 0N (λ ξ N t η σa) z R 02(λ ξ 2 t ησ1) z R 01(λ ξ 1 t) a=1 N 1 X NX S q(t) = S 1(ξ 1 t) S 2(ξ 2 t ησ1) z... S N (ξ N t η σa) z and S z = Eigenvalues and eigenvectors of the (periodic) 8-vertex transfer matrix (Baxter, 1973) Restrictions: Lattice size N = 2n even Lη = m 1π m 2πω L N, m 1, m 2 Z (cyclic SOS model) a=1 n=1 σ z n

12 The SOS model: Dynamical Yang-Baxter algebra Felder, Varchenko (1996) : representations of E τ,η(sl 2) N 1 X M 0(λ t) = R 0N (λ ξ N t η σa) z R 02(λ ξ 2 t ησ1) z R 01(λ ξ 1 t) «a=1 A(u t) B(u t) = C(u t) D(u t) [0] satisfy R 00 (λ 00 t ηs z ) M 0(λ 0 t) M 0 (λ 0 t ησ0) z = M 0 (λ 0 t) M 0(λ 0 t ησ z 0 ) R 00 (λ 00 t) with Sz = P N n=1 σz n introduce dynamical operators bτ and T b τ such that T b τ bτ = (bτ η) T b τ so as to simplify the commutation relations: «! A(u) B(u) bt τ 0 M 0(u) = = M C(u) D(u) 0(u bτ) [0] 0 T b 1 τ [0] satisfy R 00 (λ 00 bτ ηs z ) M 0(λ 0) M 0 (λ 0 ) = M 0 (λ 0 ) M 0(λ 0) R 00 (λ 00 bτ) operator algebra: A(u), B(u), C(u), D(u) act on H = V N D V N = N n=1v n (with V n C 2 ) : quantum spin space D : representation space of dynamical operators algebra

13 The SOS model: Representation spaces of the operator algebra A(u), B(u), C(u), D(u) act on H = V N D V N = N n=1v n (with V n C 2 ) : (2 N -dimensional) quantum spin space D : representation space of dynamical operators algebra basis: t(j), j Z (or j Z/LZ if Lη = m 1π m 2πω: cyclic case) with ˆτ t(j) = t(j) t(j), t(j) = t 0 ηj ˆT τ t(j) = t(j 1) Periodic transfer matrix: T (u) = tr 0{M 0(u)} = A(u) D(u) commuting family on the subspace H (0) V N [0] D of H associated to the zero eigenvalue of S z = P N n=1 σz n Antiperiodic transfer matrix: T (u) = tr 0{σ0M x 0(u)} = B(u) C(u) commuting family on the subspace H(0) of H associated to the zero eigenvalue of S τ ηs z 2ˆτ

14 The SOS model: Representation spaces of the operator algebra A(u), B(u), C(u), D(u) act on H = V N D V N = N n=1v n (with V n C 2 ) : (2 N -dimensional) quantum spin space D : representation space of dynamical operators algebra basis: t(j), j Z (or j Z/LZ if Lη = m 1π m 2πω: cyclic case) with ˆτ t(j) = t(j) t(j), t(j) = t 0 ηj ˆT τ t(j) = t(j 1) Periodic transfer matrix: T (u) = tr 0{M 0(u)} = A(u) D(u) commuting family on the subspace H (0) V N [0] D of H associated to the zero eigenvalue of S z = P N n=1 σz n Antiperiodic transfer matrix: T (u) = tr 0{σ0M x 0(u)} = B(u) C(u) commuting family on the subspace H(0) of H associated to the zero eigenvalue of S τ ηs z 2ˆτ

15 From the periodic SOS model to the periodic 8-vertex model with N even: solution by Bethe ansatz Periodic SOS transfer matrix: T (u) = tr 0{M 0(u)} = A(u) D(u) commuting family on the subspace H (0) V N [0] D Fun(V N [0]) of H associated to the zero eigenvalue of S z For generic values of η, the physical space of states H (0) model is: infinite-dimensional (unrestricted SOS model) for N = 2n even zero-dimensional for N odd except if Lη = m 1π m 2πω (cyclic case considered by Baxter): dim H (0) = L dim V N [0] of the periodic SOS The eigenstates of T (u) can be constructed by (algebraic) Bethe ansatz (Baxter 1973, Felder and Varchenko 1996) by means of the vertex-irf transformation, one can construct the eigenstates of the periodic 8-vertex transfer matrix in the case N = 2n even ( Baxter 1973 for Lη = m 1 π m 2 πω) Completeness?

16 From the antiperiodic SOS model to the (quasi-periodic) 8-vertex model: solution by SOV Antiperiodic SOS transfer matrix: T (u) = B(u) C(u) commuting family on the subspace H(0) of H = V N D associated to the zero eigenvalue (or more generally to the eigenvalue xπ yπω, x, y {0, 1}) of S τ ηs z 2ˆτ basis of the physical space of states H(0) : ` N n=1 n, h n t h, h (h 1,..., h N ) {0, 1} N h n = where t h = t 0 η NX k=1 Remark: η generic, (x, y) (0, 0) if N even ( 0 if spin 1 if spin h k with t 0 = η 2 N x π 2 y π ω, x, y {0, 1} 2 H(0) has dimension 2 N and is isomorphic to the space of states V N of the 8-vertex model construction of the eigenstates of the antiperiodic SOS model by separation of variables (Felder and Schorr 99, Niccoli 13, Levy-Bencheton Niccoli V.T. 15) complete set of eigenstates of the periodic (N odd, x, y = 0) and quasi-periodic ((x, y) (0, 0)) 8-vertex transfer matrices

17 From the antiperiodic SOS model to the (quasi-periodic) 8-vertex model: solution by SOV Antiperiodic SOS transfer matrix: T (u) = B(u) C(u) commuting family on the subspace H(0) of H = V N D associated to the zero eigenvalue (or more generally to the eigenvalue xπ yπω, x, y {0, 1}) of S τ ηs z 2ˆτ basis of the physical space of states H(0) : ` N n=1 n, h n t h, h (h 1,..., h N ) {0, 1} N h n = where t h = t 0 η NX k=1 Remark: η generic, (x, y) (0, 0) if N even ( 0 if spin 1 if spin h k with t 0 = η 2 N x π 2 y π ω, x, y {0, 1} 2 H(0) has dimension 2 N and is isomorphic to the space of states V N of the 8-vertex model: P : ` N n=1 n, h n t h ` N n=1 n, h n construction of the eigenstates of the antiperiodic SOS model by separation of variables (Felder and Schorr 99, Niccoli 13, Levy-Bencheton Niccoli V.T. 15) complete set of eigenstates of the periodic (N odd, x, y = 0) and

18 From the antiperiodic SOS model to the (quasi-periodic) 8-vertex model: solution by SOV Antiperiodic SOS transfer matrix: T (u) = B(u) C(u) commuting family on the subspace H(0) of H = V N D associated to the zero eigenvalue (or more generally to the eigenvalue xπ yπω, x, y {0, 1}) of S τ ηs z 2ˆτ basis of the physical space of states H(0) : ` N n=1 n, h n t h, h (h 1,..., h N ) {0, 1} N h n = where t h = t 0 η NX k=1 Remark: η generic, (x, y) (0, 0) if N even ( 0 if spin 1 if spin h k with t 0 = η 2 N x π 2 y π ω, x, y {0, 1} 2 H(0) has dimension 2 N and is isomorphic to the space of states V N of the 8-vertex model construction of the eigenstates of the antiperiodic SOS model by separation of variables (Felder and Schorr 99, Niccoli 13, Levy-Bencheton Niccoli V.T. 15) complete set of eigenstates of the periodic (N odd, x, y = 0) and quasi-periodic ((x, y) (0, 0)) 8-vertex transfer matrices

19 Sklyanin s quantum Separation of Variables (SOV): idea of the method! e Suppose that the monodromy matrix of the model M(λ) e A(λ) B(λ) e ec(λ) D(λ) e is such that e B(λ) is a (usual, trigonometric, elliptic... ) polynomial of degree N and is diagonalizable with simple spectrum, then the operator zeroes Y n, 1 n N, of e B(λ) can be used to define a basis y 1,..., y N, (y 1,..., y N ) Λ 1 Λ N, (Λ i Λ j = if i j) Y n y 1,..., y N = y n y 1,..., y N of the space of states in which the action of e A(λ) and e D(λ) is quasi-local, in particular ea(y n) y 1,..., y n,..., y N = a(y n) y 1,..., y n η,..., y N ed(y n) y 1,..., y n,..., y N = d(y n) y 1,..., y n η,..., y N

20 Sklyanin s quantum Separation of Variables (SOV): idea of the method Suppose that the monodromy matrix of the model M(λ) e A(λ) e B(λ) e ec(λ) D(λ) e such that e B(λ) is a (usual, trigonometric, elliptic... ) polynomial of degree N and is diagonalizable with simple spectrum, then the operator zeroes Y n, 1 n N, of e B(λ) can be used to define a basis y 1,..., y N, (y 1,..., y N ) Λ 1 Λ N, (Λ i Λ j = if i j) Y n y 1,..., y N = y n y 1,..., y N of the space of states in which the action of e A(λ) and e D(λ) is quasi-local The multi-dimensional spectral problem for the transfer matrix et (λ) = A(λ) e D(λ) e can be reduced to a set of N one-dimensional spectral problems by separation of variables: T e (λ) Ψt = t(λ) Ψ t iff Ψ t = X y 1,...,y N where each Q (n) t ψ t(y 1,..., y N ) y 1,..., y N with ψ t(y 1,..., y N ) = is solution of a discrete finite-difference equation NY n=1! is Q (n) t (y n)

21 Solution of the antiperiodic SOS model by SOV: Spectrum and eigenstates of the antiperiodic transfer matrix SOV basis of H(0) : h, h = (h 1,..., h N ) {0, 1} N For any fixed N-tuple of inhomogeneities (ξ 1,..., ξ N ) C N satisfying ɛ { 1, 0, 1}, ξ j ξ k ɛη / πz πωz if j k, the spectrum Σ (SOS) of the antiperiodic SOS transfer matrix T (λ) in H(0) is simple and coincides with the set of functions of the form t(λ) = NX k=j e iy(ξ j λ) θ(t 0 λ ξ j ) θ(t 0 ) Y k j θ(λ ξ k ) θ(ξ j ξ k ) t(ξ j ), which satisfy the discrete system of equations, j {1,. (.., N}, t(ξ j ) t(ξ j η) = ( 1) xyxy a(λ) = Q N n=1 a(ξ j ) d(ξ j η), with θ(λ ξnη) d(λ) = a(λ η) The T (λ)-eigenstate associated with the eigenvalue t(λ) Σ (SOS) is Ψ (SOS) t = X NY Q t(ξ j ηh j ) h h {0,1} N j=1 where the coefficients Q t(ξ j ηh j ) are (up to an overall normalization) characterized by Q t(ξ j η) = t(ξ j ) Q t(ξ j ) d(ξ j η) = a(ξ j ) ( 1)xyxy t(ξ j η)..

22 From antiperiodic SOS transfer matrix to quasi-periodic 8-vertex transfer matrix: the case (x, y) (0, 0) When (x, y) (0, 0), the vertex-irf transformation S q P S q(ˆτ) is an isomorphism from H(0) (antip. SOS space of states) to V N (8V space of states). Define the (x, y)-twisted 8-vertex transfer matrix: h i T (8V) (x,y) (λ) = tr0 (σ0) x y (σ0) z x M (8V) 0 (λ) It has the following action on the states v of H(0) : T (8V) (x,y) (λ) Sq v = ( 1)x i xy S q T (SOS) (λ) v. Complete characterization of the (x, y)-twisted 8-vertex transfer matrix spectrum and eigenstates: Let (x, y) (0, 0). If Ψ (SOS) t H(0) is an eigenvector of the antiperiodic SOS transfer matrix T (SOS) (λ) with eigenvalue t(λ), then S q Ψ (SOS) t V N is an eigenvector of the quasi-periodic 8-vertex transfer matrix T (8V) (x,y) eigenvalue t (8V) (x,y) (λ) ( 1)x i xy t(λ), and conversely. (λ) with

23 From antiperiodic SOS transfer matrix to quasi-periodic 8-vertex transfer matrix: the case (x, y) (0, 0) When (x, y) (0, 0), the vertex-irf transformation S q P S q(ˆτ) is an isomorphism from H(0) (antip. SOS space of states) to V N (8V space of states). Define the (x, y)-twisted 8-vertex transfer matrix: h i T (8V) (x,y) (λ) = tr0 (σ0) x y (σ0) z x M (8V) 0 (λ) It has the following action on the states v of H(0) : T (8V) (x,y) (λ) Sq v = ( 1)x i xy S q T (SOS) (λ) v. Complete characterization of the (x, y)-twisted 8-vertex transfer matrix spectrum and eigenstates: Let (x, y) (0, 0). If Ψ (SOS) t H(0) is an eigenvector of the antiperiodic SOS transfer matrix T (SOS) (λ) with eigenvalue t(λ), then S q Ψ (SOS) t V N is an eigenvector of the quasi-periodic 8-vertex transfer matrix T (8V) (x,y) eigenvalue t (8V) (x,y) (λ) ( 1)x i xy t(λ), and conversely. (λ) with

24 From antiperiodic SOS transfer matrix to quasi-periodic 8-vertex transfer matrix: the case (x, y) (0, 0) When (x, y) (0, 0), the vertex-irf transformation S q P S q(ˆτ) is an isomorphism from H(0) (antip. SOS space of states) to V N (8V space of states). Define the (x, y)-twisted 8-vertex transfer matrix: h i T (8V) (x,y) (λ) = tr0 (σ0) x y (σ0) z x M (8V) 0 (λ) It has the following action on the states v of H(0) : T (8V) (x,y) (λ) Sq v = ( 1)x i xy S q T (SOS) (λ) v. Complete characterization of the (x, y)-twisted 8-vertex transfer matrix spectrum and eigenstates: Let (x, y) (0, 0). If Ψ (SOS) t H(0) is an eigenvector of the antiperiodic SOS transfer matrix T (SOS) (λ) with eigenvalue t(λ), then S q Ψ (SOS) t V N is an eigenvector of the quasi-periodic 8-vertex transfer matrix T (8V) (x,y) eigenvalue t (8V) (x,y) (λ) ( 1)x i xy t(λ), and conversely. (λ) with

25 From the antiperiodic SOS transfer matrix to the periodic 8-vertex transfer matrix in the case N odd When (x, y) = (0, 0), S q is not bijective from H(0) to V N but, from the symmetry of the periodic (resp. antiper.) transfer matrices: Γ z T (8V) (0,0)(λ) Γz = T(8V) (0,0) (λ), Γz T (SOS) (λ) Γ z = T (SOS) (λ), with Γ z = N n=1σn z it is possible to define a second vertex-irf transformation S q = Γ z S q Γ z, such that T (8V) (0,0) (λ) Sq v = Sq T (SOS) (λ) v T (8V) (0,0) (λ) Sq v = Sq T (SOS) (λ) v the spectrum and eigenstates of T (SOS) (λ) can be decomposed into a part, with eigenvalues t (λ) and eigenstates Ψ (SOS) t, a part, with eigenv. t (λ) = t (λ) and eigenst. Ψ (SOS) t = Γ z Ψ (SOS) t ker S q ker S (0) q = {0} and ker S q is generated by the type eigenstates Ψ (SOS) t ker S q is generated by the type eigenstates Ψ (SOS) t

26 From the antiperiodic SOS transfer matrix to the periodic 8-vertex transfer matrix in the case N odd When (x, y) = (0, 0), S q is not bijective from H(0) to V N but, from the symmetry of the periodic (resp. antiper.) transfer matrices: Γ z T (8V) (0,0)(λ) Γz = T(8V) (0,0) (λ), Γz T (SOS) (λ) Γ z = T (SOS) (λ), with Γ z = N n=1σn z it is possible to define a second vertex-irf transformation S q = Γ z S q Γ z, such that T (8V) (0,0) (λ) Sq v = Sq T (SOS) (λ) v T (8V) (0,0) (λ) Sq v = Sq T (SOS) (λ) v the spectrum and eigenstates of T (SOS) (λ) can be decomposed into a part, with eigenvalues t (λ) and eigenstates Ψ (SOS) t, a part, with eigenv. t (λ) = t (λ) and eigenst. Ψ (SOS) t = Γ z Ψ (SOS) t ker S q ker S (0) q = {0} and ker S q is generated by the type eigenstates Ψ (SOS) t ker S q is generated by the type eigenstates Ψ (SOS) t

27 From the antiperiodic SOS transfer matrix to the periodic 8-vertex transfer matrix in the case N odd When (x, y) = (0, 0), S q is not bijective from H(0) to V N but, from the symmetry of the periodic (resp. antiper.) transfer matrices: Γ z T (8V) (0,0)(λ) Γz = T(8V) (0,0) (λ), Γz T (SOS) (λ) Γ z = T (SOS) (λ), with Γ z = N n=1σn z it is possible to define a second vertex-irf transformation S q = Γ z S q Γ z, such that T (8V) (0,0) (λ) Sq v = Sq T (SOS) (λ) v T (8V) (0,0) (λ) Sq v = Sq T (SOS) (λ) v the spectrum and eigenstates of T (SOS) (λ) can be decomposed into a part, with eigenvalues t (λ) and eigenstates Ψ (SOS) t, a part, with eigenv. t (λ) = t (λ) and eigenst. Ψ (SOS) t = Γ z Ψ (SOS) t ker S q ker S (0) q = {0} and ker S q is generated by the type eigenstates Ψ (SOS) t ker S q is generated by the type eigenstates Ψ (SOS) t

28 From the antiperiodic SOS transfer matrix to the periodic 8-vertex transfer matrix in the case N odd Complete characterization of the periodic 8-vertex transfer matrix spectrum and eigenstates for N odd: The spectrum of the periodic 8-vertex transfer matrix T (8V) (0,0)(λ) for N odd is Σ (8V) (SOS) (0,0) = Σ, (1) (SOS) where Σ is the part of the antiperiodic SOS transfer matrix spectrum Σ (SOS). Each of the 2 N 1 T (8V) (0,0)(λ)-eigenvalues t(λ) Σ(8V) (0,0) = Σ (SOS) is doubly degenerated, with two linearly independent T (8V) (0,0)(λ)-eigenvectors given by S q Ψ (SOS) t and S q Γ z Ψ (SOS) t, (2) where Ψ (SOS) t denotes the T (SOS) (λ)-eigenvector with eigenvalue t(λ).

29 Going further: from the discrete characterization of the spectrum to a functional T Q equation SOV characterization of the spectrum/eigenstates of the transfer matrix: eigenvalue t(λ) characterized by its functional form ( elliptic polynomial of a certain type) the fact that it satisfies a discrete system of equations at the (shifted) inhomogeneity parameters ξ n ηh n, h n {0, 1}, i.e. that, for each n! {1,..., N}, there exists a non-zero vector Q (n) t(ξ n ηh n) q (hn) n q(0) n q (1) n s.t. = ( 1) xyxy a(ξ n ηh n) q (hn1) n d(ξ n ηh n) q (hn 1) n, h n {0, 1} The corresponding eigenvector Ψ (SOS) t is constructed in terms of Q (n) Question: Does it exist, for each t(λ) Σ (SOS), an entire function Q(λ) s.t. for each n {1,..., N}, Q(ξ n ηh n) = q (hn) n t(λ) Q(λ) = ( 1) xyxy a(λ) Q(λ η) d(λ) Q(λ η)? If yes, what is the functional form of Q(λ) ( Bethe equations )?

30 From the discrete characterization of the spectrum to a functional T Q equation Question: Does it exist, for each t(λ) Σ (SOS), an entire function Q(λ) s.t. t(λ) Q(λ) = ( 1) xyxy a(λ) Q(λ η) d(λ) Q(λ η) (3) and (Q(ξ n), Q(ξ n η)) (0, 0)? If yes, what is the functional form of Q(λ) ( Bethe equations )? Remark 1. The algebraic construction of the Q-operator (and the knowledge of the functional form of its eigenvalues) provides in principle a solution to this problem Remark 2. From analyticity/periodicity arguments, one can guess the functional form of Q(λ). Y N Ansatz: Q(λ) = e αλ θ X (λ λ j ) j=1 8 >< θ λ 1` ω if x = 0, 2 2 with θ X (λ) = θ 1(λ 2ω) if y = 0, >: e i λ 2 θ1` λ 2 ω θ1` λππω 2 ω if x = y. restrictions on α and P i λ i. Is it possible to prove the completeness of this solution?

31 From the discrete characterization of the spectrum to a functional T Q equation Let t(λ) be an eigenvalue of the antiperiodic SOS transfer matrix T (λ). Then, if N is even, there exists a unique function Q(λ) of the form NY Q(λ) = θ X (λ λ j ), (4) j=1 for some set of roots λ 1,..., λ N C, such that t(λ) and Q(λ) satisfy the T -Q functional equation t(λ) Q(λ) = ( 1) xyxy a(λ) Q(λ η) d(λ) Q(λ η). In (4), the notation θ X (λ) stands for the function 8 >< θ λ 1` ω if (x, y) = (0, 1), 2 2 θ X (λ) = θ 1(λ 2ω) if (x, y) = (1, 0), >: e i λ 2 θ1` λ ω θ λππω 2 1` ω if (x, y) = (1, 1). 2 Remark. In the XXZ limit (y = 1, ω i ) we have shown the completeness of the solution NY λ λj Q(λ) = sin 2 j=1 for N even or odd (5)

32 From the discrete characterization of the spectrum to a functional T Q equation Let t(λ) be an eigenvalue of the antiperiodic SOS transfer matrix T (λ). Then, if N is even, there exists a unique function Q(λ) of the form NY Q(λ) = θ X (λ λ j ), (4) j=1 for some set of roots λ 1,..., λ N C, such that t(λ) and Q(λ) satisfy the T -Q functional equation t(λ) Q(λ) = ( 1) xyxy a(λ) Q(λ η) d(λ) Q(λ η). In (4), the notation θ X (λ) stands for the function 8 >< θ λ 1` ω if (x, y) = (0, 1), 2 2 θ X (λ) = θ 1(λ 2ω) if (x, y) = (1, 0), >: e i λ 2 θ1` λ 2 ω θ1` λππω 2 ω if (x, y) = (1, 1). Complete characterization of the (SOS and hence 8V) spectrum and eigenstates in terms of the solutions of the Bethe equations: NY ( 1) xyxy θ X (λ j λ k η) NY θ X (λ j λ k η) a(λ j ) d(λ j ) = 0, 1 j N θ X (λ j λ k ) θ X (λ j λ k ) k=1 k j k=1 k j (5)

33 Conclusion the periodic 8-vertex/XYZ model with an even number of sites can be solved by relating the periodic 8-vertex transfer matrix with the periodic SOS transfer matrix solution by Bethe ansatz (cf. Baxter s work... ) the periodic 8-vertex/XYZ model with an odd number of sites, as well as the twisted cases, can be solved by relating the (periodic or twisted) 8-vertex transfer matrix with the antiperiodic SOS transfer matrix solution by Separation of Variables complete description of the transfer matrix spectrum and eigenstates in terms of solutions of discrete equations evaluated at the inhomogeneity parameters of the model it is possible to reformulate this description in terms of some particular classes of solutions of a functional T -Q equation description in terms of Bethe-type equations enabling one to study the homogeneous/thermodynamic limit (completeness proven for N even)