Thessaloniki, October 2010
Motivation Integrable b.c. interesting for integrable systems per ce, new info on boundary phenomena + learn more on bulk behavior. Examples of integrable b.c. that modify the bulk. Bigger picture, strong motivations nowadays: recent developments within the AdS/CFT context (Minahan, Zarembo 03). Important to study both quantum and classical integrable models. Further: recent results on open spin chains and open string theories (Agarwal, Hofman, Maldacena,...)
Outline Brief overview on quantum integrability (mathematical and physical description). Review periodic and open boundary conditions. Classical discrete integrable models: Review general setting. Lax pair and algebraic description. Generalize the boundary case. Rigorous universal results on IM and Lax pairs based on the underlying algebra. Examples: Discrete-self-trapping (DST) model (discrete NLS). Classical continuum integrable models: similar investigations; IM and boundary Lax pairs from the algebraic setting. Examples: NLS and sine-gordon models.
Review quantum integrability The Yang-Baxter equation (Baxter 72) R 12 (λ 1 λ 2 ) R 13 (λ 1 ) R 23 (λ 2 ) = R 23 (λ 2 ) R 13 (λ 1 ) R 12 (λ 1 λ 2 ) R acts on V V, YBE on V V V, and R 12 = R I, R 23 = I R etc. R physically describes scattering, YBE the factorization of multi-particle scattering. V associated to reps of underlying (deformed) Lie algebras.
Generalize the YBE to include generic reps of (deformed) Lie algebras (Faddeev, Takhtajan Reshetikhin): R 12 (λ 1 λ 2 ) L 1n (λ 1 ) L 2n (λ 2 ) = L 2n (λ 2 ) L 1n (λ 1 ) R 12 (λ 1 λ 2 ) L End(V A), A defined by fundamental algebraic relation above: deformed or quantum algebras. This describes physically integrable models with periodic bc.
Periodic integrable models Tensorial reps of the fundamental algebraic relation (Faddeev, Takhtajan 80 s): T 0 (λ) = L 0N (λ) L 0N 1 (λ)... L 01 (λ) T End(V A N ) satisfies the fundamental algebraic relation. The trace over the auxiliary space, defines the transfer matrix : t(λ) = tr 0 T 0 (λ) Via the RTT relations integrability is shown: [t(λ), t(λ )] = 0, λ, λ
Open integrable models Introduce the reflection or boundary YBE (Cherednik, Sklyanin 80s) 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 12 (λ 1 λ 2 ) K End(V ), the reflection matrix. Physically describes the reflection of a particle-like excitation with the boundary of the system.
Tensor reps of reflection equation (Sklyanin 83) T 0 (λ) = T 0 (λ) K 0 (λ) T 1 0 ( λ) Define the open transfer matrix t(λ) = tr 0 {K + 0 (λ) T 0(λ)} Using the reflection equation we show integrability Generating function of IM. [t(λ), t(λ )] = 0, λ, λ
Classical limit Now focus on classical models. Consider the classical limit of the R matrix as: R = 1 + r + O( 2 ) Then the r matrix satisfies the classical YBE (Semenov-Tian-Shansky 83) [r 12, r 13 ] + [r 12, r 23 ] + [r 13, r 23 ] = 0. The classical limit obtained by setting: 1 [, ] = {, } {L a (λ), L b (λ )} = [r ab (λ λ ), L a (λ)l b (λ )] Systematic study of classical limit (Avan, Doikou, Sfetsos 10).
Discrete integrable classical models: periodic b.c. Lax pair (L, A) for discrete integrable models, and the associated auxiliary problem ψ n+1 = L n ψ n ψ n = A n ψ n From the latter equations one obtains the discrete zero curvature condition: L n = A n+1 L n L n A n
L-operator and T satisfy the quadratic classical algebraic relation (Faddeev, Takhtajan 87): {L a (λ), L b (λ )} = [r ab (λ λ ), L a (λ)l b (λ )] We formulate: {T a (λ), L bn (λ )} = T a (N, n + 1; λ)r ab (λ λ )T a (n, 1; λ)l bn (λ ) L bn (λ )T a (N, n; λ)r ab (λ λ )T a (n 1, 1; λ) where T a (m, n; λ) = L am (λ)... L an (λ).
Take the trace over the auxiliary space and the log then: ) {ln t(λ), L(λ )} = t 1 tr a (T a (N, n + 1; λ)r ab (λ )T a (n, 1; λ) ) t 1 L bn (λ )tr a (T a (N, n; λ)r ab (λ )T a (n 1, 1; λ) λ = λ λ t is the generating function of all I.M.: ln t(λ) gives rise to all local Hamiltonians i.e. ln t(λ) = n H (n) λ n L bn The time evolution of L is given as: {ln t(λ), L(λ )} = L(λ )
The Lax pair obtained comparing with the zero curvature condition as: L n and ) A n (λ) = t 1 (λ)tr a (T a (N, n; λ)r ab (λ λ )T a (n 1, 1; λ) expansion in powers of λ provides all A s associated to all I.M. Expansion of ln t provides all local Hamiltonians. A n (λ) = n A (m) n λ m One to one correspondence between Lax pairs and I.M.
Discrete integrable classical models: open b.c. The underlying algebra: {T a (λ), T b (λ )} = r ab (λ )T a (λ)t b (λ ) T a (λ)t b (λ )r ba (λ ) + T a (λ)r ba (λ + )T b (λ ) T b (λ )r ab (λ + )T a (λ) λ ± = λ ± λ. Recall the tensorial rep of the algebra T a (λ) = T a (λ) K a (λ) T 1 ( λ) K a c-number rep of the reflection algebra.
Extract the associated boundary Lax pair (Avan, Doikou 07). Formulate {T a (λ), L bn (λ )} =... {T 1 a ( λ), L bn (λ )} =... Recall the generic rep of the reflection algebra and show that: {t(λ), L bn (λ )} =...
We read of the boundary quantity A n, which satisfies the zero curvature condition together with L: A n = tr a (K a + (λ)t a (N, n; λ)r ab (λ )T a (n 1, 1; λ)ka (λ) ˆT a (λ) +K a + (λ)t a (λ)ka (λ) ˆT a (1, n 1; λ)r ba (λ + ) ˆT ) a (n, N; λ) Special care at the boundary points n = 1, n = N, recall T (N, N + 1) = T (0, 1) = ˆT (1, 0) = ˆT (N + 1, N) = 1
Examples Focus next to simple models associated to the sl 2 -Yangian. The classical r matrix is: r(λ λ ) = P λ λ P is the permutation operator P (a b) = b a. Focus on the Discrete-Self-Trapping (DST) model L(λ) = (λ xx ) e 11 + b e 22 + b x e 12 X e 21 (e ij ) kl = δ ik δ jl, and {x n, X m } = δ nm
Expand the t(λ) and keep the first charge, which is the Hamiltonian of the model. Focus on the simplest case: K ± I then: H (2) = 1 2 N xnx 2 n 2 b n=1 N 1 n=1 x n+1 X n b2 2 x 2 1 1 2 X 2 N The associated equation of motion: L = {H, L}
The expansion of A n will provide the relevant boundary lax operator (Avan, Doikou 07): ( A (2) λ bx n = n X n 1 0 ( A (2) λ bx1 1 = bx 1 0 ), n {2,..., N} ), A (2) N 1 = ( λ XN X N 0 ) The associated equations of motion given as L n = A (i) n+1 L n L n A (i) n
For the specific example the relevant equations of motion are: ẋ n = xnx 2 n + bx n+1, Ẋ n = x n Xn 2, n {2,..., N 1} ẋ 1 = x1 2 X 1 + bx 2, Ẋ 1 = x 1 X1 2 bx 1 ẋ N = xn 2 X N + X N, Ẋ N = x N The Toda chain also obtained for the DST model. Specifically, set: X n e q n, x n e q n (b 1 + p n )
The harmonic oscillator algebra (x n, X n, x n X n ) reduces to the Euclidean Lie algebra (e ±qn, p n ) and the Lax operator: ( ) λ pn e L(λ) = qn e q n 0 The Hamiltonian is then: H (2) = 1 2 N i=1 N 1 pn 2 e q n+1 q n 1 2 e2q 1 1 2 e 2q N n=1
The relevant equations of motion for the open Toda chain: p n = q n, q n = e q n+1 q n e qn qn 1, n {2,..., N 1} p 1 = q 1, q 1 = e q 2 q 1 e 2q 1 p N = q N, q N = e 2q N e q N q N 1
Continuum integrable classical models on the full line Let U, V be the continuum Lax pair, and Ψ be the solution of the following set of equations: Ψ = U(x, t, λ) Ψ x Ψ = V(x, t, λ) Ψ t Compatibility condition of the above set gives the zero curvature condition U V + [U, V] = 0 Solution of the 1st equ. (monodromy): T (x, y; λ) = P exp{ x y U(x, t, λ)dx }
T satisfies the fundamental quadratic relation. Formulate (Faddeev, Takhtajan 87): {T a (L, L, λ), U(x, λ)} = M(x, λ, λ ) +[M(x, λ, λ ), U b (x, λ)] x where It then follows: M = T a (L, x, λ) r ab (λ λ ) T a (x, L, λ) {ln t(λ), U(x, λ)} = V x + [V(x, λ, λ ), U(x, λ)] V identified as: V(x, λ, λ ) = t 1 (λ)tr a (M ba )
Continuum integrable classical models on the interval Recalling that the rep of the reflection algebra is T a (x, y, λ) = T a (x, y, λ) K a (λ) ˆT a (x, y, λ) Formulating {T, U}, { ˆT, U} (Avan, Doikou 07): {ln t(λ), U(x, λ )} = V(x, λ, λ ) x + [V(x, λ, λ ), U(x, λ, λ )] and ) V(x, λ, λ ) = t 1 (λ)tr a (K a + (λ) M a (x, λ, λ )
The boundary quantity M is M = T (0, x, λ)r ab (λ λ )T (x, L, λ)k (λ) ˆT (0, L, λ)) + T (0, L, λ)k (λ) ˆT (x, L, λ)r ba (λ + λ ) ˆT (0, x, λ) pay particular attention at the boundary points (key point!) x = 0, L; take into account T (x, x, λ) = ˆT (x, x, λ) = I. Systematic derivation independent of the choice of model, as opposed to by hand construction in earlier investigations e.g. ATFT (Bowcock, Corrigan, Dorey, Rietdijk 95)
Example: boundary NLS model Consider the NLS model. The associated r matrix Recall the Lax pair r(λ) = P λ where U = U 0 + λu 1, V = V 0 + λv 1 + λ 2 V 2 U 1 = 1 2i (e 11 e 22 ), U 0 = ψe 12 + ψe 21 V 0 = i ψ 2 (e 11 e 22 ) i ψ e 12 + iψ e 21 V 1 = U 0, V 2 = U 1
The fields ψ, ψ are canonical {ψ(x), ψ(y)} = δ(x y) From the zero curvature conditions the equations of motion for NLS: ψ(x, t) i = 2 ψ(x, t) t 2 + 2 ψ(x, t) 2 ψ(x, t) x Idea: via the process described integrable boundary conditions (system on the half line).
Choose diagonal reflection matrix: K(λ) = λ(e 22 e 11 ) + iξi Consider the ansatz: ( ) ( ) 1 T (x, y, λ) = 1 + W (x, λ) e Z(x,y,λ) 1 + W (y, λ) Z is purely diagonal, W off diagonal. Determine Z, W from = UT, and find the associated integrals of motion. d dxt
Expanding the transfer matrix obtain the classical integrals of motion for NLS in the interval (Doikou, Fioravanti, Ravanini, 07): N = H = 0 L 0 L dx ψ(x) ψ(x) dx ( ) ψ(x) 4 + ψ (x) ψ (x) ψ(0) ψ (0) ψ (0) ψ(0) ξ + ψ(0) ψ(0) +ψ( L) ψ ( L) + ψ ( L) ψ( L) + ξ ψ( L) ψ( L) Only odd charges are conserved! E.g. momentum is not a conserved quantity anymore.
The corresponding equations of motion obtained from: ψ(x, t) = {H, ψ(x, t)}, x L x 0 ψ(x, t) x = {H, ψ(x, t)} And are of the form: ψ(x, t) i = 2 ψ(x, t) t 2 + 2 ψ(x, t) 2 ψ(x, t) x ( ψ(x) ) ξ ± ψ(x) x = 0. x=0, L Mixed boundary conditions.
Following the generic formulation described we identify the associated boundary Lax pair (Avan, Doikou 07): V b (x b, t) = V(x b, t) + V(x b, t) ( ) = V(x b, t) + i ψ(x b, t) 2 e 22 + λ ψ(xb, t)e 12 + ψ(x b, t)e 21 x b = 0, L. Obtain the equations of motion, and continuity arguments at the boundary point lead to V = 0 boundary conditions
Example: boundary sine-gordon The Lax pair for the sine Gordon model (u = e λ ) U(x, t, u) = β 4i π(x)+mu 4i e iβ 4 φσ 3 σ 2 e iβ 4 φσ 3 mu 1 e iβ 4 φσ 3 σ 2 e iβ 4 φσ 3 4i V(x, t, u) = β 4i φ (x)+ mu 4i e iβ 4 φσ 3 σ 2 e iβ 4 φσ 3 + mu 1 e iβ 4 φσ 3 σ 2 e iβ 4 φσ 3 4i
The classical r-matrix for the sine-gordon model (Jimbo 86 ) r(λ) = cosh λ sinh λ 2 i=1 e ii e ii + 1 sinh λ 2 i j=1 e ij e ji Choose the reflection matrix (Ghoshal, Zamolodchikov 94) ( sinh(λ + iξ) x K(λ) = + κ sinh(2λ) x κ sinh(2λ) sinh( λ + iξ) ), x x + = 1
The boundary Hamiltonian via the process described obtained (MacIntyre 95 ), also at quantum level (Ghoshal, Zamolodchikov 94): H = 0 L dx + 4Pm β 2 ( β 4i (π2 + φ 2 ) + m2 β 2 (1 cos βφ) ) βφ(0) cos 4Qm 2 β 2 sin βφ(0) 2 P = sin ξ 2iκ, Q = cos ξ 2iκ Generalize to ATFT s, full classification of integrable b.c., classical I.M. (Doikou 08)
The boundary Lax pair reads (Avan, Doikou 08): V (b) (0, t, u) = V(0, t, u) + V(0, t, u) V(0, t, u) = β 4i φ (0)σ 3 m 8κ cos(ξ + β 2 φ(0))σ 3 Different choices of reflection matrices modify the boundary conditions: Dirichlet, Neumann or mixed Generalize the construction of boundary Lax pairs to ATFT s (Avan, Doikou 08)
From the zero curvature condition and continuity requirement V = 0 we get the E.M and the boundary conditions: φ(x, t) φ (x, t) = m2 sin(βφ(x, t)) β βφ (0) = m 2iκ cos(ξ + β 2 φ(0)) Mixed boundary conditions.
Discussion Results have been obtained for models associated to higher rank Lie algebras e.g. vector NLS model (Doikou, Fioravanti, Ravanini 07) and ATFT (Doikou 08, and Avan, Doikou 08). Full classification of integrable boundary conditions in these models. Two distinct types of boundary conditions emerge: the soliton preserving, and the soliton non-preserving. Dynamical boundaries, e.g. coupled harmonic oscillator at the ends. Full classification of integrable boundary conditions at the quantum level as well (Doikou 00, and Arnaudon, Avan, Crampe, Doikou, Frappat, Ragoucy 03, 04).