Quantum integrable systems and non-skew-symmetric classical r-matrices. T. Skrypnyk

Transcription

1 Quantum integrable systems and non-skew-symmetric classical r-matrices. T. Skrypnyk Universita degli studi di Milano Bicocca, Milano, Italy and Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

2 Misjudgments about classical r-matrices: All classical r-matrices are connected with quasiclassical expansions of quantum groups or related structures. Quantum integrable systems are associated with skew-symmetric r-matrices. Non-skew-symmetric r-matrices are pertinent only to classical integrable systems. Quantum integrable systems associated with classical r-matrices have no physical applications and are only of academical interest.

3 Aims of the talk To demonstrate non-skew-symmetric classical r-matrices not connected with quantum groups or related structures. To show that quantum integrable systems are associated not only with skew-symmetric but also with non-skewsymmetric classical r-matrices. To present important physical applications of the constructed quantum integrable systems. Overall purpose of the research: to widen the class of quantum integrable systems and their physical applications.

4 Plan of the talk 1. Non-skew-symmetric classical r-matrices: definitions.. Examples of non-skew-symmetric classical r-matrices. 3. Examples of associated quantum integrable systems. 4. Algebraic Bethe ansatz (sl() case). 5. Applications to BCS models (sl() case). 6. Review of further developments. 7. Conclusion and open problems.

5 Classical r-matrices Let g be a simple Lie algebra over C or reductive Lie algebra gl(n), X a, a = 1, dimg be a basis in g with the commutation relations: [X a, X b ] = dimg c=1 C c abx c. (1) Let (, ) be a bilinear symmetric invariant form on g, let g ab = (X a, X b ) be its components, where {X a } is a dual basis. We will use the following definition (Maillet 1986): Definition 1. A function of two complex variables r(u 1, u ) with values in g g is called a classical r-matrix if it satisfies the generalized Yang-Baxter equation : [r 1 (u 1, u ), r 13 (u 1, u 3 )] = [r 3 (u, u 3 ), r 1 (u 1, u )] [r 3 (u 3, u ), r 13 (u 1, u 3 )], () where r 1 (u 1, u ) dimg a,b=1 r ab (u 1, u )X a X b 1 etc.

6 Remark. Let us note that if the matrix r(u 1, u ) is skew-symmetric, i.e. r 1 (u 1, u ) = r 1 (u, u 1 ) equation () pass into usual classical Yang-Baxter equation (Sklyanin 1979): [r 1 (u 1, u ), r 13 (u 1, u 3 )] = [r 3 (u, u 3 ), r 1 (u 1, u ) + r 13 (u 1, u 3 )]. (3) We will assume that the parameters u and v are such that in some open region U C the r matrix r(u, v) possesses the decomposition: r(u, v) = Ω (u v) + r 0(u, v) (4) where Ω g g is the tensor Casimir: Ω = dimg a,b=1 g ab X a X b and r 0 (u, v) is a regular in U g g-valued function i.e. is decomposed into a Taylor power series in u and v.

7 Example: Shifted rational r-matrices Let us consider rational r-matrix of the following form: r 1 (u, v) = Ω u v + c 1, (5) where c 1 is the constant g g-valued solution of the generalized classical Yang-Baxter equation: [c 1, c 13 ] = [c 3, c 1 ] [c 3, c 13 ], (6) In the case when c 1 = 0 the corresponding r-matrix coincides with a standard rational r-matrix. We will call this r-matrix to be a shifted rational r-matrix. This r-matrix is a non-skew-symmetric generalization of the so-called class zero skew-symmetric rational r-matrices (Belavin-Drinfeld 198, Stolin 1990). It is connected with graded loop algebra in homogeneous grading and non-standard Kostant-Adler decompositions.

8 Example: twisted rational r-matrices Let σ be an automorphism of the Lie algebra g of the order p, dim g j i.e σ p = 1. Let Ω (j) = X j,α X j,α are tensor Casimirs α=1 restricted to the subspaces g j, where the subspaces g j are defined as follows: σ g j = ɛ πij p gj. It is possible to define the following r-matrix (Avan 1990): r(u, v) = v p Ω (0) 1 u p v + v p 1 uω (1) 1 p u p v p vup 1 Ω (p 1) 1, u p v p which we call σ-twisted rational r-matrix. It is connected with loop algebras in intermediate gradings which are defined by the automorphism σ.

9 Example: deformed rational r-matrices Let us consider the map Φ(u) = 1 + u k Φ k, where linear maps Φ k : g g are such that: k=1 [Φ(u)(X ), Φ(u)(Y )] = Φ(u)([X, Y ] + uδφ 1 (X, Y )), where (7) δφ 1 (X, Y ) = [X, Φ 1 (Y )] + [Φ 1 (X ), Y ] Φ 1 [X, Y ]. In this case one defines the next r-matrix (Skrypnyk 006): r 1 (u, v) = Φ(u) (Φ 1 (v)) Ω 1 (u v). (8) We call this r-matrix deformed rational r-matrix. Example. Let g = gl(n), so(n), sp(n). The map Φ(u): Φ(u)(X ) = 1 + AuX 1 + Au satisfies (7), where A is numerical matrix such that Φ 1 (X ) = 1 (AX + XA) g.

10 Let X ij, i, j 1, n be a basis in the corresponding matrix algebra, X ji be a dual basis. In this case the r-matrix (8) acquires the following form (Skrypnyk 006): n 1 + AuXij 1 + Au ( 1 + Av) 1 X ji ( 1 + Av) 1 i,j=1 r(u, v) = (u v) (9) In the case A = diag(a 1, a,..., a n ), making the substitution of variables u = λ 1, v = µ 1, one comes to hyperelliptic r-matrix (Skrypnyk 005):. r(λ, µ) = n i,j=1 where λ i = λ a i, µ i = µ a i, i 1, n. λ i λ j µ i µ j X ij X ji (λ µ), (10)

11 Generalized Gaudin systems Let Ŝ a (l), a 1, dimg, l 1, N be quantum operators that constitute a representation of the Lie algebra g N, i.e.: [Ŝ (l) a dimg, Ŝ (k) b ] = δkl c=1 C c abŝ (k) c. Let ν k, ν k ν l, k, l 1,..., N be some fixed points in the complex plane belonging to the open region U in which the r-matrix r(u, v) possesses the decomposition (4). Then the operators Ĥl of the following explicit form: Ĥ l = dimg ( a,b=1 k=1,k l r ab (ν k, ν l )Ŝ (k) a Ŝ (l) b +rab 0 (ν l, ν l ) (Ŝ (l) a Ŝ (l) b (l) +Ŝ b Ŝ (l) (11) constitute an abelian (commutative) algebra (Skrypnyk 006). a ) )

12 Generalized Gaudin systems in magnetic field We will need the following definition (Skrypnyk 007): Definition. A g-valued function of one complex variable c(u) = dimg c a (u)x a is called a generalized shift element if it a=1 solves the following equation: [r 1 (u, v), c 1 (u)] [r 1 (v, u), c (v)] = 0. (1) Let c(u) be a generalized shift element which is regular at the points ν k, k 1, N. Then the operators Ĥl of the form: Ĥ l = + dimg a,b=1 k=1,k l dimg a,b=1 r ab 0 (ν l, ν l ) r ab (ν k, ν l )Ŝ a (k) Ŝ (l) b + (Ŝ a (l) Ŝ (l) b dimg + Ŝ (l) b Ŝ a (l) ) + a=1 c a (ν l )Ŝ (l) a. (13) constitute an abelian (commutative) algebra (Skrypnyk 007).

13 Algebraic Bethe ansatz: general case Let g = sl(). Let {X 3, X +, X }, be the root basis in sl() with the commutation relations: [X 3, X ± ] = ±X ±, [X +, X ] = X 3. Let ˆL(u) = ˆL 3 (u)x 3 + ˆL + (u)x + + ˆL (u)x be quantum Lax operator that satisfies the linear r-matrix bracket [ˆL 1 (u), ˆL (v)] = [r 1 (u, v), ˆL 1 (u)] [r 1 (v, u), ˆL (v)], (14) with some r-matrix satisfying the generalized Yang-Baxter equation and possessing the regularity property (4). In this case it is possible to show (Skrypnyk 007) that the generating functions: ˆτ(u) = (ˆL 3 (u)) + (ˆL + (u)ˆl (u) + ˆL (u)ˆl + (u)). (15) commute: [ˆτ(u), ˆτ(v)] = 0.

14 Let us now consider only U(1)-invariant r-matrices: r(u, v) = ( 1 r (u, v)x + X + 1 r + (u, v)x X + +r 3 (u, v)x 3 X 3 ). We consider irreducible representation of the Lax algebra in a space H. We assume that there exist vacuum vector 0 : ˆL (u) 0 = 0, ˆL 3 (u) 0 = Λ 3 (u) 0. The vectors v 1 v v M = ˆL + (v 1 )ˆL + (v ) ˆL + (v M ) 0 are eigen-vectors of ˆτ(u) (Skrypnyk 007) with the eigen-values Λ(u {v i }) = (Λ 3 (u) Λ 3 (v i ) M r 3 (v i, u)) i=1 M r + (v i, u)r (v i, u) i=1 + (r + 0 (u, u) + r 0 (u, u))λ 3(u) + u Λ 3 (u), where M j=1,j i r 3 (v j, v i ) = r0 3 (v i, v i ) (r + 0 (v i, v i ) + r 0 (v i, v i )).

15 Bethe ansatz: case of Gaudin-type models The Lax matrices of the Gaudin models in magnetic field are: ˆL(u) = (r 3 (ν k, u)ŝ (k) 3 X 3 + r + (ν k, u) k=1 Ŝ (k) X + + r (ν k, u) Ŝ (k) + X ) + c 3 (u)x 3. The corresponding Gaudin Hamiltonians acquire the form: Ĥ l = = k=1,k l (r 3 (ν k, ν l )Ŝ (k) 3 Ŝ (l) (ν k, ν l ) 3 +r+ Ŝ (k) Ŝ (l) + + r (ν k, ν l ) Ŝ (k) + Ŝ (l) ) +r 3 0 (ν l, ν l )(Ŝ (l) 3 ) (r + 0 (ν l, ν l )+r 0 (ν l, ν l ))(Ŝ (l) Ŝ (l) + +Ŝ (l) + Ŝ (l) )+c 3 (ν l )Ŝ where Ĥ l = 1 res u=ν l ˆτ(u) = 1 res u=ν l trˆl (u).

16 The Casimir operators have the form: Ĉ l = (Ŝ (l) 3 ) + 1 (l) (Ŝ Ŝ (l) + + Ŝ (l) + Ŝ (l) ). Let us consider a finite-dimensional irreducible representation of the algebra sl() N in a space H = V λ 1 V λ V λ N, where V λ k is an irreducible finite-dimensional representation of the k-th copy of sl() with the spin λ k, where λ k 1 N. The representation V λ k contains the highest weight vector vλk : Ŝ (k) + v λk = 0, Ŝ (k) 3 v λk = λ k v λk, The space V λ k is spanned by v m λk = (Ŝ (k) ) m v λk, m 0, λ k. The vacuum vector in the space H has the form: 0 = v λ1 v λ v λn. The vector 0 is an eigen-vector for the function ˆτ(u).

17 The spectrum of the hamiltonians Ĥl vectors has the form: h l ({v i }) = λ l ( m=1,m l r 3 (ν m, ν l )λ m M r 3 (v i, ν l )+ + r 3 0 (ν l, ν l )λ l + 1 (r + 0 (ν l, ν l ) + r 0 (ν l, ν l )) + c 3 (ν l )), (16) where v i are solutions of Bethe equations (Skrypnyk 009): r 3 (ν k, v i )λ k k=1 M j=1,j i i=1 r 3 (v j, v i ) = c 0 (v i ) c 3 (v i ), i 1, M, (17) c 0 (v) = r 3 0 (v, v) 1 (r + 0 (v, v) + r 0 (v, v)), and c 3(v) = cc 0 (v). The spectrum of Casimir function Ĉ k is λ k (λ k + 1).

18 Application to BCS models Let us consider the fermionic creation-anihilation operators c j,σ,c i,σ, i, j 1, N, σ, σ {+, } with the following anti-commutation relations: c i,σ c j,σ + c j,σ c i,σ = δ σσ δ ij, c i,σ c j,σ + c j,σ c i,σ = 0, Then the following formulas: c i,σ c j,σ + c j,σ c i,σ = 0. Ŝ (j) + = c j, c j,+, Ŝ (j) = c j,+ c j,, Ŝ (j) 3 = 1 (1 c j,+ c j,+ c j, c j, ), j 1, (18) provide the realization of the Lie algebra sl() N with the highest weight λ 1 = λ =... = λ N = 1. The vacuum vector 0 in the representation of the spin algebra sl() N coincides with the fermion vacuum i.e.: c j,σ 0 = 0, j 1, N, σ +,.

19 Let us consider linear combination of the generalized Gaudin hamiltonians in an external magnetic field: Ĥ η l Ĥ l. (19) l=1 In the case of U(1)-invariant r-matrices and for the diagonal shift element we obtain the following expression for (19): + Ĥ = m,l=1 l=1 η l ( c3 (ν l ) + 1 (r 0 (ν l, ν l ) + r + 0 (ν l, ν l )) ) Ŝ (l) 3 + ( ( r + (ν m, ν l )η l + r (ν l, ν m )η m ) Ŝ (m) Ŝ (l) + +η l r 3 (ν m, ν l )Ŝ (m) 3 where r α (ν m, ν l ) r α (ν m, ν l ) if m l, r α (ν m, ν l ) r α 0 (ν m, ν l ) if m = l and α {3, +, }. Ŝ (l) 3 ),

20 In the terms of the fermion operators we further obtain: Ĥ = l=1 ɛ l ɛ +, + c l,ɛ c l,ɛ + m,l=1 U ml m,l=1 ɛ,ɛ +, g ml c m,+c m, c l, c l,+ + c m,ɛc m,ɛ c l,ɛ c l,ɛ + E 0, (0) where ɛ l = 1 4( ηl (c 3 (ν l ) + r 0 (ν l, ν l ) + r + 0 (ν l, ν l ))+ + ( r 3 (ν m, ν l )η l + r 3 (ν l, ν m )η m ) ), m=1 g ml = 1 ( ηl r + (ν m, ν l ) + η m r (ν l, ν m ) ), U ml = η l 4 r 3 (ν m, ν l ), E 0 = 1 ) η l (c 3 (ν l ) + r 0 4 (ν l, ν l ) + r + 0 (ν l, ν l ) + r 3 (ν m, ν l ). l=1 m=1

21 BCS hamiltonian of Richardson Let us consider the case of rational skew-symmetric r-matrix: r 1 (u v) = 1 u v X 1 3 X 3 + (u v) (X + X + X X + ). The shift function in this case is c(u) = cx 3. Subtracting from the hamiltonian (19) with η l = ν l the sum of the second Casimir operators 1 Ĉ l, using that l=1 Ŝ3 l = c 1 N Ĥ l is an integral of motion, subtracting it and l=1 l=1 adding one half of its second power to the hamiltonian (19) one obtains (after fermionization) the famous BCS hamiltonian of Richardson: Ĥ BCS = ɛ l (c l,+ c l,+ + c l, c l, ) g c m,+c m, c l, c l,+, l=1 m,l=1 (1)

22 where ɛ l ν l, g c 1. The connection of this hamiltonian with the rational Gaudin model was discovered in 1997 by Cambaggio, Rivas, Saracena. This model was actively used in the theory of small metallic grains and quantum dots by Amico et al, Dukelsky et all etc. The spectrum of the hamiltonian (1) has the form: h BCS = M E i, where E i v i are solutions of Bethe-Richardson s equations: j=1 1 ɛ j E i M j=1,j i i=1 = 1, i 1, M. E j E i g

23 p x + ip y BCS hamiltonian Let us consider the non-skew-symmetric r-matrix of the form: r 1 (u, v) = u u v X uv 3 X 3 + (u v ) (X + X +X X + ), The shift function in this case is c(u) = cx 3. In this case, putting in the hamiltonian (19) η l = νl and subtracting from it the sum of the second Casimir operators νl Ĉ l one obtain, after the fermionization and division by 1 l=1 (c 1/), the following BCS hamiltonian (Skrypnyk 009): Ĥ GBCS = ɛ l (c l,+ c l,++c l, c l, ) g l=1 where g = ( 1 c) 1, ɛ l = ν l. m,l=1 ɛm ɛ l c m,+c m, c l, c l,+, ()

24 Rescaling the spin operators Ŝ (p) ± p x ± 1p y Ŝ (p) p x + py ±, Ŝ (p) 3 Ŝ (p) 3, and putting ɛ p = p x + p y one arrives to p x + ip y BCS hamiltonian (Sierra et al 009), mean-field hamiltonian for which is p x + ip y hamiltonian of Green and Read. The spectrum of the hamiltonian () has the form: h BCS = M E i, i=1 where E i v i are solutions of Bethe-type equations: 1 k=1 ɛ k ɛ k E i M j=1,j i E j = 1, i 1, M. (3) E j E i g

25 Review of further developments Construction of generalized Fuchsian systems and Schlesinger equations (isomonodromy equations) associated with non-skew-symmetric r-matrices (Skrypnyk JMP 010). Construction of generalized Knizhnik-Zamolodchikov equations associated with non-skew-symmetric r-matrices (Skrypnyk JMP 010). Off-shell Bethe ansatz for U(1)-invariant non-skew-symmetric sl() sl()-valued classical r-matrices (Skrypnyk NPB 010). Applications of generalized Gaudin systems based on higher rank Lie algebras to BCS-type models (Skrypnyk NPB 01, Skrypnyk JPA 01 - to appear). Construction of boson and spin-boson integrable models associated with general non-skew-symmetric r-matrices (Skrypnyk JPA 010, Skrypnyk JSTAT 011, Skrypnyk NPB 01).

26 Conclusion There are a lot of non-skew-symmetric classical r-matrices not connected with quantum groups or related structures. With all classical r-matrices one can associated quantum integrable models, in particular spin chains of Gaudin-type. The constructed integrable models have important physical applications, in particular in a theory of superconductivity (BCS models).

27 Main open problems Classification of all solutions of the generalized classical Yang-Baxter equation. Development of methods of diagonalization of the integrable quantum hamiltonians based on non-skew-symmetric classical r-matrices.