1Quantum integrable systems and non-skew-symmetric ...€¦ · I To show that quantum integrable systems are associated not only with skew-symmetric but also with non-skew-symmetric

Quantum integrable systems andnon-skew-symmetric classical

r-matrices.

T. SkrypnykUniversita degli studi di Milano Bicocca, Milano, Italy andBogolyubov Institute for Theoretical Physics, Kyiv, Ukraine

Aims of the talk Plan of the talk Classical r -matrices. Examples of integrable systems. Algebraic Bethe ansatz Application to BCS models Review of further developments Conclusion and Open problems

Misjudgments about classical r-matrices:

I All classical r -matrices are connected with quasiclassicalexpansions of quantum groups or related structures.

I Quantum integrable systems are associated withskew-symmetric r -matrices. Non-skew-symmetricr -matrices are pertinent only to classical integrablesystems.

I Quantum integrable systems associated with classicalr -matrices have no physical applications and are only ofacademical interest.


Aims of the talk

I To demonstrate non-skew-symmetric classical r -matricesnot connected with quantum groups or related structures.

I To show that quantum integrable systems are associatednot only with skew-symmetric but also with non-skew-symmetric classical r -matrices.

I To present important physical applications of theconstructed quantum integrable systems.

Overall purpose of the research: to widen the class ofquantum integrable systems and their physical applications.


Plan of the talk

1. Non-skew-symmetric classical r -matrices: definitions.

2. Examples of non-skew-symmetric classical r -matrices.

3. Examples of associated quantum integrable systems.

4. Algebraic Bethe ansatz (sl(2) case).

5. Applications to BCS models (sl(2) case).

6. Review of further developments.

7. Conclusion and open problems.


Classical r-matricesLet g be a simple Lie algebra over C or reductive Lie algebragl(n), Xa, a = 1, dimg be a basis in g with the commutationrelations:

[Xa, Xb] =

dimg∑c=1

C cabXc . (1)

Let ( , ) be a bilinear symmetric invariant form on g, letg 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(u1, u2)with values in g⊗ g is called a classical r -matrix if it satisfiesthe “generalized Yang-Baxter equation”:

[r12(u1, u2), r13(u1, u3)] = [r23(u2, u3), r12(u1, u2)]−− [r32(u3, u2), r13(u1, u3)], (2)

where r12(u1, u2) ≡dimg∑a,b=1

r ab(u1, u2)Xa ⊗ Xb ⊗ 1 etc.


Remark. Let us note that if the matrix r(u1, u2) is“skew-symmetric”, i.e. r12(u1, u2) = −r21(u2, u1) equation (2)pass into usual classical Yang-Baxter equation (Sklyanin 1979):

[r12(u1, u2), r13(u1, u3)] = [r23(u2, u3), r12(u1, u2) + r13(u1, u3)].(3)

We will assume that the parameters u and v are such that insome open region U ⊂ C2 the r matrix r(u, v) possesses thedecomposition:

r(u, v) =Ω

(u − v)+ r0(u, v) (4)

where Ω ∈ g⊗ g is the tensor Casimir: Ω =dimg∑a,b=1

g abXa ⊗ Xb

and r0(u, v) is a regular in U g⊗ g-valued function i.e. isdecomposed into a Taylor power series in u and v .


Example: Shifted rational r-matricesLet us consider rational r -matrix of the following form:

r12(u, v) =Ω

u − v+ c12, (5)

where c12 is the constant g⊗ g-valued solution of thegeneralized classical Yang-Baxter equation:

[c12, c13] = [c23, c12]− [c32, c13], (6)

In the case when c12 = 0 the corresponding r -matrix coincideswith a standard rational r -matrix. We will call this r -matrix tobe a “shifted” rational r -matrix. This r -matrix is anon-skew-symmetric generalization of the so-called “classzero” skew-symmetric rational r -matrices (Belavin-Drinfeld1982, Stolin 1990). It is connected with graded loop algebra inhomogeneous grading and non-standard Kostant-Adlerdecompositions.


Example: “twisted” rational r-matricesLet σ be an automorphism of the Lie algebra g of the order p,

i.e σp = 1. Let Ω(j) =dim gj∑α=1

Xj ,α ⊗ X−j ,α are tensor Casimirs

restricted to the subspaces gj , where the subspaces gj aredefined as follows:

σ · gj = ε2πijp gj .

It is possible to define the following r -matrix (Avan 1990):

r(u, v) =vpΩ

(0)12

up − vp+

vp−1uΩ(1)12

up − vp+ ... +

vup−1Ω(p−1)12

up − vp,

which we call σ-twisted rational r -matrix. It is connected withloop algebras in “intermediate” gradings which are defined bythe automorphism σ.


Example: “deformed” rational r-matrices

Let us consider the map Φ(u) = 1 +∞∑

k=1

ukΦk , where linear

maps Φk : g → g are such that:

[Φ(u)(X ), Φ(u)(Y )] = Φ(u)([X , Y ] + uδΦ1(X , Y )), where(7)

δΦ1(X , Y ) = [X , Φ1(Y )] + [Φ1(X ), Y ]− Φ1[X , Y ].

In this case one defines the next r -matrix (Skrypnyk 2006):

r12(u, v) = Φ(u)⊗ (Φ−1(v))∗Ω12

(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

Φ1(X ) =1

2(AX + XA) ∈ g.


Let Xij , i , j ∈ 1, n be a basis in the corresponding matrixalgebra, Xji be a dual basis. In this case the r -matrix (8)acquires the following form (Skrypnyk 2006):

r(u, v) =

n∑i ,j=1

√1 + AuXij

√1 + Au ⊗ (

√1 + Av)−1Xji(

√1 + Av)−1

(u − v).

(9)In the case A = diag(a1, a2, ...., an), making the substitution ofvariables u = −λ−1, v = −µ−1, one comes to “hyperelliptic”r -matrix (Skrypnyk 2005):

r(λ, µ) =n∑

i ,j=1

λiλj

µiµj

Xij ⊗ Xji

(λ− µ), (10)

where λ2i = λ− ai , µ2

i = µ− ai , i ∈ 1, n.


Generalized Gaudin systemsLet S

(l)a , a ∈ 1, dimg, l ∈ 1, N be quantum operators that

constitute a representation of the Lie algebra g⊕N , i.e.:

[S (l)a , S

(k)b ] = δkl

dimg∑c=1

C cabS

(k)c .

Let νk , νk 6= νl , k , l ∈ 1, ..., N be some fixed points in thecomplex plane belonging to the open region U in which ther -matrix r(u, v) possesses the decomposition (4). Then the

operators Hl of the following explicit form:

Hl =

dimg∑a,b=1

( N∑k=1,k 6=l

r ab(νk , νl)S(k)a S

(l)b +

r ab0 (νl , νl)

2(S (l)

a S(l)b +S

(l)b S (l)

a ))

(11)constitute an abelian (commutative) algebra (Skrypnyk 2006).


Generalized Gaudin systems in magnetic fieldWe will need the following definition (Skrypnyk 2007):Definition 2. A g-valued function of one complex variable

c(u) =dimg∑a=1

ca(u)Xa is called a “generalized shift element” if it

solves the following equation:

[r12(u, v), c1(u)]− [r21(v , u), c2(v)] = 0. (12)

Let c(u) be a generalized shift element which is regular at the

points νk , k ∈ 1, N . Then the operators Hl of the form:

Hl =

dimg∑a,b=1

N∑k=1,k 6=l

r ab(νk , νl)S(k)a S

(l)b +

+

dimg∑a,b=1

r ab0 (νl , νl)

2(S (l)

a S(l)b + S

(l)b S (l)

a ) +

dimg∑a=1

ca(νl)S(l)a . (13)

constitute an abelian (commutative) algebra (Skrypnyk 2007).


Algebraic Bethe ansatz: general caseLet g = sl(2). Let X3, X+, X−, be the root basis in sl(2)with the commutation relations:

[X3, X±] = ±X±, [X+, X−] = 2X3.

Let L(u) = L3(u)X3 + L+(u)X+ + L−(u)X− be quantum Laxoperator that satisfies the linear r -matrix bracket

[L1(u), L2(v)] = [r12(u, v), L1(u)]− [r21(v , u), L2(v)], (14)

with some r -matrix satisfying the generalized Yang-Baxterequation and possessing the regularity property (4).In this case it is possible to show (Skrypnyk 2007) that thegenerating functions:

τ(u) = (L3(u))2 + 2(L+(u)L−(u) + L−(u)L+(u)). (15)

commute:[τ(u), τ(v)] = 0.


Let us now consider only U(1)-invariant r -matrices:

r(u, v) = (1

2r−(u, v)X+⊗X−+

1

2r+(u, v)X−⊗X++r 3(u, v)X3⊗X3).

We consider irreducible representation of the Lax algebra in aspace H. We assume that there exist “vacuum” vector |0〉:

L−(u)|0〉 = 0, L3(u)|0〉 = Λ3(u)|0〉.

The vectors |v1v2 · · · vM〉 = L+(v1)L+(v2) · · · L+(vM)|0〉 are

eigen-vectors of τ(u) (Skrypnyk 2007) with the eigen-values

Λ(u|vi) = (Λ3(u)−M∑i=1

r 3(vi , u))2 −M∑i=1

r+(vi , u)r−(vi , u)

+ (r+0 (u, u) + r−0 (u, u))Λ3(u) + ∂uΛ3(u), where

Λ3(vi)−M∑

j=1,j 6=i

r 3(vj , vi) = r 30 (vi , vi)−

(r+0 (vi , vi) + r−0 (vi , vi))

2.


Bethe ansatz: case of Gaudin-type modelsThe Lax matrices of the Gaudin models in magnetic field are:

L(u) =N∑

k=1

(r 3(νk , u)S(k)3 X3+

r+(νk , u)

2S

(k)− X++

r−(νk , u)

2S

(k)+ X−)

+ c3(u)X3.

The corresponding Gaudin Hamiltonians acquire the form:

Hl =

=N∑

k=1,k 6=l

(r 3(νk , νl)S(k)3 S

(l)3 +

r+(νk , νl)

2S

(k)− S

(l)+ +

r−(νk , νl)

2S

(k)+ S

(l)− )

+r 30 (νl , νl)(S

(l)3 )2+

1

4(r+

0 (νl , νl)+r−0 (νl , νl))(S(l)− S

(l)+ +S

(l)+ S

(l)− )+c3(νl)S

(l)3 ,

where

Hl =1

2resu=νl

τ(u) =1

2resu=νl

trL2(u).


The Casimir operators have the form:

C l2 = (S

(l)3 )2 +

1

2(S

(l)− S

(l)+ + S

(l)+ S

(l)− ).

Let us consider a finite-dimensional irreducible representationof the algebra sl(2)⊕N in a space H = V λ1 ⊗V λ2 ⊗ · · · ⊗V λN ,where V λk is an irreducible finite-dimensional representation ofthe k-th copy of sl(2) with the spin λk , where λk ∈ 1

2N. The

representation V λk contains the highest weight vector vλk:

S(k)+ vλk

= 0, S(k)3 vλk

= λkvλk,

The space V λk is spanned by vmλk

= (S(k)− )mvλk

, m ∈ 0, 2λk .The “vacuum” vector in the space H has the form:

|0〉 = vλ1 ⊗ vλ2 ⊗ · · · ⊗ vλN.

The vector |0〉 is an eigen-vector for the function τ(u).


The spectrum of the hamiltonians Hl vectors has the form:

hl(vi) = λl(N∑

m=1,m 6=l

r 3(νm, νl)λm −M∑i=1

r 3(vi , νl)+

+ r 30 (νl , νl)λl +

1

2(r+

0 (νl , νl) + r−0 (νl , νl)) + c3(νl)), (16)

where vi are solutions of Bethe equations (Skrypnyk 2009):

N∑k=1

r 3(νk , vi)λk −M∑

j=1,j 6=i

r 3(vj , vi) = c0(vi)− c3(vi), i ∈ 1, M ,

(17)c0(v) = r 3

0 (v , v)− 12(r+

0 (v , v) + r−0 (v , v)), and c3(v) = cc0(v).

The spectrum of Casimir function C k2 is λk(λk + 1).


Application to BCS modelsLet us consider the fermionic creation-anihilation operatorsc†j ,σ′ ,ci ,σ, i , j ∈ 1, N , σ, σ′ ∈ +,− with the followinganti-commutation relations:

c†i ,σcj ,σ′ + cj ,σ′c†i ,σ = δσσ′δij , c†i ,σc†j ,σ′ + c†j ,σ′c†i ,σ = 0,

ci ,σcj ,σ′ + cj ,σ′ci ,σ = 0.

Then the following formulas:

S(j)+ = cj ,−cj ,+, S

(j)− = c†j ,+c†j ,−, S

(j)3 =

1

2(1−c†j ,+cj ,+−c†j ,−cj ,−), j ∈ 1, N ,

(18)provide the realization of the Lie algebra sl(2)⊕N with thehighest weight λ1 = λ2 = ... = λN = 1

2. The vacuum vector

|0〉 in the representation of the spin algebra sl(2)⊕N coincideswith the fermion vacuum i.e.:

cj ,σ|0〉 = 0, j ∈ 1, N , σ ∈ +,−.


Let us consider linear combination of the generalized Gaudinhamiltonians in an external magnetic field:

H ≡N∑

l=1

ηl Hl . (19)

In the case of U(1)-invariant r -matrices and for the diagonalshift element we obtain the following expression for (19):

H =N∑

l=1

ηl

(c3(νl) +

1

2(r−0 (νl , νl) + r+

0 (νl , νl)))S

(l)3 +

+N∑

m,l=1

((r+(νm, νl)ηl + r−(νl , νm)ηm)

2S

(m)− S

(l)+ +ηl r

3(νm, νl)S(m)3 S

(l)3 ),

where rα(νm, νl) ≡ rα(νm, νl) if m 6= l , rα(νm, νl) ≡ rα0 (νm, νl)

if m = l and α ∈ 3, +,−.


In the terms of the fermion operators we further obtain:

H =N∑

l=1

εl

∑ε∈+,−

c†l ,εcl ,ε +N∑

m,l=1

gmlc†m,+c†m,−cl ,−cl ,++

+N∑

m,l=1

Uml

∑ε,ε′∈+,−

c†m,εcm,εc†l ,ε′cl ,ε′ + E0, (20)

where εl = −1

4

(ηl(2c3(νl) + r−0 (νl , νl) + r+

0 (νl , νl))+

+N∑

m=1

(r 3(νm, νl)ηl + r 3(νl , νm)ηm)),

gml =1

2

(ηl r

+(νm, νl) + ηm r−(νl , νm)), Uml =

ηl

4r 3(νm, νl),

E0 =1

4

N∑l=1

ηl

(2c3(νl) + r−0 (νl , νl) + r+

0 (νl , νl) +N∑

m=1

r 3(νm, νl)).


BCS hamiltonian of RichardsonLet us consider the case of rational skew-symmetric r -matrix:

r12(u − v) =1

u − vX3 ⊗X3 +

1

2(u − v)(X+ ⊗X− + X− ⊗X+).

The shift function in this case is c(u) = cX3.Subtracting from the hamiltonian (19) with ηl = νl the sum of

the second Casimir operators 12

N∑l=1

C 2l , using that

N∑l=1

S l3 = c−1

N∑l=1

Hl is an integral of motion, subtracting it and

adding one half of its second power to the hamiltonian (19)one obtains (after fermionization) the famous BCShamiltonian of Richardson:

HBCS =N∑

l=1

εl(c†l ,+cl ,+ + c†l ,−cl ,−)− g

N∑m,l=1

c†m,+c†m,−cl ,−cl ,+,

(21)


where εl ≡ νl , g ≡ c−1.The connection of this hamiltonian with the rational Gaudinmodel was discovered in 1997 by Cambaggio, Rivas, Saracena.This model was actively used in the theory of small metallicgrains and quantum dots by Amico et al, Dukelsky et all etc.The spectrum of the hamiltonian (21) has the form:

hBCS = 2M∑i=1

Ei ,

where Ei ≡ vi are solutions of Bethe-Richardson’s equations:

N∑j=1

1

εj − Ei−

M∑j=1,j 6=i

2

Ej − Ei= − 1

g, i ∈ 1, M .


“px + ipy” BCS hamiltonianLet us consider the non-skew-symmetric r -matrix of the form:

r12(u, v) =u2

u2 − v 2X3⊗X3 +

uv

2(u2 − v 2)(X+⊗X−+X−⊗X+),

The shift function in this case is c(u) = cX3.In this case, putting in the hamiltonian (19) ηl = ν2

l andsubtracting from it the sum of the second Casimir operators

12

N∑l=1

ν2l C

2l one obtain, after the fermionization and division by

−(c − 1/2), the following BCS hamiltonian (Skrypnyk 2009):

HGBCS =N∑

l=1

εl(c†l ,+cl ,++c†l ,−cl ,−)−g

N∑m,l=1

√εmεlc

†m,+c†m,−cl ,−cl ,+,

(22)where g = (1

2− c)−1, εl = ν2

l .


Rescaling the spin operators

S(p)± → px ±

√−1py√

p2x + p2

y

S(p)± , S

(p)3 → S

(p)3 ,

and putting εp = p2x + p2

y one arrives to “px + ipy” BCShamiltonian (Sierra et al 2009), mean-field hamiltonian forwhich is “px + ipy” hamiltonian of Green and Read.The spectrum of the hamiltonian (22) has the form:

hBCS = 2M∑i=1

Ei ,

where Ei ≡ v 2i are solutions of Bethe-type equations:

1

2

N∑k=1

εk

εk − Ei−

M∑j=1,j 6=i

Ej

Ej − Ei=

1

g, i ∈ 1, M . (23)


Review of further developmentsI Construction of generalized Fuchsian systems and

Schlesinger equations (isomonodromy equations)associated with non-skew-symmetric r -matrices (SkrypnykJMP 2010).

I Construction of generalized Knizhnik-Zamolodchikovequations associated with non-skew-symmetric r -matrices(Skrypnyk JMP 2010).

I Off-shell Bethe ansatz for U(1)-invariantnon-skew-symmetric sl(2)⊗ sl(2)-valued classicalr -matrices (Skrypnyk NPB 2010).

I Applications of generalized Gaudin systems based onhigher rank Lie algebras to BCS-type models (SkrypnykNPB 2012, Skrypnyk JPA 2012 - to appear).

I Construction of boson and spin-boson integrable modelsassociated with general non-skew-symmetric r -matrices(Skrypnyk JPA 2010, Skrypnyk JSTAT 2011, SkrypnykNPB 2012).


Conclusion

I There are a lot of non-skew-symmetric classical r -matricesnot connected with quantum groups or related structures.

I With all classical r -matrices one can associated quantumintegrable models, in particular spin chains of Gaudin-type.

I The constructed integrable models have importantphysical applications, in particular in a theory ofsuperconductivity (BCS models).


Main open problems

I Classification of all solutions of the generalized classicalYang-Baxter equation.

I Development of methods of diagonalization of theintegrable quantum hamiltonians based onnon-skew-symmetric classical r -matrices.

1Quantum integrable systems and non-skew-symmetric ...€¦ · I To show that quantum integrable systems are associated not only with skew-symmetric but also with non-skew-symmetric

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