On osp(M|2n) integrable open spin chains

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On osp(M|2n) integrable open spin chains

Daniel Arnaudon, Jean Avan, Nicolas Crampé, Anastasia Doikou, Luc Frappat, Eric Ragoucy

To cite this version:

Daniel Arnaudon, Jean Avan, Nicolas Crampé, Anastasia Doikou, Luc Frappat, et al.. On osp(M|2n) integrable open spin chains. Czechoslovak Journal of Physics, Springer Verlag, 2004, 54, pp.1153.

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ccsd-00002967, version 1 - 29 Sep 2004

On osp(M |2n) integrable open spin chains

D. Arnaudon, N. Cramp´ e, A. Doikou, L. Frappat, ´ E. Ragoucy Laboratoire d’Annecy-le-Vieux de Physique Théorique LAPTH CNRS, UMR 5108, associée ` a l’Université de Savoie LAPP, BP 110, F-74941

Annecy-le-Vieux Cedex, France J. Avan

Laboratoire de Physique Théorique et Modélisation Université de Cergy, 5 mail Gay-Lussac, Neuville-sur-Oise F-95031 Cergy-Pontoise Cedex Pacs : 02.20.Uw, 03.65.Fd, 75.10.Pq

LAPTH-Conf-1054/04

Abstract

We consider open spin chains based on osp(m|2n) Yangians. We solve the reflection equations for some classes of reflection matrices, including the diagonal ones. Having then integrable open spin chains, we write the analytical Bethe Ansatz equations. More details and references can be found in [1, 2].

1 RT T presentation for osp(M |2n) Yangians

Let us consider an M + 2n dimensional

Z₂

-graded vector space, with the M first indices bosonic and the 2n last ones fermionic.

Define

R(u) =

I

+ P

u − Q u + κ .

P being the super permutation, and Q = P

^t¹

= P

^t²

being P partially transposed.

The R-matrix R(u) satisfies the super Yang–Baxter equation R

₁₂

(u) R

₁₃

(u + v) R

₂₃

(v) = R

₂₃

(v) R

₁₃

(u + v) R

₁₂

(u) if 2κ = (M − 2n − 2), with a graded tensor product .

One defines the Yangian of osp(M |2n) by the generators T (u) = X

n∈Z≥0

T

_(n)

u

⁻ⁿ

T (0) =

I

(3)

and the relations

R

₁₂

(u − v) T

₁

(u) T

₂

(v) = T

₂

(v) T

₁

(u) R

₁₂

(u − v) T

^t

(u − κ) T (u) =

I

(i.e. RTT=TTR relations and “orthogonality relation”) [3].

2 Closed chain integrability

The closed chain monodromy matrix is defined by T

_a

(u) = R

_aL

(u)R

_a,L−1

(u) · · · R

_a2

(u)R

_a1

(u)

Using the Yang–Baxter equation, one proves that the closed chain transfer matrices, given by the super trace t(u) = Tr

a

T

a

(u), commute for all values of the spectral parameter u:

h

t(u), t(v) i

= 0, ∀u, v

3 Reflection equation and open chain integrability

We consider K

⁻

(u) ∈ End(

C^M+2n

), solution of the reflection equation:

R

_ab

(u

a

− u

_b

) K

_a⁻

(u

a

) R

_ba

(u

a

+ u

_b

) K

_b⁻

(u

b

) = K

_b⁻

(u

b

) R

_ab

(u

a

+ u

_b

) K

_a⁻

(u

a

) R

_ba

(u

a

− u

_b

) Let

T

_a

(u) = R

_aL

(u)R

_a,L−1

(u) · · · R

_a2

(u)R

_a1

(u) and

T ˆ

_a

(u) = R

_1a

(u)R

_2a

(u) · · · R

_L−1,a

(u)R

_La

(u) The open chain monodromy matrix is the super trace

t(u) = Tr

_a

K

_a⁺

(u) T

_a

(u) K

_a⁻

(u) ˆ T

_a

(u) ,

where K

⁺^t

(−λ − iκ) is another solution of the reflection equation. Again,

as was first proved by Cherednik and Sklyanin using the Yang–Baxter and

reflection equations, [4, 5], [t(u), t(v)] = 0, ∀u, v.

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4 Solutions of the reflection equation

4.1 Diagonal solutions

We solve the reflection equation for K of the form K (u) = diag

k

₁

(u), · · · , k

_M

(u) ; k

_M₊₁

(u), · · · , k

_M_+n

(u)

There are three families of generic diagonal solutions and two particular cases

D1: Solutions of sl(M + 2n) type, with one free parameter, for M even k

_i

(u) = 1 ,

k

_¯_ı

(u) = 1 + cu

1 − cu , ∀ i ∈ {1, ..., M

2 ; M + 1, ..., M + n}

This solution has no extension to odd M.

D2: Solutions with three different values of k

_l

(u), depending on one free parameter

k

₁

(u) = 1 + c

₁

u

1 − c

₁

u , k

_M

(u) = 1 + c

_M

u

1 − c

_M

u , k

_j

(u) = 1 ∀ j 6= 1, M where (κ − 1)c

₁

c

_M

+ c

₁

+ c

_M

= 0. This solution does not hold for M = 0, 1.

D3: Solutions without any free continuous parameter, but with two integer parameters m

₁

, n

₁

, and c =

_κ−(2m ²

1−2n1−1)

k

_i

(u) = k

_¯_ı

(u) = 1 ∀ i ∈ {1, ..., m

₁

; M + 1, ..., M + n

₁

} k

_i

(u) = k

_¯_ı

(u) = 1 + cu

1 − cu otherwise

D4: In the particular case of so(4), the solution takes the more general form:

K(u) = diag

1 , 1 + c

₂

u

1 − c

₂

u , 1 + c

₃

u

1 − c

₃

u , 1 + c

₂

u 1 − c

₂

u

1 + c

₃

u 1 − c

₃

u

This solution contains the three generic solutions D1 (c

₂

c

₃

= 0), D2 (c

₂

+ c

₃

= 0) and D3 (c

₂

= c

₃

= ∞).

D5: In the particular case of so(2), any function-valued diagonal matrix is

solution.

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4.2 Antidiagonal and mixed solutions

The classification of such solutions is best shown by a few examples.

One finds the two following solutions for osp(4|2) : so diagonal :





 1

1 −1

−1

k

₅

ℓ

₅

ℓ

₆

−k

₅







where k

₅²

+ ℓ

₅

ℓ

₆

= 1.

sp diagonal :







1 0

0 ℓ

₂

ℓ

⁻¹₂

0 0 1

1 1







For osp(2|4) the two solutions take the form so diagonal :





 1

−1

k

₃

ℓ

₃

k

₄

ℓ

₄

ℓ

₅

−k

₄

ℓ

₆

−k

₃







where k

₃²

+ ℓ

₃

ℓ

₆

= 1 and k

²₄

+ ℓ

₄

ℓ

₅

= 1.

sp diagonal :







0 ℓ

₁

ℓ

⁻¹₁

0

1 −1

−1 1







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5 Pseudovacuum and one eigenvalue of the transfer matrix for the open chain

We now choose an appropriate pseudo-vacuum, which is an exact eigenstate of the transfer matrix t(u) of the open chain; it is the state with all “spins”

up, i.e.

|ω

₊

i =

L

O

i=1

|+i

_i

where |+i =





 1 0 .. . 0







∈

C^M+2n

.

Then t(λ) | ω

₊

i = Λ

⁰

(λ) |ω

₊

i , where Λ

⁰

(λ) = a(λ)

^2L

g

₀

(λ) + b(λ)

^2L

2n+M−2

X

l=1

(−1)

^[l+1]

g

_l

(λ) + c(λ)

^2L

g

_2n+M₋₁

(λ) with

a(λ) = (λ + i)(λ + iκ), b(λ) = λ(λ + iκ), c(λ) = λ(λ + iκ − i) the functions g(λ) being written as (in the case osp(2m + 1|2n), with K

^±

=

I

)

g

_l

(λ) = λ(λ +

^iκ₂

−

₂ⁱ

)(λ + iκ)

(λ +

^iκ₂

)(λ +

^il₂

)(λ +

^i(l+1)₂

) , l = 0, . . . , n − 1, g

_l

(λ) = λ(λ +

^iκ₂

−

₂ⁱ

)(λ + iκ)

(λ +

^iκ₂

)(λ + in −

^il₂

)(λ + in −

^i(l+1)₂

) ,

l = n, . . . , n + m − 1 g

_n+m

(λ) = λ(λ + iκ)

(λ + i

^n−m₂

)(λ + i

^n−m+1₂

) if M = 2m + 1 g

_l

(λ) = g

_2n+M_−l−1

(−λ − iκ), l = 0, 1, ..., M + 2n

The dressing consists in the insertion of factors A

_l

(λ) to get the other eigenvalues Λ(λ) of the transfer matrix

Λ(λ) = a(λ)

^2L

g

₀

(λ)A

0

(λ) + b(λ)

^2L

2n+M−2

X

l=1

(−1)

^[l+1]

g

_l

(λ)A

l

(λ)

+ c(λ)

^2L

g

_2n+M₋₁

(λ)A

2n+M−1

(λ)

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The factors A

_l

take the form A

₀

(λ) =

M⁽¹⁾

Y

j=1

λ + λ

⁽¹⁾_j

−

₂ⁱ

λ + λ

⁽¹⁾_j

+

₂ⁱ

λ − λ

⁽¹⁾_j

−

₂ⁱ

λ − λ

⁽¹⁾_j

+

₂ⁱ

,

A

_l

(λ) =

M⁽^l⁾

Y

j=1

λ + λ

^(l)_j

+

^il₂

+ i λ + λ

^(l)_j

+

^il₂

λ − λ

^(l)_j

+

^il₂

+ i λ − λ

^(l)_j

+

^il₂

×

M⁽^l⁺¹⁾

Y

j=1

λ + λ

^(l+1)_j

+

^il₂

−

₂ⁱ

λ + λ

^(l+1)_j

+

^il₂

+

₂ⁱ

λ − λ

^(l+1)_j

+

^il₂

−

₂ⁱ

λ − λ

^(l+1)_j

+

^il₂

+

₂ⁱ

l = 1, . . . , n − 1

Analyticity around the poles introduced in the factors A

_l

now imposes the so-called Bethe equations in the λ

_i

:

e

₁

(λ

⁽¹⁾_i

)

^2L

= Y

ǫ=±1 M⁽¹⁾

Y

j=1,j6=i

e

₂

(λ

⁽¹⁾_i

− ǫλ

⁽¹⁾_j

)

M⁽²⁾

Y

j=1

e

₋₁

(λ

⁽¹⁾_i

− ǫλ

⁽²⁾_j

)

1 = Y

ǫ=±1 M⁽^l⁾

Y

j=1,j6=i

e

₂

(λ

^(l)_i

− ǫλ

^(l)_j

) Y

τ=±1 M⁽^l⁺^τ⁾

Y

j=1

e

₋₁

(λ

^(l)_i

− ǫλ

^(l+τ)_j

) l = 2, . . . , n + m − 1, l 6= n

1 = Y

ǫ=±1 M⁽ⁿ⁺¹⁾

Y

j=1

e

₁

(λ

⁽ⁿ⁾_i

− ǫλ

⁽ⁿ⁺¹⁾_j

)

M⁽ⁿ⁻¹⁾

Y

j=1

e

₋₁

(λ

⁽ⁿ⁾_i

− ǫλ

⁽ⁿ⁻¹⁾_j

)

1 = Y

ǫ=±1 M⁽ⁿ⁺^m⁾

Y

j=1,j6=i

e

₁

(λ

^(n+m)_i

− ǫλ

^(n+m)_j

)

M⁽ⁿ⁺^m⁻¹⁾

Y

j=1

e

₋₁

(λ

^(n+m)_i

− ǫλ

^(n+m−1)_j

)

with

e

_x

(λ) = λ +

^ix₂

λ −

^ix₂

.

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Acknowledgments

This work has been financially supported by the TMR Network EUCLID:

“Integrable models and applications: from strings to condensed matter”, contract number HPRN-CT-2002-00325.

References

[1] D. Arnaudon, J. Avan, N. Cramp´e, A. Doikou, L. Frappat, and E. Ragoucy. ´ Classification of reflection matrices related to (super) Yan- gians and application to open spin chain models. Nucl. Phys. B668:469 (2003) and

math.QA/0304150.

[2] D. Arnaudon, J. Avan, N. Cramp´e, A. Doikou, L. Frappat, and E. Ragoucy. ´ Bethe Ansatz equations and exact S matrices for the osp(M|2n) open super spin chain. Nucl. Phys. B687:257 (2004) and

math-ph/0310042.

[3] D. Arnaudon, J. Avan, N. Cramp´e, L. Frappat and ´ E. Ragoucy, R matrix presentation for (super)-Yangians Y (g), J. Math. Phys. 44 (2003) 302 and

math.QA/0111325.

[4] I.V. Cherednik, Factorizing particles on a half line and root systems, Theor. Math. Phys. 61 (1984) 977.

[5] E.K. Sklyanin, Boundary conditions for integrable quantum systems,

J. Phys. A21 (1988) 2375.