Baxter’s Qoperator for the homogeneous XXX spin chain
S.É. Derkachov
[0.8cm]
Department of Mathematics, St.Petersburg Technology Institute,
St.Petersburg, Russia
[0.2cm]
Submitted to J.Phys.A:Math.Gen.
[1cm]
Abstract:
[10pt]
Applying the PasquierGaudin procedure we construct the
Baxter’s Qoperator for the homogeneous XXX model as integral
operator in standard representation of .
The connection between Qoperator and local Hamiltonians is discussed.
It is shown that operator of Lipatov’s duality symmetry arises
naturally as leading term of the asymptotic expansion of
Qoperator for large values of spectral parameter.
Contents
1 Introduction.
The modern approach to the theory of integrable systems is given by the quantum inverse scattering method(QISM) [4, 7]. In the framework of QISM, eigenstates are obtained by the algebraic Bethe ansatz (ABA) method as an excitations over the vacuum state and the spectral problem is reduced to the set of algebraic Bethe equations (BE) for the parameters . In fact the ABA is equivalent to the construction of the eigenfunctions in a special representation as polynomials of some suitable variables.
The alternative approach is the method of Qoperator [1] proposed by Baxter:there exists the operator which obeys the Baxter equation. The set of the Bethe equations is equivalent to the Baxter equation for the eigenvalue of the Qoperator. This second order finitedifference equation is the simple consequence of the Baxter relation for the transfer matrix and the Qoperator [1].
The ABA and method of Qoperator are equivalent when eigenfunctions and therefore are polynomials. In more general ”nonpolynomial” situation one could use the method of Qoperator. The Qoperator for the periodic Toda chain was constructed in the work of Pasquier and Gaudin [2]. The application of the Qoperator for the construction of eigenstates with arbitrary complex values of conformal wights in the case XXX spin chain was considered in the work of Korchemsky and Faddeev [7]. In the present paper we construct Qoperator for the homogeneous XXX spin chain using the PasquierGaudin procedure.
The presentation is organized as follows. Section 2 introduces definitions and the standard facts about Baxter equation and construction of local Hamiltonians. In Section 3 we construct the Qoperator and study some properties of the obtained Qoperator in the simplest case of homogeneous chain. In Section 4 we obtain the connection between Qoperator and local Hamiltonian. In Section 5 we consider the asymptotic expansion of Qoperator for large spectral parameter. The operator of duality symmetry introduced by L.N.Lipatov [9] appears naturally as leading term in this asymptotic. Finally, in Section 6 we summarize.
2 XXX spin chain
In this section we collect some basic facts about XXX spin chain.
2.1 matrix and YangBaxter equation
The main object is the so called  matrix which is the solution of the YangBaxter equation:
(2.1.1) 
The operator depends on some complex variable – spectral parameter and two sets of generators and acting in different vector spaces and .
Fixing the representations of the spins and in the vector spaces and we obtain the following matrices.

in space and arbitrary representation for the :
This matrix is used for the construction of the Lax operator:
(2.1.2) 
The equivalent representations in spaces and [3]:
(2.1.3) where is the permutation and is the ”twoparticle” Casimir in . This fundamental matrix is the building block for the construction of the local Hamiltonians.
2.2 Baxter equation for XXXmodel.
The ”usual” quantum monodromy matrix is defined as product of the matrices in the common twodimensional auxiliary space. is a matrix with operator entries acting in the quantum space .
(2.2.1) 
The quantum transfer matrix is obtained by taking trace of in the auxiliary space:
(2.2.2) 
Due to the YangBaxter equation the family of operators is commuting, its expansion begins with power and provides commuting operators :
(2.2.3) 
It is possible to show that transfer matrix is invariant
Therefore there exists the ”full” set of commuting operators: operators and operator . Due to invariance the subspace of the eigenvectors of operator with eigenvalue is the  module generated by highest weight vector ,i.e. vector space spanned by linear combinations of monomials in the applied to vector . The highest weight vector is defined by the equation .
We shall work in standard discreteseries representation of the group :
where generators are realised as differential operators:
(2.2.4) 
acting in the space of polynomials of the variable . Here ”spin” is arbitrary number. In this representation the commuting operators are ”local” differential operators acting in the space of polynomials of the n variables and there exists the vacuum vector :
so that we can use the Algebraic Bethe Ansatz(ABA) method and reduce the problem of the common diagonalization of the operators and :
to the solution of the Bethe equation [4, 6]. The vacuum vector is the common highest vector of the local representations of :
and
so that
(2.2.5) 
Let us look now the eigenvector in the form:
It is possible to show that vector is eigenvector of operator with eigenvalue:
(2.2.6) 
on condition that the parameters obey the Bethe equations:
(2.2.7) 
It appears also that Bethe vectors are the highest weight vectors:
In the representation (2.2.4) the highest weight vector is represented by homogeneous, translation invariant polynomial degree l() of n variables :
(2.2.8) 
One can obtain Bethe equation from the formula for by taking residue at and using the fact that polynomial is regular at this point. Finally we see that equations (2.2.6,2.2.7) are equivalent to the Baxter equation for the polynomial :
(2.2.9) 
where
(2.2.10) 
2.3 Local Hamiltonians
Let us consider the homogeneous XXXchain of equal spins: and and fix the same representation in auxiliary space. In this case the quantum monodromy matrix is the product of the fundamental matrices (2.1.3):
The transfer matrix is obtained by taking trace of in the auxiliary space and due to the YangBaxter equation the families of operators and are commuting:
The expansion of the provides Hamiltonians :
(2.3.1) 
where the th operator describes the interaction between nearest neighbours on the chain. Due to the evident equalities (see (2.1.3)):
one obtains the following expression for the first ”twoparticle” Hamiltonian :
where is logarithmic derivative of . It is convenient to work with the ”shifted” Hamiltonian:
(2.3.2) 
where the ”shift” constant is defined by the requirenent:
Let us calculate the eigenvalues of the operator . Operator is invariant
and its highest weight eigenfunctions have the simple form in the representation (2.2.4):
The twoparticle Casimir is the second order differential operator:
and its eigenvalues and eigenvalues of operator can be easily calculated:
Finally we obtain the eigenvalues of the operator :
In the representation (2.2.4) the operator can be realized as some ”twoparticle” integral operator acting on the variables and :
(2.3.3) 
Note that these integral operators arise naturally in QCD [8]. To prove the equality (2.3.3) it is sufficient to show that eigenvalues of integral operator coincide with the eigenvalues :
The expression for the eigenvalues of the full Hamiltonian can be found by the ABA method [5]:
It is possible to rewrite this expression in terms of the function (2.2.10) as follows [7]:
(2.3.4) 
There exists an additional commuting with transfer matrix operator – shift operator :
(2.3.5) 
Eigenvalues of the shift operator can be found by the ABA method [5] also:
(2.3.6) 
In the next sections we shall construct the Baxter’s Qoperator and show that Baxter equation (2.2.9) and equations (2.3.4,2.3.6) arise from the corresponding relations for the Qoperator.
3 Baxter’s Qoperator.
The Baxter’s  operator is the operator with the properties [1]:
Operators and have the common set of eigenfunctions:
(3.0.1) 
and eigenvalues of these operators obey the Baxter equation (2.2.9). Note that function (2.2.10) has the natural interpretation as the eigenvalue of the Qoperator.
We construct the operator in the standard representation of the group in the following form:
(3.0.2) 
where is the transformation of inversion. The scalar product here is the standard invariant scalar product for functions of the one variable:
(3.0.3) 
and is ”integration” or ”dumb” variable. In (3.0.2) the scalar product over all variables is assumed. The generators are conjugated with respect to this scalar product:
Using the evident identities:
we obtain the following rules for transposition:
(3.0.4) 
In fact the construction of the Qoperator repeates the similar construction from the paper Pasquier and Gaudin [2].
The operator , where
is invariant with respect to transformation of the local matrices [1]:
where are the matrices with scalar elements. Simple calculation shows that matrix elements of the transformed matrix
have the form
This expression for operator suggests to consider the function:
The operators act on this function as follows:
Let us fix the dependence on the variables in the kernel of operator in the form:
where  is the set of arbitrary parameters now. Then we have:
After multiplication of these triangular matrices and calculation of the trace we obtain the ”right” Baxter’s relation:
Next step we fix the dependence on the variables in the kernel of operator to obtain the ”left” Baxter’s relation:
The rules (3.0.4) allow to move generators from the function to the kernel of the Qoperator:
where
and means transposition. The trace of the product of matrices can be calculated
and finally we obtain:
Therefore the dependence of the kernel on the  and variables have the same form.
In the sequel we shall concentrate on the case of homogeneous XXXchain.
3.1 Qoperator for the homogeneous XXXchain
In this section we consider the homogeneous XXXchain of equal spins: . The kernel:
has the ”true”  and dependences and therefore operator can be defined as follows:
There exists some useful integral representations for obtained Qoperator.
3.2 The representation for the Qoperator
Let us consider the operator:
and transform the integral using the following identity:
(3.2.1) 
To prove this identity we use the Feynman formula:
(3.2.2) 
and transform the product:
The remaining integral can be easily calculated:
Finally we obtain the useful integral representation (representation) for the operator:
(3.2.3) 
where and
Let us consider the eigenvalue problem for the Qoperator:
where polynomial belongs to the space of homogeneous polynomials degree (2.2.8):
(3.2.4) 
The Qoperator transforms polynomial to homogeneous polynomial degree whose coefficients are polynomials in degree . Therefore eigenvalues of the Qoperator are polynomials in degree .
For the proof we use obtained representation. Let us consider the action of Qoperator on polynomial :
The expression for the integral have the form:
where coefficients
are polynomials in degree because of evident equality:
There are similar expressions for the remaining integrals and we obtain that Qoperator transforms polynomial to homogeneous polynomial degree whose coefficients are polynomials in degree .
There exists some another useful representation for the Qoperator(trepresentation):
(3.2.5) 
This formula is obtained from the (3.2.3) by the following change of variables:
3.3 invariance of the Qoperator. Commutativity
We shall prove the two important properties of obtained Qoperator: invariance and commutativity. Let us begin from the invariance:
The simplest way is to use the representation (3.2.5). We start from :
and make the change of variable in integral:
After this change of variable the integral is transformed to the integral of required form:
It is worth to emphasize that all factors like are cancelled in the whole product.
The second important property of Qoperator is commutativity:
(3.3.1) 
It follows that there exists a unitary operator independent on which diagonalizes simultaneously for all values of and therefore due to the Baxter relation operators and commute also:
(3.3.2) 
It is useful to visualize the Qoperator itself and the product of two Qoperators as shown in figure: the line with index between the points and represents the function where and . The integration (3.0.3) in any fourpoint vertex is supposed.