Q-operator for the quantum NLS model
N. Belousov, S. Derkachov
TL;DR
This paper demonstrates that Tsvetkov's operator functions as a valid Q-operator for the quantum nonlinear Schrödinger (NLS) model and connects it to the XXX spin chain in the continuum limit.
Contribution
It establishes the properties of Tsvetkov's operator as a Q-operator for the quantum NLS model and links it to the XXX spin chain at large spin.
Findings
Tsvetkov's operator satisfies all properties of a Q-operator.
The Q-operator for the XXX spin chain converges to Tsvetkov's operator in the continuum limit.
Provides a bridge between spin chain models and quantum NLS models.
Abstract
In this paper we show that the operator introduced by A. A. Tsvetkov enjoys all the needed properties of a Q-operator. It is shown that the Q-operator of the XXX spin chain with generic values of spin turns into Tsvetkov's operator in the continuum limit for large spin.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Nonlinear Photonic Systems
-operator for the quantum NLS model
N. M. Belousov , S. E. Derkachov
St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences,
Fontanka 27, 191023 St. Petersburg, Russia
Abstract
In this paper we show that the operator introduced by A. A. Tsvetkov enjoys all the needed properties of a -operator. It is shown that the -operator of the XXX spin chain with spin turns into Tsvetkov’s operator in the continuum limit as .
**Dedicated to M. A. Semenov-Tain-Shansky
on the occasion of his 70-th birthday**
Contents
1 Introduction
In the paper [1] A. A. Tsvetkov studied an infinite one-dimensional boson system with pair interaction. The Hamiltonian of the system is given by
[TABLE]
and the fields and obey the canonical commutation relations
[TABLE]
In the paper [1] it was shown that under the certain conditions on the potential the one-parameter family of operators
[TABLE]
turns out to be commutative . By we denote the normal ordered form of an operator wherein all creation operators are to the left of all annihilation operators . As an example, the potential satisfies found conditions. This potential corresponds to the quantum NLS model.
E. K. Sklyanin proposed that the operator introduced by A. A. Tsvetkov is the -operator for the quantum NLS model. The aim of this paper is to prove this conjecture of E. K. Sklyanin.
The paper is composed of two parts. In the first part we consider the XXX spin chain of spin with the quantum space carrying infinite dimensional representations. We discuss in detail the construction of the local Hamiltonian and the -operator for this model. It is well known [7, 9] that the monodromy matrix and the Hamiltonian of the quantum NLS model can be obtained from the monodromy matrix and the local Hamiltonian of the XXX spin chain in the continuum limit as . Here we investigate the continuum limit of the -operator for the XXX spin chain constructed in [11, 13]. We show that A. A. Tsvetkov’s operator emerges naturally in this limit.
In the second part we derive independently all the needed identities for the -operator of the quantum NLS model. For the sake of completeness we prove the main statements of the quantum NLS model with more emphasis on functional methods [14, 15, 16] than usual. Also we tried to stress an analogy between the formulas from the spin chain and from the NLS model.
Acknowledgements.
We are grateful to E. K. Sklyanin for the formulation of the problem and very useful discussions. This work is supported by the Russian Science Foundation (project no.14-11-00598).
2 XXX spin chain
2.1 Monodromy matrix and algebraic Bethe ansatz
In the XXX spin chain of spin the monodromy matrix is defined as the product of the -operators [5, 3, 7, 8]
[TABLE]
Each -operator in equation (2.3) as a matrix acts in auxiliary linear space . The matrix elements of the operator are generators of the Lie algebra . They act in the local quantum space associated with the th site. Here we deal with irreducible representation of the generators parameterized by spin
[TABLE]
where , are the creation and annihilation operators: . Hence is a Fock space generated by acting with the creation operators on a vacuum vector: . In the following we assume that the spin parameter is an arbitrary complex number. The entries of the monodromy matrix , the operators , act in the global quantum space .
A convenient and particularly natural choice would be to take , . Therefore becomes the space of polynomials in one complex variable , and the vacuum vector is represented by the constant: . The global quantum space coincides with the space of polynomials in variables , and the entries of the monodromy matrix are differential operators acting in this space.
The -operators satisfy the following local relation
[TABLE]
Here all operators act in a tensor product of three spaces , where and are auxiliary spaces, and is the local Fock space. By we denote permutation operator: , , .
Clearly, from the local relation we can derive the global relation for the monodromy matrices and
[TABLE]
Let us rewrite the relation (2.10) in terms of matrices by using the standard basis in the tensor product
[TABLE]
With respect to the chosen basis the matrices and have the form
[TABLE]
Similarly, the operator is given by
[TABLE]
Substituting these formulas into expression (2.10), we get the system of quadratic relations for the operators and . Note that, as it follows from relation (2.10), traces of the monodromy matrices, transfer matrices , commute: . The eigenvectors of the transfer matrix can be found within the framework of the algebraic Bethe ansatz method [5, 3, 7, 8, 17]. The following relations are at the heart of this method
[TABLE]
The algebraic Bethe ansatz method requires the existence of the vacuum vector such that
[TABLE]
where are polynomials of degree in spectral parameter . In our case, each local quantum space contains the vacuum vector . This means that the vector is the eigenvector for the operator , while the operator annihilates it
[TABLE]
The global vector is constructed from the local ones: . Whence we obtain the explicit expressions for the functions : .
In these terms, the eigenvectors of the transfer matrix have the form
[TABLE]
All information about parameters is accumulated in the polynomial
[TABLE]
Using (2.12) and (2.13) it can be shown [5, 3, 7, 8] that the vector is the eigenvector of the transfer matrix with the eigenvalue
[TABLE]
if the parameters obey the Bethe equations:
[TABLE]
The Bethe equations have a simple interpretation: is a polynomial, but each term on the right-hand side of the formula (2.14) has a pole at the point . Thus the residues of these terms should cancel each other.
2.2 Local Hamiltonian
To construct the local Hamiltonian, we need the -operator that interchanges spectral parameters in the product of -operators [3]
[TABLE]
The -operator can be represented in the following form [12] ()
[TABLE]
Calculating the derivatives over of the both sides of expression (2.15) and putting , we get
[TABLE]
At the same time from the explicit formula for the -operator we obtain
[TABLE]
where is the logarithmic derivative of the Euler gamma function. Finally, we obtain the commutation relation [3]
[TABLE]
Let us define the operator acting in the global quantum space . Therefore from the last expression it follows that this operator commutes with the transfer matrix . Here we assume that periodic boundary condition holds . Nevertheless, it is particularly convenient to work with slightly changed operator
[TABLE]
In this notation, the vacuum state corresponds to the zero eigenvalue of the pair Hamiltonian . Let us remark that this Hamiltonian is local. In other words, each term in the sum describes the interaction between the nearest neighbors only.
2.3 -operator
Recall that all information about the eigenvector is accumulated in the polynomial . Also, the eigenvalues of the transfer matrix are expressed in terms of the function . Baxter [15] suggested to interpret the polynomial as an eigenvalue of some operator, what we now call the -operator. In the general case,
[TABLE]
where the constant depends on parameters and doesn’t depend on spectral parameter . Usually there exists the special point such that the -operator turns into the identity operator at this point: . Therefore
[TABLE]
By definition the -operator satisfies the following properties
- •
commutativity
[TABLE]
- •
finite-difference Baxter equation
[TABLE]
These properties are sufficient to reproduce all formulas with the polynomial obtained in the framework of the algebraic Bethe ansatz. So, the -operator provides another approach to solve the model. Below we construct the -operator for the considered case [11, 13] using the most straightforward approach.
2.3.1 -operator
In the representation (2.9) the -operator has a convenient factorized form
[TABLE]
The main building block in the construction of the -operator is an -operator defined by the relation
[TABLE]
where
[TABLE]
Notice that this -operator differs from the operator defined by (2.15) (the old one interchanges parameters and ).
The -operator acts in the space of polynomials in variables and and could be represented as a function of operator argument [12]
[TABLE]
or as an integral operator
[TABLE]
Two forms of the -operator are connected through the integral representation of the Euler beta function
[TABLE]
and the explicit expression for the acting of the operator :
[TABLE]
In our case, , .
2.3.2 Baxter equation
On the left-hand side of the Baxter equation (2.16) there is a product of the transfer matrix and the -operator. The building blocks of the transfer matrix are the -operators. Similarly, we will construct the -operator using the -operators as its building blocks. The global relation involving the -operator and the transfer matrix will be derived from the corresponding local relation for their building blocks. The needed local relation is the defining equation for the -operator (2.18). Using the factorized form (2.17) of and and the commutativity , let us rewrite (2.18)
[TABLE]
Note that the dependence of the -operator on the parameter enters by a shift of the spectral parameter , and Z_{k}\equiv\left(\begin{array}[]{cc}1&0\\ z_{k}&1\\ \end{array}\right). Computing the product of the matrices on the right-hand side, we get
[TABLE]
[TABLE]
Clearly, at the point the matrix on the right-hand side becomes upper triangular. So, we put in the derived matrix relation and slightly change it by choosing the second space to be the local quantum space at site and the first space to be the local quantum space at site
[TABLE]
Actually this particular local relation leads to the Baxter equation. Let’s turn to the global objects: appending one additional site , we take the product over all sites
[TABLE]
[TABLE]
In this product matrices and () become neighbors and therefore cancel each other. Then we calculate the trace over the two-dimensional space , use commutativity of all -operators and with in order to move to the left and finally identify sites 0 and . Thus we obtain the Baxter equation
[TABLE]
for operator
[TABLE]
It is also possible to check the commutativity properties of [11, 13]. Now, in order to visualize the constructed operator we will derive the explicit formula for the action of on polynomials [13].
2.3.3 Explicit formula for the action on polynomials
Let us derive a very simple formula for the action of on the global generating function . This formula contains in transparent form all information about the action of the operator on polynomials. The global problem reduces to the local one
[TABLE]
[TABLE]
The explicit formula for the action of the -operator on a local generating function
[TABLE]
could be obtained using Feynman’s formula
[TABLE]
so that the action on the global generating function is
[TABLE]
[TABLE]
Finally, we put and change the norm of the -operator
[TABLE]
The action of the renormalized operator looks very simple
[TABLE]
[TABLE]
The operator maps any monomial to polynomial with respect to variables and the spectral parameter
[TABLE]
This property guarantees that eigenvalues of are polynomials in . Using formula (2.19), we get , so that we can interpret the polynomial as the eigenvalue of the operator . Finally, we obtain the canonical form of the Baxter equation for the renormalized operator
[TABLE]
So, the whole construction of the -operator is based on the local relations.
3 Continuum limit
3.1 Monodromy matrix in the continuum limit
The monodromy matrix of the continuous model satisfies the differential equation (4.3). In order to formulate a natural recipe for taking the limit [7, 17, 18], let us derive a finite-difference version of this differential equation. First we extract the matrix from the -operator (2.8)
[TABLE]
and consider the difference of the operators
[TABLE]
Hence for the following operator
[TABLE]
we obtain a finite-difference equation of the form
[TABLE]
Now, we define renormalized creation and annihilation operators
[TABLE]
so that we could interpret the commutation relations (3.5) as a discrete version of the canonical commutation relations for the fields. Let be the lattice spacing; then , become continuous variables as , and
[TABLE]
Taking into account (3.5), we get . The difference on the left-hand side of expression (3.1) turns into the derivative so that
[TABLE]
In the limit we obtain
[TABLE]
This equation for the monodromy matrix coincides with (4.3) after the renormalization of the spectral parameter .
3.2 Local Hamiltonian in the continuum limit
To find a natural interpretation of the emerged parameter , we shall take the limit in the local Hamiltonian [6]
[TABLE]
First recall that becomes a continuous variable as ; secondly, and the same for . Using these formulas, the asymptotic expansion of the logarithmic derivative of the gamma function
[TABLE]
and replacing sums by integrals , we get [6]
[TABLE]
where we have written all terms to order . Terms with total derivatives are canceled due to periodic boundary conditions. Instead of the original local Hamiltonian we can consider the operator ()
[TABLE]
which also commutes with the transfer matrix and describes the interaction between the nearest neighbors. By the same argument, we get the expansion
[TABLE]
Thus in the continuum limit we obtain the standard Hamiltonian of the continuous model [2, 9, 17]
[TABLE]
and is the coupling constant.
3.3 -operator in the continuum limit
We know the explicit connection between the discrete and continuous models so that it is possible to calculate the limit in the -operator
[TABLE]
expressed in terms of the discrete fields , . Before we plunge into the calculation, recall that the eigenvalues of the operator depend on spin
[TABLE]
So, before taking the limit we should normalize the -operator appropriately. One way of doing it
[TABLE]
Here and further we mean the summation over all sites of chain. The operator counts the number of -parameters in a given state , and we put the parameter in the denominator for the sake of convenience of the final result.
To take the continuum limit we use the integral form of the -operators
[TABLE]
The last factor in this expression acts nontrivially on the variable only
[TABLE]
Together with the normalization part from (3.6) the product of these operators acts on functions of all variables ()
[TABLE]
It is useful to rewrite this operator in a normal ordered form. Consider the following operator
[TABLE]
Under the sign of the normal ordering the operators and commute. Acting on the monomial
[TABLE]
for an arbitrary function of we get
[TABLE]
Note also that for the neighboring operators
[TABLE]
since and commute with . By taking appropriate coefficients , we find the normal ordered form of the original operators
[TABLE]
Thus we have the following formula for the renormalized -operator
[TABLE]
where
[TABLE]
Under the sign of the normal ordering the operators and commute, hence in what follows all fields are treated as classical. The continuous model is defined on the finite interval. Denote by and the upper and lower endpoints of this interval so that . Since the length of the interval in the limit remains constant, the number of chain sites . In other words, the number of integrals tends to infinity. Below we will show that, in fact, in the continuum limit this multiple integral boils down to some functional integral.
Let us change integration variables in each integral
[TABLE]
and pass from the original operators and to the renormalized ones
[TABLE]
Then we have
[TABLE]
where the factor in front of the integral equals
[TABLE]
Using Stirling’s formula for the beta function, it’s easy to calculate its asymptotic behaviour
[TABLE]
The sums in the exponent can be interpreted as some discretized functional of the original fields and and one auxiliary field . In the limit each sum turns into the corresponding integral. For example, for the first sum we have
[TABLE]
where we have used the fact that . For the remaining sums everything is similar so that in the continuum limit we obtain
[TABLE]
Here we introduced the notation for the integration measure
[TABLE]
In order to check the norm, one could consider a simpler integral:
[TABLE]
Note that we can simplify the obtained formula by changing integration variables
[TABLE]
Moreover, this functional integral is Gaussian, so we can compute it
[TABLE]
Additional terms appearing after integration by parts in the first term cancel each other due to periodic boundary conditions.
Recall that in the continuum limit we obtained the following equation for the monodromy matrix
[TABLE]
which coincides with (4.3) after the renormalization of the spectral parameter . In these terms, the -operator for the continuous model has the following form
[TABLE]
4 Continuous model
4.1 Monodromy matrix
The classical monodromy matrix is defined by the differential equation [2, 9]
[TABLE]
which can be solved using the ordered exponential
[TABLE]
The entries of the monodromy matrix
[TABLE]
are functionals of the fields , , hence the full notation for the monodromy matrix should be . To make formulas more readable, we omit the dependence on in the expressions containing functional derivatives with respect to the fields, and similarly, we drop the dependence on the fields in the equations with no functional derivatives.
In the quantum NLS model [2, 7, 9, 17, 18] the fields obey the canonical commutation relations
[TABLE]
The global quantum space is a Fock space , and the vacuum vector is defined in a standard way
[TABLE]
The Hamiltonian of the model is given by
[TABLE]
where is the coupling constant.
The quantum monodromy matrix is defined as [2, 7, 17]
[TABLE]
In other words, a normal (or Wick) symbol of the quantum monodromy matrix coincides with the classical monodromy matrix. The quantum monodromy matrix acts in a tensor product of two spaces: one auxiliary space and the Fock space ,
[TABLE]
This implies that the entries of the quantum monodromy matrix are the operators acting in the Fock space, and as a matrix acts nontrivially in the auxiliary space .
Commutation relations between the entries of the quantum monodromy matrices and can be written compactly as [2, 7]
[TABLE]
Here all operators act in a tensor product of three spaces , where and are two auxiliary spaces. The operator as a matrix acts nontrivially in the first space . Its matrix elements act in the Fock space. In the second auxiliary space the operator acts as an identity matrix. In the same way, the operator acts nontrivially in the and . Finally the -matrix is defined as
[TABLE]
where by we denote a permutation operator (2.11). The proof of the commutation relations (4.13) is given in Appendix C.
4.2 -operator
4.2.1 Commutation relations with the monodromy matrix
Consider the operator (3.7)
[TABLE]
Below we prove that this operator satisfies all the needed properties of a -operator.
In what follows we use the so-called universal notations [14] and omit the obvious integral symbol, the arguments of the fields, etc. The example is given in formula (4.15), where the exponent is written using universal notation, and below it is written in a full form.
So, first let us prove that the operator commutes with the transfer matrix. To do this, we want to rewrite the product of the -operator and the monodromy matrix in the normal ordered form. Representing the quantum monodromy matrix (4.12) as
[TABLE]
and using identity (A.2)
[TABLE]
we get
[TABLE]
Here we use the universal notation again. As an example, we write two terms in the full form
[TABLE]
and hope that further the notation will be clear. Let us remark that the underlined term in the exponent leads to the additional local terms corresponding to higher functional derivatives, for instance,
[TABLE]
In expression for the monodromy matrix (4.7) the strict order of parameters (such as , etc.) is required. Hence the local terms don’t contribute to the result.
Calculating the functional derivatives
[TABLE]
we obtain
[TABLE]
where the monodromy matrix under the sign of the normal ordering is defined by the differential equation
[TABLE]
It is easy to check that the matrix can be transformed to the much simpler matrix by the appropriate gauge transformation
[TABLE]
Then
[TABLE]
where is a solution of the equation
[TABLE]
In the case of periodic boundary conditions: , . Hence if we take the trace of both sides in formula (4.17), the matrices and cancel each other, and for the product of the -operator and the transfer matrix we get
[TABLE]
Now consider the product of the -operator and the transfer matrix in the reverse order. As above, we use the identity (A.2) and obtain a shift of the field
[TABLE]
By differentiating with respect to the sources
[TABLE]
we find
[TABLE]
where the matrix is a solution of the equation
[TABLE]
We use the gauge transformation again
[TABLE]
so that
[TABLE]
where satisfies the differential equation
[TABLE]
As before, the matrices and in (4.19) cancel each other within the trace due to periodic boundary conditions. Thus for the product of the -operator and the transfer matrix in the reverse order we get
[TABLE]
Notice that the matrices and are connected by the similarity transformation
[TABLE]
hence a similar formula holds for the matrices and . So we have . This proves that the -operator (4.15) commutes with the transfer matrix
[TABLE]
Now we derive the Baxter equation. Actually we already have all the needed formulas. Consider the product of the -operator and the transfer matrix with the same spectral parameter
[TABLE]
where is a solution of the equation
[TABLE]
For the calculation of the trace, we need the diagonal elements of the matrix only. Since the matrix is triangular, we find
[TABLE]
and thus we derive the Baxter equation
[TABLE]
Here we took into account that and used formula (A.5).
Finally we prove the commutation relation [1]
[TABLE]
Recall from the previous section that the -operator can be expressed as a functional integral
[TABLE]
Using this formula and equation (A.3), we rewrite the product of the -operators in the normal ordered form
[TABLE]
Clearly, the last expression is symmetric under the permutation .
So, we have shown in two ways, by taking the continuum limit as of the known -operator for the XXX spin chain and directly in the continuous model, that the -operator for the quantum NLS model is given by expression (4.15). It will be interesting to study a connection between the -operator and the Backlund transformation [4] for this model, hopefully we will return to it in future.
Appendix
Appendix A Normal symbols of operators
All formulas in the Appendix are written in the universal notation. The general formula for a product of two normal ordered operators can be expressed in two forms: using functional derivatives
[TABLE]
or a functional integral
[TABLE]
Applying any of these formulas it is not difficult to show that
[TABLE]
Similarly, some general relations
[TABLE]
can be proved using formula (A.1). In the first formula , are any linear operators acting on the fields and .
By putting in (A.4) , and , we obtain formulas for the -operator
[TABLE]
Appendix B Ordered exponentials
A fundamental solution of the differential equation
[TABLE]
can be expressed with the help of the ordered exponential
[TABLE]
The main formulas with ordered exponentials
[TABLE]
The last one defines perturbation series for a solution of the equation (B.1) with the slightly changed matrix ; here is the solution of the original equation (B.1) with the matrix . In order to derive this result, consider an equation for the function
[TABLE]
Let us look for a solution in the form . Then for we get
[TABLE]
hence the solution can be expressed in terms of (B.2) substituting . Using this and formula (B.3), we get
[TABLE]
Multiplying both sides by , we obtain formula (B.4). Note also that same formula (B.4) could be used to calculate the variation . Substituting ,
[TABLE]
In this work we are interested in the variations of a form , where is a scalar function and a matrix does not depend on . The general formula (B.5) leads to the explicit expressions for the derivatives
[TABLE]
and so on.
B.1 Gauge transformations
Suppose we want to represent a solution of the differential equation
[TABLE]
in the form , where is some known matrix. One could easily derive the equation for
[TABLE]
Using ordered exponentials, we can rewrite these formulas as
[TABLE]
Appendix C Commutation relations for the monodromy
matrix in the continuous model
In this Appendix we prove relation (4.13). Let us remark that this formula was proved by E. K. Sklyanin in his paper [2] in two ways. In fact, our proof repeats the main steps of E. K. Sklyanin’s proof, but with more emphasis on functional methods, and, of course, is of purely methodical interest.
The proof is based on explicit formulas for products of normal ordered operators. Let us rewrite the product of the monodromy matrices on the left-hand side of the relation (4.13) in the normal ordered form
[TABLE]
Note that expression (C.1) is symmetric under the permutation of the variables , so we get rid of the factor by passing to integration over the fundamental domain .
In order to compute functional derivatives, let us substitute in (B.5). Hence we obtain the formulas similar to (B.6), (B.7)
[TABLE]
Then we rewrite the product of these expressions in the following form
[TABLE]
where . In the same way we compute the higher derivatives, for example,
[TABLE]
In fact, the sum in expression (C.1) has the same form as the perturbation series (B.4) with substitutions and . The product is a solution of the equation
[TABLE]
with initial condition . However applying the normal ordering operation, we obtain the perturbation series (B.4) for , where is a solution of the equation
[TABLE]
with initial condition . Thus
[TABLE]
Similarly, we derive the formula for the inverse product of the monodromy matrices
[TABLE]
where solves the equation
[TABLE]
with initial condition . So the global relation (4.13) corresponds to the local one
[TABLE]
which can be easily proved. Note also that taking into account the explicit formula for the -matrix (4.14), it actually remains to check the simpler relation
[TABLE]
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