On the boundaries of quantum integrability for the spin-1/2 Richardson-Gaudin system

We discuss a generalised version of Sklyanin's Boundary Quantum Inverse Scattering Method applied to the spin-1/2, trigonometric sl(2) case, for which both the twisted-periodic and boundary constructions are obtained as limiting cases. We then investigate the quasi-classical limit of this approach leading to a set of mutually commuting conserved operators which we refer to as the trigonometric, spin-1/2 Richardson-Gaudin system. We prove that the rational limit of the set of conserved operators for the trigonometric system is equivalent, through a change of variables, rescaling, and a basis transformation, to the original set of trigonometric conserved operators. Moreover we prove that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit.

1 Introduction boundary constructions are equivalent in the trigonometric case, but not in the rational limit. Some aspects of these equivalences have been previously identified in [12]. Here our aim is to detail a more comprehensive account.
In Section 2 we begin by introducing a generalised version of Sklyanin's construction using the trigonometric six-vertex solution of the Yang-Baxter Equation, which extends the approach of Karowski and Zapletal [25] to include inhomogeneities in the transfer matrix. The algebraic Bethe Ansatz is applied to determine the transfer matrix eigenvalues and associated Bethe Ansatz Equations. This formulation is dependent on a parameter ρ such that Sklyanin's construction is obtained by setting ρ = 0. In the limit ρ → ∞ the twisted-periodic transfer matrix is recovered. We refer to this as the attenuated limit, since it has the effect of collapsing the double-row transfer matrix to the single-row transfer matrix. We also discuss the rational limit, and illustrate the general framework for the well-known case of the Heisenberg XXZ and XXX models. In Section 3 we turn our attention to a detailed analysis of the quasi-classical limit of this construction. We initially study the Bethe Ansatz Equations in this limit, and establish that several equivalences emerge through appropriately chosen changes of variable. We then show that these same equivalences extend to the conserved operators of the system by identifying appropriate rescalings and basis transformations. Concluding remarks are provided in Section 4. For completeness, we confirm in the Appendix that the equivalences hold at the level of eigenvalue expressions for the conserved operators.

Boundary Quantum Inverse Scattering Method
In this section we discuss a generalisation of Sklyanin's Boundary Quantum Inverse Scattering Method (BQISM) [39]. A key element is the R-matrix, which is an invertible operator R(u) ∈ End(V ⊗ V ) (in this paper V = C 2 ) depending on the spectral parameter u ∈ C and satisfying the Yang-Baxter Equation (YBE) [4,49] It is an equation in End(V ⊗V ⊗V ), with the subscripts indicating the spaces in which the corresponding R-matrix acts non-trivially. In this paper we will work with the trigonometric 2 R-matrix associated with the XXZ model [4] R(u) = 1 sinh(u + η) where η ∈ C is the quasi-classical parameter. Note that (2) satisfies the regularity property, i.e., R(0) = P , where P is the permutation operator. Also, it is symmetric, i.e., R 12 (u) = R 21 (u) and satisfies the unitarity property: R 12 (u)R 12 (−u) = I ⊗ I. Noting that R t1 21 (u) is invertible, we introduce an additional operator where t 1 denotes the partial transpose over the first space in the tensor product. One can check that for the trigonometric R-matrix R(u) ∝ R(−u − 2η).
In the BQISM framework we require that in addition to the YBE (1) the R-matrix satisfies two reflection equations in End(V ⊗ V ) [10] 2 While it is conventional to refer to the R-matrix as trigonometric, for convenience we adopt the hyperbolic parametrisation.
for some operators K ± ∈ End(V ), referred to as the reflection matrices or the K-matrices. One can check that the following reflection matrices satisfy equations (3) together with the trigonometric R-matrix (2): Introduce the double row monodromy matrix acting in V a ⊗ V ⊗L , where V a is called the auxiliary space (in our case a copy of C 2 ) and V ⊗L is the quantum space, T a (u) = R aL (u − ε L )...R a1 (u − ε 1 )K − a (u + ρ/2)R a1 (u + ε 1 + ρ)...R aL (u + ε L + ρ), where ρ, ε j ∈ C are complex parameters. The parameters ε j are known as inhomogeneities. These are typically set to be zero in the construction of one-dimensional quantum lattice models, but are retained as generic parameters in Richardson-Gaudin systems.
Using (1) one can check that the monodromy matrix T (u) given by (5) satisfies the following reflection type Remark 2.2. We are implementing a modification of Sklyanin's formulation, following Karowski and Zapletal [25]. This consists of introducing an additional parameter ρ, which provides a shift in the parameters: u → u+ρ/2, ε l → ε l +ρ/2. It will allow us to interpolate between the boundary and the twisted-periodic cases. The limit ρ → 0 reduces to the boundary formulation, while the limit ρ → ∞, as we will see later, yields the twisted-periodic construction.
The next step is to introduce the transfer matrix Using (6) one can prove that the transfer matrices given by (7) commute for any two values of the spectral parameter: This is a fundamental property of the transfer matrix that allows it to be used it as a generating function for the conserved operators. For future calculations it is convienient to introduce another shift u → u − η/2 in the spectral parameter and to redefine all functions taking this into account. It is also convenient to introduce the Lax operator obtained as a scaling of the (shifted) R-matrix: It satisfies the YBE Also, we need to rescale the K-matrices (4): The monodromy matrix is now and the transfer matrix is, correspondingly, One can write the monodromy matrix (10) as an operator valued 2 × 2-matrix in the auxiliary space: It is convenient to work withÃ(u) = sinh(2u + ρ)A(u) − sinh ηD(u) instead of A(u). Then, using (6), one can show that the following commutation relations hold: The transfer matrix (11) can be written in the form To find it's eigenstates and eigenvalues we follow the algebraic Bethe Ansatz as described in [39]. We start with a reference state Ω ∈ V ⊗L , s.t.
where a(u) and d(u) are scalar functions, so that Ω is an eigenstate for A(u) and D(u) simultaneously and, hence, also forÃ(u):Ã(u)Ω =ã(u)Ω, whereã(u) = sinh(2u + ρ)a(u) − sinh ηd(u). Thus, it is also an eigenstate ofť(u), which is a linear combination ofÃ(u) and D(u). It is an analogue to a "lowest weight" state in the representation theory of gl (2). We next look for other eigenstates in the form Using relations (12) one can prove that the state Φ given by (13) is an eigenstate ofť(u) with the eigenvaluě if Φ = 0 and the Bethe Ansatz Equations (BAE) are satisfied: 5 One can check that Ω = 0 1 ⊗L is a reference state. Theň where we follow the tradition that * denotes an operator which does need to be known to continue calculations. From here one can derive the formulae forã(u) and d(u): In the following, we look to take various limits of quantities such as the operatorsǨ ± (u) andĽ(u), the transfer matrix, its eigenvalues and the BAE. For readability we have chosen not to introduce new notation for each limiting object, but will ensure that it is clear which expression is being affected.

Attenuated limit
Setting ρ = 0 above, the construction reduces to the regular form of the BQISM with inhomogeneities. In this section we show that the limit ρ → ∞ reduces to the twisted-periodic QISM formulation, where the twist is sector dependent. We refer to this limit as the attenuated limit, since the double row transfer matrix reduces to a single row transfer matrix as ρ → ∞. This approach was used in [25] to construct twisted-periodic one-dimensional quantum lattice models in a manner which preserved certain Hopf-algebraic symmetries.
Consider a matrixN j = 1 0 0 0 j acting on the jth V space from the tensor product V ⊗L . We then have l . A transfer matrix eigenstate Φ is also an eigenstate of the operatorN with eigenvalue equal to N , the number of C-operators applied to the reference state in order to obtain Φ = C(v 1 )...C(v N )Ω. In this manner it is seen that the transfer matrix has a block diagonal structure wherebyN takes a constant value on each block. Furthermore, We then havě SinceN is a conserved operator, it commutes with both A 1 and D 1 . Thus, Remark 2.3. The twisted-periodic transfer matrix has the form [40] Thus, to obtain the twisted-periodic transfer matrix (19) from the attenuated limit (18) of the boundary transfer matrix (11), we need to impose that γ depends on N : From (16) we can compute that In the limit as ρ → ∞: Thus, the BAE (15) in this limit reduce to In a similar manner we obtain the limit of (14) aš Remark 2.4. We recognise that (21) subject to (20) are the BAE for (19), as required; e.g. see [15,47]. We also recognise that (22) subject to (20) are the eigenvalues of (19).

Rational limit
In this section we show that there is a relationship between the rational twisted-periodic system and the rational boundary system that is similar to the trigonometric case that we have just discussed in the previous section. By introducing a parameter ν (the so-called rational parameter) as a scaling factor in the argument of the hyperbolic functions, and using lim ν→0 sinh(νx) ν = x, one can obtain the rational limit of the relevant operatorsĽ(u) of equation (8) and theǨ ± (u) of equations (9) as follows: We observe that in this same limit, the BAE (15) become and the expression for the eigenvalues given in (14) reduces tǒ The transfer matrix (11) in the rational limit, particularly in the form (17), is readily obtained by employing the expressions (23), (24) and (25) above. To then determine the attenuated limit of this rational transfer matrix, we first observe that from (23) above,Ľ(u) → I as u → ∞. This implies that the termsĽ aj (u + ε j + ρ) occuring to the right ofǨ − a (u + ρ/2) in (17) all simplify to the identity as ρ → ∞. Without loss of generality, we moreover suppose that ξ − does not depend on ρ, in which case taking the attenuated limit of (24) giveš Furthermore, we set ξ + = ζρ, where ζ ∈ C, from which we obtain the attenuated limit of equation (25) above: Thus, we have the attenuated limit of the rational transfer matrix in the form (17) being given by where the operatorsĽ aj (u − ε j ) and, correspondingly, the operators A 1 and D 1 are in the rational limit. Finally, imposing the condition that ζ = ±1/2 to avoid any technical issues of divergence, for convenience we rescaleǨ to match this limiting expression forť(u) with that of the twisted-periodic case given in equation (19) above. This is achieved by setting In the attenuated limit (i.e. ρ → ∞), the rational BAE (26) become It is evident that by setting we may identify (29) with the rational limit of (21). It is also worth pointing out that (30) is consistent with (28). Finally, the expression for the eigenvalues (27) in the attenuated limit iš By once again applying (28), we may identify the expression (31) with the rational limit of (22). In other words, we have shown that the rational and attenuated limits commute, subject to appropriate scaling of relevant quantities. A convenient way to summarise our discussions so far in Section 2 is to provide a diagram highlighting the connections we have made between the various trigonometric, hereafter denoted Trig., and rational, hereafter denoted Rat., limits. We will also use the notations BQISM to denote the general construction, and QISM for the attentuated limit. Note below that Trig. BQISM and Rat. BQISM are merely the respective Trig. BQISM and Rat. BQISM with ρ included explicitly in all expressions. We do not consider these to be fundamentally different systems (consider variable change #1 in the diagram, denoted simply by #1, which is just v k → v k + ρ/2, ε l → ε l + ρ/2), but make the distinction as a convenience to highlight our utilisation of the methods of Karowski and Zapletal [25] via the attenuated limit.

Heisenberg model
In this section we show how the Heisenberg model can be obtained as a special case from the general construction outlined so far. Here we will omit the shift u → u − η/2 and the scalings described in equations (8) - (11), in order to obtain the standard form of the Heisenberg model.
Consider the transfer matrix (7) with ε j = 0: If we take ρ → 0 we obtain the open chain Heisenberg model transfer matrix: The Hamiltonian is constructed from t(u) given by (33) as follows: where (0) are derivatives of the corresponding operators at u = 0. The explicit form of the Hamiltonian (34) in terms of Pauli matrices may be found in [39]. Now if we consider ρ → ∞, the transfer matrix (32) will tend to we can match it with the transfer matrix for the closed chain, namely Here again where H j(j+1) = P j(j+1) R j(j+1) (0) and X = e −ηγ 0 0 e ηγ . In the rational limit (XXX model), the calculations are completely analogous to Section 2.2, so we omit the details.
As in Section 2.2, we may summarise the analogous connections for the Heisenberg model in the following diagram: It is worth highlighting the fact that for the Heisenberg case, since we have set the parameters ε j = 0, it is not possible to implement the variable change #1 discussed in the previous section.
3 Quasi-classical limit and the spin-1/2 Richardson-Gaudin system Here we develop the main results of the current article. We investigate the quasi-classical limit of the system described in Section 2, which involves expanding all expressions in η as η → 0 and taking the first non-trivial term. In the quasi-classical limit, unlike the special case of the Heisenberg model above, we are able to implement variable change #1. Moreover, we gain the capability of implementing two additional variable changes. It is through these variable changes that we are able to make unexpected connections between various systems in the quasi-classical limit. We find that the following commutative diagram, in contrast to those presented in Section 2, illustrates the connections we shall make in this section for the BAE and the conserved operators:

Rat. QISM
The connections that have been established previously still hold in the quasi-classical limit. Dashed arrows represent the connections that are yet to be established. In the diagram we adopt the notation where #1 denotes variable change #1, #2 is used for variable change #2 combined with some other operations, and #3 represents variable change #3 with different operations, all of which are specified explicitly in the text below.

Bethe Ansatz Equations
We start by considering the BAE. Substituting the expressions (16) forã(u) and d(u) into the BAE (15) gives If we set η = 0 in (35), the expression reduces to Furthermore, we assume that ξ ± depend on η in such a way that (36) holds as η → 0. We impose the following choice which is consistent with that property: The expansion up to first order in η for the right hand side of the BAE (35) with (37) is given by Also, up to first order in η, the expansion of the left hand side of (35) is Let us denote δ = −(α + β + 1). Then, in the limit as η → 0, the BAE in the case Trig. BQISM are given by

Variable change #1
It is a straightforward matter to see that Trig. BQISM (38) turns into Trig. BQISM as ρ → 0: Variable change #1 reverses this effect:

Variable change #2
It can be seen that we may transform from Rat. BQISM (43) to Trig. BQISM (39) by a suitable variable change. Application of Now, in order to transform this expression into Trig. BQISM (39) we make a change of variables Thus, the mapping from Rat. BQISM (43) Analogously, including ρ gives the mapping from Rat. BQISM (42) to Trig. BQISM (38): Generally, we refer to equations (47) as the variable change #2, and note that (46) is merely a specialisation of (47) with ρ = 0. This results in the variable change given by Equivalently, we may take which gives the same. We refer to the (48) as variable change #3.

Reduction to the rational, twisted-periodic case
One can obtain Rat. QISM by taking the rational limit of Trig. QISM (41). Introduce the rational parameter ν into (41): Then, denoting δ = γ/ν, multiply through by ν and consider ν → 0. In such a case we obtain Rat. QISM in the quasi-classical limit: We can also obtain Rat. QISM (49) by taking the attenuated limit from Rat. BQISM (42): .

Rat. QISM
It turns out that the limit labelled Rat. QISM is not equivalent to any of the other five nodes in the diagram above. This is deduced by knowledge of a particular solution of the BAE. For the BAE (44), it was identified in [24] that when δ = N − 1 there is a solution for which y k = 0 for all k. Results from [36] show that such a solution where all roots are equal does not exist for the BAE (49). Consequently (44) and (49) cannot be equivalent.
The most unexpected aspect of the above calculations concerns the parameter ξ. Recall that this parameter arises in the expansion of the variables ξ ± , as given by (37), where ξ ± are the free parametrising variables of the reflection matrices (9). The above calculations show that ξ is a spurious variable which can be removed by appropriate variable changes. In the next section we will show that it is also possible to remove the ξ-dependence from the conserved operators, but this requires an appropriate rescaling and basis transformation in conjunction with the variable changes.

Conserved operators
In the quasi-classical limit, the conserved operators τ j are constructed as follows from the transfer matrix: To calculate these conserved operators, we first set ρ = 0, and impose the conditions (37) on ξ ± that appear in the reflection matrices given in equations (9). ExpandingǨ ± (u) in η as η → 0 then giveš where we define .
It is easily verified thatĽ(u) given by (8) can be represented as follows: where Here we have introduced the representation matrices of su(2) corresponding to the fundamental (i.e. twodimensional) representation. Specifically, they are the matrices It is worth remarking that the connections that we make in the current article are only concerning this twodimensional local Hilbert space. These are what we refer to as the spin-1/2 Richardson-Gaudin system.
Using the expressions of equations (51) and (52) above, we may take the expression (17) for the transfer matrix and expand (50) explicitly as In the above expression each K-matrix acts on the auxiliary space, however we have suppressed the subscripts "a" for ease of notation. Finally, after computing the traces, we obtain We rescale and denote τ trig Thus, {τ trig j , j = 1, ..., L} are the mutually commuting conserved operators for Trig. BQISM.

Rational limit
The rational limit of the conserved operators for Trig. BQISM (53) gives the conserved operators for Rat. BQISM: We rewrite this expression as Using S + S − = I/2 + S z , S − S + = I/2 − S z we obtain, after simplification and rescaling by ε j :

Rational BQISM and trigonometric QISM equivalence
Separating the terms with ξ from the rest in equation (57) we obtain the following equivalent expression: Now it is seen that if we set ξ = 0 we obtain The variable change ε j → exp ε j gives Trig. QISM (55) with γ = −(α + β + N − L/2) (up to a constant term −3I/4): Now let us start with Trig. QISM (55) (with a change of variables ε j = ln z j ): Using and (S z ) 2 = I/4 for the spin-1/2 representation, we obtain A change of variable z j → ε 2 j − ξ 2 gives the following conserved operators: Note that up to this point, all we have done is apply the change of variables given in (45) on the ε j . We further rescale each conserved operator τ (2) j by the factor ε 2 j ε 2 j − ξ 2 : Consider a local transformation on the jth space in the tensor product Under these transformations we have Under the global transformation U = U 1 U 2 ...U L we find We begin with the the form (57) of Rat. BQISM, multiplied by ε j : Then, rescale by Finally, we haveτ which is the same as τ trig j (58) up to the constant term:

Variable change #3, rescaling, and a basis transformation
As in the case of the BAE, variable change #3 is defined as the composition which leads to (48). Combined with the appropriate composition of basis transformations and rescalings described above, this leads to the following mappings for the conserved operators: Trig. QISM where the arrow labels refer to the subsections where the corresponding operations are described.

Conclusion
In this work we have studied the spin-1/2 Richardson-Gaudin system as the quasi-classical limit of a formulation provided by a generalised BQISM. In this manner we uncovered some surprising features, viz. that the rational limit of the boundary trigonometric system is equivalent to the original boundary trigonometric system. Additionally we found that the twisted-periodic and boundary constructions are equivalent in the trigonometric case, but not in the rational limit. One consequence of this finding is that for the spin-1/2 Richardson-Gaudin system the BQISM formalism does not extend the integrable structure beyond that provided by the QISM formalism. This is an unexpected result, in contrast to the Heisenberg model. There are several directions for future studies. One is to investigate the analogous system obtained by implementing non-diagonal solutions of the reflection equations. Due to the breaking of u(1) symmetry in this instance, there is the possibility to make connection with elliptic parametrisations. The construction of conserved operators for this case has previously been undertaken in [50], and we have already initiated an analysis of this problem. Higher spin versions of the Richardson-Gaudin system is another option. The BQISM formulation of these systems appears in the work [12]. Whether a basis transformation exists to establish the equivalence between the Rat. BQISM and Trig. QISM conserved operators in this case remains an open problem, but examination of the associated BAE in [12] is suggestive that it does exist. Models based on higher rank algebras are also worthy of investigation. In this regard, a systematic construction of conserved operators has been undertaken in [44,45] which unifies previous particular case studies. Supersymmetric analogues, such as the osp(1|2) Richardson-Gaudin system [26], provide another avenue for future research.

A Eigenvalues of the conserved operators
In this article we have shown, in the quasi-classical limit, the explicit connections between the BAE and conserved operators associated with the rational limit of the BQISM for Richardson-Gaudin systems, and the corresponding twisted-periodic trigonometric systems. We can also verify analogous connections between the eigenvalues of the conserved operators. While this necessarily follows from the equivalence of the conserved operators, it is useful as a consistency check as well as having the potential to provide some alternative insights into the methods used. The summary diagram for the BAE, with the same variable changes, also holds on the level of eigenvalue formulae.
The eigenvalues λ j in the quasi-classical limit are constructed from (14) as follows (set ρ = 0): It gives the eigenvalues for Trig. BQISM up to a factor of sinh 2 ε j sinh(ε j + ξ) sinh(ε j − ξ) as follows: where δ = −(α + β + 1). We can check that the constant terms agree. To do this, we need to check that the action of τ trig j on the state Ω, where Ω = 0 1 ⊗L , is equal to the constant term in (61). Namely, that Indeed, by making repeated use of the identity sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y) and other similar identities for hyperbolic functions, we may easily check that .

Rational limit
The rational limit of Trig. BQISM (61) gives Rat. BQISM: Or, multiplied by ε j : Equivalence of the rational BQISM and the trigonometric QISM Set ξ = 0 in Rat. BQISM (65): Making a change of variables ε j → exp ε j , we obtain Trig. QISM (63) up to a constant term −3/4: coth(ε j − v i ). Now, we want to turn Trig. QISM (63) back into Rat. BQISM (65). We start with Trig. QISM (63) (with a change of variables ε j = ln z j , v i = ln y i ) Make the change of variables Then, rescale by ε 2 j ε 2 j − ξ 2 : Choose γ = −(α + β + N − L/2), which leads to γ + N − L 2 + 1 2 = −(α + β) + 1 2 = −(α + β + 1) + 3 2 = δ + 3 2 . Thus, is the same as Rat. BQISM (65) up to a constant term. Hence, Trig. QISM is equivalent to Rat. BQISM in the quasi-classical limit also on the level of the eigenvalue formula. The difference of the constants in the eigenvalues is the same as the action of the difference of the conserved operators on the reference state: Variable change #2 Here we want to transform the eigenvalue formula Rat. BQISM (64) back into Trig. BQISM (61). We start with Rat. BQISM in the form (65), multiplied by ε j : We follow similar steps as in the case of the conserved operators, without the basis transformation. Start with the change of variables This gives

Now rescale by
Using the identity (z 2 j − z −2 j ) (z j − z −1 j ) 2 − (z k − z −1 k ) 2 = 1 2 The variable change 3 is obtained in the same way as for the BAE and conserved operators, described in Sections 3.1 and 3.2.