Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation

We propose commuting sets of matrix-valued difference operators in terms of trigonometric ${\rm GL}(N|M)$-valued $R$-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric $R$-matrices are used for this purpose. The first is the one related to the affine quantized algebra ${\hat{\mathcal U}}_q({\rm gl}(N|M))$. The second is a graded version of the standard $\mathbb Z_n$-invariant $A_{n-1}$ type $R$-matrix. We show that being properly normalized the latter graded $R$-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the ${\rm gl}(N|M)$-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.


Introduction
The Haldane-Shastry long-range spin chain [11] is a quantum integrable model describing interactions between L spins distributed equidistantly on a circle.The Hamiltonian is where x k = k/L, k = 1, ..., L and P ij is the permutation operator representing exchange interaction between spins; for the gl n model it acts on the Hilbert space H = (C n ) ⊗L by permuting the i-th and the j-th tensor components.The Haldane-Shastry model is naturally extended to the supersymmetric case [12,3]; the Hamiltonian (1.1) keeps the same form but the permutation operator is defined on the graded space H = (C N |M ) ⊗L (see (2.3) below).
The non-supersymmetric model has several important generalizations.The first one provides anisotropic analogues [28,24] of (1.1) likewise it happens in the short-range spin chains (magnets), where the isotropic XXX model is extended to the anisotropic XXZ and XYZ models.For example, the XXZ version of the Haldane-Shastry model (1.1) for the gl 2 case is where σ α denotes the Pauli σ α in the i-th tensor component.The second type of generalization is a q-deformation, which is similar to the relativistic generalization of the Calogero-Moser-Sutherland differential operators to the Ruijsenaars-Schneider or Macdonald difference operators.For the gl 2 Haldane-Shastry model (1.1) the q-deformation was suggested in [30].Then it was studied in [17,18], where the Hamiltonian was presented in the following form (q = e πi ):  (1.4) and Ri,j = R (x i − x j ) is the quantum R-matrix for the affine quantized algebra Ûq (gl 2 ), see (B.2).
In (1.4) e ij stands for the standard matrix units.This result was extended in [23,24] to a wide class of GL n R-matrices including the elliptic Baxter-Belavin R-matrix and its trigonometric degenerations.
The idea was first to define a set of matrix-valued commuting difference operators D k , k = 1, ..., L generalizing the (scalar) commuting difference Ruijsenaars-Macdonald operators [26,22].The commutativity property [D i , D j ] = 0 was shown to be equivalent to a set of R-matrix identities, which was then proved by analytical methods using explicit form of R-matrices in the fundamental representation of GL n .Next, using the Polychronakos freezing trick [25,19], the commuting Hamiltonains of q-deformed Haldane-Shastry type models were found.A similar idea will be used in this paper for the case of supersymmetric R-matrices.There are extensions of the Haldane-Shastry model to different root systems [4] although we do not discuss them in this paper.
The goal of the paper is three-fold.First, we describe two types of supersymmetric trigonometric R-matrices which we deal with.One is the widely known R-matrix related to the affine quantized algebra Ûq (gl(N |M )) [5].Another one is a supersymmetrization of Z n -invariant A n−1 type R-matrix [7].We suggest an answer for it and could not find it in the literature although we believe it is known, since the graded extensions of trigonometric R-matrices were extensively studied [16].In order to relate our R-matrix to known results we construct a Drinfeld-type twist [8], which transforms it into the Ûq (gl(N |M )) case.For the graded Z n -invariant R-matrix we also show that (after being properly normalized) it satisfies the so-called associative Yang-Baxter equation [9].This result is obtained as a by-product.At the same time it is important by itself since it extends the classification of trigonometric solutions of the associative Yang-Baxter equation [27] to the graded case.Also, solutions of the associative Yang-Baxter equation are used for different constructions in integrable systems [9,20,13,10,28,29] including the above mentioned R-matrix identities [23].
Second, we prove that the main statement of [23] concerning construction of commuting matrixvalued difference operators is valid for the graded R-matrices under consideration.In this way we obtain two sets (corresponding to two possible choices of R-matrices) of commuting graded matrix-valued difference operators.The operators can be viewed as Hamiltonians of the graded (or supersymmetric) spin Ruijsenaars-Schneider model or its anisotropic versions.The spin Ruijsenaars-Schneider model was introduced at the classical level in [14] and much progress was achieved in studies of corresponding Poisson structure and its quantization [2].The supersymmetric version of the quantum trigonometric Ruijsenaars-Schneider model was introduced in [6].The quantization in [6] was based on the usage of Cherednik operators.Our approach is different since our difference operators commute not for any R-matrix but only for those which satisfy a certain set of identities.It is also interesting to compare our results with the classical construction of the spin Ruijsenaars-Schneider model from [15].Finally, we use the Polychronakos freezing trick [25,19] and obtain supersymmetric generalizations of q-deformed Haldane-Shastry type models.The first Hamiltonian has the form: where It is valid for both graded R-matrices which we consider, i.e. for (B.1) and (B.4).For example, in the supersymmetric case Ûq (gl(1|1)) related to the R-matrix of (B.3) one gets the same expression as in (1.3) The paper is organized as follows.In the next Section the supersymmetric trigonometric Rmatrices are described.In Section 3 we formulate the main properties of these R-matrices including the associative Yang-Baxter equation (and the quantum Yang-Baxter equation) for the graded version of Z n -invariant R-matrix and the associative Yang-Baxter equation with additional term for the Rmatrix of the affine quantized algebra Ûq (gl(N |M )).The twist transformation relating two R-matrices is given as well.In Section 4 the construction of commuting spin (or matrix) difference operators is given.Finally, in Section 5 we apply the freezing trick to obtain new families of integrable long-range spin chains based on supersymmetric R-matrices.In Appendix A we give a sketch of the proof that the graded version of Z n -invariant R-matrix satisfies the associative Yang-Baxter equation.Appendix B contains the list of explicit expressions for the normalized graded trigonometric R-matrices.

Notations
In the supersymmetric case consider the Z 2 -graded vector space Denote by e ij the elementary matrix units acting on the space and the graded permutation operator acting on C N |M ⊗ C N |M is given by (−1) p j e ij ⊗ e ji . (2.3)
Let us remark that the parameter in a quantum R-matrix plays the role of the Planck constant providing the semiclassical limit → 0: R 12 At the same time it plays the role of the relativistic deformation parameter in the sense that the limit → 0 corresponds also to transition from Lie group to Lie algebra.For example, quantum R-matrices can be used at the classical level for constructing "relativistic" integrable systems (those described by quadratic Poisson brackets of the Sklyanin type) with relativistic parameter .A direct relation to the Ruijsenaars-Schneider model with the relativistic parameter can be established, see [21,13].
Example Ûq (gl(1|1)): (2.7) In the non-graded case (M = 0) it was introduced in [7] and obtained in the trigonometric limit from the elliptic R-matrix in [1].(2.9) The normalized version of the latter R-matrix (B.6)) leads to the graded version of the q-deformed XXZ (anisotropic) model.

Yang-Baxter equations and other properties of R-matrices
Any quantum R-matrix, by definition, satisfies the quantum Yang-Baxter equation: For the R-matrix (2.4) this property is known.Below we prove it for (2.7).

Associative Yang-Baxter equation
In this subsection we discuss the associative Yang-Baxter equation [9]: The Yang-Baxter equations (3.1) and (3.2) have different sets of solutions.This is easy to see from the scalar (i.e.gl 1 ) case.Indeed, the quantum Yang-Baxter equation (3.1) becomes an identity in this case, while (3.2) is a nontrivial functional equation.It has solution given by the following function1 : However, there is a class of R-matrices satisfying both Yang-Baxter equations (3.1) and (3.2).It includes the elliptic GL n -valued R-matrix (in the fundamental representation of GL n ) and its degenerations.Trigonometric GL n solutions of (3.2) were classified in [27].The non-graded R-matrix (2.7) with M = 0 is contained in this classification (see [13]) as well as more general (the so-called non-standard) trigonometric R-matrix found previously in [1].
As for the R-matrix related to Ûq (gl n ), it was mentioned in [13] that the non-graded R-matrix (2.4) with M = 0 satisfies the associative Yang-Baxter equation with additional term: The r.h.s. of (3.4) vanishes for n ≤ 2 so that (2.5) satisfies (3.2).But for n > 2 the expression on the r.h.s. of (3.4) is nontrivial although it is independent of spectral parameters z 1 , z 2 , z 3 .
Theorem 1 The supersymmetric extension of the Z n -invariant R-matrix (2.7) satisfies the associative Yang-Baxter equation (3.2).

Theorem 2
The supersymmetric R-matrix (2.4) of the affine quantized algebra Ûq (gl(N |M )) satisfies the associative Yang-Baxter equation (3.2) with additional term: where the right-hand side is independent of the spectral parameters.For N + M ≤ 2 (3.5) turns into (3.2).
The proofs of both theorems are given in Appendix A. The result of Theorem 1 extends the classification [27] of trigonometric solutions of (3.2) to the graded case.As we previously mentioned, the nonsupersymmetric version of (2.7) has generalizations satisfying (3.2).For example, it is true for the 7-vertex trigonometric R-matrix and its higher rank analogues.The question whether these type Rmatrices have supersymmetric counterparts deserves further elucidation.Different applications of the associative Yang-Baxter equation (see [9,27,20,13,10,28,29]) can be studied for the graded R-matrix (2.7).

Quantum Yang-Baxter equation
We first mention that both R-matrices (2.4) and (2.7) obey the unitarity property and the skew-symmetry property This is simply verified by straightforward calculation.

Theorem 3
The supersymmetric extension of the Z n -invariant R-matrix (2.7) satisfies the quantum Yang-Baxter equation (3.1).
Proof.Of course, one can verify the statement by direct computation.But there is a simpler way to do it using the statement of Theorem 1.It is known (see e.g.Section 4 in [20]) that any unitary (3.We use two different normalizations of R-matrices.The one (3.6)means that R-matrix is normalized to the function φ( , z).R-matrices normalized as in (3.6) are used in the the R-matrix identities.Another normalization is the standard one.The normalized R-matrices are defined as follows: The short notation Rij = R ij (z i − z j ) is used for the R-matrix acting non-trivially on the i-th and j-th tensor components of the Hilbert space H = (C N |M ) ⊗L .The quantum Yang-Baxter equation for any distinct integers 1 ≤ i, j, k, l ≤ N .

Twist transformation
Here we describe a relation between the Ûq (gl(N |M )) R-matrix R 12 (u − v) in (2.4), and the one in (2.7), R 12 (u − v).It is given by the following Drinfeld twist.

Theorem 4
The relation between the R-matrices (2.4) and (2.7) is as follows: where ) and .15)This statement may be confirmed by direct computation.
Notice that for the case N + M = 2 the twist matrix F becomes trivial: That is, the R-matrices (2.5) and (2.8) as well as (2.6) and (2.9) are related by a gauge transformation with G(u) = diag(1, e πıu ).Obviously, the relation (3.12) remains the same for the normalized R-matrices from Appendix B.

Graded spin Ruijsenaars-Macdonald operators
Let p i be the shift operator defined by its action on a (sufficiently smooth) function f (z 1 , . . ., z L ): Denote by D k the trigonometric Ruijsenaars-Macdonald operators [26,22]: where the sum is taken over all subsets I of {1, . . ., L} of size k, and φ(z) ≡ φ( , z) 3) The commutativity of operators (4.2) holds: Namely, it was shown in [26] that the commutativity is equivalent to a set of relations for the function φ, which can be considered as a set of functional equations.For the function (3.3) these relations become identities.
Following ideas of [30,18] we introduced in [23] a set of (spin or matrix-valued) difference operators.The graded (spin) Ruijsenaars-Macdonald operators are defined in the same way (but with the graded R-matrices): where k = 1, ..., L, Rij = R ij (z i − z j ).The arrows in (4.5) mean the ordering in the R-matrix products.
It was shown in [23] that the operators D k in (4.5) commute with each other if and only if the following set of identities holds true: where and In the scalar case M = 1, the identities (4.7) coincide with the identities for φ-functions underlying commutativity of the Ruijsenaars-Macdonald operators (4.2).
The expressions D k (4.5) are matrix-valued difference operators.For example, That is, D k in (4.5) are End(H)-valued difference operators.The commutativity of operators (4.5) was proved (in the case M = 0) for the trigonometric Ûq (gl 2 ) R-matrix in [18].In [23] a proof was given based on the identities (4.7) for the elliptic R-matrix (including some trigonometric and rational degenerations).The Ûq (gl n ) R-matrix was considered separately in [24].
In order to prove that commutativity of operators (4.5) holds true for the graded R-matices (2.4) and (2.7) we need to prove the identities (4.7) for these R-matrices.In fact, the proof is almost the same as in [23,24].Let us give a sketch of the proof.The aim is to show that the l.h.s. of (4.7) (denote it by F) is independent of η.Then we can put η = 0 and the statement becomes simple: F| η=0 = 0 due to the unitarity property (3.6) of the R-matrix.More precisely, F| η=0 = IdF scalar , where F scalar is the same expression as F with R-matrices being replaced by φ-functions (4.3).In this case the statement (F scalar = 0) was proved in [26].In order to prove that the l.h.s. of (4.7) is a constant as a function of η, we make two steps.The first step is to show that F has no poles in the variable η = z i − z j .The proof is given in [23] (see Proposition 4.1) and can be performed in the same way for the graded R-matrices with the property Res u=0 R 12 (u) = P 12 , where P 12 is the graded permutation.The second step is to show that the function F has no poles at η = z i − z j + m, where m ∈ Z.For this purpose we use the quasi-periodic property of R-matrix (2.7): where Notice that Q is independent of the spectral parameter z.Therefore, we have Res The absence of poles at η = z i − z j + m guarantees that F is a constant function of η, see Appendix C in [24].This proof works also for the case of the Ûq (gl(N |M ))-valued R-matrix (2.4), which is periodic: R 12 (z + 1) = R 12 (z).In fact, the proof in this case repeats the one from Appendix C in [24] for the non-supersymmetric case.The only difference is that the permutation operator is now graded.
We also mention that the identity (4.7) for k = 1 follows from the associative Yang-Baxter equation (see [23]), so that for the R-matrix (2.7) the relation (4.7) with k = 1 follows from (3.2).Another possible application of the associative Yang-Baxter equation is in the construction of classical analogues of integrable models related to the presented difference operators.These are models of interacting tops.The underlying Lax representations and R-matrix structures are discussed in [13,10,28,29].

Long-range spin chains
Here we derive commuting Hamiltonians of long-range spin chains by applying the Polychronakos freezing trick [25] to the difference operators (4.5).The derivation is performed similarly to the one presented in [24].

Hamiltonians of the spin chain
Consider expansion of the scalar difference operators D k (4.2) in the variable η near η = 0: (5.1) Using also the expansion of (4.1) we conclude that D [m] k are some m-th order differential operators.For example, and Consider now a similar expansion of the spin operators (4.5) D k in variable η (near η = 0): , due to the unitarity (3.9), we have where Id = 1 ⊗L N +M is the identity matrix in End(H).For the set of D (5.7) In the general case we also have where Hk ∈ End(H) are some matrix-valued functions, which contain R-matrix derivatives but do not contain differential operators.
By restricting the operators Hk to the points z k = x k = k/L we obtain the set of Hamiltonians of long-range spin chains: (5.9)

Commutativity of these Hamiltonians
[H i , H j ] = 0 , i, j = 1, ..., L − 1 (5.10) follows from the set of identities for the function φ: and for any k = 1, ..., L. See [24] for the proof in the elliptic case.This proof is the same for the trigonometric case.The set of equidistant points x k is an equilibrium position in the underlying classical spinless model which is the trigonometric (spinless) Ruijsenaars-Schneider model.Moreover, it is the equilibrium position for all flows of this model.
Introduce the following compact notations: and Fij = F ij (x i − x j ) . (5.13) Let us write down expressions for the first two Hamiltonians.