A novel class of translationally invariant spin chains with long-range interactions

We introduce a new class of open, translationally invariant spin chains with long-range interactions depending on both spin permutation and polarized spin reversal operators, which includes the Haldane-Shastry chain as a degenerate case. The new class is characterized by the fact that the Hamiltonian is invariant under"twisted"translations, combining an ordinary translation with a spin flip at one end of the chain. When the spin-spin interactions depend on the inverse square of the distance, we are able to compute in closed form the model's partition function and hence study several statistical properties of its spectrum. In particular, we show that this model possesses a highly degenerate spectrum, which suggests the existence of an underlying twisted Yangian symmetry.


Introduction
Exactly solvable quantum dynamical models in one dimension and spin chains with longrange interactions have given rise to interesting applications as prototypes of strongly correlated systems exhibiting generalized exclusion statistics [1][2][3][4], as well as in connection with such diverse phenomena as the quantum Hall effect [5,6], quantum transport in mesoscopic systems [7,8] or quantum simulation of long-range magnetism [9]. Moreover, recent experiments involving optical lattices of ultracold Rydberg atoms and trapped ions have opened up the exciting possibility of experimentally probing various properties of such low-dimensional quantum spin chains in a very precise way [10][11][12][13][14][15]. In the context of high-energy physics, quantum spin chains with long-range interactions have played a key role in the computation of the spectrum of the dilation operator in N = 4 super Yang-Mills theories [16][17][18]. Intriguing connections between (1+1)-dimensional conformal field theory (CFT) and exactly solvable models with long-range interactions have also been uncovered in recent years [19][20][21][22]. In the same vein, it has been found that the Casimir equation for conformal blocks in d dimensions can be transformed into the eigenvalue problem for a hyperbolic Calogero-Sutherland model associated with the BC N reflection group [23,24].
The translationally invariant Haldane-Shastry (HS) spin-1 2 chain [25,26], whose spins are equispaced on a circular lattice and exhibit two-body interactions inversely proportional to the square of their chord distances, is the prototypical example of an exactly solvable lattice model with long-range interactions. The HS spin chain features many interesting properties, a few of which we shall briefly review next. Indeed, its exact ground state coincides with the U → ∞ limit of Gutzwiller's variational wavefunction for the Hubbard model, and with the one-dimensional version of Anderson's "resonating-valence-bond state" [25,[27][28][29]. Moreover, the spinon excitations of this chain [30] (in the antiferromagnetic case) can be described through a generalized Pauli exclusion principle, as they can be regarded as anyons with exclusion parameter g = 1/2 (so called semions) [1]. The original -su(2)-HS spin chain and its su(m) generalization also exhibit Yangian quantum group symmetry, even for a finite number of lattice sites [2,31]. As a consequence, the spectrum of these integrable spin chains can be described by means of finite sequences of the binary digits 0 and 1, known as "motifs" in the literature [2], related to certain finite-dimensional representations of the Yangian algebra [32,33]. In addition, the complete energy spectrum of these chains, including the degeneracy of each level, can be generated from the energy function of an associated one-dimensional classical vertex model [34]. On the other hand, the su(m) HS chain can also be obtained by taking the freezing limit [35] of the su(m) spin Sutherland model [36], whose particles possess both dynamical and spin degrees of freedom. More precisely, in the limit of large coupling constant the spin part of the latter model's Hamiltonian decouples from the dynamical one and yields the Hamiltonian of the HS chain. In fact, the partition function of the su(m) HS chain can be computed in closed form [37] by taking advantage of this decoupling of the spin and dynamical degrees of freedom of the spin Sutherland model, which is the essence of Polychronakos's freezing trick.
In view of these remarkable properties, several methods have been proposed for constructing integrable or exactly solvable variants of the HS chain. In particular, the connection between quantum integrable models with long-range interactions and different (extended) root systems has proved a useful tool in this endeavor [38,39]. The main idea in this respect is replacing the A N −1 root system associated with the spin Sutherland model, which yields the original HS chain, by other (extended) root system like, e.g., BC N , B N and D N . The spin Sutherland models associated with all of these systems have in fact been studied in the literature, and used to construct spin chains of HS type by taking their strong coupling limits [40][41][42][43]. In particular, the exact partition function of each of these chains has been computed in closed form applying the freezing trick to its parent spin dynamical model. It should be noted, however, that none of these integrable variants of the HS chain retains the translational invariance characteristic of the original model. Several extensions of the HS spin chain with lattice sites arbitrarily distributed on a circle and exact ground state wavefunctions have also been recently constructed using infinite matrix product states (MPSs) [19] related to certain rational CFTs. More precisely, it has been observed that the correlators associated with the chiral vertex operators of such CFTs play the role of infinite MPSs yielding the ground state wavefunctions of these HS-like chains [19][20][21][22]. Using the null field techniques related to rational CFTs, one can explicitly construct the Hamiltonians of these chains and study interesting physical properties thereof, like, e.g., the spin-spin correlation functions. However, these models lack two of the key properties of the original HS chain, namely, they do not exhibit translational invariance nor are they integrable in general. More recently, exact ground state wavefunctions of some HSlike spin chains, with open boundary conditions and lattice points arbitrarily distributed on a half-circle, have been constructed using infinite MPSs related to suitable boundary CFTs and corresponding null fields [44]. Again, these CFT-inspired generalizations of the HS chain with open boundary conditions become integrable [44] and, in fact, exactly solvable [45], only for some special choices of the lattice points, but none of them possess the translational invariance of the original HS chain.
The aim of this work is to construct a new type of solvable spin chain with longrange interactions possessing two of the essential properties of the su(m) HS chain, namely translational invariance and uniform spacing of the sites. Our construction is still based on applying Polychronakos's freezing trick to an appropriate (solvable) translationally invariant spin dynamical model [46]. However, unlike the original HS chain and its BC N , B N and D N versions mentioned above, neither this dynamical model nor the resulting chain are associated with an extended root system in the standard fashion. More precisely, the spin chain we shall construct is translationally invariant, in the sense that (like the original HS chain of A N −1 type) the interactions between sites i and j depends only on the difference i − j. However, in contrast with the latter chain, its Hamiltonian includes both spin permutation and spin flip operators generating the full Weyl group of D N type. In particular, the presence of the spin flip operators in the Hamiltonian also distinguishes the new model from the spin chains constructed from infinite MPSs in the above cited references. On the other hand, we shall see that the new chain's sites are uniformly arranged on a half-circle, like in uniformly spaced HS chains of BC N type, so that the model is naturally regarded as an open chain. As such, the usual translation operator does not commute with the Hamiltonian due to boundary terms. Remarkably, however, the new model is exactly invariant under the cyclic group of "twisted" translations, suitably combining standard translations with spin flips. This symmetry is indeed one of the hallmarks of the model, which we believe sets it apart from all the solvable spin chains with long-range interactions discussed above. In fact, our construction can be substantially extended in two different directions. In the first place, we can replace the standard spin flip operators in the Hamiltonian by partially polarized versions thereof without compromising any of the key properties discussed above, namely translational invariance of the interactions, uniform spacing of the sites, symmetry under twisted translations and solvability. Secondly, we can allow for much more general spin-spin interactions such that the resulting models still possess the characteristic symmetry under twisted translations discussed above. This yields a wide class of new open, translationally invariant spin chains with long-range interactions and uniformly spaced sites, whose solvability properties are certainly worth investigating. In particular, this class includes a new model with elliptic interactions reminiscent of the well known Inozemtsev [47] chain.
In this paper we shall focus on the simplest of these new models, with inverse square interactions and ordinary (minimally polarized) spin flip operators, which formally reduces to the HS chain if the latter operators are replaced by 1. We shall take advantage of the connection between the new chain and its associated dynamical spin model to evaluate the chain's partition function in closed form. As it turns out, the structure of this partition function depends critically on the parity of the number m of spin degrees of freedom. In particular, for even m the partition function factorizes as the product of the partition functions of the su(m/2) and su(1|1)-supersymmetric HS chain. This fact is certainly surprising, since the new model is built from standard (non-supersymmetric) permutation operators, and deserves further study. On a more concrete level, the formula for the partition function in the even m case straightforwardly yields a description of the spectrum in terms of suitable motifs, which could be used to study the thermodynamics of the model using the transfer matrix method developed, e.g., in Refs. [48][49][50].
The paper's organization is as follows. The new models are introduced in Section 2, where we also discuss their characteristic invariance under twisted translations. In Section 3 we first show how these models arise from a suitable spin dynamical model of Calogero-Sutherland type through Polychronakos's freezing trick. We then establish the prerequisites needed for evaluating the partition function of the new chain with inverse square interactions. To this end, we introduce an auxiliary scalar operator obtained replacing the spin permutation and reversal operators in the corresponding spin dynamical model by their counterparts acting on the spins' spatial coordinates. We finally show how this operator can be explicitly triangularized exploiting its connection with a suitable family of Dunkl operators. Using this fact, in Section 4 we compute the spectrum of the spin dynamical model associated with the novel chain with inverse square interactions and of its scalar version. These spectra are then used in Section 5 to derive an exact closed-form expression for the partition function of the novel spin chain. Section 6 contains a brief discussion of the general properties of the spectrum of this model, and a comparison with related models like the A N −1 and D N HS chains. In Section 7 we present our conclusions, and briefly discuss several avenues for further research suggested by our results. The paper ends with three technical appendices in which we list the main algebraic relations satisfied by the coordinate permutation and shift operators K ij and K i , evaluate a sum used in the computation of the new chain's partition function in Section 5 and compute the standard deviation of the new chain's spectrum in closed form.

The model
In this paper we shall study the su(m) spin chain with Hamiltonian where S ij is the operator permuting the i-th and j-th spins and S ij = S ij S i S j , with S k a local operator at site k satisfying S 2 k = 1 for all k. More precisely, if we label the su(m) internal degrees of freedom by the half-integers −M, −M + 1, . . . , M := (m − 1)/2 the action of the operator S ij on the canonical basis states C m is given by The operators S ij and S i S j (with 1 i < j N ) generate the Weyl group of the D N classical root system, algebraically determined by the non-trivial relations where the indices i, j, k are all distinct and it is understood that S ij = S ji , for i > j. The model, however, does not have the standard form 1 of a spin chain with sites ξ k associated with the D N root system [42,51]. Nor is it clearly associated with the A N root system like, e.g., the original HS chain and its rational [35,52], hyperbolic [53] and elliptic [47] counterparts, due to the presence of the operators S i in the Hamiltonian. Thus the chain Hamiltonian (2.1) is not directly associated with an extended root system, unlike most chains of HS type considered so far in the literature (see, e.g., [40,42,43,54] for other instances of HS-like chains of BC N , B N and D N types). For each k = 1, . . . , N the operator S k generates the cyclic group Z 2 of order 2, which has two inequivalent irreducible representations (necessarily one-dimensional, as Z 2 is abelian) R ± for which S k = ±1. If the internal space C m decomposes under the action of each operator S k as C m = R l + ⊕ R m−l − , we shall say that the degree of polarization of the corresponding m-dimensional representation of Z 2 is d = |2l − m| ∈ {ε, ε + 1, . . . , m} (where ε ∈ {0, 1} is the parity of m). When d = m is maximal, i.e., for S k = ±1, S ij = S ij and consequently the Hamiltonian (2.1) reduces to the Hamiltonian H HS of the HS spin chain with J = 2. Thus the model (2.1) can be formally regarded as an extension of the Haldane-Shastry spin chain. On the other hand, in the remaining cases where the polarization is not maximal (i.e., for d < m), the spin-spin interactions associated with the operators S ij and S ij do not coincide with each other. As a result, the Hamiltonian (2.1) describes a novel class of spin chains whose sites are uniformly spaced on the upper half of a circular lattice.
In this paper we shall focus exclusively on the simplest of these new models, corresponding to the case of minimal polarization d = ε. It is then convenient to decompose C m into (m−ε)/2 copies of the two-dimensional (reducible) representation R + ⊕R − plus ε copies of R ± . We shall choose the canonical spin basis so that each space R + ⊕ R − is spanned by the basis vectors {| − i , |i } with i = 1, . . . , M , so that (if ε = 1, i.e., m is odd) the remaining one-dimensional space R ± is spanned by |0 . A moment's thought reveals that we can also choose 2 the basis so that S k |i = | − i for i = 1, . . . , M and (if ε = 1) S k |0 = |0 . Thus in this case S k acts on the internal space as a spin flip operator: Except in the trivial case with maximum polarization d = m, the two different spin-spin interactions in the Hamiltonian (2.1) can be regarded as inverse square interactions between two spins or one spin and its image, as we shall now explain. Indeed, the first term in eq. (2.1) can be interpreted as a two-body interaction with strength inversely proportional to the square of the chord distance between the i-th and j-th spins, provided that the chain sites ξ k are located at the points ξ k = e 2iθ k in the complex plane. Since θ k = (kπ)/(2N ) with k = 1, . . . , N , the chain sites lie on uniformly spaced points in the unit half -circle |z| = 1, Im z 0, with angular coordinates arg ξ k = 2θ k . Thus it is natural to regard the model (2.1) as an open chain, in spite of its translationally invariant interactions. If we now define the image ξ * k of the chain site ξ k as its reflection with respect to the origin, i.e., ξ * k = −ξ k , we have ξ k − ξ * j = e 2iθ k + e 2iθ j = 2 cos(θ j − θ k ). Thus the second term in the Hamiltonian (2.1) can be interpreted as a two-body interaction of each spin in the chain with the image of any other spin, whose strength is again inversely proportional to the square of their distance.
We shall next discuss the remarkable behavior of the Hamiltonian (2.1) under translations along the chain sites, which, as mentioned in the Introduction, is one of the model's distinctive features. To this end, let T denote the left translation operator defined by T |s 1 , . . . , s N := |s 2 , . . . , s N , s 1 , Since the interaction strength between sites i and j in the Hamiltonian (2.1) depends only on i − j, it may naively seem that H commutes with the translations generator T when we use eqs. (2.4) to deal with boundary terms. This is actually not the case due to the open nature of the model (except, of course, in the trivial case d = m), which is also apparent from the fact that the interaction strengths of site 1 with sites 2 and N differ. More precisely, taking eq. (2.4) into account we obtain .
From the identities and similarly cos 2 (θ j−1 − θ N ) = sin 2 (θ j − θ 1 ), it then follows that Thus H does not commute with the left translation T unless S j = 1 for all j, i.e., in the trivial case of maximal polarization when the model (2.1) reduces to the HS spin chain. However, this problem is easily cured introducing the "twisted" (left) translation operator Indeed, taking into account that and similarly for T † S ij T (since S i+1,j+1 or S i+1,j+1 commutes with S 1 unless j = N ) and proceeding as above we obtain We thus see that the chain Hamiltonian (2.1) commutes with the elements of group of twisted translations generated by T .
It follows that T 2N = 1, so that the latter group is a cyclic group of order 2N (i.e., twice that of the group of standard translations). Of course, for d = m (i.e, when the model (2.1) reduces to the HS chain), and only in this case, the twisted translation operator reduces to the standard one T . This again underscores the fundamentally new character of the new model in the case of non-maximal polarization d < m. In particular, in the case of minimal polarization d = ε the operator T acts on the canonical spin basis as Thus in this case, which shall be extensively dealt with in the remaining sections, the twisted translation consists of a translation combined with a flip of the first site's spin.
In fact, the symmetry under twisted translations discussed above leads to a natural generalization of the chain (2.1) to a much wider class of models. To see this, consider the Hamiltonian where f is an even function of one variable. A straightforward computation along the previous lines leads to the identity We thus see that H will commute with T provided that the following two identities between the unknown functions f and f are satisfied: The first identity relates f to f , while the second one is then equivalent to the condition that f be 2N -periodic. These considerations lead to the Hamiltonian with f an even 2N -periodic function. This periodicity condition makes it natural to regard the latter model as a uniformly spaced semi-circular (open) chain, with sites ξ k = e ikπ/N ≡ e 2iθ k . The model (2.1) is then obtained requiring that f (i − j) be the square of the inverse distance between sites i and j.
Remark 1. The new chain (2.1) is also related to the (non-translationally invariant) HS chain of D N type. To see this, note first of all that the Hamiltonian (2.1) can be written as where the chain sites ξ k = e 2iθ k = −ξ * k are the coordinates of the unique (up to an overall translation) equilibrium of the potential If, on the other hand, we define the image ξ * k by ξ * k = ξ k (where the bar denotes complex conjugate), then Taking θ k as the coordinates of the unique equilibrium (in the set

Spin dynamical model and Dunkl operators
As mentioned above, from now on we shall deal exclusively with the inverse square model (2.1) in the case of minimal polarization d = ε, when the operators S i act simply as ordinary spin flip operators. One of the main aims of this paper is to show that the partition function of the Hamiltonian (2.1) can be evaluated in closed form by exploiting its connection with a non-standard spin dynamical model introduced in ref. [55], whose spectrum we shall compute in closed form in Section 4. The model (2.1) is therefore a simple but nontrivial example of a solvable spin chain with long-range two-body interactions and twisted translation invariance. Note in this respect that the rational and hyperbolic chains of HS type mentioned above are solvable but not translationally invariant, while the Inozemtsev chain [47] (an HS-like spin chain with elliptic two-body interactions) is translationally invariant but not solvable.

Spin dynamical model
More precisely, the solvability of the spin chain (2.1) relies on its connection with the solvable spin dynamical model with Hamiltonian where ∆ := i ∂ 2 ∂x 2 i and a > 1/2. The key idea, which is originally due to Polychronakos [56], is to take the large coupling constant limit a → ∞ in eq. (3.1). In this limit the eigenfunctions of H become sharply peaked around the coordinates of the equilibrium position of the scalar potential (2.7) in the configuration space A of the model (3.1). Note that this configuration space can be taken as due to the impenetrable nature of the singularities of the scalar potential (2.7), explicitly given by The unique global minimum of U in the open set A is the point whose coordinates x k = kπ/(2N ) ≡ θ k coincide with the sites of the chain (2.1), apart from an irrelevant overall translation [57]. Thus in the limit a → ∞ the particles' dynamic and spin degrees of freedom effectively decouple. In fact, since and H spin (θ 1 , . . . , θ N ) = H, for a → ∞ the energies of the spin dynamical model (3.1) are approximately given by where E i and E j denote respectively two arbitrary eigenvalues of H sc and H. From the above relation we immediately obtain the following exact formula for the partition function Z of the chain (2.1): where Z and Z sc are respectively the partition functions of the Hamiltonians H and H sc .
Remark 2. In fact, the general spin chain Hamiltonian (2.5) can also be derived from a spin dynamical model akin to eq. (3.1). Indeed, let where F (x) is an even 2τ -periodic function 4 . The spin chain Hamiltonian constructed from H applying the freezing trick discussed above is given by (up to a proportionality factor which can be chosen at will) where (ξ 1 , . . . , ξ N ) =: ξ are the coordinates of the equilibrium of the scalar potential To see that ξ is uniquely defined (up to a rigid translation), note first of all that u is even and τ -periodic. Indeed, It then follows that u is odd and τ -periodic. Furthermore, u (τ /2) = 0, since Reasoning as in Theorems 1-2 of ref. [57] we conclude that For instance, taking F (x) = sin −2 x we have 2τ = π and f (k) = sin −2 (kπ/2N ), yielding the model (2.1). Another interesting possibility is F (x) = sn −2 (x; k) ≡ sn −2 x, so that τ = K(k) (the complete elliptic integral of the first kind), These elliptic models (or their equivalent counterparts constructed using Weierstrass elliptic functions) appear to be new and deserve further study in their own right. In fact, it should be noted that the Hamiltonian H does not reduce in the maximally polarized case , whose associated spin chain is the Inozemtsev chain.

Auxiliary Hamiltonian
The solvability of the spin dynamical model H in eq. (3.1) and of its scalar version H sc can be established by studying the scalar auxiliary Hamiltonian as we shall next explain. In the latter equation K ij denotes the coordinate permutation operator acting on a scalar function f (x) by Note that the operator K l is obviously related to the star operator defined in the previous section by z * l = e 2i(K l x l ) . Before proceeding with the details of the computation, a comment on the domain of the operator H is in order. Indeed, the presence of the operators K ij and K i K j in H entails that we have to enlarge the natural configuration space A of the operators H sc and H to a region invariant under the action of the latter operators. To begin with, the invariance under the permutation operators K ij leads to the region However, for the operators K ij and K i K j to generate the Weyl group of D N type, it is necessary that (K i K j ) 2 = 1 for all i = j. Thus it may seem that the appropriate configuration space for H is the region for each pair of basis vectors e i , e j with i < j. In fact, as will become clear in the sequel, the stronger condition K 2 i = 1 for all i is actually needed in what follows. In view of this condition, we shall take as configuration space of H the region A with π-periodic boundary conditions x ≡ x + πe i in each coordinate 5 . It is therefore natural to take as domain of H an appropriate dense subspace P of the Hilbert space H = L 2 ([−π/2, π/2] N ∩ A ) (see figure 1 for a sketch of the configuration space [−π/2, π/2] N ∩ A in two and three dimensions). More precisely, we shall choose P as the span of the π-periodic trigonometric functions 6 ϕ n (x) = µ(x)e 2in·x , n := (n 1 , . . . , n N ) ∈ Z N , (3.8) where the factor (which, as we shall see, is the ground state of H ) is included for later convenience. Indeed, the span of the functions (3.8) is dense in L 2 ([−π/2, π/2] N ), and hence in H . It is also important to realize that the periodic boundary conditions K 2 i = 1 we are imposing are equivalent to quantizing the system's total momentum, which causes its spectrum on H to be discrete. 5 It is straightforward to check that A is invariant under the operators KiKj if we take into account the periodicity conditions x ≡ x + πei for all i. 6 We shall see below that the functions ϕn actually belong to the domain of H , i.e., that H ϕn is regular on the singular hyperplanes xi = xj and |xi − xj| = π/2 with i = j. In the next section we shall explicitly compute the spectrum of H on H by expressing it in terms of a family of commuting Dunkl operators that we shall introduce below. This result will then be used to derive the spectra of the scalar and spin dynamical models H sc and H in the CM frame.

Dunkl operators
We start by introducing the differential-difference Dunkl operators [55,58] 10) where l = 1, . . . , N and K lj = K jl for l > j. The Dunkl operators J l are related to the analogous operators J 0 l (z) in ref. [55] by where z := (z 1 , . . . , z N ) with z k = e 2 ix k and the inessential parameter m in the latter reference has been set to 0. Note also that the action of the operator K i on the variable z i is simply It should be noted that for N 3 it is essential for the validity of this result that the operators K i satisfy K 2 i = 1, as opposed to the weaker condition (K i K j ) 2 = 1 for all i = j stemming from the D N Weyl group algebra. Moreover, a similar calculation proves that the auxiliary operator H can be expressed in terms of the Dunkl operators J i as Again, for N 3 the above identity requires that K 2 i = 1 for all i.

Spectrum of the auxiliary Hamiltonian
The starting point in the computation of the spectrum of H on H is the fact that the operators J l leave invariant the (infinite-dimensional) subspace P n ⊂ P spanned by the functions ϕ n (x) with |n| := i n i = n, for an arbitrary (possibly negative) integer n. In fact, the operators J l also admit the finite-dimensional invariant subspaces R p (p = 0, 1, . . . ) spanned by the latter monomials with multiindices satisfying the conditions |n i | p for i = 1, . . . , N . By eq. (3.12), the auxiliary operator H will also leave invariant the subspaces P n and R p , and hence their intersection. We shall construct a non-orthonormal (i.e., Schauder) basis B of H by appropriately ordering the set {ϕ n (x) : n ∈ Z N }, and then show that the operator H is upper triangular with respect to this basis. Finally, we shall derive the spectrum of H by explicitly computing its diagonal matrix elements in the basis B.
In order to simplify the calculation, we shall apply the change of variables z k = e 2 ix k and the pseudo-gauge transformation with gauge factor µ to the Dunkl operators J l , and thus work with the operators where sgn(x) is the sign of x. Defining where n k ∈ Z, we then have It shall also be necessary for what follows to define a partial order in the set of monomials φ n . This is done in four stages, as we shall now explain. To begin with, we order the multiindices n ∈ Z N by p(n) := max{|n i | : i = 1, . . . , N }, and then (for p(n) = p(n )) by n := i n i . We next define [n] ∈ Z N as the rearrangement of the multiindex n ∈ Z N whose components are weakly decreasing, i.e., such that that [n] 1 [n] 2 · · · [n] N . The multiindices with given p(n) and n are then partially ordered using their rearrangements [n], i.e., defining that n precedes n if the first nonzero difference [n] i − [n ] i is negative. The above prescription clearly defines a total order among weakly decreasing multiindices and a partial order in Z N which we shall denote by ≺ , with 0 ≡ (0, . . . , 0) as first element. We shall then say that φ n ≺ φ n if and only if n ≺ n . We shall also use the notation n n to indicate that n ≺ n (i.e., either n ≺ n or [n] = [n ]), and similarly for φ n φ n . It is obvious from the previous definition that the partial order thus defined is invariant under coordinate permutations, i.e., where W is an arbitrary element of the permutation group generated by the operators K ij . Of course, the partial order ≺ can be equivalently defined on the functions ϕ n by setting ϕ n ≺ ϕ n if and only if n ≺ n . Note, finally, that ≺ can be promoted to a total order by setting the precedence of two multiindices which differ by a permutation of their components in an arbitrary way. For instance, if we choose lexicographic order then (−1, . . . , −1) is the second element (this is always the case due to the partial order), (−1, . . . , −1, 0) is the third, (−1, . . . , −1, 0, −1) the fourth, etc. We start by computing J * i φ n on a monomial φ n . To this end, note first of all that where by definition sgn(0) = 0. Since the exponents in the numerator are nonnegative we can carry out the division, thus easily obtaining Similarly, (3.15) and Note, in particular, that the right-hand side of eqs. (3.14)-(3.16) obviously belongs to P * n := µ −1 P n with n = |n|, i.e., the space spanned by monomials φ n with integer exponents n k (k = 1, . . . , N ) of total degree |n| = n. Hence J * i P * n ⊂ P * n , or equivalently J i P n ⊂ P n , as we had previously claimed. In the same way it is established that J * i R * p ⊂ R * p := µ −1 R p , and hence J i R p ⊂ R p . It is also easy to convince oneself that all the terms in the sums in eqs. (3.14)-(3.15) precede the monomial z n i i z n j j with respect to the partial order defined above, since the exponents of z i and z j in each of them range over min(n i , n j ) + 1 and max(n i , n j ) − 1 and their sum equals n i + n j . This is easily seen to imply that for appropriate coefficients c n n ∈ C N . The last observation can be considerably strengthened if we assume that n ∈ Z N , i.e., if n 1 n 2 · · · n N . Indeed, in this case and therefore Multiplying both sides of the latter equation by k;k =i,j z n k k we finally obtain where l.o.t. consists of monomials φ n ≺ φ n and with |A| denoting the cardinal of a set A. The above formula can be slightly simplified by introducing the notation l(n i ) = min{j : n j = n i } , #(n i ) = |{j : n j = n i }|, in terms of which We shall next make use of eqs. (3.19)-(3.21) above to obtain the spectrum of the operator H . More precisely, we shall see that the action of H is upper triangular on the (non-orthonormal) basis B = {ϕ n : n ∈ Z} of H , ordered with any ordering compatible with ≺, and shall explicitly compute its diagonal matrix elements.
We start by evaluating H ϕ n when the multiindex n belongs to [Z N ]. To this end, note first of all that Since by assumption n ∈ [Z N ], by eq. (3.20) we have where in the second equality we have used eq. (3.19). Hence in this case The coefficient of ϕ n in the latter formula can be simplified by noting that We shall next compute H on a basis function ϕ n with an arbitrary n ∈ Z N . To this end, let us denote by W any permutation such that ϕ n = W ϕ [n] . Since H commutes with W (cf. Appendix A) we have where we have used the fact that the partial order ≺ is invariant under permutations. We conclude that H is indeed represented by an upper triangular matrix in the basis B of H , with eigenvalues E([n]) with n ∈ Z N . Moreover, since H preserves the subspace P n for all integers n, H admits a basis of (unnormalized) eigenfunctions of H of the form ψ n = ϕ n + n ≺n, |n |=|n| c n n ϕ n , n ∈ Z N . (3.24) Note that the sum over n in the previous equation is actually finite, since max i |n i | p(n) by the invariance of the subspaces R p under H . On the other hand, the auxiliary operator H (as well as its counterparts H and H sc in the previous section) is translationally invariant, and thus commutes with the total momentum operator Since obviously P ϕ n = 2|n |ϕ n , the states (3.24) are also eigenfunctions of P with eigenvalue 2|n|. It follows that the set {ψ n : n ∈ Z N } is a (non-orthonormal) basis of H consisting of common eigenfunctions of H and P , with H ψ n = E([n])ψ n , P ψ n = 2|n|ψ n . (3.25)

Spectrum of the scalar and spin dynamical models
In this section we shall compute the spectra of the Hamiltonians H and H sc introduced in Section 2 by exploiting their connection with the auxiliary operator H . We shall first work on appropriate subspaces of H or H ⊗ Σ on which the action of H respectively coincides with that of H sc or H, and then boost the energies of the eigenfunctions of the latter operators in these subspaces to the center of mass (CM) frame.

Spectrum with periodic boundary conditions
Consider, to begin with, the scalar Hamiltonian H sc . It is clear from its definition that H sc coincides with H on the subspace of H in which i.e., whose elements are symmetric under permutations and translations by π/2 (modulo π) of an even number of coordinates. This subspace is therefore where Λ s and Λ 0 respectively denote the symmetrizer with respect to coordinate permutations and translations by π/2 (modulo π) of an even number of coordinates. It is important to note that the spectrum of H sc on H sc is actually the same as its spectrum on the more natural space L 2 ([−π/2, π/2] ∩ A). Indeed, what we are basically doing here is enlarging the original configuration space A to A by applying the Weyl group generated by the operators K ij , K i K j , and then restricting ourselves to wavefunctions defined on this larger space which have a certain well-defined symmetry under the action of the latter operators (see, e.g., ref. [42] for a more technical explanation). The symmetrizer Λ 0 can be expressed in terms of the projectors Λ ± 0 onto states symmetric ("+") or antisymmetric ("−") under the action of the operators K i by noting that where W + i (respectively W − i ) denotes the product of an even (respectively odd) number of operators K i . We thus have Λ 0 = Λ + 0 + Λ − 0 , and therefore 7 Since it is clear that either Λ ε 0 (ϕ n ) = ϕ n , if all the components of the multiindex n have the same parity (1 − ε)/2, or Λ ε 0 (ϕ n ) = 0 if this is not the case. Hence the functions ϕ n,δ = Λ s ϕ 2n+δ1 , n 1 · · · n N , n k ∈ Z , (4.1) where δ ∈ {0, 1} and 1 := (1, . . . , 1), form a (non-orthonormal) basis of H sc = Λ s Λ 0 (H ). Moreover, from eqs. (3.14)-(3.16) it immediately follows that J i , and hence H , preserves the subspace P n,δ of P n spanned by the functions ϕ 2n+δ1 with |2n + δ1| = n and δ equal to either 0 or 1. Thus when n is a multiindex whose components have the same parity δ we can assume that all the components of the multiindices n appearing in eq. (3.24) have also parity δ. Defining ψ n,δ = Λ s ψ 2n+δ1 we thus have ψ n,δ = ϕ n,δ + n ≺n, |n |=|n| c 2n +δ1,2n+δ1 ϕ n ,δ , so that the functions ψ n,δ with n 1 · · · n N and δ ∈ {0, 1} are a basis of H sc . Since H commutes with all the operators K ij and K i , it commutes with the symmetrizers Λ s and Λ 0 . Moreover, by construction we have and hence An analogous argument shows that P ψ n,δ = 2(2|n| + N δ) ψ n,δ .
Thus the functions ψ n,δ with n ∈ Z N , n 1 · · · n N and δ = 0, 1 are a basis of H sc consisting of common eigenfunctions of H sc and P .
Consider, next, the Hamiltonian H of the spin dynamical model (3.1). This operator obviously coincides with H in the subspace H of H ⊗ Σ defined by the relations Thus 7 Note that Λ ± 0 commutes with all Kij, and hence with Λs.
where (with a slight abuse of notation) Λ ± 0 denotes the projector onto even ("+") or odd ("−") states with respect to the operators K i S i , and Λ a stands for the antisymmetrizer with respect to simultaneous permutations of the coordinates and spin variables. As in the scalar case, the spectrum of H on H actually coincides with its spectrum in its natural Hilbert space L 2 ([−π/2, π/2] ∩ A) ⊗ Σ. A basis of H consists of the states Φ ε n,s = Λ a Λ ε 0 (ϕ n (x)|s ) , (4.2) where n ∈ Z N and s ∈ {−M, −M + 1, . . . , M } N are appropriately restricted so that the latter states are linearly independent. For instance, it suffices that Indeed, the first two conditions simply take into account the antisymmetry of Φ ε n,s with respect to permutations. As to the third one, note first of all that acting on Φ ε n,s with K i S i (which does not change the physical state, since by construction K i S i Φ ε n,s = ε Φ ε n,s ) we can reverse the sign of s i if necessary. Moreover, if s i = 0 we have so that (−1) n i ε = 1. By the same token, the states Ψ ε n,s = Λ a Λ ε 0 (ψ n (x)|s ) , where the quantum numbers n and s satisfy conditions 1-3 above, are also a basis of H. Since, again by construction, Similarly, P Ψ ε n,s = 2|n| Ψ ε n,s . Thus the states Ψ ε n,s , with ε = ± and n, s satisfying the previous three conditions, are a basis of H of common eigenfunctions of H and P .

Spectrum in the CM frame
As mentioned above, in order to compute the spectrum of the Hamiltonians H sc and H in the CM frame we simply boost their respective eigenfunctions ψ n,δ and Ψ ε n,s in the Hilbert spaces Λ s Λ 0 H and Λ a Λ 0 (H ⊗ Σ) to the latter frame and evaluate their resulting energies. To this end, it suffices to note that if a state Ψ ∈ H ⊗ Σ is a simultaneous eigenfunction of H and P with respective eigenvalues E and p, then e ic|x| Ψ (where |x| := k x k ) is an eigenfunction of these operators with eigenvalues E + 2cp + N c 2 and p + N c. In particular, e − ip N |x| Ψ is an eigenfunction of H with zero momentum and energy E − p 2 N . The same result holds for simultaneous eigenfunctions of H sc and P .
Consider, first, the scalar Hamiltonian H sc , whose eigenfunctions are a basis of H sc . We can, to begin with, take δ = 0, since ψ k,1 and ψ k,0 differ by a momentum boost e 2iδ|x| and can thus be regarded to represent the same physical state in two different reference frames. Secondly, the wavefunction ψ 2k can be expressed as for a suitable l ∈ Z. Finally, the state is equivalent (up to a momentum boost) to Λ s (ψ 2n ), whose energy and momentum are respectively E(2n) and 4|n|. Boosting the latter state to the CM frame we conclude that the states are a basis of eigenfunctions of H sc in the CM frame, with energy Consider next the Hamiltonian H of the spin dynamical model (3.1). As we saw in the previous subsection its eigenfunctions Ψ ε k,s , where ε = ±1 and the quantum numbers k ∈ Z N and s ∈ −M, −M + 1, . . . , M satisfy the conditions 1-3 above, are a basis of H. Proceeding as before, we write Ψ ε k,s = Λ a Λ ε 0 (e 2il|x| ψ n (x)|s ) = e 2il|x| Λ a Λ (−1) l ε 0 ψ n |s , with n 1 · · · n N = 0 and l ∈ Z. Since the factor e 2il|x| is just a momentum boost, we need only consider eigenfunctions of H of the form Ψ ± n,s with n as above, whose energy and momentum are respectively E(n) and 2|n|. As a consequence, the boosted states Λ a Λ ± 0 (ψ n |s ) , n 1 · · · n N = 0 , is the Hamiltonian of the su(m/2) spin Sutherland model [36] and E 0,S (a) = E 0 /4 its ground energy [37], but with twice the degeneracy for each level.

Partition function
We shall next apply the results of the previous section to evaluate in closed form the partition function of the spin chain (2.1) through the freezing trick formula (3.2). We shall start by computing the a → ∞ limit of the partition function Z sc of the scalar Hamiltonian H sc . To this end, we expand in powers of a the energies E n in eq. (4.4), obtaining is the ground-state energy of H sc in the CM frame. We thus have 1) where the sum was evaluated in ref. [37] in connection with the large a limit of the partition function of the scalar Sutherland model [59,60]. This connection is not surprising, since the change of variables y k = 2x k transforms H sc into four times the Hamiltonian of the latter model.
We shall next evaluate the partition function of the spin dynamical model (3.1) in the large a limit. As it turns out, the calculation depends in an essential way on the parity of the number m of internal degrees of freedom.
where P N is the set of compositions (i.e., ordered partitions) of the integer N . The sum i p i (N + 1 − 2i) in the latter equation can be evaluated as in ref. [37], with the result where ν i := ν i − ν i+1 > 0 and the dispersion relation E is defined by (see Appendix B for the details). Since the variables ν i = 1, 2, . . . are unconstrained, from eq. (5.4) we easily obtain and therefore Alternatively, this formula can be deduced by exploiting the connection between the spectra of H and the Hamiltonian of the spin Sutherland model described above. Indeed, from this connection it follows that is the partition function of the Hamiltonian (4.7). The a → ∞ of the function f (a) was evaluated in ref. [37], obtaining as a result the sum in eq. (5.5).
Using eqs. (5.1) and (5.5) in the freezing trick formula (3.2) we finally obtain the following explicit formula for the partition function of the chain (2.1) in the case of even m: Remarkably, the latter formula can be written as where Z HS and Z (m/2) HS respectively denote the partition functions of the su(1|1) and su(m/2) antiferromagnetic HS spin chain [37,61]. In other words, the model (2.1) with an even number m of internal degrees of freedom is isomorphic to the direct product of an su(1|1) and an su(m/2) (antiferromagnetic) HS chain. This property, which is far from obvious from the definition (2.1) of the Hamiltonian, makes it possible to describe the model's spectrum in terms of Haldane motifs and their corresponding Young tableaux, which greatly facilitates the study of its thermodynamics. Work on these and related topics is in fact in progress, and will be reported elsewhere.
In particular, in the simplest m = 2 case the sum in eq. (5.6) contains only the term with = 1 N , so that Thus the spin 1/2 chain (2.1) is equivalent to the su(1|1) supersymmetric HS chain, up to the overall energy shift N (N 2 − 1)/6. This is again remarkable, since in principle there is no connection between the spin flip operators S i appearing in the Hamiltonian (2.1) and the su(1|1) supersymmetric permutation operator S (1|1) ij .

Odd m
According to condition 3 in Section 4.1, when m is odd the components of the spin vector s i can take the value 0 only if the corresponding component n i of the multiindex n has parity (1 − ε)/2. Hence in this case s i can take (m + (−1) n i ε)/2 values instead of m/2, and the intrinsic degeneracy of each energy (4.6) is thus given by As a consequence, eq. (5.3) now becomes Introducing again the unconstrained variables ν i = ν i − ν i+1 ∈ N (with i = 1, . . . , r − 1), in terms of which ν i = r−1 j=i ν j , and using eq. (5.4) we obtain The above expression can be further simplified by setting (5.9) and hence eq. (5.8) is equivalent to Using eq. (5.1) and the freezing trick relation (3.2) we finally obtain the following explicit formula for the partition function of the chain (2.1) in the odd m case: with d δ ( ) defined by eq. (5.9).

Spectrum and degeneracy
The explicit formulas for the partition function of the chain (2.1) derived in the last section make it possible to exactly compute its spectrum for relatively high values of N for given m. Since, as remarked above, the model with m = 2 is essentially the su(1|1) HS chain, we shall restrict in this section to m > 2. More specifically, we shall briefly report several general properties of the spectrum suggested by our analysis in the su(3) and su(4) cases.
To begin with, it is clear from eqs. (5.6) and (5.10) that the partition function of the model (2.1) has a different structure when m is even or odd. In fact, while in the former case the factorization (5.7) yields a description of the spectrum using Haldane's motifs, this does not seem to be possible in the odd m case due to the complicated structure of the partition function (5.10). In spite of this fact, our results indicate that the spectrum of the model (2.1) has very similar properties for odd or even m.
The first conclusion we can draw from our evaluation of the partition function of the chain (2.1) for several values of N is that, regardless of the parity of m, the spectrum exhibits a huge degeneracy. In order to put this statement into perspective, we have compared the degeneracy of the new chain's spectrum with the degeneracy of the spectra of the A N −1 and D N HS chains, which is very high due to the Yangian (or twisted Yangian) invariance of the latter models. For each of these chains, we have chosen the normalization parameter J so that their average energy coincides with that of the chain (2.1). More precisely, the average energy H of the latter model can be computed taking into account that tr S ij = tr S ij = m N −1 , (6.1) so that where the last sum is evaluated in ref. [62]. The average energy H HS of the HS chain can be computed in a similar way, with the known result [37] H HS = which coincides with H if we take J = 2. This is consistent with the fact that the Hamiltonian (2.1) formally reduces to the HS chain Hamiltonian (2.3) if we set S i = 1 for all i. As to the HS chain of D N type, proceeding as before and using the variables x i = cos(2θ i ) (with x i a zero of the Jacobi polynomial P −1,−1 N ) after a straightforward computatation we obtain The latter sum can be evaluated using the summation formula [63] j;j =i Requiring that H HS,D agrees with H to leading order in N we conclude that in this case we should take J = 1/2. By way of example, in fig. 2 we have plotted the degeneracy of the spectrum of the new chain (2.1) (computed using the exact formula (5.6)-(5.10) for its partition function) and of the HS chains of A N −1 and D N types (with J = 2 and J = 1/2, respectively) for N = 16 spins and m = 3 (left) or m = 4 (right). It is apparent that in both cases the new model's spectrum has a very high degeneracy, similar to the degeneracy of the spectrum of the D N -type HS chain but somewhat smaller than the corresponding degeneracy of the A N −1 HS chain's spectrum, even in the odd m case (when the spectrum cannot be described in terms of motifs). To make this observation more quantitative, we have computed the average degeneracy of the spectra of the latter models, where n denotes the number of distinct energy levels, for m = 3, 4 and N = 8, . . . , 16 spins. The results, which are are presented in fig. 3, again clearly indicate that for sufficiently high N the spectrum of the new model has a degeneracy similar to that of the D N -type HS chain and is somewhat less degenerate than that of the original HS chain. It is also apparent from these plots that the degeneracy of the new chain grows exponentially with N , as is typically the case for Yangian-invariant models [64]. All of these facts strongly suggest that the model (2.1) possesses some kind of Yangian symmetry, as is known to be the case for the original (A N −1 ) HS chain [2]. In fact, given the open character of the new model it is more likely that its underlying symmetry group is a twisted Yangian, as is the case for similar open chains with long-range interactions like the BC N , B N and D N HS chains or the Simons-Altshuler model [40,44,65] and its integrable generalizations [45]. The plots in fig. 2 also seem to indicate that the level density of the new model (2.1) is Gaussian for sufficiently high N , as for all spin chains of HS type [41,43,51,66]. That this is indeed the case can be clearly seen for instance from fig. 4, where we have plotted both the histogram of the energy levels and the cumulative level density F (E) of the chain (2.1), defined by where d(E i ) denotes the degeneracy of the i-th level E i , for N = 16 spins and m = 4 (the case m = 3 is completely similar). It is patent from fig. 4 that the latter density is virtually indistinguishable from the cumulative Gaussian distribution with parameters µ and σ respectively equal to the mean H of the spectrum, given by eq. (6.2), and its standard deviation ( H 2 − H 2 ) 1/2 . As shown in appendix C, the latter quantity can also be evaluated in closed form, with the result (6.4) where π(m) is the parity of m. Note that (6.4) differs from its counterpart for the HS chain with J = 2 computed in ref. [37], although both quantities do coincide at leading order O(N 5 ).

Conclusions and outlook
In this paper we have introduced a new class of open, translationally invariant spin chains with long-range interactions, which includes the original Haldane-Shastry as a particular (degenerate) case. The Hamiltonian of the new models, which contains both spin permutation and (polarized) spin reversal operators, turns out to be invariant under "twisted" translations combining an ordinary translation with a spin flip at one end of the chain. This remarkable invariance is one of the key properties shared by all members of the new class, regardless of the form of the spin-spin interactions. In fact, the models in this class are fundamentally different from all spin chains of Haldane-Shastry type studied so far, since their Hamiltonian cannot be constructed in the usual way from an extended root system. We have studied in detail what is perhaps the simplest model of the new type, in which the spin-spin exchange interaction is proportional to the inverse square of the distance. We have computed in closed form the partition function of this model applying Polychronakos's freezing trick to a related spin dynamical model, whose spectrum we have also determined. With the help of the partition function, we have explicitly computed the spectrum of the new chain for several values of m and N , and studied some of its statistical properties. Our results evidence that the model's spectrum has a huge degeneracy, comparable to that of well-known Yangian or twisted Yangian-invariant models like the HS chain and its open versions. This fact is a strong indication that the new chain possesses an underlying twisted Yangian symmetry even for a finite number of spins. Moreover, its level density appears to be Gaussian for sufficiently large N , as is the case with spin chains of HS type.
Our results open up several lines for future research. To begin with, the structure of the partition function of the model explicitly solved in this paper in the even m case indicates that its spectrum can be expressed in terms of suitable Haldane's motifs. It would therefore be possible in this case to study the thermodynamics of the model through the inhomogeneous transfer matrix approach used successfully to compute in closed form the thermodynamic functions of similar chains [48][49][50]. Another open problem in connection with the new model is to rigorously establish its twisted Yangian invariance, which as mentioned above is suggested by its open nature and the huge degeneracy of its spectrum, and thus prove its integrability. It would also be of interest to find other solvable and/or integrable members of the new class introduced in this paper. For instance, the version of the chain studied here with arbitrarily polarized spin reversal operators, as well as its supersymmetric counterpart, should also be solvable with the method developed in this paper. Another interesting problem would be studying the (possibly partial) solvability of models with other spin-spin interactions, like the elliptic chain introduced in section 3.10.
As mentioned in the Introduction, the Hamiltonians of several spin chains with longrange interactions, including the original HS chain, are the parent Hamiltonians of infinite MPSs states (usually the ground state) constructed from certain rational CFTs like the su(m) 1 WZNW model. Although the models introduced in this work share many properties with the latter spin chains, a crucial difference between them is the presence of spin reversal operators in the Hamiltonian of the new models. It would therefore be worth investigating whether the new models can also be constructed, at least in some cases, from appropriate CFTs, and if so what kind of CFTs would appear in this context. On a more speculative note, it could also be of interest to explore whether any of the new models are relevant in connection with the AdS-CFT conjecture, as has proved to be the case with other translationally invariant spin chains like the Inozemtsev chain.

A Algebraic relations satisfied by the operators K ij and K i
In this appendix we shall list the algebraic relations satisfied by the operators K ij , K i , as well as their commutation relations with the Dunkl operators J i and the auxiliary operator H . These relations play an important role in the computation of the spectrum of H carried out in Section 3.
To begin with, from the definition (3.6)-(3.7) we immediately obtain the basic identities where the indices i, j, k, l are all distinct. A straightforward calculation using these identities and eq. (3.10) yields the relations k=min(i,j)+1 where again i = j. Note that, since the gauge factor µ in eq. (3.9) clearly commutes with the operators K ij and K i , the commutation relations (A.2)-(A.4) are still valid if we replace J l by µ −1 J l µ. By repeatedly applying the previous relations it is straightforward to show that if W is an arbitrary element of the Weyl group of D N type W generated by the operators K ij and K i K j we have where W is an appropriate element of the group algebra of W. In the latter equation the action of W on the index i follows from eqs. (A.2)-(A.4), namely K ij (j) = i , K ij (l) = K i (l) = l for any three distinct indices i, j, l. Finally, it is clear from its definition (3.5) that the auxiliary operator H commutes with K ij and K j for all i = j, and thus [H , W ] = 0 , ∀W ∈ W.
In particular, H commutes with permutations, a property which plays an essential role in the computation of the spectrum of this operator in Section 3.

B Evaluation of the sum (5.4)
In this appendix we shall compute the sum (5.4) used in Section 5 for the calculation of the partition function of the chain (2.1).
To begin with, let us define (with ν r := 0, cf. eq. (5.2)), and note that where the inner sum can be easily evaluated:

C Evaluation of the standard deviation of the spectrum of the chain (2.1)
In this appendix we shall compute the standard deviation of the spectrum of the chain (2.1) when the operators S k are taken as spin flip operators. To begin with, note that The average of H is easily computed using eq. (6.1), with the result On the other hand, from the identities [41] tr S ij S kl = tr S ij S kl = m N −2+2(δ ik δ jl +δ il δ jk ) , after a long but straightforward calculation we obtain where π(m) is the parity of m. The last sum is easily evaluated using eq. (6.2):