On symmetries of generalized Calogero model and Polychronakos-Frahm chain

The symmetry of the generalized Polychronakos-Frahm chain is obtained from the Dunkl-operator deformation of the unitary algebra, which describes the symmetry of the generalized Calogero model.

extracting the conserving quantities for the discrete system from the deformed unitary generators, wherein the diagonal elements correspond to the previously constructed integrals. Algebraic relations between the reduced generators are derives also. The symmetric combinations of the invariants produce integrals of the Polychronakos-Frahm chain. We finish by inspecting the Coulomb confining potential in this context. The analog of the Runge-Lenz vector vanishes at the discrete level and does not lead to a conservation law.

II. GENERALIZED CALOGERO MODEL
The Calogero model is an integrable system of one-dimensional identical particles interacting by inverse-square potential and bound by external harmonic force. It is given by the Hamiltionain [4] Herep i = −ıh∂ i is the momentum operator, g is a coupling constant, and the −(+) sign in the potential corresponds to the bosons (fermions). We use a conventional notation for the particle distance,x ij =x i −x j . The oscillator frequency is set to unity.
Most properties of the Calogero model (and its various extensions), like (super)integrability, spectrum, wavefunctions, and conservation laws are conditioned by its slightly modified version known as a generalized Calogero Hamiltonian [18,19] It includes a permutation operator M ij , which exchanges the coordinates of i-th and j-th particles. At a first glance, such a modification may look rather strange due to its nonlocality but for the identical particles it just reduces to the standard Calogero Hamiltonian (1). The advantage of the above Hamiltonian is the representation in terms of the deformed momentum operator with derivative replaced with the Dunkl operator: The latter can be considered as a kind of flat nonlocal covariant derivative with the following algebra [17]: At the g → 0 limit when the Calogero inverse-square potential is absent, the operatorsπ i ,Ŝ ij are mapped to thep i andhδ ij , correspondingly, and above relations are reduced to the Heisenberg algebra commutation rules. A Dunkl-operator analog of lowering-rising operators is defined in the standard way [7,18,19]: The generalized Hamiltonian can be expressed in terms of them: Here by S the rescaled invariant of the permutation group algebra is denoted: As a result, the lowering-rising operators obey a standard spectrum generating relations [7,19]: So, any bilinear combinationÊ ij =â + iâ j (12) satisfies the conservation law: The elementsÊ ij are Dunkl-operator deformations of the unitary group generators. Together with permutations M ij , they provide entire algebra of symmetries for the generalized Calogero model. In addition, the following quadratic relation takes place amongÊ ij andŜ ij [20]: The latter implies, in particular, the following commutation relation: As a consequence, the diagonal elements are closed under commutation. Unlike the Cartan algebra, they are not abelian but obey a simple commutation, The above algebra ensures that the power sums form a system of Liouville integrals of the Calogero system [18]: The generalized Hamiltonian itself is expresses in terms of the first member in this family, Moreover, it is a unique Casimir element (up to a nonessential constant term) of the Dunkl-deformed u(N ) algebra [20]. Remember that the same algebra (16) describes also the symmetries of the generalized Hamiltonian related to the Sutherland model [18], an analog of the Calogero-Moser system with trigonometric interactions [3].
The antisymmetric combinations ofÊ ij yield the Dunkl angular momentum components [13,29], Together with permutations they produce a deformation of so(N ) algebra with a unique Casimir element given by the Dunkl angular momentum squareL 2 = i<jL 2 ij shifted by a permutation invariant term [20], It can be considered as a generalized angular Calogero Hamiltonian, which is reduced to the angular part of the Calogero model for identical particles [30], The symmetric combinations of the deformed u(N ) generators produce a Dunkl-operator deformation for the wellknown Fradkin tensor [31],Î Remember that the angular momentum and Fradkin tensor describe, respectively, the dynamical and hidden symmetries of the N -dimensional isotropic oscillator [32]. The diagonal algebra (16) has an abelian basis obtained by applying a shift to its elements. The shift is a tail composed of exchange operators [26],D whereŜ 1 = 0 is supposed. Together with permutations, the elementsD i satisfy the defining relations of degenerate affine Hecke algebra: Note that the tails S i satisfy the same relations, i.e. above equations remain true upon the substitutionD i →Ŝ i . As a result, the modified diagonal elements can be considered as an analog of Liouville integrals for the generalized Hamiltonian (2), which may be expressed via them using the representation (9), The higher-order power sums define higher Hamiltonians, The second member,D 2 , corresponds to the generalized Calogero-Sutherland model. Note that in contrast to the previous integrals (17), the permutation invariance is not evident but can be verified. More familiar are the monomials given by the generating function i (u −D i ) [26]. In general, any symmetric polynomial inD i is permutation invariant and reduced to the constant of motion of the Calogero model (1) for indistinguishable particles. More symmetric polynomial may include the nondiagonal components of the generatorsÊ ij . The permutations can be used in such expressions as well. Here are some simplest examples of such integrals:

III. GENERALIZED POLYCHRONAKOS-FRAHM CHAIN
Let us set the interaction constant to unity, g = 1, and consider the generalized Calogero model at the equilibrium pointx = x, where the (classical) confining Calogero potential takes its minimal value: The coordinates are given by the roots of the N th-order Hermite polynomial [6]. Here and in the following, in order to simplify notations, we omit the hat over the operators taken at the potential minimum. All roots differ, so there are N ! equivalent minima connecting by the coordinate exchanges M ij . These are only allowed evolutions in the freezed system. Consider the expansion of the Hamiltonian (2) in powers of Plancks constant: The first-order term can be considered as a generalized Polychronakos-Frahm Hamiltonian, where the spin exchange operator is replaced by the coordinate exchange [5]: The SU (n) symmetric spin chain is recovered from the above Hamiltonian after the replacement of the coordinate permutations with spin exchange operators so that Here P ij permutes the ith and jth spins, which take values in the fundamental representation of the SU (n) group. Both Hamiltonians become identical on the bosonic (fermionic) states provided that the particles are endowed with additional spin degrees of freedom. Then the entire wavefunction must be symmetric (antisymmetric) under simultaneous exchanges of coordinates and spins for bosons (fermions). A permutation of spatial coordinates M ij can be replaced by the spin exchange operators P ij and −P ij in the bosonic and fermionic cases respectively. Note that the projection inverts the order of permutations so that the operator M ij M kl must be substituted by P kl P ij . Let us now construct the constants of motion of the chain Hamiltonian (30) by applying theh expansion to the integrals of the original dynamical model (2) at the equilibrium point. They do not provide directly the invariants of the SU (n) spin chain (31). In order to get them, one needs to carry out symmetrization over all particles prior to the projection. For instance, H (1) stays invariant under a selected coordinate exchange, M ij , but H PF does not preserve its spin counterpart, P ij . However, both Hamiltonians preserve the symmetrised version given by the element S (10).
The Dunkl momentum (4) and permutation matrix (6) can be presented as follows: Here and in the following, the superscript is omitted in the zeroth-order term of any operator, so that π 0 i = π i . In that limit, the canonical commutation relations resemble their original form (5): Recall now that the particle coordinates are set by the roots of the Hermite polynomial, which imposes certain algebraic relations on them (see, for example, [27,33]). As a result, the discrete Dunkl momenta are not independent any more but undergo additional algebraic constraints. In particular, the following relations hold among the fixed phase space variables: In general, all relations between the operators of the dynamical Calogero system are preserved at theh = 0 limit. In particular, the freezed lowering-rising operators obey a rule similar to the commutations of the deformed Heisenberg algebra (8) [5]: Due to the minimum condition (28), the spectrum generating relation (11) remains valid for the generalized Polychronakos-Frahm chain too [5]: However, unlike the dynamical case, the discrete Hamiltonian is not expressed via lowering-rising operators [see eq. (9)]. The constants of motion of the generalized Calogero model (12) have terms up to the second order in their expansion, The relations (39) imply conservation of the constant terms, For the Dunkl angular momentum (19), theh 2 part vanishes, while theh term corresponds to the usual angular momentum operator in quantum mechanics: A similar expansion for the Fradkin tensor is more complex: The first order operator-valued coefficient is given by As was discussed above, the algebraic relations between the symmetry generators of the dynamical system remain true at the freezing limit. In particular, the most general relation (14) and its consequences (15), (16) are reduced, respectively, to the following equations: The power sums of diagonal elements yield Liouville integrals of the Polychronakos-Frahm chain [5]: The first element of this set is rather trivial, , as is easy to get using the equations (36) and (37). The higher rank E k have more complicated expressions. For the dynamical system, the quadratic relations (14) are only constraints which the symmetry generatorsÊ ij obey [20]. However, there are a lot of other restrictions on them at the equilibrium point. For example, the eqs.(35) imply the sum vanishing rules: In the dynamical case, the angular Calogero Hamiltonian (20) plays an important role among constants of motion. In the absence of oscillator potential, it maps the Liouville set to additional integrals. However, in the equilibrium limit, the angular part does not produce a new integral but just is expressed via trivial ones. Using the relations (35), (36), (19), and (51), it is easy to verify that the operatorL 2 is a scalar at the equilibrium. Itsh-linear coefficient reproduces the chain Hamiltonian as was argued earlier [30]: with r 2 = 1 2 N (N − 1) (36). Let us calculate also a chain analog of the Fradkin's tensor square. It corresponds to the second (k = 2) member of the first sequence presented in (27): Here M ijk is a cyclic permutation of the marked three coordinates. It also is expressed through the invariants of the permutation group algebra. Nevertheless, we expect that higher degree power sums from (27) at the equilibrium give rise to nontrivial integrals of motion for the chain Hamiltonians (30), (31). Finally, consider the shifted diagonal elements (23). At the freezing level, the also commute: Together with permutations, they form also degenerate affine Hecke algebra (24) (with g = 1). Contrary to the nonshifted case (41), (49), the related symmetric polynomials, D k = i D k i , are scalars (multiples of identity) and do not lead to a conservation law. At the same time, theh-linear terms of the dynamical integrals (26) form a family of commuting nontrivial integrals (49), where D ii . The first element of the family describes the chain Hamiltonian: D (1) . This remarkable property was established, first, for the Haldane-Shastry chain using the Yangian represtation [28], and has been extended later to the Polychonakos-Frahm chain [27].

IV. RELATION TO GENERALIZED CALOGERO-COULOMB SYSTEM
One can start from another radial confining potential in the dynamical system (2) in order to get its discrete analog at the semiclassical limit (30). Apart from the oscillator case, the Coulomb potential is preferable among others since it leads to a superintegrable system as well [16],Ĥ Coul =π So, there is a hope to extract more integrals for the chain system as a reduction of the recently obtained Dunkl-operator analog of the Runge-Lenz vector [22],Â Together with the Dunkl angular momentumL ij , it yields the Dunkl-operator deformation of the SO(N + 1) group generators, which describe the symmetry of the generalized Calogero system with Coulomb potential (55) [22]. The equilibrium point is defined now by solution of the equations In order to separate it from the oscillator minimum (28), the Coulomb point is marked by y. It can be obtained from the oscillator point, given by zeros of Hermite polynomial (28), by the map y i = r 3 x i . The latter implies that r y = r 4 . Therefore, passing from the oscillator to Coulomb potential, the chain Hamiltonian (30) just undergoes a rescaling H 1) . Evidently, the Dunkl momentum transforms as r −3 π i → π i . The relations (35), (36) stay unchanged apart from the first equation in (36), which must be replaced now by Using them, it is easy to verify that the Dunkl deformed Runge-Lenz vector vanishes trivially at the Coulomb equilibrium point: This fact eliminates the power sumsÂ k = iÂ k i at the equilibrium, A k = 0, but does not necessarily implies the vanishing of the first-order operator in powers ofh. Actually, the first two such operators are A (1) as can be derived from the definition (56). Note that the square of the Dunkl Runge-Lenz vector may be shifted certain way to become compatible with the Dunkl angular momentum. Moreover, it provides a deformation of the well known relation between the angular momentum and Runge-Lenz squares [22]: Applying the Taylor expansion with the coefficients written in terms of y coordinates, we obtain the vanishing condition, A ′(1) 2 = 0, which is equivalent to the second equation in (60).

ACKNOWLEDGMENTS
The author is grateful to Armen Nersessian for stimulating discussions. This work was supported by the Armenian State Committee of Science Grants SFU-02, 18RF-002, and 18T-1C106. It was fulfilled within the ICTP Affiliated Center Program AF-04.