${\cal PT}$ deformation of Calogero-Sutherland models

Calogero-Sutherland models of $N$ identical particles on a circle are deformed away from hermiticity but retaining a $\cal PT$ symmetry. The interaction potential gets completely regularized, which adds to the energy spectrum an infinite tower of previously non-normalizable states. For integral values of the coupling, extra degeneracy occurs and a nonlinear conserved supersymmetry charge enlarges the ring of Liouville charges. The integrability structure is maintained. We discuss the $A_{N-1}$-type models in general and work out details for the cases of $A_2$ and $G_2$.


Introduction and summary
Recently, integrable systems have been subjected more intensely to non-hermitian deformations, as has been reviewed in [1]. Specifically, PT deformations of rational Calogero models and their spherical reductions have been analyzed in some detail [2,3,4,5,6]. It was found that the mathematical structures and tools pertaining to integrability are compatible with PT deformations, as long as the latter respects the Coxeter reflection symmetries of the models. A bonus of certain PT deformations is the complete regularization of the coincident-particle singularities of Calogero models, which leads to an enhancement of the Hilbert space of physical states by previously non-normalizable wave functions. For integral values of the Calogero coupling(s), most of the new states are energy-degenerate with old ones, and a conserved nonlinear supersymmetry charge arises on top of the Liouville integrals of motion.
Here, we carry our analysis of PT deformed spherically reduced (or angular) Calogero models [6] over to the trigonometric or Calogero-Sutherland case. These models were completely solved more than 20 years ago [7,8,9] and describe N interacting identical particles on a circle or, equivalently, one particle moving on an N-torus subject to a particular external potential. The latter has inverse-square singularities on the hyperplanes corresponding to incident-particle locations. We shall see that, also here, there exists a deformation which renders the potential nonsingular and retains the integrable structure, adding an infinite tower of new states to the energy spectrum and allowing for a nonlinear supersymmetryoperator mapping between 'new' and 'old' states. Employing PT -deformed Dunkl operators, we construct the deformed intertwiners (shift operators increasing the coupling by unity) and analyze their action on the deformed energy eigenstates. We then find a set of deformed Liouville charges which intertwine homogeneously (like the Hamiltonian), so that their eigenvalues are preserved by the shift. Details are worked out for the three-body cases based on the A 2 and G 2 groups.
Our analysis can straightforwardly be extended to any (higher-rank) Coxeter group, but explicit expressions quickly become rather lengthy. While Dunkl operators and Liouville charges are known for all models (and PT -deformed effortlessly), we are not aware of a general classification of Weyl anti-invariant polynomials, which will be needed to extend the ring of Liouville charges to include all intertwiners. Also interesting is the exploration of the hyperbolic models and, finally, the elliptic ones. We have employed the most simple PT deformation compatible with the symmetries of the model, but there exist other options. The two we shortly discuss in the context of the A 2 model do not fully regularize the potential, but there may be other ones more suitable. A classification will be most welcome.
The paper is organized as follows. After defining the A N −1 Calogero-Sutherland model and introducing a suitable PT deformation in Section 2, we describe the energy spectrum including the eigenstates in Section 3. The following Section 4 is devoted to the construction of deformed conserved charges and i ntertwiners with the help of Dunkl operators. Sections 5 and 6 work out the details for the A 2 and G 2 cases, respectively. Explicit low-lying wave functions and energy degeneracies are listed in a Appendices.

PT deformation of Calogero-Sutherland models
The N-particle model is governed by a rank-N Lie algebra g. Translation invariance implies that g = A 1 ⊕ g ⊥ , where the A 1 part represents the center of mass. It may be decoupled, but we retain it for the time being.
For A-type models, g ⊥ = A N −1 . They describe N identical particles on a circle of circumference L, mutually interacting via a repulsive inverse-square two-body potential. We label the L-periodic particle coordinates as but it proves more convenient to pass to multiplicative coordinates with the useful relations The A N −1 Calogero-Sutherland model is defined by the Hamiltonian We remark on the invariance under g → 1−g. Rather than an N-body problem on a circle, this system may also be interpreted as a single particle moving on an N-torus T N and subject to a particular external potential. The latter's singularities on the walls of the Weyl alcove restrict the particle motion to a fundamental domain in the A 1 ⊕ A N −1 weight space. For later use we introduce the shorthand notation as well as the totally antisymmetric degree-zero homogeneous rational function (2.6) To establish PT symmetry, it is necessary to identify two involutions, a unitary P and an anti-unitary T , such that the deformed Hamiltonian is invariant under their product. While for the latter we take the standard choice of complex conjugation, the former leaves various possibilities. In this paper we shall choose P to be parity flip of all coordinates, 1 thus 1 For N =3, we shall ponder on some other choices later on.
The Hamiltonian (2.4) is parity symmetric, so a PT -symmetric way of deforming can be induced by a PT -covariant complex coordinate change. The obvious option is Thus, the multiplicative coordinate x ǫ i is PT invariant for any value of {ǫ i }. If we do not want to deform the center-of-mass degree of freedom we must impose the restriction i ǫ i = 0.
This deformation generically removes the singularities in the potential because never vanish for ǫ ij ≡ ǫ i −ǫ j different from zero. Therefore, the deformed Hamiltonian no longer restricts the particle motion to a single Weyl alcove but allows it to range over the entire T N . This space still being compact, the energy sprectum will remain discrete. Only in the L → ∞ limit we recover the rational Calogero model with its continous spectrum. In the following, we drop the superscript 'ǫ' but understand to have a generic deformation turned on with ǫ ij = 0.

The energy spectrum
So far, the PT deformation (2.8) is fully compatible with the integrability of the A-type Calogero-Sutherland model. It merely hides in the substitution x i → x ǫ i . This remains true for the energy spectrum: the known energy levels are unchanged under the deformation, and the eigenstates are obtained from the undeformed ones simply by again deforming the coordinates. However, due to the disappearance of the singularities in the potential, previously non-normalizable eigenstates become physical, adding extra states to the spectrum! One popular way to completely label the energy eigenstates is by an N-tupel n = (n 1 , n 2 , . . . , n N ) with n 1 ≥ n 2 ≥ . . . ≥ n N ≥ 0 (3.1) of quasiparticle excitation numbers. After removing the center-of-mass energy by boosting to its rest system one obtains Due to translation invariance, this expression is invariant under a common shift n k → n k + c.
In order to remove this redundancy, we put n N = 0, so that the sums over k run from 1 to N−1 only. The energy is bounded from below, with the ground-state value but a different lower bound (minimally zero) for g < 0.
To study the degeneracy, we rewrite (3.2) as a sum of squares, Any collection n of quantum numbers uniquely yields an element λ = (λ 1 , λ 2 , . . . , λ N −1 ) in a particular Weyl chamber of the A N −1 weight space Λ N −1 , and the energy of the corresponding state is given by the radius-squared of a circle in Λ N −1 centered at µ g. Since µ lies outside the Weyl chamber in question, for positive g the minimal distance from the circle center to the physical states is given by | µ| and represents the nonzero ground-state energy E 0 (g). So the degeneracy of a given energy level may be found by counting the number of physical weight lattice points on the appropriate "energy shell". The eigenfunctions of the Hamiltonian are given by Jack polynomials, on which there exists an extensive literature. They are of the form and P (g) n is a homogeneous permutation-symmetric polynomial of degree | n| in x. The rational function R (g) n is homogeneous of degree zero, but in the center-of-mass frame we have R  Before the PT deformation, ∆ ∝ i<j x ij vanishes at coinciding coodinate values (the Weyl-alcove walls), which renders the wave functions (3.6) non-square-integrable when g < 0. Therefore, the physical spectrum is empty there. However, due to the g ↔ 1−g symmetry of the Hamiltonian, we should consider the two "mirror values" of g together to form a single Hilbert space H g . Then, for a given value of g > 1 2 , a generic deformation (with all ǫ ij nonzero) will abruptly add a second infinite set of energy eigenstates to the spectrum, given by replacing g with 1−g. Their energies are given by E n (1−g) from (3.2) or (3.4) for a second set of quantum numbers n. This produces a second "energy shell", which may carry states all the way down to zero energy (if µ(1−g) is located in the physical Weyl chamber). For particular (typical integer) values of g the two shells may possess simultaneous states, leading to an enhancement of energy degeneracy. We shall illustrate these features in the examples below.

Conserved charges and intertwiners
A key tool in the construction of the spectrum and conserved charges is the Dunkl operator 2 where the reflection s ij acts on its right by permuting labels i and j. It obeys a simple commutation relation, effects a cyclic permutation of the labels i, j, k.
The importance of the Dunkl operator is twofold. First, any permutation-invariant (in general: Weyl-invariant) polynomial of some degree k in {D i } will, when restricted to totally symmetric functions, give rise to a Liouville charge C k , i.e. a conserved quantity in involution. A simple basis of this ring is provided by the Newton sums, where 'res' denotes the restriction to totally symmetric functions, giving The total momentum and the Hamiltonian itself are the prime examples, In the center of mass, P = 0 of course. Only the first N charges are functionally independent; any I k>N can be expressed in terms of these. The I k for 3 ≤ k ≤ N may be employed to lift the degeneracy of the state labelling by energy alone.
Second, the symmetric restriction of any anti-invariant Dunkl polynomial of some degree k will yield an intertwining operator (or shift operator) M k (g), obeying The simplest such intertwiner is where the sum is over all permutations of thek factors in the product. Comparing (4.6) with (3.4) it can be inferred that the action of Mk(g) on the states is which will vanish if the target quantum numbers no longer respect the restriction in (3.1). The shift operator translates the energy shell by the vector µ. Its repeated action will eventually get the state | n g to the edge of the physical Weyl chamber. Therefore, any state gets mapped to zero after a certain number of shifts. The adjoint intertwiner M † k (g) = M k (−g) has the opposite action, n → n+ δ while g → g−1.
Since M k has a nonzero kernel, M † k is not surjective. The Liouville charges I ℓ together with the intertwiners M k form a larger algebra, which is of interest. Beyond the total momentum and the Hamiltonian, the higher conserved charges (4.3) do not intertwine homogeneously but mix when M k is passed through them, with some coefficients c km (g) polynomial in g. However, it may be possible to find another basis {C k (g)} which intertwines nicely, meaning that the shift effected by M k will map simultaneous eigenstates of the whole set {C k } to each other. The composition M † k (g)M k (g) is by construction an element of the Liouville ring and thus can be expressed in terms of the I k (g) (or C k (g)).
When g ∈ N, the energy levels are degenerate with some at coupling 1−g. In this case, there exists an extra, odd, conserved charge 3 mapping H g to itself after fusing the spectra at couplings g and 1−g. We note that the action of Q k is well defined only after applying the PT deformation, since the undeformed spectrum is empty for negative g values. The operator Q maps between 'even' and 'odd' states in the joined spectrum, which arise from the originally positive and negative g values, respectively.

Details of the A 2 model
In this section we work out the details of the simplest nontrivial case, which describes three particles on a circle interacting according to the A 2 structure. For simplicity we put L = π from now on; the dimensions can easily be reinstated. The Hamiltonian in the center-of-mass frame then reads (deformation superscript 'ǫ' suppressed) The other two conserved charges are but I 4 is already dependent, The energy formulae (3.2) and (3.4) specialize to and the ground state for g ≥ 0 is Here, we introduced (for later purposes) the antisymmetric basis functions, so These Laurent polynomials (in x 1/3 i ) form a ring whose structure we detail in Appendix A.2. In Appendix A.3 we list the explicit wave functions (see (3.6)) for small values of n 1 . Each eigenstate |n 1 , n 2 corresponds to a point in a π 3 wedge around the negative λ 1 axis. The circles determining the energy eigenstates for couplings g and 1−g are centered at in λ-space, respectively. This is illustrated in Figure 1. Since µ 2 = 0, we have an obvious energy degeneracy for |n 1 , n 2 g and |n 1 , n 1 −n 2 g (5.10) except for n 1 =2n 2 , of course. For g ≥ 0 there rarely appears higher degeneracy, 4 but at g < 0 energy levels are up to 12-fold degenerate! This plethora of states becomes physical only after the PT deformation and greatly enlarges the Hilbert space H g for any g>0. Figure 2 displays the energy spectra with degeneracies for low levels and small integral values of g. The basic intertwiner for the A 2 model is of order three, with the obvious notation D ij = D i −D j . It computes to In terms of the multiplicative variables, the shift operator takes the form (5.13) It action on the states is M 3 (g) |n 1 , n 2 g ∝ |n 1 −2, n 2 −1 g+1 (5.14) conserving the energy. In weight space it moves (λ 1 , λ 2 ) → (λ 1 +2, λ 2 ). For homogeneous intertwining relations, we redefine This imples that the eigenvalue of C 3 is also conserved under the shift action. Indeed, it is readily verified that C 1 |n 1 , n 2 g = 0 , C 2 (g) |n 1 , n 2 g = −2 (n 1 +2g) 2 + 1 3 (n 1 −2n 2 ) 2 |n 1 , n 2 g , C 3 (g) |n 1 , n 2 g = 8 9 i (n 1 −2n 2 )(2n 1 −n 2 +3g)(n 1 +n 2 +3g) |n 1 , n 2 g , which is compatible with the shift (5.14). The degeneracy reflection n 2 → n 1 −n 2 flips the sign of C 3 , so two such states can be discriminated by their C 3 eigenvalues. As expected, the composition of the intertwiner with its adjoint yields an expression in the Liouville charges, Let us take a look at the extra degeneracy between even (g>0) and odd (g≤0) states appearing when g ∈ Z. The odd operator Q 3 (g) mapping one to the other and defined in (4.11) is of order 3(2g−1) and shifts the quantum numbers as n 1 , n 2 → n 1 −4g+2, n 2 −2g+1 , (5.19) which produces a rather large kernel. Q 3 commutes with all conserved charges I k , so it keeps their eigenvalues. In weight space, it maps between the 'even' and 'odd' energy shells. Finally, we briefly discuss two other kinds of PT deformations in the A 2 -model context, which we denote as 'angular' and 'radial', respectively. Different from the parity transformation in (2.7), which amounts to the outer conjugation automorphism of A N −1 , the angular and radial deformations are compatible with an elementary Coxeter reflection (or particle permutation), e.g. and while T remains complex conjugation. The angular PT deformation is homogeneous in the q i coordinates, in contrast to the constant complex coordinate shift (2.8). It is induced by a complex orthogonal coordinate change, q → Γ ǫ q with Γ ǫ ∈ SO(3, C) modulo SO(3, R), described in [6]. Explicitly, with (i, j, k) being a cyclic permutation of (1, 2, 3). This deformation does not entirely remove the singular loci of the potential given by 0 = sin q ǫ ij = cosh( √ 3 sinh ǫ q k ) sin(cosh ǫ q ij ) + i sinh( √ 3 sinh ǫ q k ) cos(cosh ǫ q ij ) ⇔ q ij = ℓ π cosh ǫ ∧ q k = 0 for ℓ = 0, 1, 2, . . . , where again (i, j, k) are cyclic and we went to the conter-of-mass frame, so q ik + q jk = −3q k . For small enough ǫ, only the origin {q i = 0} remains singular, but with growing value of ǫ extra singularities appear inside the Weyl alcove. The radial PT deformation is a nonlinear one, and (i, j, k) being cyclic once more. The remaining singularities occur ar q ij = ℓ π ∧ q k = 0 for ℓ = 0, 1, 2, . . . , (5.24) and in addition one should average the potential, V → V ǫ + V −ǫ with V ǫ (q) = V (q ǫ ). Both cases can be parametrized jointly by writing for the angular and radial PT deformation, respectively.

Details of the G 2 model
For a more complicated and non-simply-laced example, we turn to the G 2 model [10,11] for three particles on a circle and apply the constant-shift PT deformation (2.8) but suppress it notationally. The G 2 model adds to the previous two-body potential of the A 2 case (5.1) a specific three-body interaction, where the index 'k' complements i and j to the triple (1,2,3), there are two independent real couplings g S and g L , and we again put L = π for simplicity. The potential can be viewed as a sum of two copies of the A 2 potential, with a relative coordinate rotation between them. The singular walls appear for bounding the G 2 Weyl chambers. The Weyl group is enhanced from S 3 to D 6 , generated by and permutations, which for the x i coordinates translates to The Hamiltonian (6.1) yields eigenvalues E n 1 ,n 2 (g) = 4 3 (n 2 1 + n 2 2 − n 1 n 2 ) + 4g S n 1 + 4g L (2n 1 − n 2 ) + 4(g 2 S + 3g 2 L + 3g S g L ) = (n 1 + 2g S + 3g L ) 2 + 1 3 (n 1 − 2n 2 + 3g L ) 2 with n 1 ≥ 2n 2 ≥ 0 , (6.5) and the ground-state wave function for g S ≥ 0 and g L ≥ 0 is where we introduced (6.7) In addition to the permutation symmetry inherited from the A 2 model, we also have to impose (anti-)invariance under the additional (even) Coxeter element  in (6.4), which implements an inversion in x space and flips the sign of the roots by a π rotation in the relative q space. Noting that  2 = 1 and [, s ij ] = 0 and one sees that j flips the sign of both ∆ S and ∆ L , hence We also deduce that the energy-degenerate states |n 1 , n 2 and |n 1 , n 1 −n 2 of the A 2 model are related by the action of j. Therefore, only their sum or difference will be a G 2 -model state, so the range of n 2 can be restricted to n 2 ≤ 1 2 n 1 , as already claimed in (6.5). The excited states then take the form where the R (g S ,g L ) n 1 ,n 2 are again particular Weyl-symmetric rational functions of degree zero. Appendix A.5 contains a list of low-lying wave functions.
Each eigenstate |n 1 , n 2 corresponds to a point in a π 6 wedge above the negative λ 1 axis, in accord with one G 2 Weyl chamber. The circles determining the energy eigenstates for couplings (g S , g L ), (1−g S , g L ), (g S , 1−g L ) and (1−g S , 1−g L ) are centered at in λ-space, respectively. This is illustrated in Figure 3. After the PT deformation, the Hilbert space H g S ,g L comprises the four towers obtained from the four circles in Figure 3. Again, for integral values of the couplings, the towers have matching energy levels, which greatly increases their degeneracy.
The G 2 Dunkl operator is an extension of the A 2 one (again with {i, j, k} = {1, 2, 3}), The first two Newton sums in this Dunkl operator yield the conserved momentum and energy, 14) because i q i and i q 2 i are not only permutation-symmetric but also invariant under the rotation  from (6.3). This, however, is not the case for i q r i when r≥3, but one can find a Weyl-invariant combination at order six, which generates another Liouville charge, where symmetrization means Weyl ordering of every summand, and the coefficients c s 1 s 2 s 3 (x) are given in Appendix A.6.
From (6.8) we see that the G 2 model enjoys two separate intertwiners, which independently shift by unity the couplings g S and g L , respectively, Their explicit form is . (6.20) A better basis for the Liouville charges is and (6.21) obeying homogeneous intertwining relations This is also signified by the action C 6 |n 1 ,n 2 g S ,g L = − 64 81 (3g L +n 1 −2n 2 ) 2 (6g L +3g S +2n 1 −n 2 ) 2 (3g L +3g S + n 1 +n 2 ) 2 |n 1 ,n 2 g S ,g L . (6.24) The intertwining with their corresponding conjugates produces two polynomials in the Liouville charges, (6.25) The intertwining operators also enable odd conserved charges when the couplings take integer values, in the form of the chain of operators which in the simplest non-trivial cases squares to the form of the polynomials in (6.25), 3 + lower terms , Q 2 3,L (g S , 2) = 81C 6 + 4C 2 1 (C 2 1 − 9 2 C 2 ) 2 3 + lower terms .

A Appendix
A.1 Potential-free frame We display some relations with the potential-free frame for the A N −1 model. By conjugating the Hamiltonian one can trade the potential for a first-order derivative term, (A.1)
The relation between the parameters is To remove the redundancy of the labelling, we stipulate that Even assuming m 1 ≥ n 1 (without loss of generality), the right-hand side may produce contributions [k 1 , k 2 ] ± with k 1 < k 2 or with k 2 < 0, which we outlawed. However, it is easy to see that so one may employ the first relation in the first case and the second one in the second case to obtain an admissible result. Some examples are (A.8)

A.3 Wave functions for the A 2 model
The A 2 wave functions take the form where x = (x 1 , x 2 , x 3 ) and P (g) n 1 ,n 2 is a homogeneous permutation-symmetric Jack polynomial of degree n 1 +n 2 in x. Passing to the more convenient variables The first few Laurent polynomials are R 0,0 = Q + 0,0 = 6 , (A.12) From the paper of Lapointe and Vinet [7], one can see that the Jack polynomials can be also constructed in terms of modified Dunkl operators In terms of the combinations the Jack polynomials are given by With the normalization Ψ (g) n 1 ,n 2 = 1 (g) n 1 −n 2 (g) n 2 (2g+n 1 −n 2 ) n 2 P (g) n 1 ,n 2 (x 1 x 2 x 3 ) −(n 1 +n 2 )/3 ∆ g (A.16) employing the Pochhammer symbol (a) n , the action of the intertwiner takes the form M 3 (g) Ψ (g) n 1 ,n 2 = n 2 (n 1 +g)(n 1 −n 2 )Ψ (g+1) Clearly, the n 2 =0 and n 2 =n 1 states are annihilated by M 3 (g).

A.4 Wave functions for the G 2 model
In the variables y = (y i ) = (x 1/3 i ) the G 2 wave functions take the form y|n 1 , n 2 g S ,g L ≡ Ψ (g S ,g L ) n 1 ,n 2 (y) = R (g S ,g L ) n 1 ,n 2 (y) Ψ given in (6.6) and (6.7) and R (g S ,g L ) n 1 ,n 2 being a degree-zero rational function in y. It turns out that for n 1 +n 2 not divisible by three Ψ does not in general factorize as R times Ψ 0 in the ring of Laurent polynomials, so R is of a more general class. However, the full wave function Ψ can be expressed in terms of the symmetric and antisymmetric basis polynomials Some low-lying factorizable states are listed below. Beyond Ψ g S ,g L 0,0 In addition, one can infer that (A.21) For g S , g L ∈ {0, 1}, the model is free, so the states take a very simple form: In the last two lines, ∆ L can only be factored off in case n 1 +n 2 is a multiple of three.