A multispecies Calogero model

We study a multispecies one-dimensional Calogero model with two- and three-body interactions. Using an algebraic approach (Fock space analysis), we construct ladder operators and find infinitely many, but not all, exact eigenstates of the model Hamiltonian. Besides the ground state energy, we deduce energies of the excited states. It turns out that the spectrum is linear in quantum numbers and that the higher-energy levels are degenerate. The dynamical symmetry responsible for degeneracy is SU(2). We also find the universal critical point at which the model exhibits singular behaviour. Finally, we make contact with some special cases mentioned in the literature.


Introduction
The ordinary Calogero [1] model describes N indistinguishable particles on the line which interact through an inverse-square two-body interaction and are subjected to a common confining harmonic force. The model is completely integrable in both the classical and quantum case [2], the spectrum is known and the wave functions are given implicitly. The model and its various descendants (also known as Calogero-Sutherland-Moser systems [3]) are connected with a number of physical problems, ranging from condensed matter physics [4] to gravity and black hole physics [5]. The algebraic structure of the Calogero model, studied earlier using group theoretical methods [2,6], has recently been reconsidered by a number of authors in the framework of the S N (permutational) algebra [7]. This operator approach is considerably simpler than the original one, yields an explicit expression for the wavefunctions and emphasizes the interpretation in terms of generalized statistics [8], especially Haldane's exclusion statistics [9]. In Haldane's formulation there is the possibility of having particles of different species with a mutual statistical coupling parameter depending on the i th and j th species coupled. On the level of the Calogero model, this corresponds to the generalization of the ordinary 1D Calogero model with identical particles to the 1D Calogero model with non-identical particles. This can be done by allowing particles to have different masses and different couplings to each other. In this way we obtain a 1D multispecies Calogero model. Very little is known about its spectra and wavefunctions [10][11][12][13].
In the present Letter, which is in a sense a continuation of our investigation of the ordinary Calogero model [14], we use an algebraic (operator) method to find some of the salient features of the multispecies Calogero model on the line with two-and three-body interactions. In Section 2 we present the S N invariant Hamiltonian H of the model, together with its ground state wavefunction and ground state energy. After performing a certain similarity transformation, we get a much simpler HamiltonianH, which we separate into parts describing the center-of-mass motion and the relative motion of particles. We expressH in terms of generators of the SU(1, 1) algebra. All analysis is made in Hilbert space. Section 3 contains our most important results. By applying Fock space analysis, we find some of the excited states ofH, their energies and degeneracies. Closer inspection of the Fock space that corresponds to the relative motion of particles reveals the existence of the universal critical point at which the system exhibits singular behaviour. This result generalizes that mentioned in [14]. We also establish the necessary conditions for the equivalence of the two multispecies Calogero models. In Section 4 we briefly repeat the main points of the paper and make contact with the models studied in [10][11][12][13].
We particularly discuss the necessary conditions for vanishing of the three-body interaction in the starting Hamiltonian H.

A multispecies Calogero model with a threebody interaction
Let us consider the most general Calogero type ground state for the N-body quantum mechanical problem on the line (h = 1): where the prefactor ∆ is given by and ν ij are symmetric statistical parameters between particles (i, j). The harmonic frequency ω is taken to be the same for all particles. Masses of the particles ( m i ) are, in general, not equal.
The Hamiltonian which possesses the above state (1) as the ground state is The symbol (i, j, k =) in the last term denotes the summation over all triples of mutually distinct indices.
The Hamiltonian in Eq. (2) is invariant under the group of permutation of N elements, S N . The elementary generators K ij of the symmetry group S N exchange labels i and j in all quantities, according to the rules: In a sense, one may think of (2) as a smooth, sufficiently small deformation of the ordinary Calogero model.
A few additional remarks concerning the Hamiltonian (2) are in order.
(1) It describes distinguishable particles on the line, interacting with harmonic, twobody and three-body potentials. As far as we know, it is the first time that such Hamiltonian is considered in full generality. An earlier attempt to solve the similar, but less general Hamiltonian ( with all masses m i equal) can be find in [11] (see also Ref. [13]).
(2) The asymptotic behaviour of its eigenstates should be Ψ ∝ ( (3) A well-known stability condition demands that the two-body couplings Setting ν ij = ν, ∀i, j and m i = m, ∀i, we recover the ordinary N-body Calogero model [1]. In that case, owing to the identity cycl.
1 (x i −x j )(x i −x k ) = 0 which holds in 1D, the three-body term in Eq.(2) trivially vanishes. When ν ij are mutually different but m i = m, ∀i, we recover the model treated in [11,13]. Finally, when ν ij = αm i m j , α being constant, we obtain the model mentioned in [10,12]. We comment on these cases in Section 4.
In the following we analyse the most general case, namely the Hamiltonian of Eq. (2) without restrictions (4), given above.
We findH as . (4) Notice that in Eq. (4) two-body and three-body interactions apparently disappeared but they are hidden in the last term of Eq. (4).
It is convenient to introduce the variables (X, ξ i ) and the linear combinations of derivatives ( ∂ ∂ξ i , ∂ ∂X ) Note that the variables ξ i ( as well as ∂ ∂ξ i ) are not linearly independent, i.e.
In terms of the variables just introduced, the HamiltonianH, Eq.(4), separates into parts which describe its center-of-mass motion (CM) and its relative motion (R), The Hamiltonian H, Eq. (2), can also be decomposed into H CM and H R .

Ladder operators and Fock space representation
Now we introduce pairs of creation and annihilation operators {A + 1 , A − 1 } and {A + 2 , A − 2 }: which satisfy the following commutation relations: [H, They act on the Fock vacuum |0 ∝Ψ 0 (x 1 , x 2 , · · · x N ) as The excited states are built as The repeated action of the operators A + 1 ( A + 2 ) on the vacuum |0 reproduces in the coordinate representation Hermite polynomials ( hypergeometric function ), respectively.
The states |n 1 , n 2 are eigenstates ofH, Eq. (4), with the eigenvalues ( cf. Eq. (10) ) The energy spectrum is linear in quantum numbers n 1 , n 2 . This result is universal, i.e. it holds for all parameters m i and ν ij in the HamiltonianH (or H ). Notice that the energy of the ground state and the energy of the first excited state are non-degenerate whereas the higher energy levels are degenerate. The structure of degeneracy is as follows: n 1 n 2 n = n 1 + 2n 2 Degenerate states 0 0 0 It is evident that for n = even, the degeneracy is ( n 2 + 1) and for n = odd, the degeneracy is ( n+1 2 ). In order that the two models described by Hamiltonian (2), with statistical parameters ν ij and ν ′ ij , have the same tower of states (11) and the same spectrum (12), the necessary conditions are i<j ν ij = i<j ν ′ ij , ω = ω ′ and the number of particles should be the same, i.e. N = N ′ .
The dynamical symmetry algebra of the model is defined as maximal algebra commuting with the HamiltonianH. The dynamical symmetry of the ordinary Calogero model is complicated polynomial algebra denoted by C N (ν) in [15]. In our case, owing to the fact thatH (10) can be rewritten in terms of two independent, uncoupled oscillators (see bellow Eqs. (13) and (14)), this polynomial algebra can be linearized to the ordinary SU(2) algebra. This is the minimal symmetry that remains in the generic case, i.e. for general ν ij and m i . In fact, this is the same dynamical symmetry underlying the two-body Calogero model [15,16].
We point out that one can construct the creation and annihilation operators In terms of A ± 1 and B ± 2 , the above-mentioned SU (2) Here,N 1 andN 2 are number operators counting A 1 -and B 2 -modes, respectively.
The benefit of the construction (13) is that the operators A ± 1 , corresponding to the CM motion, decouple completely (cf. Eq.(10)), i.e.
At this point it is useful to make a contact with the Fock space of the ordinary Calogero model. In the ordinary Calogero model the S N -symmetric subspace of the whole Fock space is spanned by the states A +n 1 1 |0 CM and (B +n 2 2 · · · B +n N N )|0 R [7,16], where The so-called Dunkl-Polychronakos operators a i satisfy the algebra [7] [a i , a Our operators A 1 and B 2 correspond exactly to the operators A 1 and B 2 in the ordinary Calogero model. Within our algebraic treatment, we are unable to construct the eigenstates of (2) which correspond to the Calogero-states (B +n 3 Reducing the problem (2) to the (4) and (14), i.e. H →H →H R , has an interesting consequence, namely the existence of the universal critical point, defined by the null-vector The above relation (16) follows directly from Eq. (14) by demanding R 0 |H R |0 R = E 0R = 0. More generally, from Eq. (14) (see also Ref. [17] ) immediately follows that Since Eq. (16) implies that φ(1) = E 0R ω = 0, it is obvious that the critical point (Eq.(16)) is unique, i.e. there are no similar critical points when norms of states involving higher powers of the operators B 2 are involved.
At the critical point the system described byH R collapses completely. This means that the relative coordinates, the relative momenta and the relative energy are all zero at this critical point. There survives only one oscillator, describing the motion of the centre-of-mass. This singular behaviour was first noticed in [14] for the case ν ij = ν and m i = m. Of course, for the initial Hamiltonian H, which is not unitary ( i.e. physically ) equivalent toH, this corresponds to some ν ij < 0 and the norm of the wave function (1) blows up at the critical point. For ν ij negative but greater than the critical values (16), the wave function is singular at coincidence points but still quadratically integrable.

Discussion and outlook
In this Letter we have studied the most general multispecies Calogero model on the line, Eq. (2) , with a three-body interaction and an extended S N invariance.
By applying the similarity transformation ∆, we have obtained the Hamiltonian (4), on which we have performed the Fock space analysis (9,10) and found some of its (but not all) excited states, Eq. (11), and their energies E n 1 ,n 2 , (12). It turns out that the energy (12) is linear in quantum numbers n 1 and n 2 and there is a dynamical SU(2) symmetry responsible for the degeneracy of higher-energy levels with n = n 1 + 2n 2 ≥ 2. By splitting the Fock space into the CM-Fock space and the R-Fock space (14), we have detected the universal critical point (16) at which the system exhibits singular behaviour.
To conclude this analysis, let us consider the last term in (2) more closely, namely If we put m j = m = const. in (18), ∀j, symmetrize under the cyclic exchange of the indices (i → j → k → i) and reduce the sum to a common denominator using the identity cycl. 1 we obtain that the necessary condition for vanishing of the three-body interaction is ν ij = ν = const., ∀i, j. In this way, the problem (2) is reduced to the ordinary N-body Calogero model with two-body interactions only [1].
For the general ν ij and m j , the above procedure yields the following necessary conditions for the absence of the three-body interaction (18): The unique solution of these conditions is ν ij = α m i m j , α being some universal constant. This particular connection between masses and interaction parameters was also displayed in [10,12]. In [12], the condition (19) arose from the demand that the asymptotic Bethe ansatz should be applicable to the ground state of a multispecies many-body quantum system obeying mutual statistics, while in [10] its origin was not obvious. In our approach, it has the simplest possible interpretation.
The results presented here are easily applied to the model with F distinct families of particles. For example, for α = 1, the ground state energy becomes Two systems characterized by {ω, m a , N a } and {ω ′ , m ′ a , N ′ a } are identical if N a = N ′ a , ω = ω ′ and E 0 = E ′ 0 .
The open problem that still remains is the construction of (generalized) Dunkl-Polychronakos operators a i and a † i (15), which may help in finding the complete set of eigenstates of the Hamiltonian (2) (or (4)), in one-to-one correspondence with the ordinary Calogero model.