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abstract: 'A generalized spin Sutherland model including a three-body potential is proposed. The problem is analyzed in terms of three first-order differential-difference operators, obtained by combining SUSYQM supercharges with the elements of the dihedral group $D_6$. Three alternative commuting operators are also introduced.'
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[**Three-body generalization of the Sutherland model**]{}

[**with internal degrees of freedom**]{}

PACS: 03.65.Fd, 02.20.Df, 11.30.Pb

In recent years, the Sutherland one-dimensional $N$-particle model [@sutherland] and its rational limit, the Calogero model [@calogero], have received considerable attention in the literature because they are relevant to several important physical problems (for a list of references, see e.g. [@cq]).

The Sutherland problem can be analyzed in terms of a set of $N$ commuting first-order differential-difference operators [@poly], related to the root system of the ${\cal A}_{N-1}$ algebra [@perelomov] and known in the mathematical literature as Dunkl operators [@dunkl]. Use of the latter leads to a Hamiltonian with exchange terms, connected with an extension of the model for particles with internal degrees of freedom, referred to as the spin Sutherland problem [@ha; @bernard].

A similar type of approach can be employed [@buchstaber] for other integrable models related to root systems of Lie algebras [@perelomov].

In the present letter, we shall deal with a generalized Sutherland three-particle problem including an extra three-body trigonometric potential. Such a problem is related to the exceptional Lie algebra $G_2$, whose Weyl group, of order 12, is the dihedral group $D_6$ [@perelomov]. In contrast with the approaches used elsewhere [@poly; @dunkl; @ha; @bernard; @buchstaber], our starting point will be an analysis of the problem in supersymmetric quantum mechanics (SUSYQM), thereby emphasizing the link between Dunkl operators and SUSYQM.

Let us consider a system of three particles on a circle of length $\pi/a$ interacting via long-range two- and three-body potentials. Its Hamiltonian is defined by $$H = - \sum_{i=1}^3 \partial_i^2  
       + g a^2 \sum_{\scriptstyle i,j=1 \atop \scriptstyle i\ne j}^3 \csc^2
       \left(a(x_i-x_j)\right)
       + 3f a^2  \sum_{\scriptstyle i,j,k=1 \atop \scriptstyle i\ne j\ne k\ne i}^3 
       \csc^2  \left(a(x_i+x_j-2x_k)\right), \label{eq:H}$$ where $x_i$, $i=1$, 2, 3, $0\le x_i\le \pi/a$, denote the particle coordinates, $\partial_i \equiv \partial/\partial x_i$, and $g$, $f$ are assumed not to vanish simultaneously and to be such that $g > -1/4$, $f > -1/4$. In the case where $g\ne
0$ and $f = 0$, Hamiltonian (\[eq:H\]) reduces to the Sutherland Hamiltonian [@sutherland].

Throughout this paper, we shall use the notations $x_{ij} \equiv
x_i - x_j$, $i\ne j$, and $y_{ij} \equiv x_i + x_j - 2x_k$, $i\ne j\ne k\ne i$, where in the latter, index $k$ is suppressed as it is entirely determined by $i$ and $j$. Except where otherwise stated, we shall assume that the particles are distinguishable. In the case of indistinguishable particles, an additional symmetry requirement has to be imposed on the wave functions.

For distinguishable particles, the unnormalized ground-state wave function of Hamiltonian (\[eq:H\]), is given by $\psi_0(\xb) = \prod_{i\ne j}
\left|\sin(a x_{ij})\right|^{\kappa} \left|\sin(a y_{ij})\right|^{\lambda}$, and corresponds to the eigenvalue $E_0 = 8 a^2 (\kappa^2 + 3 \kappa \lambda + 3
\lambda^2)$, where $$\kappa \equiv \left\{\begin{array}{ll}
                                     \case{1}{2} (1 + \sqrt{1 + 4g}) & \mbox{if $g\ne 0$}
                                          \\[0.1cm]
                                     0                                              & \mbox{if $g=0$}  
                                   \end{array}\right., \qquad
  \lambda \equiv \left\{\begin{array}{ll}
                                     \case{1}{2} (1 + \sqrt{1 + 4f}) & \mbox{if $f\ne 0$}
                                          \\[0.1cm]
                                     0                                              & \mbox{if $f=0$}  
                                     \end{array}\right.,
  \label{eq:kappa-lambda}$$ or, equivalently, $g = \kappa (\kappa - 1)$, $f = \lambda (\lambda - 1)$. The proof of this result is based upon the trigonometric identities $$\begin{aligned}
  \sum_{\scriptstyle i,j,k \atop \scriptstyle i\ne j\ne k\ne i} \cot(a x_{ij})
       \cot(a x_{jk}) & = &  \sum_{\scriptstyle i,j,k \atop \scriptstyle i\ne j\ne k\ne 
       i} \cot(a y_{ij}) \cot(a y_{jk}) = 2, \nonumber \\
   \sum_{\scriptstyle i,j,k \atop \scriptstyle i\ne j\ne k\ne i} \cot(a x_{ij})
       \cot(a y_{jk}) & = & 4.        \label{eq:trig2}  \end{aligned}$$

The three-particle Hamiltonian (\[eq:H\]) can be alternatively considered as that of a particle in three-dimensional space. By using the Andrianov [*et al*]{} generalization of SUSYQM for multidimensional Hamiltonians [@andrianov], $H - E_0$ can therefore be regarded as the $H^{(0)}$ component of a supersymmetric Hamiltonian $\hat H = {\mathop{\rm diag}\nolimits}\left(H^{(0)}, H^{(1)}, H^{(2)}, H^{(3)}\right)$ with supercharge operators $\hat Q^+$, $\hat Q^- = \left(\hat Q^+\right)^{\dagger}$. The matrix elements of the latter can be expressed in terms of six differential operators $$Q^{\pm}_i = \mp \partial_i - \kappa a \sum_{j\ne i} \cot(a x_{ij}) 
  - \lambda a \left(\sum_{j\ne i} \cot(a y_{ij}) - {\sum_{\scriptstyle j,k \atop \scriptstyle i\ne j\ne k\ne i}}\cot(a y_{jk})\right),
  \qquad i = 1,2,3,      \label{eq:charges}$$ which are obtained from the ground-state wave function by using the recipe $Q_i^{\pm} = \mp \partial_i + \partial_i \chi(\xb)$, where $\chi(\xb) = - \ln
\psi_0(\xb)$. In terms of such operators, $H^{(0)} = Q^+_i Q^-_i$ while $H^{(3)} =
Q^-_i Q^+_i$. Apart from some additive constant, the latter turns out to be given by (\[eq:H\]) with $g = \kappa (\kappa - 1)$, $f = \lambda (\lambda - 1)$ replaced by $g = \kappa (\kappa + 1)$, $f = \lambda (\lambda + 1)$, respectively.

Let us now transform the supercharge operators $Q^-_i$ of Eq. (\[eq:charges\]) into some differential-difference operators $$D_i = \partial_i - \kappa a \sum_{j\ne i} \cot(a x_{ij}) K_{ij} 
  - \lambda a \left(\sum_{j\ne i} \cot(a y_{ij}) L_{ij} - {\sum_{\scriptstyle j,k \atop \scriptstyle i\ne j\ne k\ne i}}\cot(a y_{jk}) L_{jk}
  \right),      \label{eq:D}$$ by inserting some finite group elements $K_{ij}$ and $L_{ij} \equiv K_{ij} I_r$. Here $K_{ij}$ are particle permutation operators, while $I_r$ is the inversion operator in relative-coordinate space. In the centre-of-mass coordinate system to be used in the remainder of this paper, they satisfy the relations $$\begin{aligned}
  K_{ij} & = & K_{ji} = K_{ij}^{\dagger}, \qquad K_{ij}^2 = 1, \qquad K_{ij} K_{jk} =
          K_{jk} K_{ki} = K_{ki} K_{ij}, \nonumber \\
  K_{ij} I_r & = & I_r K_{ij}, \qquad I_r = I_r^{\dagger}, \quad I_r^2 = 1,      
          \label{eq:K-Ir1} \\
  K_{ij} x_j & = & x_i K_{ij}, \quad K_{ij} x_k = x_k K_{ij}, \quad I_r x_i = - x_i
          I_r,       \label{eq:K-Ir2}\end{aligned}$$ for all $i\ne j\ne k\ne i$. The operators 1, $K_{ij}$, $K_{ijk} \equiv K_{ij} K_{jk}$, $I_r$, $L_{ij}$, and $L_{ijk} \equiv K_{ijk} I_r$, where $i$, $j$, $k$ run over the set {1, 2, 3}, are the 12 elements of the dihedral group $D_6$ [@hamermesh].

From their definition and Eqs. (\[eq:K-Ir1\]), (\[eq:K-Ir2\]), it is obvious that the differential-difference operators $D_i$ are both antihermitian and $D_6$-covariant, i.e., $D_i^{\dagger} = - D_i$, $K_{ij} D_j = D_i K_{ij}$, $K_{ij}D_k = D_k K_{ij}$, and $I_r D_i = - D_i I_r$, for all $i\ne j\ne k\ne i$. After some straightforward, although rather lengthy, calculations using again the trigonometric identities (\[eq:trig2\]), one obtains that their commutators are given by $$\left[D_i, D_j\right] = - a^2 \left(\kappa^2 + 3 \lambda^2 - 4 \kappa \lambda I_r
  \right) \sum_{k\ne i,j} \left(K_{ijk} - K_{ikj}\right), \qquad i\ne j, 
  \label{eq:D-com}$$ and that $$\begin{aligned}
  -\sum_i D_i^2 & = & - \sum_i \partial_i^2 + a^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\csc^2(a x_{ij}) \kappa
       (\kappa - K_{ij}) + 3 a^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\csc^2(a y_{ij}) \lambda (\lambda - L_{ij})
       \nonumber\\
  & & - 6 a^2 \left(\kappa^2 + 3 \lambda^2\right) - a^2 \left(\kappa^2 + 
       3 \lambda^2 + 12 \kappa \lambda I_r\right) \left(K_{123} + K_{132}\right).
       \label{eq:D-square}     \end{aligned}$$

From Eq. (\[eq:D-com\]), it is clear that the operators $D_i$ do not commute among themselves, except in the $a \to 0$ limit, i.e., in the rational case considered many years ago by Wolfes [@wolfes], and by Calogero and Marchioro [@marchioro]. Furthermore, Eq. (\[eq:D-square\]) shows that the generalized Hamiltonian with exchange terms $$H_{exch} = - \sum_i \partial_i^2 + a^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\csc^2(a x_{ij}) \kappa
  (\kappa - K_{ij}) + 3 a^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\csc^2(a y_{ij}) \lambda (\lambda - L_{ij})
  \label{eq:Hexch}$$ only differs by some exchange operators from the Hamiltonian $\sum_i \pi_i^2$, written in terms of the generalized momenta $\pi_i = \pi_i^{\dagger} = - i D_i$. In those subspaces of Hilbert space wherein $\left(K_{ij}, L_{ij}\right) = (1,1)$, $(1,-1)$, $(-1,1)$, or $(-1,-1)$, $H_{exch}$ reduces to Hamiltonian (\[eq:H\]) corresponding to $(g,f) = (\kappa (\kappa-1), \lambda
(\lambda-1))$, $(\kappa (\kappa-1), \lambda (\lambda+1))$, $(\kappa (\kappa+1),
\lambda (\lambda-1))$, or $(\kappa (\kappa+1), \lambda (\lambda+1))$, respectively.

As in the case of the Sutherland problem [@bernard], we can try to reformulate the present one in terms of some commuting, albeit non-covariant, differential operators ${\hat D}_i$. Let $${\hat D}_i = D_i + i \kappa a \sum_{j\ne i} \alpha_{ij} K_{ij} + i \lambda a \left(
  \sum_{j\ne i} \beta_{ij} L_{ij} - {\sum_{\scriptstyle j,k \atop \scriptstyle i\ne j\ne k\ne i}}\beta_{jk} L_{jk}\right),    \label{eq:hatD}$$ where $\alpha_{ij}$ and $\beta_{ij}$ are some real constants. With such a choice, the transformed operators remain antihermitian, i.e., ${\hat D}_i^{\dagger} = - {\hat D}_i$. We shall assume in addition that $\alpha_{ji} = - \alpha_{ij}$ and $\beta_{ji} =
\beta_{ij}$. This assumption is justified by the fact that for $\lambda =
0$ [@bernard], the operators (\[eq:hatD\]) with $\alpha_{ij} = - \alpha_{ji} =
- 1$, $i<j$, do fulfil the required property $\left[{\hat D}_i, {\hat D}_j\right] = 0$.

From definition (\[eq:hatD\]) and the $D_6$-covariance of $D_i$, the new operators ${\hat D}_i$ have the following transformation properties under $D_6$: $$\begin{aligned}
  K_{ij} {\hat D}_j - {\hat D}_i K_{ij} & = & - i \kappa a \left(2 \alpha_{ij} + \sum_{k\ne i,j}
       \left(\alpha_{ik} - \alpha_{jk}\right) K_{ijk}\right) \nonumber \\
  & & - i \lambda a \sum_{k\ne i,j} \left(\beta_{ik} - \beta_{jk}\right) I_r 
       \left(K_{ijk} + 2 K_{ikj}\right), \qquad i\ne j,      \label{eq:hatD-cov1} \\
  \left[K_{ij}, {\hat D}_k\right] & = & i a \left[\kappa \left(\alpha_{ik} - \alpha_{jk}
       \right) - \lambda \left(\beta_{ik} - \beta_{jk}\right) I_r\right] \left(K_{ijk}
       - K_{ikj}\right), \nonumber \\
  & & i\ne j\ne k\ne i,       \label{eq:hatD-cov2} \\
  \left\{I_r, {\hat D}_i\right\} & = & 2i \kappa a \sum_{j\ne i} \alpha_{ij} L_{ij}
       + 2i \lambda a \left(\sum_{j\ne i} \beta_{ij} K_{ij} - {\sum_{\scriptstyle j,k \atop \scriptstyle i\ne j\ne k\ne i}}\beta_{jk} K_{jk}
       \right).          \label{eq:hatD-cov3}\end{aligned}$$ Hence, they can only be covariant provided $\alpha_{ij} = \beta_{ij} = 0$, i.e., when they coincide with the $D_i$’s.

By using various properties of the operators $D_i$, as well as the relation $\sum_i
D_i = 0$ valid in the centre-of-mass coordinate system, it is straightforward to show that if the constants $\alpha_{ij}$, $\beta_{ij}$ fulfil the three relations $$\alpha_{12} \alpha_{23} + \alpha_{23} \alpha_{31} + \alpha_{31} \alpha_{12}
  =  -1, \qquad \beta_{12}\beta_{23} + \beta_{23} \beta_{31} + \beta_{31} 
  \beta_{12} = -1,        \label{eq:cond1}$$ $$(\alpha_{23} - \alpha_{31}) \beta_{12} + (\alpha_{31} - \alpha_{12}) \beta_{23}
  + (\alpha_{12} - \alpha_{23}) \beta_{31} = 4,      \label{eq:cond2}$$ then the operators ${\hat D}_i$ commute among themselves, and $H_{exch}$ only differs by some additive constant from the Hamiltonian $\sum_i \hat{\pi}_i^2$, where $\hat{\pi}_i = \hat{\pi}_i^{\dagger} = - i {\hat D}_i$. Furthermore, this additive constant can be set equal to zero by normalizing $\alpha_{ij}$ and $\beta_{ij}$ in such a way that the relations $$\alpha_{12}^2 + \alpha_{23}^2 + \alpha_{31}^2 = 3, \qquad 
  \beta_{12}^2 + \beta_{23}^2 + \beta_{31}^2 = 3,      \label{eq:norm}$$ are satisfied.

The constants $\alpha_{ij} = - \alpha_{ji} = -1$, $i<j$, used in Ref. [@bernard], do fulfil the first relation in both Eqs. (\[eq:cond1\]) and (\[eq:norm\]). It should be stressed however that such equations admit other solutions too. It is then easy to prove that the remaining three relations in (\[eq:cond1\])–(\[eq:norm\]) admit four, and only four, solutions for the $\beta_{ij}$’s, compatible with this choice for the $\alpha_{ij}$’s: $(\beta_{12},
\beta_{23}, \beta_{31}) = (-1,1,1)$, $(-1,1,-1)$, $(- 5/3, 1/3, 1/3)$, and $(- 1/3,
5/3, - 1/3)$.

A relation can be established between $H_{exch}$ and a Hamiltonian ${\cal
H}^{(\kappa,\lambda)}$ describing a one-dimensional system of three particles with $SU(n)$ “spins” (or colours in particle physics language), interacting via spin-dependent two and three-body potentials, $${\cal H}^{(\kappa,\lambda)} = - \sum_i \partial_i^2 + a^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\csc^2(a x_{ij})
  \kappa (\kappa - P_{ij}) + 3 a^2 {\sum_{\scriptstyle i,j \atop \scriptstyle i\ne j}}\csc^2(a y_{ij}) \lambda (\lambda -
  \tilde P_{ij}).      \label{eq:spinH}$$ Here each particle is assumed to carry a spin with $n$ possible values, and $P_{ij}$, $\tilde P_{ij} \equiv P_{ij} \tilde P$ are some operators acting only in spin space. The operator $P_{ij}$ is defined as the operator permuting the $i$th and $j$th spins, while ${\tilde P}$ is a permutation-invariant and involutive operator, i.e., ${\tilde P}\sigma_i = \sigma_i^* {\tilde P}$, for some $\sigma^*_i$ such that $P_{jk} \sigma_i^* = \sigma_i^* P_{jk}$ for all $i$, $j$, $k$, and $\sigma_i^{**} =
\sigma_i$. For $SU(2)$ spins for instance, $\sigma_i = \pm 1/2$, $P_{ij} =
(\sigma^a_i \sigma^a_j + 1)/2$, where $\sigma^a$, $a=1$, 2, 3, denote the Pauli matrices, ${\tilde P}$ may be taken as 1 or $\sigma^1_1 \sigma^1_2 \sigma^1_3$, and accordingly $\sigma_i^* =
\sigma_i$ or $-\sigma_i$. The operators $P_{ij}$ and ${\tilde P}$ satisfy relations similar to those fulfilled by $K_{ij}$ and $I_r$ (cf. Eqs. (\[eq:K-Ir1\]) and (\[eq:K-Ir2\])), with $x_i$ and $-x_i$ replaced by $\sigma_i$ and $\sigma_i^*$ respectively. Hence 1, $P_{ij}$, $P_{ijk} \equiv P_{ij} P_{jk}$, ${\tilde P}$, $\tilde P_{ij}$, and $\tilde P_{ijk} \equiv P_{ijk} \tilde P$ realize the dihedral group $D_6$ in spin space. Such a realization will be referred to as $D_6^{(s)}$ to distinguish it from the realization $D_6^{(c)}$ in coordinate space, corresponding to $K_{ij}$ and $I_r$.

The Hamiltonian ${\cal H}^{(\kappa,\lambda)}$ remains invariant under the combined action of $D_6$ in coordinate and spin spaces (to be referred to as $D_6^{(cs)}$), since it commutes with both $K_{ij} P_{ij}$ and $I_r {\tilde P}$. Its eigenfunctions corresponding to a definite eigenvalue therefore belong to a (reducible or irreducible) representation of $D_6^{(cs)}$. For indistinguishable particles that are bosons (resp. fermions), only those irreducible representations of $D_6^{(cs)}$ that contain the symmetric (resp. antisymmetric) irreducible representation of the symmetric group $S_3$ should be considered. There are only two such inequivalent representations, which are both one-dimensional and denoted by $A_1$ and $B_1$ (resp. $A_2$ and $B_2$) [@hamermesh]. They differ in the eigenvalue of $I_r
{\tilde P}$, which is equal to $+1$ or $-1$, respectively.

In such representations, for an appropriate choice of the parameters $\kappa$, $\lambda$, ${\cal H}^{(\kappa,\lambda)}$ can be obtained from $H_{exch}$ by applying some projection operators. Let indeed $\Pi_{B\pm}$ (resp. $\Pi_{F\pm}$) be the projection operators that consist in replacing $K_{ij}$ and $I_r$ by $P_{ij}$ (resp. $-P_{ij}$) and $\pm {\tilde P}$, respectively, when they are at the right-hand side of an expression. It is obvious that $\Pi_{B\pm}(H_{exch}) = {\cal H}^{(\kappa,\pm\lambda)}$, and $\Pi_{F\pm}(H_{exch}) = {\cal H}^{(-\kappa,\pm\lambda)}$. If $H_{exch}$ has been diagonalized on a basis of functions depending upon coordinates and spins, then its eigenfunctions $\Psi(\xb,\sigmab)$ are also eigenfunctions of ${\cal
H}^{(\kappa,\pm\lambda)}$ (resp. ${\cal H}^{(-\kappa,\pm\lambda)}$) provided that $(K_{ij} - P_{ij}) \Psi(\xb,\sigmab) = 0$ (resp. $(K_{ij} + P_{ij}) \Psi(\xb,\sigmab)
= 0$) and $(I_r \mp {\tilde P}) \Psi(\xb,\sigmab) = 0$. We shall not pursue the determination of the eigenfunctions of ${\cal H}^{(\kappa,\lambda)}$ any further, leaving a detailed derivation for a forthcoming publication.

In conclusion, in the present letter we did propose a three-body generalization of the Sutherland problem with internal degrees of freedom, related to a corresponding problem with exchange terms. For the latter, we did construct both $D_6$-covariant, but non-commuting, and commuting, but non-$D_6$-covariant differential-difference operators, interms of which the Hamiltonian can be expressed in a very simple way. We did show that whereas the former operators can be derived in a well-defined way from SUSYQM supercharges, there is some freedom in the choice of the latter.

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