The physics and mathematics of Calogero particles

Abstract

We give a review of the mathematical and physical properties of the celebrated family of Calogero-like models and related spin chains.

Generalizations of MICZ-Kepler system

“…Similarly, by the use of the replacement (8) we can equip the two-body system with the magnetic charges, without changing the qualitative properties of the initial system. Similar phenomenon was observed since its inception in the Calogero model [22] describing the N one-dimensional identical particles with pair-wise inverse-square and harmonic interaction: the energy spectrum of Calogero system coincides with the one of N free harmonic oscillators, apart from the coupling dependent shift of the ground-state energy (for the recent review see [23]). This structure of spectrum has led Calogero to suggest the possibility of a map between Calogero model and the system of decoupled oscillators, which has been found very recently [24] (see also [25]).…”

Exact partition function of supersymmetric Haldane–Shastry spin chain

“…However, it has been found that the NNS distribution for SU(m) bosonic HS spin chain does not follow either Wigner's law or Poisson's law within a wide range of N [9]. Instead, the cumulative NNS distribution for this bosonic spin chain can be well approximated by a function likẽ 12) where v = u/u max with u max being the largest normalized spacing, α and β are two free parameters taking values within the range 0 < α, β < 1, and the value of γ is fixed by requiring that the average normalized spacing be equal to 1. In our study we also find that, NNS distribution in the spectrum of SU(m|n) HS spin chain does not follow either Wigner's law or Poisson's law within a wide range of N. In particular, it is observed that the slope of cumulative NNS distribution diverges for both u → 0 and u → u max , which can not be explained from Wigner's or Poisson's distribution.…”

“…where H 0 is the Hamiltonian of spinless CS model given by [19] 12) and H (m|n) is obtained from H (m|n) (2.9) by the replacement ξ j → x j . Now the decoupling of the dynamical degrees of freedom of H * (2.10) from its spin degrees of freedom can be achieved by using the freezing trick [10][11][12].…”

The exactly solvable spin Sutherland model of type and its related spin chain

“…In particular, this type of systems have appeared as paradigms of various condensed matter systems exhibiting generalized exclusion statistics [18][19][20], the quantum Hall effect [21], and quantum electric transport phenomena [22,23]. More recently, quantum integrable spin chains with long-range interactions have played a key role in calculating higher-loop effects in the spectra of trace operators of planar N = 4 super Yang-Mills theory [24][25][26][27].…”