4
F. Finkel et al.
explain the construction of these chains starting from a spin dynamical model based
on those studied by Olshanetsky and Perelomov and the evaluation of their partition
function. In this contribution we shall outline the main steps of this construction,
and conclude with some recent applications to the study of the thermodynamics of
these chains.
2 Calogero–Sutherland Models
The study of integrable systems, in the classical sense of the Liouville–Arnol’d
theory, has occupied the activity of many researchers working in different fields
of physics and mathematics. However, many-body systems satisfying the required
integrability properties are rarely found in physics. Thus the construction of onedimensional models of many particles by Calogero and Sutherland in the sixties
was a breakthrough, and represented a major contribution to this field, as shown by
their impressive number of applications of Calogero–Sutherland models in so many
areas of physics. 1 We will introduce in this section a short account of these models,
thus paving the way for the construction of long-range integrable spin chains based
on them.
2.1 The Calogero Model
The Calogero model [5]
H = −
i
∂
2
x i
+ ω
2
i
x
2
i +
i =j
a(a − 1)
(x i − x j ) 2
(1)
describes a system of N particles on the line with an inverse-square interaction
potential. In the previous formulas all sums are understood to run from 1 to N ,
x ≡ (x 1 , . . . , x N ) and a > 1/2 is the system’s coupling constant. Its classical
version
H =
i
p
2
i + ω
2
i
x
2
i +
a(a − 1)
(x i − x j ) 2
is integrable (in fact, superintegrable [26]), as can be shown by Moser using Lax pair
techniques [20]. In the quantum case, the ground state is given by the Jastrow-type
expression
1 Bill Sutherland, Francesco Calogero, and Michel Gaudin were recently awarded the 2019 Dannie
Heineman Prize of the American Institute of Physics and the American Physical Society for their
seminal contributions to statistical mechanics and many-body physics.
F. Finkel et al.
explain the construction of these chains starting from a spin dynamical model based
on those studied by Olshanetsky and Perelomov and the evaluation of their partition
function. In this contribution we shall outline the main steps of this construction,
and conclude with some recent applications to the study of the thermodynamics of
these chains.
2 Calogero–Sutherland Models
The study of integrable systems, in the classical sense of the Liouville–Arnol’d
theory, has occupied the activity of many researchers working in different fields
of physics and mathematics. However, many-body systems satisfying the required
integrability properties are rarely found in physics. Thus the construction of onedimensional models of many particles by Calogero and Sutherland in the sixties
was a breakthrough, and represented a major contribution to this field, as shown by
their impressive number of applications of Calogero–Sutherland models in so many
areas of physics. 1 We will introduce in this section a short account of these models,
thus paving the way for the construction of long-range integrable spin chains based
on them.
2.1 The Calogero Model
The Calogero model [5]
H = −
i
∂
2
x i
+ ω
2
i
x
2
i +
i =j
a(a − 1)
(x i − x j ) 2
(1)
describes a system of N particles on the line with an inverse-square interaction
potential. In the previous formulas all sums are understood to run from 1 to N ,
x ≡ (x 1 , . . . , x N ) and a > 1/2 is the system’s coupling constant. Its classical
version
H =
i
p
2
i + ω
2
i
x
2
i +
a(a − 1)
(x i − x j ) 2
is integrable (in fact, superintegrable [26]), as can be shown by Moser using Lax pair
techniques [20]. In the quantum case, the ground state is given by the Jastrow-type
expression
1 Bill Sutherland, Francesco Calogero, and Michel Gaudin were recently awarded the 2019 Dannie
Heineman Prize of the American Institute of Physics and the American Physical Society for their
seminal contributions to statistical mechanics and many-body physics.