Spin Chains of Haldane–Shastry Type: A Bird’s Eye View
9
Calling k = (k 1 , . . . , k r ), the partition function of the Hamiltonian Z can be
expressed as
Z (2ωT ) = q
E 0 /(2ω)
k∈P N
p 1 >···>p r 0
D (k; m) q
r
i=1 k i p i ,
where P N denotes the set of partitions of the integer N taking order into account.
After a straightforward calculation the latter expression yields the closed formula
[10]
Z (2ωT ) =
q E 0 /(2ω)
1 − q N
k∈P N
D (k; m)
r−1
i=1
q K i
1 − q K i
,
(11)
where we have set
K i ≡
i
j =1
k i .
Similar results are obtained for the Sutherland (trigonometric) model (4) with spin
degrees of freedom.
4 Spin Chains of Haldane–Shastry Type
By a spin chain we usually understand a one-dimensional lattice whose sites are
occupied by particles with internal degrees of freedom. In this work we shall deal
exclusively with spin chains with long-range interactions, involving all the sites.
Moreover, the chain sites must be chosen in a very specific way, which is critical for
ensuring the symmetry and solvability properties of the models.
For the sake of simplicity, in what follows we shall focus on the rational Calogero
model of A N −1 type and its associated spin chain introduced below, although the
methods applied can be used for any of the models we have previously discussed.
Let us then consider the two Hamiltonians we have previously studied, namely the
scalar one H in Eq. (1) and the spin dynamical Hamiltonian H in Eq. (7). Defining
the scalar potential
U(x) =
i =j
1
(x i − x j ) 2 +
i
x
2
i
(12)
9
Calling k = (k 1 , . . . , k r ), the partition function of the Hamiltonian Z can be
expressed as
Z (2ωT ) = q
E 0 /(2ω)
k∈P N
p 1 >···>p r 0
D (k; m) q
r
i=1 k i p i ,
where P N denotes the set of partitions of the integer N taking order into account.
After a straightforward calculation the latter expression yields the closed formula
[10]
Z (2ωT ) =
q E 0 /(2ω)
1 − q N
k∈P N
D (k; m)
r−1
i=1
q K i
1 − q K i
,
(11)
where we have set
K i ≡
i
j =1
k i .
Similar results are obtained for the Sutherland (trigonometric) model (4) with spin
degrees of freedom.
4 Spin Chains of Haldane–Shastry Type
By a spin chain we usually understand a one-dimensional lattice whose sites are
occupied by particles with internal degrees of freedom. In this work we shall deal
exclusively with spin chains with long-range interactions, involving all the sites.
Moreover, the chain sites must be chosen in a very specific way, which is critical for
ensuring the symmetry and solvability properties of the models.
For the sake of simplicity, in what follows we shall focus on the rational Calogero
model of A N −1 type and its associated spin chain introduced below, although the
methods applied can be used for any of the models we have previously discussed.
Let us then consider the two Hamiltonians we have previously studied, namely the
scalar one H in Eq. (1) and the spin dynamical Hamiltonian H in Eq. (7). Defining
the scalar potential
U(x) =
i =j
1
(x i − x j ) 2 +
i
x
2
i
(12)