Spin Chains of Haldane–Shastry Type: A Bird’s Eye View
13
5 Supersymmetric Spin Chains
In the spin models discussed so far the internal degrees of freedom were all
fermionic (if = −1) or bosonic (if = 1). Moreover, in either case the permutation operator S ij belongs to the enveloping algebra of the defining representation of
su(m), so these models are naturally regarded as being of su(m) type. We shall
introduce in this section a more general class of su(m|n) supersymmetric spin
models, in which the first m internal degrees of freedom are bosonic and the last n
fermionic. We shall mainly focus on the models of this type associated with the root
system A N −1 , for which we shall briefly outline the computation of the spectrum
and the exact evaluation of the partition function [4].
5.1 The su(m|n) Supersymmetric Exchange Operator
As we did in the purely fermionic or bosonic cases, we shall start by constructing
the spin exchange operators appearing in the Hamiltonian. The possible values of
the spin now run from 1 to m + n, and the particle will be a boson (fermion) if
s i ∈ {1, . . . , m} (s i ∈ {m + 1, . . . , m + n}). The exchange operator is then defined
as
S
(m|n)
ij
|s 1 , . . . , s i , . . . , s j , . . . , s N = ij (s)|s 1 , . . . , s j , . . . , s i , . . . , s N
where
ij (s) =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
1,
s i , s j bosons
(−1) p , {s i , s j } = {fermion, boson}, with p = number of
fermions in positions i + 1, . . . , j − 1
−1,
s i , s j fermions.
By analogy with the purely bosonic or fermionic case, the Hamiltonian is taken as
H
(m|n)
0
=
i
(m|n)
ij
) ,
(23)
where the coupling constants J ij = Jg(ξ i − ξ j ) and chain sites ξ i are still defined
by Eqs. (19)–(20). As a matter of fact, it is easily checked that with this definition
H
(m|0)
0
= H 1 while H
(0|m)
0
= H −1 .
13
5 Supersymmetric Spin Chains
In the spin models discussed so far the internal degrees of freedom were all
fermionic (if = −1) or bosonic (if = 1). Moreover, in either case the permutation operator S ij belongs to the enveloping algebra of the defining representation of
su(m), so these models are naturally regarded as being of su(m) type. We shall
introduce in this section a more general class of su(m|n) supersymmetric spin
models, in which the first m internal degrees of freedom are bosonic and the last n
fermionic. We shall mainly focus on the models of this type associated with the root
system A N −1 , for which we shall briefly outline the computation of the spectrum
and the exact evaluation of the partition function [4].
5.1 The su(m|n) Supersymmetric Exchange Operator
As we did in the purely fermionic or bosonic cases, we shall start by constructing
the spin exchange operators appearing in the Hamiltonian. The possible values of
the spin now run from 1 to m + n, and the particle will be a boson (fermion) if
s i ∈ {1, . . . , m} (s i ∈ {m + 1, . . . , m + n}). The exchange operator is then defined
as
S
(m|n)
ij
|s 1 , . . . , s i , . . . , s j , . . . , s N = ij (s)|s 1 , . . . , s j , . . . , s i , . . . , s N
where
ij (s) =
⎧
⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
1,
s i , s j bosons
(−1) p , {s i , s j } = {fermion, boson}, with p = number of
fermions in positions i + 1, . . . , j − 1
−1,
s i , s j fermions.
By analogy with the purely bosonic or fermionic case, the Hamiltonian is taken as
H
(m|n)
0
=
i
(m|n)
ij
) ,
(23)
where the coupling constants J ij = Jg(ξ i − ξ j ) and chain sites ξ i are still defined
by Eqs. (19)–(20). As a matter of fact, it is easily checked that with this definition
H
(m|0)
0
= H 1 while H
(0|m)
0
= H −1 .