224
7 Bounded Quantum Systems
Case 2: The XY Model with Single-Site Anisotropy The second Hamiltonian we
consider is that for the XY model with single-site anisotropy, which can be written
ˆ
H a = − ¯
h
2 ( ˆ
σ 1x ˆ
σ 2x + ˆ
σ 1y ˆ
σ 2y )−
1
2
a ¯
h
2
[( ˆ
σ 1x )
2
+( ˆ
σ 2x )
2
−( ˆ
σ 1y )
2
−( ˆ
σ 2y )
2
]. (7.48)
Srivastava et al. (1990) studied the energy spectrum of these two quantum spin
XY models to determine how the disappearance of a second constant of the motion
manifests itself in the spectrum of these quantum systems. The first step in analyzing
the spectrum is to determine the symmetry groups of the system because the
spectrum will be decomposed in terms of invariant subspaces of any such symmetry
group.
It is fairly easy to show that both XY models are invariant under the symmetry
group D 2 ⊗ S 2 , which is the direct product of the group S 2 of permutations of two
particles and the group D 2 , which consists of the group of rotations by angle π
about the x, y, and z axes. The multiplication table for the group D 2 ⊗ S 2 is shown
in Table 7.1, where ˆ
I is the identity element, ˆ
P is the permutation operator, and ˆ
R x ,
ˆ
R y , and ˆ
R z are rotation operators defined as
ˆ
R x = e
iπ ˆ
σ x , ˆ
R y = e
iπ ˆ
σ y , and ˆ
R z = e
iπ ˆ
σ z .
Here ˆ
σ α = ˆ
σ 1α + ˆ
σ 2α is the total α-component of spin.
The group elements ˆ
A, ˆ
B, and ˆ
C are defined as ˆ
A = ˆ
R x ˆ
P , ˆ
B = ˆ
R y ˆ
P , and
ˆ
C = ˆ
R z ˆ
P . The group D 2 ⊗ S 2 is Abelian so that every element is a class and,
therefore, since the group is of order 8, there will be eight invariant subspaces. The
character table for this group (Atkins et al. 1970) is reproduced in Table 7.2.
Srivastava and Muller have studied the spectrum of the Hamiltonians in
Eqs. (7.46) and (7.48). They use linear combinations of the basis, |m 1 , m 2 =
|m 1 ⊗ |m 2 (|m i are eigenstates of ˆ
σ iz ), that are eigenstates of the symmetry
operators to construct the block diagonal matrix form of the Hamiltonian. They
then obtain the spectrum numerically.
In Fig. 7.19a, we show a quantum web constructed from the invariant subspaces
A1A and B1S for the Hamiltonian H γ (γ = 0.2, S = 1, σ = 35) and the second
Table 7.1 Multiplication
table for the group D 2 ⊗ S 2
I
R x R y R z P
A
B
C
I
I
R x R y R z P
A
B
C
R x R x I
R z R y A
P
C
B
R y R y R z I
R x B
C
P
A
R z R z R y R x I
C
B
A
P
P
P
A
B
C
I
R x R y R z
A
A
P
C
B
R x I
R z R y
B
B
C
P
A
R y R z I
R x
C
C
B
A
P
R z R y R x I
7 Bounded Quantum Systems
Case 2: The XY Model with Single-Site Anisotropy The second Hamiltonian we
consider is that for the XY model with single-site anisotropy, which can be written
ˆ
H a = − ¯
h
2 ( ˆ
σ 1x ˆ
σ 2x + ˆ
σ 1y ˆ
σ 2y )−
1
2
a ¯
h
2
[( ˆ
σ 1x )
2
+( ˆ
σ 2x )
2
−( ˆ
σ 1y )
2
−( ˆ
σ 2y )
2
]. (7.48)
Srivastava et al. (1990) studied the energy spectrum of these two quantum spin
XY models to determine how the disappearance of a second constant of the motion
manifests itself in the spectrum of these quantum systems. The first step in analyzing
the spectrum is to determine the symmetry groups of the system because the
spectrum will be decomposed in terms of invariant subspaces of any such symmetry
group.
It is fairly easy to show that both XY models are invariant under the symmetry
group D 2 ⊗ S 2 , which is the direct product of the group S 2 of permutations of two
particles and the group D 2 , which consists of the group of rotations by angle π
about the x, y, and z axes. The multiplication table for the group D 2 ⊗ S 2 is shown
in Table 7.1, where ˆ
I is the identity element, ˆ
P is the permutation operator, and ˆ
R x ,
ˆ
R y , and ˆ
R z are rotation operators defined as
ˆ
R x = e
iπ ˆ
σ x , ˆ
R y = e
iπ ˆ
σ y , and ˆ
R z = e
iπ ˆ
σ z .
Here ˆ
σ α = ˆ
σ 1α + ˆ
σ 2α is the total α-component of spin.
The group elements ˆ
A, ˆ
B, and ˆ
C are defined as ˆ
A = ˆ
R x ˆ
P , ˆ
B = ˆ
R y ˆ
P , and
ˆ
C = ˆ
R z ˆ
P . The group D 2 ⊗ S 2 is Abelian so that every element is a class and,
therefore, since the group is of order 8, there will be eight invariant subspaces. The
character table for this group (Atkins et al. 1970) is reproduced in Table 7.2.
Srivastava and Muller have studied the spectrum of the Hamiltonians in
Eqs. (7.46) and (7.48). They use linear combinations of the basis, |m 1 , m 2 =
|m 1 ⊗ |m 2 (|m i are eigenstates of ˆ
σ iz ), that are eigenstates of the symmetry
operators to construct the block diagonal matrix form of the Hamiltonian. They
then obtain the spectrum numerically.
In Fig. 7.19a, we show a quantum web constructed from the invariant subspaces
A1A and B1S for the Hamiltonian H γ (γ = 0.2, S = 1, σ = 35) and the second
Table 7.1 Multiplication
table for the group D 2 ⊗ S 2
I
R x R y R z P
A
B
C
I
I
R x R y R z P
A
B
C
R x R x I
R z R y A
P
C
B
R y R y R z I
R x B
C
P
A
R z R z R y R x I
C
B
A
P
P
P
A
B
C
I
R x R y R z
A
A
P
C
B
R x I
R z R y
B
B
C
P
A
R y R z I
R x
C
C
B
A
P
R z R y R x I