7.7 Quantum Spin Models
223
we plot the energy eigenvalues E = E n versus
E| ˆ
p 2
θ |E =
n| ˆ
p 2
θ |n. The web
formed by these quantities no longer has a regular structure because there is no
second good quantum number for the case w = 1.5.
The chaotic dynamics of the cut-circle billiard (D-billiard) has formed the basis
for development of a chip-scale, electrically pumped semiconductor laser that
demonstrates high power per mode with lower spatial coherence than conventional
laser sources. It has the unique property that it combines low spatial coherence with
high spectral radiance, which may be useful for a range of high-speed, full field
imaging applications (Redding et al. 2015).
7.7 Quantum Spin Models
The signatures of chaos can be observed in spin systems. To show this, we first
consider the XY models with exchange anisotropy and with single-site anisotropy
(Srivastava et al. 1990; Srivastava and Muller 1990), and then consider the XY
model with a constant magnetic field (Robb and Reichl 1998).
7.7.1 The XY Models with Anisotropy
We consider a system with two spin-σ particles for two different types of interaction.
In the first case, we assume the dynamics is governed by the quantum XY model
with exchange anisotropy. In the second case, we assume the dynamics is governed
by the XY model with single-site anisotropy
Case 1: The Quantum XY Model with Exchange Anisotropy The Hamiltonian for
this system can be written in the form (Srivastava et al. 1990; Srivastava and Muller
1990)
ˆ
H γ = − ¯
h
2 (1 + γ ) ˆ
σ 1x ˆ
σ 2x − ¯
h
2 (1 − γ ) ˆ
σ 1y ˆ
σ 2y .
(7.46)
where ˆ
σ iα is the αth component of the spin operator ˆ
σ i for the ith spin. This system
has a second constant of the motion,
ˆ
I γ = ¯
h
2 (1 + γ )
2
[( ˆ
σ 1x )
2
+ ( ˆ
σ 2x )
2
] + ¯
h
2 (1 − γ )
2
[( ˆ
σ 1y )
2
+ ( ˆ
σ 2y )
2
]
−2 ¯
h
2 (1 − γ
2 ) ˆ
σ 1z ˆ
σ 2z .
(7.47)
That is, [ ˆ
H γ , ˆ
I γ ] = 0.
223
we plot the energy eigenvalues E = E n versus
E| ˆ
p 2
θ |E =
n| ˆ
p 2
θ |n. The web
formed by these quantities no longer has a regular structure because there is no
second good quantum number for the case w = 1.5.
The chaotic dynamics of the cut-circle billiard (D-billiard) has formed the basis
for development of a chip-scale, electrically pumped semiconductor laser that
demonstrates high power per mode with lower spatial coherence than conventional
laser sources. It has the unique property that it combines low spatial coherence with
high spectral radiance, which may be useful for a range of high-speed, full field
imaging applications (Redding et al. 2015).
7.7 Quantum Spin Models
The signatures of chaos can be observed in spin systems. To show this, we first
consider the XY models with exchange anisotropy and with single-site anisotropy
(Srivastava et al. 1990; Srivastava and Muller 1990), and then consider the XY
model with a constant magnetic field (Robb and Reichl 1998).
7.7.1 The XY Models with Anisotropy
We consider a system with two spin-σ particles for two different types of interaction.
In the first case, we assume the dynamics is governed by the quantum XY model
with exchange anisotropy. In the second case, we assume the dynamics is governed
by the XY model with single-site anisotropy
Case 1: The Quantum XY Model with Exchange Anisotropy The Hamiltonian for
this system can be written in the form (Srivastava et al. 1990; Srivastava and Muller
1990)
ˆ
H γ = − ¯
h
2 (1 + γ ) ˆ
σ 1x ˆ
σ 2x − ¯
h
2 (1 − γ ) ˆ
σ 1y ˆ
σ 2y .
(7.46)
where ˆ
σ iα is the αth component of the spin operator ˆ
σ i for the ith spin. This system
has a second constant of the motion,
ˆ
I γ = ¯
h
2 (1 + γ )
2
[( ˆ
σ 1x )
2
+ ( ˆ
σ 2x )
2
] + ¯
h
2 (1 − γ )
2
[( ˆ
σ 1y )
2
+ ( ˆ
σ 2y )
2
]
−2 ¯
h
2 (1 − γ
2 ) ˆ
σ 1z ˆ
σ 2z .
(7.47)
That is, [ ˆ
H γ , ˆ
I γ ] = 0.