142
8 Detection of Phase Transition by Machines
First, the training data can be generated relatively easily by using the Metropolis
method. Consider a spin on a two-dimensional square lattice. The Hamiltonian H is
H [s] = −
i,j
s i,j (s i+1,j + s i,j +1 ) .
(8.7)
Here, the exchange interaction constant is taken as −1, which represents the
ferromagnetism. This model is known to undergo a paramagnetic / ferromagnetic
phase transition around T = 2.27, as mentioned in the column of Chap. 5 (see
[74, 75]).
Let us move on to the phase transition detection using neural networks. The
spin configuration that serves as the training data is generated by the following
procedure:
1. Select a temperature value T .
2. Sample spin configuration s at the temperature T generated by the Metropolis
method.
3. If T < 2.27, set d = 0. If T > 2.27, set d = 1.
4. Add (s, d) to the training data.
If this step is repeated in the range T ∈ [T low , T high ] which includes the phase
transition temperature T = 2.27, it is possible to generate a training data for a neural
network to determine whether a given spin configuration s is in the ordered phase
T < 2.27 or the disordered phase (T > 2.27). In other words, a neural network is
designed and trained as a “phase discriminator.”
Interesting applications are possible using the neural network trained with this
data. Instead of the square lattice Ising model defined by the previous H , it is known
that the triangular lattice Ising model
H 3 [s] = −
i,j
s i,j (s i+1,j + s i,j +1 + s i+1,j +1 )
(8.8)
has a similar phase structure. The phase transition temperature is not T = 2.27, but
T = 3.64. In fact, a neural network trained using H can detect the phase transition
of H 3 (Fig. 8.1). We can see that the output of the neural network jumps from 0 to 1
around T = 3.64. If this fact holds for other models, we have a new way to detect
phase transitions.
The idea of this “phase discriminator” is certainly interesting, but it has its
weaknesses when actually discovering unknown phase transitions. The point is that
when generating the training data, at least the phase transition point of a similar
model must be known. In other words, all of the phase transition detection cannot be
covered by the neural network, and it is necessary to know the phase transition point
theoretically for a model. This means that the usage is limited. The following simple
question comes to mind: can we automatically detect even the phase transition point
itself in a single model — in the present case the transition temperature T = 2.27?
8 Detection of Phase Transition by Machines
First, the training data can be generated relatively easily by using the Metropolis
method. Consider a spin on a two-dimensional square lattice. The Hamiltonian H is
H [s] = −
i,j
s i,j (s i+1,j + s i,j +1 ) .
(8.7)
Here, the exchange interaction constant is taken as −1, which represents the
ferromagnetism. This model is known to undergo a paramagnetic / ferromagnetic
phase transition around T = 2.27, as mentioned in the column of Chap. 5 (see
[74, 75]).
Let us move on to the phase transition detection using neural networks. The
spin configuration that serves as the training data is generated by the following
procedure:
1. Select a temperature value T .
2. Sample spin configuration s at the temperature T generated by the Metropolis
method.
3. If T < 2.27, set d = 0. If T > 2.27, set d = 1.
4. Add (s, d) to the training data.
If this step is repeated in the range T ∈ [T low , T high ] which includes the phase
transition temperature T = 2.27, it is possible to generate a training data for a neural
network to determine whether a given spin configuration s is in the ordered phase
T < 2.27 or the disordered phase (T > 2.27). In other words, a neural network is
designed and trained as a “phase discriminator.”
Interesting applications are possible using the neural network trained with this
data. Instead of the square lattice Ising model defined by the previous H , it is known
that the triangular lattice Ising model
H 3 [s] = −
i,j
s i,j (s i+1,j + s i,j +1 + s i+1,j +1 )
(8.8)
has a similar phase structure. The phase transition temperature is not T = 2.27, but
T = 3.64. In fact, a neural network trained using H can detect the phase transition
of H 3 (Fig. 8.1). We can see that the output of the neural network jumps from 0 to 1
around T = 3.64. If this fact holds for other models, we have a new way to detect
phase transitions.
The idea of this “phase discriminator” is certainly interesting, but it has its
weaknesses when actually discovering unknown phase transitions. The point is that
when generating the training data, at least the phase transition point of a similar
model must be known. In other words, all of the phase transition detection cannot be
covered by the neural network, and it is necessary to know the phase transition point
theoretically for a model. This means that the usage is limited. The following simple
question comes to mind: can we automatically detect even the phase transition point
itself in a single model — in the present case the transition temperature T = 2.27?