8.2 Detecting Phase Transition by a Neural Network
141
The magnetization M that appears here is called an order parameter, which
characterizes the phase transition.
The phase transition phenomenon is not only widely observed in materials such
as the magnetization transition and the gas–liquid phase transition of water, but also
in the field of elementary particle physics. For example, a vacuum phase transition
that changes the nature of the vacuum itself is related to the existence of mass, and
it is an important research subject.
Phase transitions are described by finding the symmetry that characterizes the
phase and defining the order parameter associated with the symmetry. And typically,
the order parameter is derived from a derivative of the free energy. Except for a few
models such as the two-dimensional Ising model, the phase transition temperature
has not been determined analytically, but can only be determined using numerical
calculations. It is also known that the phase transition of topological materials
does not have a local order parameter, making it difficult to characterize the phase
transition. In such a situation, it is a natural idea to find a phase transition using a
neural network. The following describes a study to directly find a phase transition
using a neural network.
8.2 Detecting Phase Transition by a Neural Network
When using a neural network for supervised learning, the neural network can do the
following:
• When a lot of data (x, d) = (input, answer) are given, it can guess what is the
result d for a new input value x .
To predict unknown phenomena, a lot of training data must be prepared. Fortunately,
in the research in computational physics, a huge amount of data is available
through numerical calculations. Particularly in the field of computational physics
using the Monte Carlo method, data can be generated almost infinitely (as long as
computational resources allow), so the situation is suitable for machine learning.
For example, consider a model with some phase transition in statistical mechanics, and generate its spin configurations by the Monte Carlo method. Can machine
learning determine which phase each spin configuration belongs to? The twodimensional Ising model is the simplest of the models that have a phase transition at
finite temperature, and the question of whether a neural network can distinguish the
phase transition of the Ising model is positively answered in the literature [106]. 2
2 Principal component analysis was used to detect a phase transition [107].
141
The magnetization M that appears here is called an order parameter, which
characterizes the phase transition.
The phase transition phenomenon is not only widely observed in materials such
as the magnetization transition and the gas–liquid phase transition of water, but also
in the field of elementary particle physics. For example, a vacuum phase transition
that changes the nature of the vacuum itself is related to the existence of mass, and
it is an important research subject.
Phase transitions are described by finding the symmetry that characterizes the
phase and defining the order parameter associated with the symmetry. And typically,
the order parameter is derived from a derivative of the free energy. Except for a few
models such as the two-dimensional Ising model, the phase transition temperature
has not been determined analytically, but can only be determined using numerical
calculations. It is also known that the phase transition of topological materials
does not have a local order parameter, making it difficult to characterize the phase
transition. In such a situation, it is a natural idea to find a phase transition using a
neural network. The following describes a study to directly find a phase transition
using a neural network.
8.2 Detecting Phase Transition by a Neural Network
When using a neural network for supervised learning, the neural network can do the
following:
• When a lot of data (x, d) = (input, answer) are given, it can guess what is the
result d for a new input value x .
To predict unknown phenomena, a lot of training data must be prepared. Fortunately,
in the research in computational physics, a huge amount of data is available
through numerical calculations. Particularly in the field of computational physics
using the Monte Carlo method, data can be generated almost infinitely (as long as
computational resources allow), so the situation is suitable for machine learning.
For example, consider a model with some phase transition in statistical mechanics, and generate its spin configurations by the Monte Carlo method. Can machine
learning determine which phase each spin configuration belongs to? The twodimensional Ising model is the simplest of the models that have a phase transition at
finite temperature, and the question of whether a neural network can distinguish the
phase transition of the Ising model is positively answered in the literature [106]. 2
2 Principal component analysis was used to detect a phase transition [107].