Chapter 8
Detection of Phase Transition by
Machines
Abstract As an approach to the important question of whether machines can
learn the discovery of physics, this chapter examines the question “Can phase
transitions be found by deep learning?”. Understanding phases is one of the most
important subjects in physics. Can machine learning really discover the thermal
phase transition in the basic physical system: Ising model?
In this chapter, we explain how to detect a phase transition in the Ising model by
machine learning. After a brief review of the phase transitions, we will explain phase
transition detection using neural networks.
8.1 What is Phase Transition?
Let us review the phase transition of the Ising model, which will be used in this
chapter as a target physical system. First, consider a spin variable s i in the total
volume V , and consider the expectation value of the spatial average of the spin
variables M[s] =
1
V
i s i in the canonical ensemble, that is, the (spontaneous)
magnetization. That is given by
M =
1
Z
{s}
e
−βH [s] M[s] .
(8.1)
On the right–hand side, the spin variable s is regarded as a random variable, and the
magnetization is defined as the spatial average M[s] of the spin variable under the
probability distribution
1
Z e −βH [s] .
If the Hamiltonian has a spin inversion symmetry H [s] = H [−s], the
magnetization M always vanishes. This can be shown in the following way. We
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
A. Tanaka et al., Deep Learning and Physics, Mathematical Physics Studies,
https://doi.org/10.1007/978-981-33-6108-9_8
139
Detection of Phase Transition by
Machines
Abstract As an approach to the important question of whether machines can
learn the discovery of physics, this chapter examines the question “Can phase
transitions be found by deep learning?”. Understanding phases is one of the most
important subjects in physics. Can machine learning really discover the thermal
phase transition in the basic physical system: Ising model?
In this chapter, we explain how to detect a phase transition in the Ising model by
machine learning. After a brief review of the phase transitions, we will explain phase
transition detection using neural networks.
8.1 What is Phase Transition?
Let us review the phase transition of the Ising model, which will be used in this
chapter as a target physical system. First, consider a spin variable s i in the total
volume V , and consider the expectation value of the spatial average of the spin
variables M[s] =
1
V
i s i in the canonical ensemble, that is, the (spontaneous)
magnetization. That is given by
M =
1
Z
{s}
e
−βH [s] M[s] .
(8.1)
On the right–hand side, the spin variable s is regarded as a random variable, and the
magnetization is defined as the spatial average M[s] of the spin variable under the
probability distribution
1
Z e −βH [s] .
If the Hamiltonian has a spin inversion symmetry H [s] = H [−s], the
magnetization M always vanishes. This can be shown in the following way. We
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
A. Tanaka et al., Deep Learning and Physics, Mathematical Physics Studies,
https://doi.org/10.1007/978-981-33-6108-9_8
139