# Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination.

@article{Hu2017DiscoveringPP, title={Discovering phases, phase transitions, and crossovers through unsupervised machine learning: A critical examination.}, author={Wenjian Hu and Rajiv R. P. Singh and Richard T. Scalettar}, journal={Physical review. E}, year={2017}, volume={95 6-1}, pages={ 062122 } }

We apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models-the square- and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-1 Ising (BSI) model, and the two-dimensional XY model-and we examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow…

## 178 Citations

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## References

SHOWING 1-10 OF 82 REFERENCES

### Learning phase transitions by confusion

- Computer Science
- 2017

This work proposes a neural-network approach to finding phase transitions, based on the performance of a neural network after it is trained with data that are deliberately labelled incorrectly, and paves the way to the development of a generic tool for identifying unexplored phase transitions.

### Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders

- Computer SciencePhysical review. E
- 2017

Unsupervised machine learning techniques to learn features that best describe configurations of the two-dimensional Ising model and the three-dimensional XY model are examined, finding that the most promising algorithms are principal component analysis and variational autoencoders.

### Discovering phase transitions with unsupervised learning

- Computer Science
- 2016

This work shows that unsupervised learning techniques can be readily used to identify phases and phases transitions of many-body systems by using principal component analysis to extract relevant low-dimensional representations of the original data and clustering analysis to identify distinct phases in the feature space.

### Machine learning phases of matter

- Computer Science
- 2016

It is shown that modern machine learning architectures, such as fully connected and convolutional neural networks, can identify phases and phase transitions in a variety of condensed-matter Hamiltonians.

### Machine Learning Phases of Strongly Correlated Fermions

- Physics, Computer SciencePhysical Review X
- 2017

This work shows that a three dimensional convolutional network trained on auxiliary field configurations produced by quantum Monte Carlo simulations of the Hubbard model can correctly predict the magnetic phase diagram of the model at the average density of one (half filling).

### Dynamics of rough surfaces generated by two-dimensional lattice spin models.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2007

An analysis of mapped surfaces obtained from configurations of two classical statistical-mechanical spin models in the square lattice: the q -state Potts model and the spin-1 Blume-Capel model, using the Monte Carlo method and a mapping of the spin configurations to a solid-on-solid growth model.

### First-order phase transition and tricritical scaling behavior of the Blume-Capel model: A Wang-Landau sampling approach.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2015

The authors' estimation of the tricritical eigenvalue exponents provides the first Wang-Landau verification of the conjectured exact values, demonstrating the effectiveness of the density-of-states-based approach in finite-size scaling study of multicritical phenomena.

### Entanglement in a simple quantum phase transition

- Physics
- 2002

What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems that can be…

### Machine learning quantum phases of matter beyond the fermion sign problem

- Physics, Computer ScienceScientific Reports
- 2017

It is demonstrated that convolutional neural networks (CNN) can be optimized for quantum many-fermion systems such that they correctly identify and locate quantum phase transitions in such systems.

### Ising‐Model Spin Correlations on the Triangular Lattice

- Physics
- 1964

A Pfaffian representation, of the partition function of the triangular lattice is used to derive expressions for various two, four, and six spin correlations in terms of Pfaffians. The pair…