Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.19995

Deep Learning of Phase Transitions for Bilinear-Biquadratic Spin-1 Chains from Correlation Aspects

Ming-Chiang Chung, Yu-Ting Kuo, Ian McCulloch, Yuan-Hong Tsai

Published Oct 14, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the role of different correlation functions in reconstructing the phase diagram of the spin-1 bilinear-biquadratic chain, parametrized by the angle $θ$ between the bilinear and biquadratic couplings, using our previously developed label-free machine-learning pipeline \cite{Ming21, Ming23}, in which an autoencoder compresses iDMRG correlation matrices into latent features and K-means clustering with silhouette analysis groups them into phases. Without any prior labels, the scheme reveals four natural phases, the Haldane, dimerized, critical trimerized, and ferromagnetic phases, and accurately locates the Takhtajan-Babujian ($θ=-π/4$), $SU(3)$-symmetric Uimin-Lai-Sutherland ($θ=π/4$), and Ferro-Dimer ($θ=5π/4$) transition points; a shallow supervised refinement further sharpens the phase boundaries. We find that different correlation functions play fundamentally different roles in phase identification. The $SU(3)$-symmetric correlation alone selects a four-cluster partition that can be associated with the four phases, whereas the spin-spin and dimer-dimer correlations must be combined to recover the complete phase diagram. In contrast, the trimer-trimer correlation behaves as a marginal correlation: it is insufficient for phase identification by itself but improves the precision of the predicted phase boundaries when combined with the other correlations. Furthermore, the latent representations and silhouette values reveal signatures that we relate to the $θ$ dependence of the correlations within each phase and, tentatively, to the underlying renormalization-group structure. These results demonstrate that machine learning can not only identify quantum phases but also uncover the physical information encoded in different correlation functions.

DOI
10.48550/arxiv.2610.19995

Related papers

Deep learning of phase transitions for quantum spin chains from correlation aspects

Ming-Chiang Chung, Guangyu Huang, Ian P. McCulloch +1 more

Using machine learning (ML) to recognize different phases of matter and to infer the entire phase diagram has proven to be an effective tool given a large dataset. In our previous proposals, we have successfully explored phase transitions for topological phases of matter at low dimensions either in a supervised or an unsupervised learning protocol with the assistance of quantum-information-related quantities. In this work, we adopt our previous ML procedures to study quantum phase transitions of magnetism systems such as the $XY$ and $XXZ$ spin chains by using spin-spin correlation functions as the input data. We find that our proposed approach not only maps out the phase diagrams with accurate phase boundaries, but also indicates some features that have not been observed in the field of machine learning before. In particular, we define so-called relevant correlation functions to some corresponding phases that can always distinguish between those and their neighbors. Based on the unsupervised learning protocol we proposed [Phys. Rev. B 104, 165108 (2021)], the reduced latent representations of the inputs combined with the clustering algorithm show the connectedness or disconnectedness between neighboring clusters (phases) just corresponding to the continuous or disrupt quantum phase transition, respectively. This property reminds us of the behavior of order parameters. Moreover, in the silhouette analysis we show that the ferromagnetic states in the $XXZ$ model with various anisotropy parameters correspond to almost the same silhouette value, while the critical or antiferromagnetic states behave quite differently. The analysis further indicates that the minima of silhouette values are close to the phase-transition points, showing strong positive correlation. These results again justify the usefulness of our proposed ML procedures, and they move us a step further toward understanding the relation between ML and quantum phase transitions from correlation function aspects.

arXiv:2301.06669•Physical review. B./Physical review. B•Jun 29, 2023•6 citations

Spin nematics correlations in bilinear-biquadratic S=1 spin chains

Andreas Martin Läuchli, Guido Schmid, Simon Trebst

We present an extensive numerical study of spin quadrupolar correlations in single and coupled bilinear-biquadratic spin one chains, using several methods such as exact diagonalization, density matrix renormalization group, and strong coupling series expansions. For the single chain, we clarify the dominant correlation function in the enigmatic gapless period-three phase for $\ensuremath{\theta}∊(\ensuremath{\pi}∕4,\ensuremath{\pi}∕2)$, which is of spin quadrupolar nature with a period three spatial structure. Then we revisit the open problem of the possible existence of a ferroquadrupolar phase between the dimerized and the ferromagnetic phases. Although an extended critical region is in principle compatible with the numerical results, a scenario with a huge crossover scale is more plausible. Finally we study the fate of the dimerized phase upon coupling two chains in a ladder geometry. The dimerized phase rapidly vanishes and an extended gapped phase takes over. This gapped phase presumably has dominant short-ranged ferroquadrupolar correlations for $\ensuremath{\theta}∊(\ensuremath{-}3\ensuremath{\pi},4,\ensuremath{-}\ensuremath{\pi}∕2)$ and---surprisingly---seems to be adiabatically connected to the plaquette single solid phase of the Heisenberg $S=1$ ladder and therefore also with the Haldane phase of isolated chains.

arXiv:cond-mat/0311082•Physical Review B•Oct 26, 2006•180 citations

Autoencoder-based learning of Quantum phase transitions in the two-component Bose-Hubbard model

Iftekher S. Chowdhury, Binay Prakash Akhouri, Haque, Shah +1 more

This paper investigates the use of autoencoders and machine learning methods for detecting and analyzing quantum phase transitions in the Two-Component Bose-Hubbard Model. By leveraging deep learning models such as autoencoders, we investigate latent space representations, reconstruction error analysis, and cluster distance calculations to identify phase boundaries and critical points. The study is supplemented by dimensionality reduction techniques such as PCA and t-SNE for latent space visualization. The results demonstrate the potential of autoencoders to describe the dynamics of quantum phase transitions.

arXiv:2409.18480•Sep 27, 2024•0 citations

Deep learning of topological phase transitions from entanglement aspects: An unsupervised way

Yuan-Hong Tsai, Kuo-Feng Chiu, Yong-Cheng Lai +5 more

Machine learning techniques have been shown to be effective to recognize different phases of matter and produce phase diagrams in the parameter space of interest, while they usually require prior labeled data to perform well. Here, we propose a machine learning procedure, mainly in an unsupervised manner, which can first identify topological/nontopological phases and then refine the locations of phase boundaries. By following this proposed procedure, we expand our previous work on the one-dimensional $p$-wave superconductor [Tsai et al., Phys. Rev. B 102, 054512 (2020)] and further on the Su-Schrieffer-Heeger model, with an emphasis on using the quantum-entanglement-based quantities as the input features. We find that our method not only reproduces similar results to the previous work with sharp phase boundaries but importantly it also does not rely on prior knowledge of the phase space, e.g., the number of phases present. We conclude with a few remarks about its potential, limitations, and explanatory power.

arXiv:2105.03870•Physical review. B./Physical review. B•Oct 5, 2021•17 citations

Machine Learning-Driven Prediction of Phase Transitions in Advanced Condensed Matter Systems: A Review of Methods, Applications, and Open Challenges

Amit Ghas et al.

The identification and characterization of phase transitions is one of the oldest and most central problems in condensed matter physics, traditionally approached through order parameters, free-energy expansions, renormalization-group analysis, and direct numerical simulation. Over the past decade, machine learning (ML) has emerged as a complementary and, in some regimes, transformative methodology for this task. Supervised classifiers can learn to distinguish phases directly from raw configurations without an a priori order parameter; unsupervised and self-supervised techniques such as principal component analysis, autoencoders, and learning-by-confusion can locate transition points with minimal or no labeled data; and generative and variational architectures, including neural-network quantum states, now approximate ground-state wavefunctions of strongly correlated systems well beyond the reach of exact diagonalization. This paper reviews the theoretical foundations of phase transitions relevant to machine-learning applications, surveys the principal families of algorithms used to detect and classify phases, and examines their application across classical spin systems, percolation, topological and quantum phases, strongly correlated electron systems, and experimentally measured spectroscopic data. Representative case studies are discussed, including convolutional and graph-neural-network classifiers of lattice configurations, explainable-ML detection of thermodynamic transitions in photoemission spectra, and deep-learning-guided discovery of high-temperature superconductors. The review closes with a critical discussion of persistent challenges interpretability, data scarcity in correlated-electron experiments, generalization across Hamiltonians, and the reliability of learned order parameters near criticality and outlines directions for future research, including physics-informed architectures, uncertainty quantification, and closed-loop integration with autonomous experimentation.

The International Journal of Multiphysics•Dec 31, 2021•0 citations

Density matrix renormalization group study of the correlation function of the bilinear-biquadratic spin-1 chain

Robert J. Bursill, Tao Xiang, G A Gehring

Using the recently developed density matrix renormalization group approach, we study the correlation function of the spin-1 chain with quadratic and biquadratic interactions. This allows us to define and calculate the periodicity of the ground state which differs markedly from that in the classical analogue. Combining our results with other studies, we predict three phases in the region where the quadratic and biquadratic terms are both positive.

arXiv:cond-mat/9406076•Journal of Physics A Mathematical and General•Apr 21, 1995•57 citations
We investigate the role of different correlation functions in reconstructing the phase diagram of the spin-1 bilinear-biquadratic chain, parametrized by the angle θθθ between the bilinear and biquadratic couplings, using our previously developed label-free machine-learning pipeline \cite{Ming21, Ming23}, in which an autoencoder compresses iDMRG correlation matrices into latent features and K-means clustering with silhouette analysis groups them into phases. Without any prior labels, the scheme reveals four natural phases, the Haldane, dimerized, critical trimerized, and ferromagnetic phases, and accurately locates the Takhtajan-Babujian (θ=−π/4θ=-π/4θ=−π/4), SU(3)SU(3)SU(3)-symmetric Uimin-Lai-Sutherland (θ=π/4θ=π/4θ=π/4), and Ferro-Dimer (θ=5π/4θ=5π/4θ=5π/4) transition points; a shallow supervised refinement further sharpens the phase boundaries. We find that different correlation functions play fundamentally different roles in phase identification. The SU(3)SU(3)SU(3)-symmetric correlation alone selects a four-cluster partition that can be associated with the four phases, whereas the spin-spin and dimer-dimer correlations must be combined to recover the complete phase diagram. In contrast, the trimer-trimer correlation behaves as a marginal correlation: it is insufficient for phase identification by itself but improves the precision of the predicted phase boundaries when combined with the other correlations. Furthermore, the latent representations and silhouette values reveal signatures that we relate to the θθθ dependence of the correlations within each phase and, tentatively, to the underlying renormalization-group structure. These results demonstrate that machine learning can not only identify quantum phases but also uncover the physical information encoded in different correlation functions.
Using machine learning (ML) to recognize different phases of matter and to infer the entire phase diagram has proven to be an effective tool given a large dataset. In our previous proposals, we have successfully explored phase transitions for topological phases of matter at low dimensions either in a supervised or an unsupervised learning protocol with the assistance of quantum-information-related quantities. In this work, we adopt our previous ML procedures to study quantum phase transitions of magnetism systems such as the XYXYXY and XXZXXZXXZ spin chains by using spin-spin correlation functions as the input data. We find that our proposed approach not only maps out the phase diagrams with accurate phase boundaries, but also indicates some features that have not been observed in the field of machine learning before. In particular, we define so-called relevant correlation functions to some corresponding phases that can always distinguish between those and their neighbors. Based on the unsupervised learning protocol we proposed [Phys. Rev. B 104, 165108 (2021)], the reduced latent representations of the inputs combined with the clustering algorithm show the connectedness or disconnectedness between neighboring clusters (phases) just corresponding to the continuous or disrupt quantum phase transition, respectively. This property reminds us of the behavior of order parameters. Moreover, in the silhouette analysis we show that the ferromagnetic states in the XXZXXZXXZ model with various anisotropy parameters correspond to almost the same silhouette value, while the critical or antiferromagnetic states behave quite differently. The analysis further indicates that the minima of silhouette values are close to the phase-transition points, showing strong positive correlation. These results again justify the usefulness of our proposed ML procedures, and they move us a step further toward understanding the relation between ML and quantum phase transitions from correlation function aspects.
We present an extensive numerical study of spin quadrupolar correlations in single and coupled bilinear-biquadratic spin one chains, using several methods such as exact diagonalization, density matrix renormalization group, and strong coupling series expansions. For the single chain, we clarify the dominant correlation function in the enigmatic gapless period-three phase for $\ensuremath{\theta}∊(\ensuremath{\pi}∕4,\ensuremath{\pi}∕2)$, which is of spin quadrupolar nature with a period three spatial structure. Then we revisit the open problem of the possible existence of a ferroquadrupolar phase between the dimerized and the ferromagnetic phases. Although an extended critical region is in principle compatible with the numerical results, a scenario with a huge crossover scale is more plausible. Finally we study the fate of the dimerized phase upon coupling two chains in a ladder geometry. The dimerized phase rapidly vanishes and an extended gapped phase takes over. This gapped phase presumably has dominant short-ranged ferroquadrupolar correlations for $\ensuremath{\theta}∊(\ensuremath{-}3\ensuremath{\pi},4,\ensuremath{-}\ensuremath{\pi}∕2)$ and---surprisingly---seems to be adiabatically connected to the plaquette single solid phase of the Heisenberg S=1S=1S=1 ladder and therefore also with the Haldane phase of isolated chains.
This paper investigates the use of autoencoders and machine learning methods for detecting and analyzing quantum phase transitions in the Two-Component Bose-Hubbard Model. By leveraging deep learning models such as autoencoders, we investigate latent space representations, reconstruction error analysis, and cluster distance calculations to identify phase boundaries and critical points. The study is supplemented by dimensionality reduction techniques such as PCA and t-SNE for latent space visualization. The results demonstrate the potential of autoencoders to describe the dynamics of quantum phase transitions.
Machine learning techniques have been shown to be effective to recognize different phases of matter and produce phase diagrams in the parameter space of interest, while they usually require prior labeled data to perform well. Here, we propose a machine learning procedure, mainly in an unsupervised manner, which can first identify topological/nontopological phases and then refine the locations of phase boundaries. By following this proposed procedure, we expand our previous work on the one-dimensional ppp-wave superconductor [Tsai et al., Phys. Rev. B 102, 054512 (2020)] and further on the Su-Schrieffer-Heeger model, with an emphasis on using the quantum-entanglement-based quantities as the input features. We find that our method not only reproduces similar results to the previous work with sharp phase boundaries but importantly it also does not rely on prior knowledge of the phase space, e.g., the number of phases present. We conclude with a few remarks about its potential, limitations, and explanatory power.
The identification and characterization of phase transitions is one of the oldest and most central problems in condensed matter physics, traditionally approached through order parameters, free-energy expansions, renormalization-group analysis, and direct numerical simulation. Over the past decade, machine learning (ML) has emerged as a complementary and, in some regimes, transformative methodology for this task. Supervised classifiers can learn to distinguish phases directly from raw configurations without an a priori order parameter; unsupervised and self-supervised techniques such as principal component analysis, autoencoders, and learning-by-confusion can locate transition points with minimal or no labeled data; and generative and variational architectures, including neural-network quantum states, now approximate ground-state wavefunctions of strongly correlated systems well beyond the reach of exact diagonalization. This paper reviews the theoretical foundations of phase transitions relevant to machine-learning applications, surveys the principal families of algorithms used to detect and classify phases, and examines their application across classical spin systems, percolation, topological and quantum phases, strongly correlated electron systems, and experimentally measured spectroscopic data. Representative case studies are discussed, including convolutional and graph-neural-network classifiers of lattice configurations, explainable-ML detection of thermodynamic transitions in photoemission spectra, and deep-learning-guided discovery of high-temperature superconductors. The review closes with a critical discussion of persistent challenges interpretability, data scarcity in correlated-electron experiments, generalization across Hamiltonians, and the reliability of learned order parameters near criticality and outlines directions for future research, including physics-informed architectures, uncertainty quantification, and closed-loop integration with autonomous experimentation.