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:2104.03574 · Physical review. B./Physical review. B

Deep learning of topological phase transitions from the point of view of entanglement for two-dimensional chiral p -wave superconductors

Ming-Chiang Chung, Tsung-Pao Cheng, Guang-Yu Huang, Yuan-Hong Tsai

Published Jul 22, 2021• 6 citations
arXivHTMLPDF

Abstract

Applying deep learning to investigate topological phase transitions (TPTs) becomes a useful method due to not only its ability to recognize patterns but also its statistical excellency to examine the amount of information carried by different types of data inputs. Among possible data types, entanglement-related quantities, such as Majorana correlation matrices (MCMs), one-particle entanglement spectra (OPES), and entanglement eigenvectors (OPEEs), have been proved effective, however, are to date mostly restricted to one dimension. Here, we propose practical input data forms based on those quantities to study TPTs and to compare the efficiency of each form on classic two-dimensional chiral $p$-wave superconductors via the deep learning approach. First, we find that different input forms, either matrices or tensors both originated from real MCMs, can affect the precise locations of the predicted transition points. Next, due to the complex nature of OPEEs, we extract three spatially dependent quantities from OPEEs, one related to the ``intensity,'' and the other two related to ``phases'' of particle and hole components. We show that similar to taking OPES directly as inputs, solely using ``intensity'' quantity can only distinguish topological phases from trivial ones, whereas using either whole MCMs or complete OPEE-extracted quantities can provide sufficient information for deep learning to distinguish between phases of matter with different $U(1)$ gauges or Chern numbers. Finally, we discuss certain characteristic features in the deep learning approach and, in particular, they reveal that our trained models indeed learn physically meaningful features, which confirms the potential use even at high dimensions.

DOI
10.1103/physrevb.104.024506

Related papers

Deep learning of topological phase transitions from entanglement aspects

Yuan-Hong Tsai, Meng-Zhe Yu, Yu-Hao Hsu +1 more

The one-dimensional $p$-wave superconductor proposed by Kitaev has long been a classic example for understanding topological phase transitions through various methods, such as examining the Berry phase, edge states of open chains, and, in particular, aspects from quantum entanglement of ground states. In order to understand the amount of information carried in the entanglement-related quantities, here we study topological phase transitions of the model with emphasis of using the deep learning approach. We feed different quantities, including Majorana correlation matrices (MCMs), entanglement spectra (ES) or entanglement eigenvectors (EE) originating from Block correlation matrices, into the deep neural networks for training, and investigate which one could be the most useful input format in this approach. We find that ES is information that is too compressed compared to MCM or EE. MCM and EE can provide us abundant information to recognize not only the topological phase transitions in the model but also phases of matter with different $U$(1) gauges, which is not reachable by using ES only.

arXiv:1909.04784•Physical review. B./Physical review. B•Aug 18, 2020•27 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

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

Entanglement modes and topological phase transitions in superconductors

T. P. Oliveira, Pedro D Sacramento

Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional $p$-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the $k$-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the $k$-space fidelity operator.

arXiv:1503.05154•Physical Review B•Mar 13, 2014•22 citations

Quaternion-based machine learning on topological quantum systems

Min-Ruei Lin, Wan-Ju Li, Shin-Ming Huang

Abstract Topological phase classifications have been intensively studied via machine-learning techniques where different forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the complexity in quantum physics, advanced mathematical architecture should be considered in designing machines. In this work, we incorporate quaternion algebras into data analysis either in the frame of supervised and unsupervised learning to classify two-dimensional Chern insulators. For the unsupervised-learning aspect, we apply the principal component analysis on the quaternion-transformed eigenstates to distinguish topological phases. For the supervised-learning aspect, we construct our machine by adding one quaternion convolutional layer on top of a conventional convolutional neural network. The machine takes quaternion-transformed configurations as inputs and successfully classify all distinct topological phases, even for those states that have different distributions from those states seen by the machine during the training process. Our work demonstrates the power of quaternion algebras on extracting crucial features from the targeted data and the advantages of quaternion-based neural networks than conventional ones in the tasks of topological phase classifications.

arXiv:2209.14551•Machine Learning Science and Technology•Mar 1, 2023•4 citations

Deep learning of phase transitions with minimal examples

A. M. Abuali, David A. Clarke, Morten Hjorth-Jensen +3 more

Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.

arXiv:2501.05547•Physical review. E•Aug 20, 2025•6 citations
Applying deep learning to investigate topological phase transitions (TPTs) becomes a useful method due to not only its ability to recognize patterns but also its statistical excellency to examine the amount of information carried by different types of data inputs. Among possible data types, entanglement-related quantities, such as Majorana correlation matrices (MCMs), one-particle entanglement spectra (OPES), and entanglement eigenvectors (OPEEs), have been proved effective, however, are to date mostly restricted to one dimension. Here, we propose practical input data forms based on those quantities to study TPTs and to compare the efficiency of each form on classic two-dimensional chiral ppp-wave superconductors via the deep learning approach. First, we find that different input forms, either matrices or tensors both originated from real MCMs, can affect the precise locations of the predicted transition points. Next, due to the complex nature of OPEEs, we extract three spatially dependent quantities from OPEEs, one related to the intensity,'' and the other two related to phases'' of particle and hole components. We show that similar to taking OPES directly as inputs, solely using ``intensity'' quantity can only distinguish topological phases from trivial ones, whereas using either whole MCMs or complete OPEE-extracted quantities can provide sufficient information for deep learning to distinguish between phases of matter with different U(1)U(1)U(1) gauges or Chern numbers. Finally, we discuss certain characteristic features in the deep learning approach and, in particular, they reveal that our trained models indeed learn physically meaningful features, which confirms the potential use even at high dimensions.
The one-dimensional ppp-wave superconductor proposed by Kitaev has long been a classic example for understanding topological phase transitions through various methods, such as examining the Berry phase, edge states of open chains, and, in particular, aspects from quantum entanglement of ground states. In order to understand the amount of information carried in the entanglement-related quantities, here we study topological phase transitions of the model with emphasis of using the deep learning approach. We feed different quantities, including Majorana correlation matrices (MCMs), entanglement spectra (ES) or entanglement eigenvectors (EE) originating from Block correlation matrices, into the deep neural networks for training, and investigate which one could be the most useful input format in this approach. We find that ES is information that is too compressed compared to MCM or EE. MCM and EE can provide us abundant information to recognize not only the topological phase transitions in the model but also phases of matter with different UUU(1) gauges, which is not reachable by using ES only.
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.
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.
Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional ppp-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the kkk-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the kkk-space fidelity operator.
Abstract Topological phase classifications have been intensively studied via machine-learning techniques where different forms of the training data are proposed in order to maximize the information extracted from the systems of interests. Due to the complexity in quantum physics, advanced mathematical architecture should be considered in designing machines. In this work, we incorporate quaternion algebras into data analysis either in the frame of supervised and unsupervised learning to classify two-dimensional Chern insulators. For the unsupervised-learning aspect, we apply the principal component analysis on the quaternion-transformed eigenstates to distinguish topological phases. For the supervised-learning aspect, we construct our machine by adding one quaternion convolutional layer on top of a conventional convolutional neural network. The machine takes quaternion-transformed configurations as inputs and successfully classify all distinct topological phases, even for those states that have different distributions from those states seen by the machine during the training process. Our work demonstrates the power of quaternion algebras on extracting crucial features from the targeted data and the advantages of quaternion-based neural networks than conventional ones in the tasks of topological phase classifications.
Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.