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Chinese Physics B

Phase transition extracted by principal component analysis in the disordered Moore–Read state

Na 娜 Jiang 江, Shuaixin 帅鑫 Fu 付, Zhengzhi 正直 Ma 马, L. M. Wang

Published Feb 5, 2025• 1 citation
Publisher

Abstract

Abstract We study the influence of disorder on the Moore–Read state by principal component analysis (PCA), which is one of the ground state candidates for the 5/2 fractional Hall state. By using PCA, the topological features of the ground state wave functions with different disorder strengths can be distilled. As the disorder strength increases, the Moore–Read state will be destroyed. We explore the phase transition by analyzing the overlaps between the random sample wave functions and the topologically distilled state. The cross-point between the amplitudes of the principal component and its counterpart is the phase transition point. Additionally, the origin of the second component comes from the excited states, which is different from the Laughlin state.

DOI
10.1088/1674-1056/adb25f

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Abstract We study the influence of disorder on the Moore–Read state by principal component analysis (PCA), which is one of the ground state candidates for the 5/2 fractional Hall state. By using PCA, the topological features of the ground state wave functions with different disorder strengths can be distilled. As the disorder strength increases, the Moore–Read state will be destroyed. We explore the phase transition by analyzing the overlaps between the random sample wave functions and the topologically distilled state. The cross-point between the amplitudes of the principal component and its counterpart is the phase transition point. Additionally, the origin of the second component comes from the excited states, which is different from the Laughlin state.
We study the behavior of two-dimensional electron gas in the fractional quantum Hall (FQH) regime in the presence of disorder potential. The principal component analysis is applied to a set of disordered Laughlin ground state model wave function to enable us to distill the model wave function of the pure Laughlin state. With increasing the disorder strength, the ground state wave function is expected to deviate from the Laughlin state and eventually leave the FQH phase. We investigate the phase transition from the Laughlin state to a topologically trivial state by analyzing the overlap between the random sample wave functions and the distilled ground state wave function. It is proposed that the cross point of the principal component amplitude and its counterpart is the critical disorder strength, which marks the collapse of the FQH regime.
For the special single-layer fractional quantum Hall system with a filling factor of 5/2, which has an even denominator, this paper uses principal component analysis (PCA) to study its behavior under the breaking of particle-hole symmetry. By introducing a model three-body potential to represent the mechanism of particle-hole symmetry breaking, the paper finds that the 5/2 system evolves into two types of special topological quantum states with non-Abelian statistics as the strength and direction of the three-body potential vary. The transition points of these states correspond to the particle-hole symmetric pure Coulomb interaction system. Our results validate the applicability of machine learning as a new research tool in fractional quantum Hall systems. Furthermore, machine learning directly analyzes the raw wave functions, without relying on prior empirical theoretical assumptions and models, making it applicable to a broader range of fractional quantum Hall systems experiencing phase transitions due to particle-hole symmetry breaking.
= 5/2 fractional quantum Hall (FQH) state, as a special single-layer system with an even-denominator filling factor, serves as a promising platform for exploring exotic topological phases. In this study, we employ principal component analysis (PCA), an unsupervised machine learning technique, to investigate the evolution of many-body wavefunctions under particle-hole (PH) symmetry breaking. By introducing a model three-body potential to represent the mechanism of PH symmetry breaking, we demonstrate that the ground state of the system transitions continuously between two distinct non-Abelian topological states, namely the Pfaffian and the anti-Pfaffian, as the strength and direction of the three-body term vary. Notably, the critical transition point corresponds to the pure Coulomb interaction, which preserves PH symmetry. Our results demonstrate that machine learning techniques, exemplified by PCA, provide a powerful and unbiased tool to identify and characterize topological phase transitions in FQH systems. Unlike traditional approaches that rely on predefined model wavefunctions, our method analyzes raw many-body wavefunctions directly, offering a model-independent framework applicable to a broader class of FQH systems undergoing topological transitions driven by symmetry breaking.
In this work, we apply a principal component analysis (PCA) method with a kernel trick to study the classification of phases and phase transitions in classical XY models of frustrated lattices. Compared to our previous work with the linear PCA method, the kernel PCA can capture nonlinear functions. In this case, the Z 2 chiral order of the classical spins in these lattices is indeed a nonlinear function of the input spin configurations. In addition to the principal component revealed by the linear PCA, the kernel PCA can find two more principal components using the data generated by Monte Carlo simulation for various temperatures as the input. One of them is related to the strength of the U (1) order parameter, and the other directly manifests the chiral order parameter that characterizes the Z 2 symmetry breaking. For a temperature-resolved study, the temperature dependence of the principal eigenvalue associated with the Z 2 symmetry breaking clearly shows second-order phase transition behavior.
We investigate the disorder-driven phase transitions in bosonic fractional quantum Hall liquids at filling factors f=1/2f=1/2f=1/2 and f=1f=1f=1 in the lowest Landau level. We use the evolution of ground-state entanglement entropy, fidelity susceptibility, and Hall conductance with increasing disorder strength to identify the underlying phase transitions. The critical disorder strengths obtained from these different quantities are consistent with each other, validating the reliability of our numerical calculations based on exact diagonalization. At f=1/2f=1/2f=1/2, we observe a clear transition from the bosonic Laughlin state to a trivial insulating phase. At f=1f=1f=1, we identify a direct phase transition from the non-Abelian bosonic Moore-Read state to a trivial insulating phase, although some signs of a disorder-induced intermediate fractional quantum Hall phase were recently reported for the f=5/2f=5/2f=5/2 fermionic cousin.
Abstract Frustration-driven quantum fluctuation leads to many exotic phases in the ground state (GS) and the study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. We study a frustrated Heisenberg J 1 − J 2 model of spin-1/2 chain with nearest exchange interaction J 1 and next nearest exchange interaction J 2 using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSCs) of GS and first excited state (FES) for different J 2 / J 1 are used as the input in PCA to construct the covariance matrix. The ‘quantified principal component’ p 1 ( J 2 / J 1 ) of the largest eigenvalue of the covariance matrix is calculated and it is shown that the nature and variation of p 1 ( J 2 / J 1 ) can accurately predict the phase transitions and degeneracies in the GS. The p 1 ( J 2 / J 1 ) calculated from the MPSC of GS only exhibits the signature of degeneracies in the GS, whereas, p 1 ( J 2 / J 1 ) calculated from the MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that the jump in p 1 ( J 2 / J 1 ) of FES at J 2 / J 1 ≈ 0.241 , indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of the first two principal components of FES shows distinct band formation for different phases. The MPSCs are obtained by using an iterative variational method (IVM) which gives the approximate eigenvalues and eigenvectors.