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Journal of Physics Condensed Matter

Unveiling particle-hole symmetry breaking and topological transitions in the v = 5/2 fractional quantum hall state via principal component analysis

Zhe Zhang, Jiajie Qiao, Baobing Zheng, Qin Jin

Published Jan 21, 2026• 0 citations
Publisher

Abstract

= 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.

DOI
10.1088/1361-648x/ae3b85

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= 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.
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.
We report on the study of the fractional quantum Hall effect at the filling factor 5/2 using exact diagonalization method with torus geometry. The particle-hole symmetry breaking effect is considered using an additional three-body interaction. Both Pfaffian and anti-Pfaffian states can be the ground state depending on the sign of the three-body interaction. The results of the low-energy spectrum, the wave function overlap, and the particle-hole parity evolution, have shown clear evidence of a direct sharp transition (possibly first order) from the Pfaffian to the anti-Pfaffian state at the Coulomb point. A quantum phase diagram is established, where one finds further transitions from the Pfaffian or anti-Pfaffian state to the nearby compressible phases induced by a change of the pseudopotential.
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.
The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a family of incompressible paired parton FQH states at filling $\ensuremath{\nu}=\frac{1}{6}$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\ensuremath{\nu}=\frac{1}{3}$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-type anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principal existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimensions.
The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.
Fractional quantum Hall systems host strongly correlated topological phases with potential applications in topological quantum computation, yet identifying their microscopic ground states remains challenging because competing phases are often nearly degenerate and Landau-level mixing is difficult to treat. Here, we introduce a symmetry-adapted neural-network wavefunction that describes Abelian and non-Abelian states while incorporating all Landau levels. The method accurately captures composite-fermion liquids and the emergence of Wigner crystallization at low filling. When the first Landau level is partially filled, finite-size overlaps and entanglement spectra provide evidence consistent with 2ˉk1k+1\bar{2}^{k}1^{k+1}2ˉk1k+1-type parton phases at ν=4/3ν=4/3ν=4/3 and 7/57/57/5, while the ν=3/2ν=3/2ν=3/2 state exhibits anti-Pfaffian-like entanglement structure. These results demonstrate that neural-network wavefunctions offer a flexible framework for studying competing topological orders in fractional quantum Hall systems.