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:2311.16039

Is the Anti-Pfaffian a PH-Pfaffian Topological Order State?

Jiangke Yang

Published Nov 27, 2023• 0 citations
arXivHTMLPDF

Abstract

Recently we have shown that the anti-Pfaffian state and a state described by the second Landau level (SLL) projection of antiholomorphic Pfaffian wavefunctions have large overlap and almost identical low-energy orbital entanglement structures, suggesting they have the same topological order. In this paper, we further show that the similar entanglement structure observed in the SLL projected state is also identifiable in other Landau level projected states, indicating it is not the result of the SLL projection but rather a "fingerprint" of the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions. Consequently, we argue that the SLL projected state, and therefore the anti-Pfaffian state, is a PH-Pfaffian topological order state. We also discuss the implications on the edge physics.

DOI
10.48550/arxiv.2311.16039

Related papers

Second Landau level projection of antiholomorphic Pfaffian wave functions

Jiangke Yang

We propose a state described by the second Landau level (SLL) projection of a generalized Moore-Read Pfaffian wave function with an antiholomorphic pairing component. Unlike the PH-Pfaffian state which is described by the lowest Landau level (LLL) projection of the same antiholomorphic Pfaffian wave function, we find the proposed state, which we call the SLL PH-Pfaffian, represents a gapped, incompressible phase, and provides an excellent description for the exact ground state of finite-size SLL Coulomb-interacting electrons over a range of the short-distance interaction strength. We also find the SLL PH-Pfaffian, when mapped to the LLL, has a very large overlap and the same low-energy orbital entanglement spectrum structures with the anti-Pfaffian state. We discuss possible interpretations of our results on the nature of the topological order of both states.

arXiv:2302.07389•Physical review. B./Physical review. B•May 15, 2023•0 citations

Mapping the Second Landau Level PH-Pfaffian State to the Lowest Landau Level

Jiangke Yang

Recently we proposed a state described by the second Landau level (SLL) projection of the antiholomorphic Pfaffian wavefunction as a candidate for the ground state of the 5/2 fractional quantum Hall effect. In this paper we provide a mathematical proof that, when mapped to the lowest Landau level (LLL), the aforementioned state, which we call the SLL PH-Pfaffian state, becomes exactly a LLL projected orbital angular momentum l = -3 pairing Pfaffian state, which we call the LLL antiholomorphic f-wave pairing Pfaffian state, or the LLL af-Pfaffian state for short. We further prove that there is also an exact mapping between the upstream neutral Majorana fermion edge mode of the SLL PH-Pfaffian state and that of the LLL af-Pfaffian state. However, we find that while there exists an upstream neutral boson edge mode in the LLL af-Pfaffian state, it has no mapped counterpart in the SLL PH-Pfaffian state.

arXiv:2503.22940•Mar 29, 2025•0 citations

Anomalous Reentrant 5/2 Quantum Hall Phase at Moderate Landau-Level-Mixing Strength

Sudipto Das, Sahana Das, Sudhansu S. Mandal

A successful probing of the neutral Majorana mode in recent thermal Hall conductivity measurements opines in favor of the particle-hole symmetric Pfaffian (PH-Pf) topological order, contrasting the theoretical predictions of Pfaffian or anti-Pfaffian phases. Here we report a reentrant anomalous quantized phase that is found to be gapped in the thermodynamic limit, distinct from the conventional Pfaffian, anti-Pfaffian, or PH-Pf phases, at an intermediate strength of Landau level mixing. Our proposed wave function consistent with the PH-Pf shift in spherical geometry rightly captures the topological order of this phase, as its overlap with the exact ground state is very high and it reproduces low-lying entanglement spectra. A unique topological order, irrespective of the flux shifts, found for this phase, possibly corroborates the experimentally found topological order.

arXiv:2206.04419•Physical Review Letters•Aug 3, 2023•9 citations

Investigating the topological order of an ansatz for the fractional quantum Hall effect in the half-filled second Landau level

John J. McCord

The Moore-Read Pfaffian and anti-Pfaffian states have been under scrupulous review as candidates which describe the fractional quantum Hall effect at filling factor 5/2. Quantum states in the universality class of the Moore-Read Pfaffian/anti-Pfaffian have non-trivial intrinsic topological order and support low-energy non-Abelian excitations that have applications in fault-tolerant topological quantum computing schemes. Both states are exact ground states of three-body Hamiltonians that explicitly break particle-hole symmetry. We study the topological order of a competing ansatz state &PSgr;2 that is the exact ground state of a two-body Hamiltonian that preserves particle-hole symmetry. In particular, we calculate the bipartite entanglement entropy and spectra in the lowest Landau level in the spherical geometry for &PSgr; 2. We perform such calculations for a finite number of electrons up to 14. We then extrapolate to the thermodynamic limit the topological entanglement entropy γ as a measure of the topological order of the ansatz and compare to the known value of the Moore-Read Pfaffian/anti-Pfaffian state. We also study the orbital entanglement spectra for &PSgr;2 and compare with the Moore-Read Pfaffian and two-body Coulomb ground states. We show that our extrapolation of γ lies within the uncertainty of the known value of γ for the Moore-Read Pfaffian state, and that the orbital entanglement spectra of &PSgr;2 assumes a similar structure to that of the two-body Coulomb interaction.

•Jan 1, 2016•0 citations

Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry

Philip Zucker, Dima E. Feldman

Numerical results suggest that the quantum Hall effect at ν=5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau-level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau-level mixing are strong. We show that the recent proposal of a particle-hole (PH)-Pfaffian topological order by Son is consistent with all experiments. The absence of particle-hole symmetry at ν=5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.

arXiv:1603.03754•Physical Review Letters•Aug 22, 2016•109 citations

A Compressed Particle-Hole Symmetric Pfaffian State for $ν= 5/2$ Quantum Hall Effect

Jian Cheng Yang

A recent thermal Hall conductance experiment [Banerjee et al., Nature {\bf559}, 205 (2018)] for $ν= 5/2$ fractional quantum Hall system appears to rule out both the Pfaffian and anti-Pfaffian and be in favor of the PH-Pfaffian topological order, while the existing numerical results without disorder have shown otherwise. In this paper we offer a possible resolution by proposing a new state, termed compressed PH-Pfaffian state by "compressing" the PH-Pfaffian state with two flux quanta removed to create two abelian Laughlin type quasiparticles of the maximum avoidance from one another (or of the maximum number of zeros). The compressed PH-Pfaffian state is not particle-hole symmetric but possesses the PH-Pfaffian topological order. In spherical geometry, the compressed PH-Pfaffian state has the same magnetic flux number $N_ϕ= 2N-3$ as the Pfaffian state, allowing a direct numerical comparison between the two states. Results of exact diagonalization of finite disorder-free systems in the second Landau level show that, by increasing the short range component of the Coulomb interaction, the ground state undergoes a phase transition from the Pfaffian state to the compressed PH-Pfaffian state before further entering into a gapless state. The low energy gapped excited states result from the breakup of the abelian Laughlin type quasiparticle into two non-abelian quasiparticles.

arXiv:2001.01915•Jan 7, 2020•0 citations
Recently we have shown that the anti-Pfaffian state and a state described by the second Landau level (SLL) projection of antiholomorphic Pfaffian wavefunctions have large overlap and almost identical low-energy orbital entanglement structures, suggesting they have the same topological order. In this paper, we further show that the similar entanglement structure observed in the SLL projected state is also identifiable in other Landau level projected states, indicating it is not the result of the SLL projection but rather a "fingerprint" of the PH-Pfaffian topological order of the original unprojected antiholomorphic Pfaffian wavefunctions. Consequently, we argue that the SLL projected state, and therefore the anti-Pfaffian state, is a PH-Pfaffian topological order state. We also discuss the implications on the edge physics.
We propose a state described by the second Landau level (SLL) projection of a generalized Moore-Read Pfaffian wave function with an antiholomorphic pairing component. Unlike the PH-Pfaffian state which is described by the lowest Landau level (LLL) projection of the same antiholomorphic Pfaffian wave function, we find the proposed state, which we call the SLL PH-Pfaffian, represents a gapped, incompressible phase, and provides an excellent description for the exact ground state of finite-size SLL Coulomb-interacting electrons over a range of the short-distance interaction strength. We also find the SLL PH-Pfaffian, when mapped to the LLL, has a very large overlap and the same low-energy orbital entanglement spectrum structures with the anti-Pfaffian state. We discuss possible interpretations of our results on the nature of the topological order of both states.
Recently we proposed a state described by the second Landau level (SLL) projection of the antiholomorphic Pfaffian wavefunction as a candidate for the ground state of the 5/2 fractional quantum Hall effect. In this paper we provide a mathematical proof that, when mapped to the lowest Landau level (LLL), the aforementioned state, which we call the SLL PH-Pfaffian state, becomes exactly a LLL projected orbital angular momentum l = -3 pairing Pfaffian state, which we call the LLL antiholomorphic f-wave pairing Pfaffian state, or the LLL af-Pfaffian state for short. We further prove that there is also an exact mapping between the upstream neutral Majorana fermion edge mode of the SLL PH-Pfaffian state and that of the LLL af-Pfaffian state. However, we find that while there exists an upstream neutral boson edge mode in the LLL af-Pfaffian state, it has no mapped counterpart in the SLL PH-Pfaffian state.
A successful probing of the neutral Majorana mode in recent thermal Hall conductivity measurements opines in favor of the particle-hole symmetric Pfaffian (PH-Pf) topological order, contrasting the theoretical predictions of Pfaffian or anti-Pfaffian phases. Here we report a reentrant anomalous quantized phase that is found to be gapped in the thermodynamic limit, distinct from the conventional Pfaffian, anti-Pfaffian, or PH-Pf phases, at an intermediate strength of Landau level mixing. Our proposed wave function consistent with the PH-Pf shift in spherical geometry rightly captures the topological order of this phase, as its overlap with the exact ground state is very high and it reproduces low-lying entanglement spectra. A unique topological order, irrespective of the flux shifts, found for this phase, possibly corroborates the experimentally found topological order.
The Moore-Read Pfaffian and anti-Pfaffian states have been under scrupulous review as candidates which describe the fractional quantum Hall effect at filling factor 5/2. Quantum states in the universality class of the Moore-Read Pfaffian/anti-Pfaffian have non-trivial intrinsic topological order and support low-energy non-Abelian excitations that have applications in fault-tolerant topological quantum computing schemes. Both states are exact ground states of three-body Hamiltonians that explicitly break particle-hole symmetry. We study the topological order of a competing ansatz state &PSgr;2 that is the exact ground state of a two-body Hamiltonian that preserves particle-hole symmetry. In particular, we calculate the bipartite entanglement entropy and spectra in the lowest Landau level in the spherical geometry for &PSgr; 2. We perform such calculations for a finite number of electrons up to 14. We then extrapolate to the thermodynamic limit the topological entanglement entropy γ as a measure of the topological order of the ansatz and compare to the known value of the Moore-Read Pfaffian/anti-Pfaffian state. We also study the orbital entanglement spectra for &PSgr;2 and compare with the Moore-Read Pfaffian and two-body Coulomb ground states. We show that our extrapolation of γ lies within the uncertainty of the known value of γ for the Moore-Read Pfaffian state, and that the orbital entanglement spectra of &PSgr;2 assumes a similar structure to that of the two-body Coulomb interaction.
Numerical results suggest that the quantum Hall effect at ν=5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau-level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau-level mixing are strong. We show that the recent proposal of a particle-hole (PH)-Pfaffian topological order by Son is consistent with all experiments. The absence of particle-hole symmetry at ν=5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.
A Compressed Particle-Hole Symmetric Pfaffian State for ν=5/2ν= 5/2ν=5/2 Quantum Hall Effect
A recent thermal Hall conductance experiment [Banerjee et al., Nature {\bf559}, 205 (2018)] for ν=5/2ν= 5/2ν=5/2 fractional quantum Hall system appears to rule out both the Pfaffian and anti-Pfaffian and be in favor of the PH-Pfaffian topological order, while the existing numerical results without disorder have shown otherwise. In this paper we offer a possible resolution by proposing a new state, termed compressed PH-Pfaffian state by "compressing" the PH-Pfaffian state with two flux quanta removed to create two abelian Laughlin type quasiparticles of the maximum avoidance from one another (or of the maximum number of zeros). The compressed PH-Pfaffian state is not particle-hole symmetric but possesses the PH-Pfaffian topological order. In spherical geometry, the compressed PH-Pfaffian state has the same magnetic flux number Nϕ=2N−3N_ϕ= 2N-3Nϕ​=2N−3 as the Pfaffian state, allowing a direct numerical comparison between the two states. Results of exact diagonalization of finite disorder-free systems in the second Landau level show that, by increasing the short range component of the Coulomb interaction, the ground state undergoes a phase transition from the Pfaffian state to the compressed PH-Pfaffian state before further entering into a gapless state. The low energy gapped excited states result from the breakup of the abelian Laughlin type quasiparticle into two non-abelian quasiparticles.