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arXiv:2001.01915

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

Jian Cheng Yang

Published Jan 7, 2020• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2001.01915

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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.
The PH-Pfaffian state having 1/2 central charge is consistent with the thermal Hall conductance measurement of 5/2 fractional quantum Hall system, but lacks support from the existing numerical results. In this paper we propose a new state described by a wavefunction obtained by multiplying a Hafnian factor to a PH-Pfaffian wavefunction. We call this new state the Hafnian PH-Pfaffian state. In spherical geometry, the Hafnian PH-Pfaffian state has the same magnetic flux number as the Pfaffian state, allowing a direct numerical comparison between the two states. Results of exact diagonalization of finite systems in the second Landau level show that the overlap of the exact ground state with the Hafnian PH-Pfaffian state exceeds that with the Pfaffian state when the short range component of the Coulomb interaction increases to a certain level, lending a numerical support to the Hafnian PH-Pfaffian state. We further show the Hafnian PH-Pfaffian state is mathematically identical to the newly proposed compressed PH-Pfaffian state [arxiv: 2001.01915 (2020)] formed 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. As the result we argue that the Hafnian PH-Pfaffian state has the same central charge as the PH-Pfaffian state, and is therefore consistent with the thermal Hall conductance measurement. Finally we present numerical results on two new wavefunctions formed by increasing the relative angular momentum by two for each of the paired composite fermions of the PH-Pfaffian state.
We numerically assess model wave functions for the recently proposed particle-hole-symmetric Pfaffian (``PH-Pfaffian'') topological order, a phase consistent with the recently reported thermal Hall conductance [M. Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $\ensuremath{\nu}=5/2$ quantum Hall plateau. We find that the most natural Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $\ensuremath{\nu}=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak ppp-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.
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.
In this work we propose a parton state as a candidate state to describe the fractional quantum Hall effect in the half-filled second Landau level. The wave function for this parton state is ${\mathcal{P}}_{\mathrm{LLL}}{\mathrm{\ensuremath{\Phi}}}_{1}^{3}{[{\mathrm{\ensuremath{\Phi}}}_{2}^{*}]}^{2}\ensuremath{\sim}{\mathrm{\ensuremath{\Psi}}}_{2/3}^{2}/{\mathrm{\ensuremath{\Phi}}}_{1}$ and in the spherical geometry it occurs at the same flux as the anti-Pfaffian state. This state has a good overlap with the anti-Pfaffian state and with the ground state obtained by exact diagonalization, using the second Landau level Coulomb interaction pseudopotentials for an ordinary semiconductor such as GaAs. By calculating the entanglement spectrum we show that this state lies in the same phase as the anti-Pfaffian state. A major advantage of this parton state is that its wave function can be evaluated for large systems, which makes it amenable to variational calculations. In the Appendix of this work, we have numerically assessed the validity of another candidate state at filling factor $\ensuremath{\nu}=\frac{5}{2}$, namely, the particle-hole-symmetric Pfaffian (PH-Pfaffian) state. We find that the proposed candidate wave function for the PH-Pfaffian state is particle-hole symmetric to a high degree but it does not appear to arise as the ground state of any simple Hamiltonian with two-body interactions.
Motivated by the experimental observation of a quantized 5/2 thermal conductance at filling $\ensuremath{\nu}=5/2$, a result incompatible with both the Pfaffian and the anti-Pfaffian states, we have pushed the expansion of the effective Hamiltonian of the 5/25/25/2-quantized Hall state to third order in the parameter $\ensuremath{\kappa}={E}_{C}/\ensuremath{\hbar}{\ensuremath{\omega}}_{c}\ensuremath{\propto}1/\sqrt{B}$ controlling the Landau-level mixing, where EC{E}_{C}EC​ is the Coulomb energy and ${\ensuremath{\omega}}_{c}$ is the cyclotron frequency. Exact diagonalizations of this effective Hamiltonian show that the difference in overlap with the Pfaffian state and the anti-Pfaffian state induced at second order is reduced by third-order corrections and disappears around $\ensuremath{\kappa}=0.4$, suggesting that these states are much closer in energy at smaller magnetic field than previously anticipated. Furthermore, we show that in this range of $\ensuremath{\kappa}$ the finite-size spectrum is typical of a quantum phase transition, with a strong reduction of the energy gap and with level crossings between excited states. These results point to the possibility of a quantum phase transition at smaller magnetic field into a phase with an emergent particle-hole symmetry that would explain the measured 5/25/25/2 thermal conductance of the 5/25/25/2-quantized Hall state.
The $\ensuremath{\nu}=5/2$ fractional quantum Hall effect is a system of intense experimental and theoretical interest as its ground state may host non-Abelian excitations, but the exact nature of the ground state is still undetermined. We present the results of an exact diagonalization study of an electron system in the disk configuration, including the effects of Landau level (LL) mixing and the finite thickness of the quantum well confining the electrons. The degeneracy between the two leading candidates for the ground state, the Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing background, in addition to the inclusion of two- and three-body interactions via LL mixing. As a result of the neutralizing background in the disk configuration, there is a phase transition from the anti-Pfaffian to the Pfaffian as LL mixing is turned on, in stark contrast to what is observed in spherical geometry. This behavior is in agreement with existing experiments, showing the appearance of the Pfaffian state at strong LL mixing before the system enters a compressible phase. The inclusion of LL mixing leads to an increased charge e/4e/4e/4 quasihole size. LL mixing interactions are also shown to overcome the effects of edge reconstruction. Due to finite thickness effects, these properties are enhanced dramatically. We also find that only the Pfaffian and anti-Pfaffian states continue to possess energy gaps at finite width, while gaps for compressible stripe states close, which is in agreement with available experimental data.