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arXiv:2210.04925 · Physical review. B./Physical review. B

Possible restoration of particle-hole symmetry in the 5/2-quantized Hall state at small magnetic field

Loïc Herviou, Frédéric Mila

Published Mar 17, 2023• 8 citations
arXivHTMLPDF

Abstract

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 $5/2$-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 ${E}_{C}$ 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 $5/2$ thermal conductance of the $5/2$-quantized Hall state.

DOI
10.1103/physrevb.107.115137

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Motivated by the experimental observation of a quantized 5/2 thermal conductance at filling \ensuremathν=5/2\ensuremath{\nu}=5/2\ensuremathν=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 \ensuremathκ=EC/\ensuremathℏ\ensuremathωc\ensuremath∝1/B\ensuremath{\kappa}={E}_{C}/\ensuremath{\hbar}{\ensuremath{\omega}}_{c}\ensuremath{\propto}1/\sqrt{B}\ensuremathκ=EC​/\ensuremathℏ\ensuremathωc​\ensuremath∝1/B​ controlling the Landau-level mixing, where EC{E}_{C}EC​ is the Coulomb energy and \ensuremathωc{\ensuremath{\omega}}_{c}\ensuremathωc​ 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 \ensuremathκ=0.4\ensuremath{\kappa}=0.4\ensuremathκ=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 \ensuremathκ\ensuremath{\kappa}\ensuremathκ 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.
Phase Diagram of the ν=5/2\nu=5/2ν=5/2 Fractional Quantum Hall Effect: Effects of Landau Level Mixing and Non-Zero Width
Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the anti-Pfaffian, offer candidate descriptions of the ν=5/2\nu = 5/2ν=5/2 fractional quantum Hall state. But the significant controversy surrounding the nature of the ν=5/2\nu = 5/2ν=5/2 state has been hampered by the fact that the competition between these and other states is affected by small parameter changes. To study the phase diagram of the ν=5/2\nu = 5/2ν=5/2 state we numerically diagonalize a comprehensive effective Hamiltonian describing the fractional quantum Hall effect of electrons under realistic conditions in GaAs semiconductors. The effective Hamiltonian takes Landau level mixing into account to lowest-order perturbatively in κ\kappaκ, the ratio of the Coulomb energy scale to the cyclotron gap. We also incorporate non-zero width www of the quantum well and sub-band mixing. We find the ground state in both the torus and spherical geometries as a function of κ\kappaκ and www. To sort out the non-trivial competition between candidate ground states we analyze the following 4 criteria: its overlap with trial wave functions; the magnitude of energy gaps; the sign of the expectation value of an order parameter for particle-hole symmetry breaking; and the entanglement spectrum. We conclude that the ground state is in the universality class of the Moore-Read Pfaffian state, rather than the anti-Pfaffian, for κ<κc(w)\kappa < {\kappa_c}(w)κ<κc​(w), where κc(w){\kappa_c}(w)κc​(w) is a www-dependent critical value 0.6≲κc(w)≲10.6 \lesssim{\kappa_c}(w)\lesssim 10.6≲κc​(w)≲1. We observe that both Landau level mixing and non-zero width suppress the excitation gap, but Landau level mixing has a larger effect in this regard. Our findings have important implications for the identification of non-Abelian fractional quantum Hall states.
Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this Letter, perturbation theory is applied to finite sizes without bias to any specific pseudopotential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in previous studies, reverts the ground state at 5/2 to the anti-Pfaffian.
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.
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.
Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this Letter, perturbation theory is applied to finite sizes without bias to any specific pseudopotential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in previous studies, reverts the ground state at 5/2 to the anti-Pfaffian.
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.