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arXiv:2408.08354 · Physical review. B./Physical review. B

Phase diagram of compressible and paired states in the quarter-filled Landau level

Misha Yutushui, David F. Mross

Published Jan 3, 2025• 7 citations
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Abstract

Quantum Hall plateaus at quarter fillings occur in GaAs wide quantum wells, hole-doped GaAs, and bilayer graphene. However, the interactions favoring incompressible states over compressible composite-Fermi liquids at such fillings are not well understood. We devise a method for the computation of the trial energies for Haldane pseudopotentials via Monte Carlo sampling. Applying it to the quarter-filled lowest Landau level, we find that tuning the third and fifth pseudopotential values can stabilize the anti-Pfaffian, Moore-Read, and $f$-wave states. The smallest deviations from pure Coulomb interactions are required by anti-Pfaffian, whose presence is indicated by daughter states in recent experiments of bilayer graphene at $\ensuremath{\nu}=\frac{3}{4}$.

DOI
10.1103/physrevb.111.035106

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Quantum Hall plateaus at quarter fillings occur in GaAs wide quantum wells, hole-doped GaAs, and bilayer graphene. However, the interactions favoring incompressible states over compressible composite-Fermi liquids at such fillings are not well understood. We devise a method for the computation of the trial energies for Haldane pseudopotentials via Monte Carlo sampling. Applying it to the quarter-filled lowest Landau level, we find that tuning the third and fifth pseudopotential values can stabilize the anti-Pfaffian, Moore-Read, and fff-wave states. The smallest deviations from pure Coulomb interactions are required by anti-Pfaffian, whose presence is indicated by daughter states in recent experiments of bilayer graphene at \ensuremathν=34\ensuremath{\nu}=\frac{3}{4}\ensuremathν=43​.
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.
Recent experiments in monolayer and trilayer graphene have observed incompressible even-denominator fractional quantum Hall states within narrow regions of parameter space associated with crossings of zeroth Landau levels. This is surprising because at half filling the Coulomb interaction in the zeroth Landau level is expected to favour a compressible composite Fermi liquid. Here we provide the first quantitative energetic study of the candidate incompressible phases and of the microscopic mechanisms that can stabilise them near such Landau-level crossings. Using exact diagonalization and infinite density-matrix renormalisation group, we first study ultra-short-range corrections to the Coulomb interaction arising from finite layer separation, lattice-scale effects, and electron-phonon coupling. We find that a sufficiently strong attractive short-range interaction drives the composite Fermi liquid first into a weakly paired ddd-wave pseudospin-singlet state and subsequently into a strongly paired state. However, microscopic estimates of these short-range corrections are far too small to stabilise either incompressible phase. We then incorporate Landau-level mixing through screening within the random-phase approximation. Screening substantially reshapes the effective interaction, suppresses the weak-pairing regime, and strongly enhances the tendency toward strong pairing. For a realistic range of microscopic parameters, the strongly paired phase is stabilised over a substantial fraction of that range. Our results identify the interplay of Landau-level mixing and ultra-short-range interactions as a plausible microscopic mechanism for the experimentally observed incompressible states in monolayer and trilayer graphene and point toward a strongly paired composite-fermion phase as their candidate ground state.
We compare ground-state energies by variational Monte Carlo of the spin unpolarized Halperin 331 and the spin polarized Moore-Read (MR) Pfaffian fractional quantum Hall states at half filling of the lowest Landau level (LLL) and the second Landau level (SLL) as a function of small deviations around the Coulomb point via the finite-thickness effect and direct alterations to the the first two Haldane pseudopotentials. In the comparison we find that in the LLL, either the 331 state or the MR Pfaffian may be lower in energy depending on the deviations. In the SLL, however, the MR Pfaffian is consistently lower in energy except for large deviations. These results suggest that even under moderate deviations in the interaction potential (through various physical processes such as finite thickness, Landau-level mixing, etc.), the MR Pfaffian description is more energetically favorable than the Halperin 331 state in the half-filled SLL (i.e., $\ensuremath{\nu}=5/2$), consistent with recent experimental investigations.
We examine the possibility of creating the Moore-Read Pfaffian in the lowest Landau level when the multicomponent Halperin 331 state (believed to describe quantum Hall bilayers and wide quantum wells at the filling factor $\ensuremath{\nu}=1/2$) is destroyed by the increase of tunneling. Using exact diagonalization of the bilayer Hamiltonian with short-range and long-range (Coulomb) interactions in spherical and periodic rectangular geometries, we establish that tunneling is a perturbation that drives the 331 state into a compressible composite Fermi liquid, with the possibility for an intermediate critical state that is reminiscent of the Moore-Read Pfaffian. These results are interpreted in the two-component BCS model for Cauchy pairing with a tunneling constraint. We comment on the conditions to be imposed on a system with fluctuating density in order to achieve the stable Pfaffian phase.
One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor $\ensuremath{\nu}=5/2$ is the Moore-Read (MR) Pfaffian state. A salient problem, however, is that it does not occur exactly at the Coulomb interaction, but rather at a modified interaction, which favors particle-hole-symmetry breaking. In an effort to find a better state, in this work, we investigate the possible connection between the paired QH state and the antisymmetrized bilayer ground state, which is inspired by the intriguing identity that the MR Pfaffian state is entirely equivalent to the antisymmetrized projection of the bilayer QH state called the Halperin (331) state, which is valid at interlayer distance, ddd, roughly equal to the magnetic length, lB{l}_{\mathrm{B}}lB​. Specifically, by using exact diagonalization in the torus geometry, we show that the exact 5/25/25/2 state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding d/lBd/{l}_{\mathrm{B}}d/lB​ via one-to-one mapping, which we call the bilayer mapping. One of the most important discoveries in this work is that the paired QH state occurring at the Coulomb interaction is mapped onto the antisymmetrized bilayer ground state at $d\ensuremath{\gg}{l}_{\mathrm{B}}$, which is equivalent to the antisymmetrized product state of two composite fermion seas at quarter filling, not the MR Pfaffian state. While maintaining high overlap with the paired QH state, the antisymmetrized bilayer ground state at $d\ensuremath{\gg}{l}_{\mathrm{B}}$ exhibits an abrupt change under the influence of small anisotropy. This suggests that the paired QH state occurring at the Coulomb interaction might be susceptible to anisotropic instability, opening up the possibility of anisotropic px{p}_{x}px​- or py{p}_{y}py​-wave pairing instead of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$-wave pairing in the MR Pfaffian/anti-Pfaffian state.
Many candidate non-Abelian quantum Hall states are accompanied by nearby “daughter” states, which are proposed to identify their topological order. Combining exact diagonalization and trial wave functions, we provide numerical evidence that daughter states reliably predict the parent topological phase. In the context of bilayer graphene and wide GaAs quantum wells, we show that the same interactions simultaneously stabilize Pfaffian, anti-Pfaffian, and their daughters, while suppressing the Jain states. The competition between Pfaffian and anti-Pfaffian, which is decided by particle-hole symmetry-breaking interactions, can likewise be deduced from their daughters. These findings strongly support the daughter-state-based identification of non-Abelian quantum Hall phases.