The Phase Diagram of the $\nu =5/2$ Fractional Quantum Hall Effect: Effects of Landau Level Mixing and Non-Zero Width

Abstract

Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the anti-Pfaffian, offer candidate descriptions of the $\nu =5/2$ fractional quantum Hall state. But the significant controversy surrounding the nature of the $\nu =5/2$ 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 $\nu =5/2$ 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 $\kappa$, the ratio of the Coulomb energy scale to the cyclotron gap. We also incorporate non-zero width $w$ of the quantum well and sub-band mixing. We find the ground state in both the torus and spherical geometries as a function of $\kappa$ and $w$. 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 $\kappa <{\kappa }_c(w)$, where ${\kappa }_c(w)$ is a $w$-dependent critical value $0.6\lesssim {\kappa }_c(w)\lesssim 1$. 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.

16 pages, 17 figures. Added: 1) Appendix D + Fig. 2 discussing planar vs spherical pseudopotentials and thermodynamic limit 2) Fig 8 showing topological gap 3) ancillary text files containing the numerical data for most of the figures

References in corpus (22)

• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Particle-hole symmetry and the Pfaffian state
• Particle-Hole Symmetry and the $\nu =5/2$ Quantum Hall State
• Entanglement entropy in fermionic Laughlin states
• Observation of neutral modes in the fractional quantum Hall regime
• Experimental studies of the fractional quantum Hall effect in the first excited Landau level
• Infinite density matrix renormalization group for multicomponent quantum Hall systems
• Finite Layer Thickness Stabilizes the Pfaffian State for the 5/2 Fractional Quantum Hall Effect: Wavefunction Overlap and Topological Degeneracy
• Density Matrix Renormalization Group Study of Incompressible Fractional Quantum Hall States
• Bipartite entanglement entropy in fractional quantum Hall states
• Fractional Quantum Hall Effect in the Second Landau Level
• Pseudopotentials for Multi-particle Interactions in the Quantum Hall Regime
• Numerical Analysis of Quasiholes of the Moore-Read Wavefunction
• Spin polarization of the $\nu =5/2$ quantum Hall state
• Orbital Landau level dependence of the fractional quantum Hall effect in quasi-two dimensional electron layers: finite-thickness effects
• Experimental probe of topological orders and edge excitations in the second Landau level
• Contrasting Behavior of the 5/2 and 7/3 Fractional Quantum Hall Effect in a Tilted Field
• Spontaneous Particle-Hole Symmetry Breaking in the $\nu =5/2$ Fractional Quantum Hall Effect
• Quantum Phase Transitions and the ν=5/2 Fractional Hall State in Wide Quantum Wells
• Pfaffian statistics through adiabatic transport in the 1D coherent state representation
• Increasing the ν = 5 / 2 gap energy: an analysis of MBE growth parameters
• Localized quasiholes and the Majorana fermion in fractional quantum Hall state at nu=5/2 via direct diagonalization

Cited by in corpus (44)

• Fractional charge and fractional statistics in the quantum Hall effects
• Transport signatures of Hall viscosity
• Valley Isospin Controlled Fractional Quantum Hall States in Bilayer Graphene
• Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond
• Abelian and Non-Abelian States in $\nu =2/3$ Bilayer Fractional Quantum Hall Systems
• Multiple Chern bands in twisted MoTe${}_2$ and possible non-Abelian states
• Competing Abelian and non-Abelian topological orders in $\nu =1/3+1/3$ quantum Hall bilayers
• Graviton Chirality and Topological Order in the Half-filled Landau Level
• Quantum Entanglement as a Diagnostic of Phase Transitions in Disordered Fractional Quantum Hall Liquids
• Partial equilibration of anti-Pfaffian edge modes at $\nu =5/2$
• Energetics of Pfaffian-AntiPfaffian Domains
• The sixteenfold way and the quantum Hall effect at half-integer filling factors
• Large-scale simulations of particle-hole-symmetric Pfaffian trial wavefunctions
• Topological Interface between Pfaffian and anti-Pfaffian Order in $\nu =5/2$ Quantum Hall Effect
• Full Classification of Transport on an Equilibrated 5/2 Edge via Shot Noise
• Energetics of the PH-Pfaffian state and the 5/2-fractional quantum Hall effect
• Phase diagram and edge states of the $\nu =5/2$ fractional quantum Hall state with Landau level mixing and finite well thickness
• Tracing non-Abelian anyons via impurity particles
• Composite fermion pairing induced by Landau level mixing
• Surprising robustness of particle-hole symmetry for composite fermion liquids
• Evolution of quantum entanglement with disorder in fractional quantum Hall liquids
• Shape of the magnetoroton at $\nu =1/3$ and $\nu =7/3$ in real samples
• An Effective Field Theory for Fractional Quantum Hall Systems near $\nu =5/2$
• Bilayer mapping of the paired quantum Hall state: Instability toward anisotropic pairing
• Onset of Quantum Criticality in the Topological-to-Nematic Transition in a Two-dimensional Electron Gas at Filling Factor $\nu =5/2$
• Nematic fluctuations balancing the zoo of phases in half-filled quantum Hall systems
• PH-Pfaffian order in a translationally and rotationally invariant system
• An Anomalous Reentrant 5/2 Quantum Hall Phase at Moderate Landau-Level-Mixing Strength
• Microscopic Diagnosis of Universal Geometric Responses in Fractional Quantum Hall Liquids
• Possible restoration of particle-hole symmetry in the 5/2 Quantized Hall State at small magnetic field
• Approximate two-body generating Hamiltonian for the PH-Pfaffian wavefunction
• Fractional Quantum Hall States of the $\mathcal{A}$ phase in the Second Landau Level
• Phase diagram of compressible and paired states in the quarter-filled Landau level
• Disorder broadening of even denominator fractional quantum Hall states in the presence of a short-range alloy potential
• Deformed Fredkin model for the $\nu =5/2$ Moore-Read state on thin cylinders
• Two-component parton fractional quantum Hall state in graphene
• Developing fractional quantum Hall states at $\nu$ = $\frac{1}{7}$ and $\frac{2}{11}$ in the presence of significant Landau level mixing
• Anomalous Bilayer Quantum Hall Effect
• Universal charge conductance at Abelian--non-Abelian quantum Hall interfaces
• Composite fermion mass enhancement and particle-hole symmetry of fractional quantum Hall states in the lowest Landau level under realistic conditions
• Enigmatic 2+6/13 Filling Factor: A Prototype Intermittent Topological State
• Torus geometry eigenfunctions of an interacting multi-Landau level Hamiltonian
• Shot noise signatures of candidate states for the fractional quantum Hall $\nu =12/5$ state
• Disorder-driven phase transitions in bosonic fractional quantum Hall liquids