Pfaffian Wave Function Research Articles

Second Landau level projection of antiholomorphic Pfaffian wave functions

We propose a new state described by the second Landau level (SLL) projection of a generalized Moore-Read Pfaffian wavefunction 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 wavefunction, 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.

Even-denominator fractional quantum Hall state in bilayer graphene

At a half-filled Landau level, composite fermions with chiral <i>p</i>-wave pairing will form a Moore-Read state which hosts charge-<i>e</i>/4 fractional excitation. This excitation supports non-Abelian statistics and has potential to enable topological quantum computation. Owing to the <i>SU</i>(4) symmetry of electron and electric-field tunability, the bilayer graphene becomes an ideal platform for exploring physics of multi-component quantum Hall state and is candidate for realizing non-Abelian statistics. In this work, high-quality bilayer graphene/hBN heterostructure is fabricated by using dry-transfer technique, and electric transport measurement is performed to study quantum Hall state behavior in bilayer graphene under electric field and magnetic field. Under strong magnetic field, the sequences of incompressible state with quantized Hall conductivity are revealed at –5/2, –1/2, 3/2 filling of Landau level. The feature of even-denominator quantum Hall state is more visible then weaker with increasing magnetic field, and this corresponds to the polarization of Landau level wave function. The experimental results indicate that the observed even-denominator fractional quantum Hall state belongs to the topological phase described by Pfaffian wavefunction.

Approximate two-body generating Hamiltonian for the particle-hole Pfaffian wave function

We present two 2-body Hamiltonians that approximate the exact PH-Pfaffian wavefunction with their ground states for all the system sizes where this wavefunction has been numerically constructed to date. The approximate wavefunctions have high overlap with the original and reproduce well the low-lying entanglement spectrum and structure factor. The approximate generating Hamiltonians are obtained by an optimisation procedure where three to four pseudopotentials are varied in the neighbourhood of second Landau level Coulomb interaction or of a non-interacting model. They belong to a finite region in the variational space of Hamiltonians where each point approximately generates the PH-Pfaffian. We diagonalize the identified Hamiltonians for up to 20 electrons and find that for them the PH-Pfaffian shift appears energetically more favorable. Possibility to interpret the data in terms of composite fermions is discussed.

Particle-hole Pfaffian intracorrelations and intercorrelations in the quantum Hall bilayer

PH Pfaffian topological phase may exist in a uniform system due to strong Landau level (LL) mixing according to theoretical predictions based on the Son - Dirac composite fermion theory. Numerical investigations in the presence of large LL mixing are limited due to numerical complexities, when taking into account at least one more LL. Because of this, we apply the same field theoretical approach to the quantum Hall bilayer at total filling factor equal to one, for which many numerical studies exist. The most recent work in the torus geometry predicts an intermediate phase (for intermediate distances between layers) with an even-odd effect. According to our approach, the intermediate phase represents a mixed negative-flux $p$-wave pairing i.e. coexisting intra (PH Pfaffian in each layer) and inter (a la PH Pfaffian) pairing correlations. This again underlines a necessity for strong entanglement with additional degrees of freedom, i.e. at least one more (additional) LL in the search for a stable PH Pfaffian phase in a single layer. Based on the analogy with the bilayer physics we propose a PH Pfaffian wave function that resides in two LLs.

Particle-hole-symmetric model for a paired fractional quantum Hall state in a half-filled Landau level

The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is: short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.

Stripe and superconducting order competing in the Hubbard model on a square lattice studied by a combined variational Monte Carlo and tensor network method

The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the $d$-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\delta$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling $U/t=10$, for the on-site interaction $U$ and the transfer $t$, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01$t$. We have identified distinct phases including a uniform $d$-wave superconducting phase for $0.17\lesssim \delta \lesssim0.22$ and a stripe charge/spin ordered phase for $\delta\lesssim0.17$ with the stripe period depending on $\delta$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\delta\lesssim0.07$ and coexisting stripe and $d$-wave superconductivity for $0.07\lesssim\delta\lesssim0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\delta=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.

Determination of many-electron basis functions for a quantum Hall ground state using Schur polynomials

A method for determining the ground state of a planar interacting many-electron system in a magnetic field perpendicular to the plane is described. The ground state wave-function is expressed as a linear combination of a set of basis functions. Given only the flux and the number of electrons describing an incompressible state, we use the combinatorics of partitioning the flux among the electrons to derive the basis wave-functions as linear combinations of Schur polynomials. The procedure ensures that the basis wave-functions form representations of the angular momentum algebra. We exemplify the method by deriving the basis functions for the 52 quantum Hall state with a few particles. We find that one of the basis functions is precisely the Moore–Read Pfaffian wave function.

Surprising robustness of particle-hole symmetry for composite-fermion liquids

We report on fixed phase diffusion Monte Carlo calculations that show that, even for a large amount of Landau level mixing, the energies of the Pfaffian and anti-Pfaffian phases remain very nearly the same, as also do the excitation gaps at $1/3$ and $2/3$. These results, combined with previous theoretical and experimental investigations, indicate that particle hole (PH) symmetry for composite fermion states is much more robust than a priori expected, emerging even in models that explicitly break PH symmetry. We provide insight into this fact by showing that the low energy physics of a generic repulsive 3-body interaction is captured, to a large extent and over a range of filling factors, by a mean field approximation that maps it into a PH symmetric 2-body interaction. This explains why Landau level mixing, which effectively generates such a generic 3-body interaction, is inefficient in breaking PH symmetry. As a byproduct, our results provide a systematic construction of a 2-body interaction which produces, to a good approximation, the Pfaffian wave function as its ground state.

Spin correlations and topological entanglement entropy in a non-Abelian spin-one spin liquid

We analyze the properties of a non-Abelian spin-one chiral spin liquid state proposed by Greiter and Thomale [Phys. Rev. Lett. 102, 207203 (2009)] using Monte Carlo. In this state the bosonic $\ensuremath{\nu}=1$ Moore-Read Pfaffian wave function is used to describe a gas of bosonic spin flips on a square lattice with one flux quantum per plaquette. For toroidal geometries there is a three-dimensional space of these states corresponding to the topological degeneracy of the bosonic Moore-Read state on the torus. We show that spin correlations for different states in this space become indistinguishable for large system size. We also calculate the Renyi entanglement entropy for different system partitions to extract the topological entanglement entropy and provide evidence that the topological order of the lattice spin-liquid state is the same as that of the continuum Moore-Read state from which it is constructed.

Engineering three-body interaction and Pfaffian states in circuit QED systems

We demonstrate a scheme to engineer the three-body interaction in circuit-QED systems by tuning a fluxonium qubit. Connecting such qubits in a square lattice and controlling the tunneling dynamics, in the form of a synthesized magnetic field, for the photon-like excitations of the system, allows the implementation of a parent Hamiltonian whose ground state is the Pfaffian wave function. Furthermore, we show that the addition of the next-nearest neighbor tunneling stabilizes the ground state, recovering the expected topological degeneracy even for small lattices. Finally, we discuss the implementation of these ideas with the current technology.

Topological Phases in the Zeroth Landau Level of Bilayer Graphene

We analyze the phase diagram of the zeroth Landau level of bilayer graphene, taking into account the realistic effects of screening of the Coulomb interaction and strong mixing between two degenerate sublevels. We identify robust quantum Hall states at filling factors ν=-1, -4/3, -5/3, -8/5, -1/2 and discuss the nature of their ground states, collective excitations, and relation to the more familiar states in GaAs using a tractable model. In particular, we present evidence that the ν=-1/2 state is non-Abelian and described by either the Moore-Read wave function or its particle-hole conjugate, while ruling out other candidates such as the 331 state.

Superconducting order parameter for the even-denominator fractional quantum Hall effect

One of the most intriguing phenomena in nature is the fractional quantum Hall effect (FQHE) observed in the half-filled second Landau level which, arising in even-denominator filling factors, $\nu=5/2$ and $7/2$, is completely different from other FQHEs in its origin, all of which, except for those two filling factors, occur in odd-denominator fractions. Usually formulated in terms of a trial wave function called the Moore-Read Pfaffian wave function, current leading theories attribute the origin of the 5/2 FQHE to the formation of Cooper pairs, not of electron, but of the true quasi-particle of the system known as composite fermion. The nature of superconductivity resulting from such Cooper pairing is particularly puzzling in the sense that it apparently coexists with strong magnetic fields, which poses an interesting dilemma since the Meissner effect is {\it the} most important defining property of superconductivity. This apparent dilemma is resolved by the fact that composite fermions do not respond to external magnetic field at even-denominator filling factors. To provide direct evidence that it is composite fermions that actually form the superconducting condensate, here, we develop a numerically exact method of creating a Cooper pair of composite fermions and explicitly compute the superconducting order parameter as a function of real space coordinates. As results, in addition to direct evidence for superconductivity, we obtain quantitative predictions for superconducting coherence length. Obtaining such theoretical predictions can serve as an important step toward fault-tolerant topological quantum computation.

Emerging bosons with three-body interactions from spin-1 atoms in optical lattices

We study two many-body systems of bosons interacting via an infinite three-body contact repulsion in a lattice: a pairs quasicondensate induced by correlated hopping and the discrete version of the Pfaffian wave function. We propose to experimentally realize systems characterized by such interaction by means of a proper spin-1 lattice Hamiltonian: spin degrees of freedom are locally mapped into occupation numbers of emerging bosons, in a fashion similar to spin-1/2 and hardcore bosons. Such a system can be realized with ultracold spin-1 atoms in a Mott insulator with a filling factor of 1. The high versatility of these setups allows us to engineer spin-hopping operators breaking the SU(2) symmetry, as needed to approximate interesting bosonic Hamiltonians with three-body hardcore constraint. For this purpose we combine bichromatic spin-independent superlattices and Raman transitions to induce a different hopping rate for each spin orientation. Finally, we illustrate how our setup could be used to experimentally realize the first setup, that is, the transition to a pairs quasicondensed phase of the emerging bosons. We also report on a route toward the realization of a discrete bosonic Pfaffian wave function and list some open problems for reaching this goal.

ν=5/2Fractional Quantum Hall Effect at 10 T: Implications for the Pfaffian State

We report on the magnetotransport (including tilt fields) around Landau level filling factor nu=5/2 in a high-purity modulation-doped GaAs/AlGaAs quantum well with twice the electron density of standard samples. A quantized 5/2 Hall plateau is observed at B approximately = 10 T, with an activation gap approximately 125 mK; the plateau can persist up to approximately 25 degree tilt field. This finding is discussed in the context of a proposed Moore-Read Pfaffian (or anti-Pfaffian) wave function being a possible ground state at 5/2. The tilt fields induce a background resistance at 5/2 that could be either isotropic or anisotropic, depending simply on the in-plane magnetic field direction with respect to the GaAs crystalline axis. Such data indicate a substantial coupling between the 5/2 collective phases and the GaAs crystal.

Evidence for Developing Fractional Quantum Hall States at Even Denominator1/2and1/4Fillings in Asymmetric Wide Quantum Wells

We report the observation of developing fractional quantum Hall states at Landau level filling factors nu = 1/2 and 1/4 in electron systems confined to wide GaAs quantum wells with significantly asymmetric charge distributions. The very large electric subband separation and the highly asymmetric charge distribution at which we observe these quantum Hall states, together with the fact that they disappear when the charge distribution is made symmetric, suggest that these are one-component states, possibly described by the Moore-Read Pfaffian wave function.

Ultimate braid-group generators for coordinate exchanges of Ising anyons from the multi-anyon Pfaffian wavefunctions

We give a rigorous and self-consistent derivation of the elementary braid matrices representing the exchanges of adjacent Ising anyons in the two inequivalent representations of the Pfaffian quantum Hall states with even and odd numbers of Majorana fermions. To this end we use the distinct operator product expansions of the chiral spin fields in the Neveu–Schwarz and Ramond sectors of the two-dimensional Ising conformal field theory. We find recursive relations for the generators of the irreducible representations of the braid group in terms of those for , as well as explicit formulae for almost all braid matrices for exchanges of Ising anyons. Finally we prove that the braid-group representations obtained from the multi-anyon Pfaffian wavefunctions are completely equivalent to the spinor representations of SO(2n + 2) and give the equivalence matrices explicitly. This actually proves that the correlation functions of 2n chiral Ising spin fields σ do indeed realize one of the two inequivalent spinor representations of the rotation group SO(2n) as conjectured by Nayak and Wilczek.

Contrasting Behavior of the52and73Fractional Quantum Hall Effect in a Tilted Field

Using a tilted-field geometry, the effect of an in-plane magnetic field on the even denominator nu=5/2 fractional quantum Hall state is studied. The energy gap of the nu=5/2 state is found to collapse linearly with the in-plane magnetic field above approximately 0.5 T. In contrast, a strong enhancement of the gap is observed for the nu=7/3 state. The radically distinct tilted-field behavior between the two states is discussed in terms of Zeeman and magneto-orbital coupling within the context of the proposed Moore-Read Pfaffian wave function for the 5/2 fractional quantum Hall effect.

Orbital Landau level dependence of the fractional quantum Hall effect in quasi-two-dimensional electron layers: Finite-thickness effects

The fractional quantum Hall effect (FQHE) in the second orbital Landau level at even-denominator filling factor of 5/2 remains enigmatic and motivates our work. We theoretically consider the effect of the quasi-two-dimensional (2D) nature of the experimental fractional quantum Hall system on a number of FQH states (filling factors: 1/3, 1/5, and 1/2) in the lowest, second, and third orbital Landau levels (LLL, SLL, and TLL, respectively) by calculating the wave-function overlap, as a function of quasi-2D layer thickness, between the exact ground state of a model Hamiltonian and the consensus variational ansatz wave functions, i.e., the Laughlin wave function for 1/3 and 1/5 and the Moore-Read Pfaffian wave function for 1/2. Using large numerical overlap as a stability (or FQHE robustness) criterion, we find that the FQHE does not occur in the TLL (for any quasi-2D layer thickness), is the most robust for zero thickness (strict 2D limit) in the LLL for 1/3 and 1/5 and for 11/5 in the SLL, and is the most robust at finite thickness (4--5 magnetic lengths) in the SLL for the mysterious even-denominator 5/2 state and the presumably more conventional 7/3 state. We do not find any FQHE at 1/2 in the LLL for any thickness for the quasi-2D models considered in our work. Furthermore, we examine the orbital effects of a nonzero in-plane (parallel) magnetic field, finding that its application effectively reduces the quasi-2D layer thickness and, therefore, could destroy the FQHE at 5/2 and 7/3, while it enhances the FQHE at 11/5, in the SLL. The in-plane field also enhances the LLL FQHE states by making the quasi-2D system more purely 2D. The in-plane field effects could thus be qualitatively different in the LLL and the SLL by virtue of magneto-orbital coupling through the finite-thickness effect. Using exact diagonalization on the torus geometry, we show the appearance of the characteristic threefold topological degeneracy expected for the Pfaffian state. This signature is enhanced by nonzero thickness, corroborating our findings from overlap calculations. Our results have ramifications for wave-function engineering, opening the possibility of creating an optimal experimental system where the 5/2 FQHE state is more likely described by the Moore-Read Pfaffian state with obvious applications in the burgeoning field of fault-tolerant topological quantum computing.

States of interacting composite fermions at the Landau level fillingν=2+3/8

There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\nu=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\nu=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.

Characterization of a series of high-quality wide GaAs quantum wells

Due to the current interest in quantum Hall states described by the Moore–Read or Pfaffian wavefunction, we have recently begun experiments involving a series of high-quality wide GaAs quantum well in an effort to search for new Pfaffian states. Here we report the characterization of these samples. The quantum well in our series have differing well widths which range from 50 to 80 nm. The self-consistently calculated electronic band edge and electron distribution function are shown for each sample. We present the magnetoresistance up to 18 T where the quantum Hall states at ν = 1 and 3 are seen to vanish with increasing L . We also discuss the observation of a fractional quantum Hall state at ν = 1 / 2 in the L = 50 nm sample.