Pfaffian State Research Articles

Parametrization and thermodynamic scaling of pair correlation functions for the fractional quantum Hall effect

The calculation of pair correlations and density profiles of quasiholes are routine steps in the study of proposed fractional quantum Hall states. Nevertheless, the field has not adopted a standard way to present the results of such calculations in an easily reproducible form. We develop a polynomial expansion that allows for easy quantitative comparison between different candidate wavefunctions, as well as reliable scaling of correlation and quasihole profiles to the thermodynamic limit. We start from the well-known expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we demonstrate, numerically unstable. We orthogonalize their basis set to obtain a new basis of modified Jacobi polynomials, whose coefficients can be stably calculated. We then apply our expansion to extract pair correlation expansion coefficients and quasihole profiles in the thermodynamic limit for a wide range of fractional quantum Hall wavefunctions. These include the Laughlin series, composite fermion states with both reverse and direct flux attachment, the Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure works for both abelian and non-abelian quasiholes, even when the density at the core is not zero. We find that the expansion coefficients for all quantum Hall states considered can be fit remarkably well using a cosine oscillation with exponentially decaying amplitude. The frequency and the decay length are related in an intuitive, but not elementary way to the filling fraction. Different states at the same filling fraction can have distinct values for these parameters. Finally, we also use our scaled correlation functions to calculate estimates for the magneto-roton gaps of the various states.

Phase transitions and topological properties of the 52 quantum Hall states with strong Landau-level mixing

We numerically study a $\frac{5}{2}$ fractional quantum Hall system with even number of electrons using the exact diagonalization where both the strong Landau-level (LL) mixing and a finite width of the quantum well have been considered and adapted into a screened Coulomb interaction. With the principal component analysis, we are able to recognize a compressible-incompressible phase transition in the parameter space made of the magnetic field and the quantum well width by the competition between the first two leading components of the ground-state wave functions, which is consistent with the low-lying spectral feature and previous works in the odd-electron system. In addition, the presence of the subdominant third component suggests an incompressible transition occurring as the LL mixing strength grows into a certain parameter region associated with the ZnO experiments. We further investigate the strongly LL-mixed phase in this emerging region with the Hall viscosity, wave function overlaps, and the entanglement spectra. Results show it can be well described as a particle-hole symmetrized Pfaffian state with the dual topological properties of the Pfaffian and the anti-Pfaffian states.

Andreev-like Reflection in the Pfaffian Fractional Quantum Hall Effect

We study tunnel transport between the edge of a Pfaffian fractional quantum Hall state and that of an integer quantum Hall state. Based on the duality argument between strong and weak tunnelings, we find that Andreev-like reflection appears for a strong tunneling regime. We calculate charge conductance in the weak and strong tunneling regimes for the low-voltage limit. In the weak tunneling limit, $\mathrm{d}I/\mathrm{d}V$ is proportional to $V^{1/\nu}$ with bias voltage $V$ and $\nu=1/2$. On the other hand, in the strong tunneling limit, $\mathrm{d}I/\mathrm{d}V$ is expressed by $(e^{2}/h)2\nu/(1+\nu)$ with correction term.

Simple analog of the black-hole information paradox in quantum Hall interfaces

The black hole information paradox has been hotly debated for the last few decades, without full resolution. This makes it desirable to find analogs of this paradox in simple and experimentally accessible systems, whose resolutions may shed light on this long-standing and fundamental problem. Here we identify and resolve an apparent "information paradox" in a quantum Hall interface between the Halperin-331 and Pfaffian states. Information carried by the pseudospin degree of freedom of the Abelian 331 quasiparticles gets scrambled when they cross the interface to enter non-Abelian Pfaffian state, and becomes inaccessible to local measurements; in this sense the Pfaffian region is an analog of black hole interior while the interface plays a role similar to its horizon. We demonstrate that the "lost" information gets recovered once the "black hole" evaporates and the quasiparticles return to the 331 region, albeit in a highly entangled form. Such recovery is quantified by the Page curve of the entropy carried by these quasiparticles, which are analogs of Hawking radiation.

Dynamics of quantum Hall interfaces

A quantum Hall (QH) interface is different from an ordinary QH edge, as the latter has its location determined by the confining potential, while the former can be unpinned and behave like a free string. In this paper, we demonstrate this difference by studying three different interfaces formed by (i) the Laughlin state and the vacuum, (ii) the Pfaffian state and the vacuum, and (iii) the Pfaffian and the anti-Pfaffian states. We find that string-like interfaces propagating freely in the QH system lead to very different dynamical properties from edges. This qualitative difference gives rise to fascinating new physics and suggests a new direction in future research on QH physics. We also discuss briefly possible analogies between QH interfaces and concepts in string theory.

Origin of the ν=1/2 fractional quantum Hall effect in wide quantum wells

The nature of the fractional quantum Hall effect at $\ensuremath{\nu}=1/2$, observed in wide quantum wells almost three decades ago, is still under debate. Previous studies have investigated it using the variational Monte Carlo method, which assumes that the transverse wave function and the gap between the symmetric and antisymmetric subbands obtained in a local density approximation at zero magnetic field remain valid even at high perpendicular magnetic fields; this method also ignores the effect of Landau level mixing. We develop in this work a three-dimensional fixed-phase diffusion Monte Carlo method, which gives, in a single framework, the total energies of various candidate states in a finite width quantum well, including Landau level mixing, directly in a large magnetic field. This method can be applied to one-component states and also to two-component states in the limit where the symmetric and antisymmetric bands are nearly degenerate. Our three-dimensional fixed-phase diffusion Monte Carlo calculations find that the one-component composite-fermion Fermi sea and the one-component Pfaffian states are very close in energy for a range of quantum-well widths and densities, suggesting that the observed 1/2 fractional quantum Hall state in wide quantum wells is likely to be the one-component Pfaffian state. We hope that this will motivate further experimental studies of this state.

Effective field theory for fractional quantum Hall systems nearν=5/2

We propose an effective field theory (EFT) of fractional quantum Hall systems near the filling fraction $\nu=5/2$ that flows to pertinent IR candidate phases, including non-abelian Pfaffian, anti-Pfaffian, and particle-hole Pfaffian states (Pf, APf, and PHPf). Our EFT has a 2+1$d$ O(2)$_{2,L}$ Chern-Simons gauge theory coupled to four Majorana fermions by a discrete charge conjugation gauge field, with Gross-Neveu-Yukawa-Higgs terms. Including deformations via a Higgs condensate and fermion mass terms, we can map out a phase diagram with tunable parameters, reproducing the prediction of the recently-proposed percolation picture and its gapless topological quantum phase transitions. Our EFT captures known features of both gapless and gapped sectors of time-reversal-breaking domain walls between Pf and APf phases. Moreover, we find that Pf$\mid$APf domain walls have higher tension than domain walls in the PHPf phase. Then the former, if formed, may transition to the energetically-favored PHPf domain walls; this could, in turn, help further induce a bulk transition to PHPf.

Topological Interface between Pfaffian and Anti-Pfaffian Order in ν=5/2 Quantum Hall Effect.

A recent thermal Hall experiment triggered renewed interest in the problem of ν=5/2 quantum Hall effect, which motivated novel interpretations based on the formation of mesoscopic puddles made of Pfaffian and anti-Pfaffian topological orders. Here, we study an interface between the Pfaffian and anti-Pfaffian states, which may play crucial roles in thermal transport, by means of state-of-the-art, density-matrix renormalization group simulations. We demonstrate that an intrinsic electric dipole moment emerges at the interface, similar to the "p-n" junction sandwiched between N-type and P-type semiconductor. Importantly, we elucidate the topological origin of this dipole moment, whose formation is to counterbalance the mismatch of guiding-center Hall viscosity of bulk Pfaffian and anti-Pfaffian states. In addition, these results imply that the formation of a dipole moment could be helpful to stabilize the puddles made of Pfaffian and anti-Pfaffian states in experimental conditions.

Automatic design of Hamiltonians

We formulate an optimization problem of Hamiltonian design based on the variational principle. Given a variational ansatz for a Hamiltonian we construct a loss function to be minimised as a weighted sum of relevant Hamiltonian properties specifying thereby the search query. Using fractional quantum Hall effect as a test system we illustrate how the framework can be used to determine a generating Hamiltonian of a finite-size model wavefunction (Moore-Read Pfaffian and Read-Rezayi states), find optimal conditions for an experiment or "extrapolate" given wavefunctions in a certain universality class from smaller to larger system sizes. We also discuss how the search for approximate generating Hamiltonians may be used to find simpler and more realistic models implementing the given exotic phase of matter by experimentally accessible interaction terms.

Pfaffian paired states for half-integer fractional quantum Hall effect

In this review, the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ([Formula: see text]-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the particle–hole (PH) Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern–Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.

Model interactions for Pfaffian paired states based on Chern-Simons field theory description

On the basis of Chern-Simons field-theoretical description we propose a simple method for derivation of model interactions for Pfaffian paired states. We verify the method in the case of Pfaffian (i.e. Moore - Read) state, and derive a general form of the model interaction in the case of PH Pfaffian. More than one Landau level is needed to establish the correlations of the PH Pfaffian, and we present the values of relevant three-body pseudo-potentials for two Landau levels.

Specific heat and pairing of Dirac composite fermions in the half-filled Landau level

A recent proposal argues that an alternate description of the half-filled Landau level is a theory of massless Dirac fermions. We examine the possibility of pairing of these Dirac fermions by numerically solving the coupled Eliashberg equations unlike our previous calculation (Wang and Chakravarty, 2016). In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian states. However, in our approximation scheme, pairing is not possible in the putative particle–hole symmetric state for ℓ=0 angular momentum. The specific heat at low temperatures of a system of massless Dirac fermions interacting with a transverse gauge field, expected to be relevant for the half-filled Landau level, is calculated. Using the Luttinger formula, it is found to be ∝TlnT in the leading low temperature limit, due to the exchange of transverse gauge bosons. The result agrees with the corresponding one in the nonrelativistic composite fermion theory of Halperin, Lee and Read of the half-filled Landau level.

Almost linear Haldane pseudopotentials and emergent conformal block wave functions in a Landau level

We consider a two dimensional (2D) model of particles interacting in a Landau level. We work in a finite disk geometry and take the particles to interact with a linearly decreasing two-body Haldane pseudo-potential. We show that the ground state subspace of this model is spanned by the wave-functions that can be written as polynomial conformal blocks (of an arbitrary conformal field theory) consistent with the filling fraction (scaling dimension). To remove degeneracies, we then add a quadratic perturbation to the Hamiltonian and show that; 1. Conformal blocks constructed using the Moore-Read construction (e.g. Laughlin, Pfaffian, and Read-Rezayi states) remain exact eigenstates of this model in the thermodynamic limit and 2. By tuning an externally imposed single-body $-L_z^2$ potential we can enforce Moore-Read conformal blocks to become exact ground states of this model in the thermodynamic limit. We cannot rule out the possibility of residual degeneracies in this limit. This model has no filling dependence and is comprised only from two-body long-range interactions and external single-body potentials. Our results provide insight into how conformal block wave-functions can emerge in a Landau level.

Disorder-Driven Transition in the ν=5/2 Fractional Quantum Hall Effect.

The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.

Prediction of a Non-Abelian Fractional Quantum Hall State with f-Wave Pairing of Composite Fermions in Wide Quantum Wells.

We theoretically investigate the nature of the state at the quarter filled lowest Landau level and predict that, as the quantum well width is increased, a transition occurs from the composite fermion Fermi sea into a novel non-Abelian fractional quantum Hall state that is topologically equivalent to f-wave pairing of composite fermions. This state is topologically distinct from the familiar p-wave paired Pfaffian state. We compare our calculated phase diagram with experiments and make predictions for many observable quantities.

Geometric responses of the Pfaffian state

We define and study the Pfaffian state on Riemann surfaces with arbitrary metrics and an inhomogeneous magnetic field and derive its universal transport coefficients. Following a path integral approach, we compute the generating functional which encodes the linear response of the system to a variation of the background metric and the magnetic field and use it to compute the leading and sub-leading corrections to the charge density in a large-$N$ expansion. We also present the first derivation of gravitational anomaly contribution at O$(k^6)$ to the static structure factor for the Pfaffian state in the long wavelength limit.

Real-time Ramsey interferometry in fractional quantum Hall states

We study the time-dependent dynamical phase of fractional quasiparticles in quantum Hall states using Ramsey interferometry. A Ramsey two-pulse voltage modulation generates an interference term in the current noise of a quantum point contact oscillating with the dynamical phase. We show that the interference pattern probes the Green's function of the fractional quasi-particles in the time domain and reveals the fractional charge. We address both the case of a point-like tunnel junction and the case of momentum-resolved tunneling which also provides information on the speed of those quasiparticles propagating along the edge. For the 5/2 fractional quantum Hall case, the Ramsey signal can differentiate between the Pfaffian and anti-Pfaffian states.

Quasielectrons in lattice Moore-Read models

Moore-Read states can be expressed as conformal blocks of the underlying rational conformal field theory, which provides a well explored description for the insertion of quasiholes. It is known, however, that quasielectrons are more difficult to describe in continuous systems, since the natural guess for how to construct them leads to a singularity. In this work, we show that the singularity problem does not arise for lattice Moore-Read states. This allows us to construct Moore-Read Pfaffian states on lattices for filling fraction 5/2 with both quasiholes and quasielectrons in a simple way. We investigate the density profile, charge, size and braiding properties of the anyons by means of Monte Carlo simulations. Further we derive an exact few-body parent Hamiltonian for the states. Finally, we compare our results to the density profile, charge and shape of anyons in the Kapit-Mueller model by means of exact diagonalization.

Quantum Hall hierarchy from coupled wires

The coupled-wire construction provides a useful way to obtain microscopic Hamiltonians for various two-dimensional topological phases, among which fractional quantum Hall states are paradigmatic examples. Using the recently introduced flux attachment and vortex duality transformations for coupled wires, we show that this construction is remarkably versatile to encapsulate phenomenologies of hierarchical quantum Hall states: the Jain-type hierarchy states of composite fermions filling Landau levels and the Haldane-Halperin hierarchy states of quasiparticle condensation. The particle-hole conjugate transformation for coupled-wire models is also given as a special case of the hierarchy construction. We also propose coupled-wire models for the composite Fermi liquid, which turn out to be compatible with a sort of the particle-hole symmetry implemented in a nonlocal way at $\nu=1/2$. Furthermore, our approach shows explicitly the connection between the Moore-Read Pfaffian state and a chiral $p$-wave pairing of the composite fermions. This composite fermion picture is also generalized to a family of the Pfaffian state, including the anti-Pfaffian state and Bonderson-Slingerland hierarchy states.

Parton construction of a wave function in the anti-Pfaffian phase

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 $\mathcal{P}_{\rm LLL} \Phi_{1}^3[\Phi_{2}^{*}]^{2}\sim\Psi^{2}_{2/3}/\Phi_{1}$ 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 $\nu=5/2$, 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.