anti-Pfaffian States Research Articles

Pairing of particle-hole symmetric composite fermions in half-filled Landau level

In a recent proposal of the half-filled Landau level, the composite fermions are taken to be Dirac particles and particle-hole symmetric. Cooper pairing of these composite fermions in different angular momentum channels, $\ell$, can give rise to different kinds of Pfaffian states. In addition to the well known Moore-Read Pfaffian and anti-Pfaffian states, a new putative particle-hole symmetric Pfaffian state, corresponding to the $s-$wave pairing channel, was also proposed. However, the possible underlying pairing mechanism is not clear at all. In this work we provide a specific pairing mechanism for realizing some of these Pfaffian states. We show that there can be nonzero pairing in angular momentum channels $|\ell|\ge 1$ depending on the magnitude of a coupling constant. There is a quantum phase transition from the Dirac composite fermi liquid state to Cooper pairing states in angular momentum channels $|\ell|\ge 1$ as the coupling constant is tuned across its critical point value. Surprisingly the particle-hole symmetric $\ell=0$ channel pairing turns out to be impossible irrespective of the size of the coupling constant.

Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry.

Numerical results suggest that the quantum Hall effect at ν=5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau-level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau-level mixing are strong. We show that the recent proposal of a particle-hole (PH)-Pfaffian topological order by Son is consistent with all experiments. The absence of particle-hole symmetry at ν=5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.

Finite frequency noise spectroscopy for fractional Hall states at ν = 5/2

.We investigate the finite frequency noise of a quantum point contact at filling factor using a weakly coupled resonant LC circuit as a detector. We show how one could spectroscopically address the fractional charged excitations inspecting separately their charge and scaling dimensions. We thus compare the behaviour of the Pfaffian and the anti-Pfaffian non-Abelian edge states models in order to give possible experimental signatures to identify the appropriate model for these fractional quantum Hall states. Finally we investigate how the temperature of the LC resonant circuit can be used in order to enhance the sensibility of the measurement scheme.

Is the Composite Fermion a Dirac Particle?

We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the $\nu=\frac12$ quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of $\pi$ around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, $\mathcal C\mathcal P$. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors $\frac n{2n+1}$ and $\frac{n+1}{2n+1}$, which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, $n$ and $n+1$, is now mapped to the same half-integer filling factor $n+\frac12$ of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as $d$-wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while $s$-wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a $\mathcal C\mathcal P$-breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.

Phase diagram and edge states of theν=5/2fractional quantum Hall state with Landau level mixing and finite well thickness

The $\nu=5/2$ fractional quantum Hall effect is a system of intense experimental and theoretical interest as its ground state may host non-abelian excitations, but the exact nature of the ground state is still undetermined. We present the results of an exact diagonalization study of an electron system in the disk configuration including the effects of Landau level (LL) mixing and the finite thickness of the quantum well confining the electrons. The degeneracy between the two leading candidates for the ground state, the Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing background, in addition to the inclusion of two- and three-body interactions via LL mixing. As a result of the neutralizing background in the disc configuration, there is a phase transition from the anti-Pfaffian to the Pfaffian as LL mixing is turned on, in stark contrast to what is observed in a spherical geometry. The LL mixing leads to an increased charge $e/4$ quasihole size. LL mixing interactions are also shown to overcome the effects of edge reconstruction. Due to finite thickness effects, these properties are enhanced dramatically. We also find that only the Pfaffian and anti-Pfaffian states continue to possess energy gaps at finite width, while gaps for compressible stripe states close.

Probing theν=52quantum Hall state with electronic Mach-Zehnder interferometry

It was recently pointed out that Halperin's 113 topological order explains the transport experiments in the quantum Hall liquid at filling factor $\nu=5/2$. The 113 order, however, cannot be easily distinguished from other likely topological orders at $\nu=5/2$ such as the non-Abelian Pfaffian and anti-Pfaffian states and the Abelian Halperin 331 state in Fabry-Perot interferometry. We show that an electronic Mach-Zehnder interferometer provides a clear identification of these candidate $\nu=5/2$ states. Specifically, the $I$-$V$ curve for the tunneling current through the interferometer is more asymmetric in the 113 state than in other $\nu=5/2$ states. Moreover, the Fano factor for the shot noise in the interferometer can reach 13.6 in the 113 state, much greater than the maximum Fano factors of 3.2 in the Pfaffian and anti-Pfaffian states and 2.3 in the 331 state.

Infinite density matrix renormalization group for multicomponent quantum Hall systems

While the simplest quantum Hall plateaus, such as the $\nu = 1/3$ state in GaAs, can be conveniently analyzed by assuming only a single active Landau level participates, for many phases the spin, valley, bilayer, subband, or higher Landau level indices play an important role. These `multi-component' problems are difficult to study using exact diagonalization because each component increases the difficulty exponentially. An important example is the plateau at $\nu = 5/2$, where scattering into higher Landau levels chooses between the competing non-Abelian Pfaffian and anti-Pfaffian states. We address the methodological issues required to apply the infinite density matrix renormalization group to quantum Hall systems with multiple components and long-range Coulomb interactions, greatly extending accessible system sizes. As an initial application we study the problem of Landau level mixing in the $\nu = 5/2$ state. Within the approach to Landau level mixing used here, we find that at the Coulomb point the anti-Pfaffian is preferred over the Pfaffian state over a range of Landau level mixing up to the experimentally relevant values.

Experimental constraints and a possible quantum Hall state atν=5/2

Several topological orders have been proposed to explain the quantum Hall plateau at $\nu=5/2$. The observation of an upstream neutral mode on the sample edge [Bid et al., Nature (London) 466, 585 (2010)] supports the non-Abelian anti-Pfaffian state. On the other hand, the tunneling experiments [Radu et al., Science 320, 899 (2008); Lin et al., Phys. Rev. B 85, 165321 (2012); Baer et al., arXiv:1405.0428] favor the Halperin 331 state which exhibits no upstream modes. We find a topological order, compatible with the results of both types of experiments. That order allows both finite and zero spin polarizations. It is Abelian but its signatures in Aharonov-Bohm interferometry can be similar to those of the Pfaffian and anti-Pfaffian states.

Effects of Landau level mixing on the fractional quantum Hall effect in monolayer graphene.

We report results of exact diagonalization studies of the spin- and valley-polarized fractional quantum Hall effect in the N = 0 and N = 1 Landau levels in graphene. We use an effective model that incorporates Landau level mixing to lowest order in the parameter κ = ((e(2)/εℓ)/(ħv(F)/ℓ)) = (e(2)/εv(F)ħ), which is magnetic field independent and can only be varied through the choice of substrate. We find Landau level mixing effects are negligible in the N = 0 Landau level for κ ≲ 2. In fact, the lowest Landau level projected Coulomb Hamiltonian is a better approximation to the real Hamiltonian for graphene than it is for semiconductor based quantum wells. Consequently, the principal fractional quantum Hall states are expected in the N = 0 Landau level over this range of κ. In the N = 1 Landau level, fractional quantum Hall states are expected for a smaller range of κ and Landau level mixing strongly breaks particle-hole symmetry, producing qualitatively different results compared to the N = 0 Landau level. At half filling of the N = 1 Landau level, we predict the anti-Pfaffian state will occur for κ ∼ 0.25-0.75.

More realistic Hamiltonians for the fractional quantum Hall regime in GaAs and graphene

We construct an effective Hamiltonian for electrons in the fractional quantum Hall regime for GaAs and graphene that takes into account Landau level mixing (for both GaAs and graphene) and subband mixing (for GaAs, due to the nonzero width of the quantum well). This mixing has the important qualitative effect of breaking particle-hole symmetry as well as renormalizing the strength of the interparticle interactions. Both effects could have important consequences for the prospect that the fractional quantum Hall effect at $\nu=5/2$ is described by states that support non-Abelian excitations such as the Moore-Read Pfaffian or anti-Pfaffian states. For GaAs, Landau level and subband mixing break particle-hole symmetry in all Landau levels and subband mixing, due to finite thickness, causes additional short-distance softening of the Coulomb interaction, further renormalizing the Hamiltonian; additionally, the Landau level and subband energy spacings are comparable so it is crucial to consider both effects simultaneously. We find that in graphene, Landau level mixing only breaks particle-hole symmetry outside of the lowest Landau level ($N\neq0$). Landau level mixing is likely to be especially important in graphene since the Landau level mixing parameter is independent of the external magnetic field and is of order one. Our realistic Hamiltonians will serve as starting points for future numerical studies.

Distinguishing particle-hole conjugated fractional quantum Hall states using quantum-dot-mediated edge transport

We study theoretically edge transport of a fractional quantum Hall liquid, in the presence of a quantum dot inside the Hall bar with well controlled electron density and Landau level filling factor \nu, and show that such transport studies can help reveal the nature of the fractional quantum Hall liquid. In our first example we study the \nu=1/3 and \nu=2/3 liquids in the presence of a \nu=1 quantum dot. When the quantum dot becomes large, its edge states join those of the Hall bar to reconstruct the edge states configuration. Taking randomness around the edges into account, we find that in the disorder-irrelevant phase the two-terminal conductance of the original \nu=1/3 system vanishes at zero temperature, while that of the \nu=2/3 case remains finite. This distinction is rooted in the fact that the \nu=2/3 state is built upon the \nu=1 state. In the disorder-dominated phase, the two-terminal conductance of \nu=1/3 system is (1/5)\frac{e^2}{h} and that of \nu=2/3 system is (1/2)\frac{e^2}{h}. We further apply the same idea to the \nu=5/2 system which realizes either the Pfaffian or the anti-Pfaffian states. In this case we study the edge transport in the presence of a central \nu=3 quantum dot. If the quantum dot is large enough for its edge states joining those of the Hall bar, in the disorder-irrelevant phase the total two-terminal conductance in the Pfaffian case is G^{Pf}_{tot}\rightarrow 2 \frac{e^2}{h} while that of anti-Pfaffian case is higher but not universal, G^{aPf}_{tot}> 2 \frac{e^2}{h}. This difference can be used to determine which one of these two states is realized at \nu=5/2. In the disorder-dominated phase, however, the total two-terminal conductances in these two systems are exactly the same, G^{Pf/aPf}_{tot}=(7/3)\frac{e^2}{h}.

Quantum Phase Transitions and theν=5/2Fractional Hall State in Wide Quantum Wells

We study the nature of the ν=5/2 quantum Hall state in wide quantum wells under the mixing of electronic subbands and Landau levels. A general method is introduced to analyze the Moore-Read pfaffian state and its particle-hole conjugate, the anti-pfaffian state, under periodic boundary conditions in a "quartered" Brillouin zone scheme containing both even and odd numbers of electrons. By examining the rotational quantum numbers on the torus, we show spontaneous breaking of the particle-hole symmetry can be observed in finite-size systems. In the presence of electronic-subband and Landau-level mixing, the particle-hole symmetry is broken in such a way that the anti-pfaffian state is unambiguously favored, and becomes more robust in the vicinity of a transition to the compressible phase, in agreement with recent experiments.

Thermoelectric Probe for Neutral Edge Modes in the Fractional Quantum Hall Regime

The ν=5/2 anti-Pfaffian state and the ν=2/3 state are believed to have an edge composed of counterpropagating charge and neutral modes. This situation allows the generation of a pure thermal bias between two composite edge states across a quantum point contact as was experimentally established by Bid etal. [Nature 466, 585 (2010)]. We show that replacing the quantum point contact by a quantum dot provides a natural way for detecting the neutral modes via the dc current generated by the thermoelectric response of the dot. We also show that the degeneracies of the dot spectrum, dictated by the conformal field theories describing these states, induce asymmetries in the thermoelectric current peaks. This in turn provides a direct fingerprint of the corresponding conformal field theory.

Quasi-Particle Tunneling in Anti-Pfaffian Quantum Hall State

We study tunneling phenomena at the edge of the anti-Pfaffian quantum Hall state at the filling factor $\nu=5/2$. The edge current in a single point-contact is considered. We focus on nonlinear behavior of two-terminal conductance with the increase in negative split-gate voltage. Expecting the appearance of the intermediate conductance plateau we calculate the value of its conductance by using the renormalization group (RG) analysis. Further, we show that non-perturbative quasi-particle tunneling is effectively described as perturbative electron tunneling by the instanton method. The two-terminals conductance is written as a function of the gate voltage. The obtained results enable us to distinguish the anti-Pfaffian state from the Pfaffian state experimentally.

Characterizing Neutral Modes of Fractional States in the Second Landau Level

Fractionally charged quasiparticles, which obey non-abelian statistics, were predicted to exist in the ν=8/3, ν=5/2, and ν=7/3 fractional quantum Hall states (in the second Landau level). Here we present measurements of charge and neutral modes in these states. For both ν=7/3 and ν=8/3 states, we found a quasiparticle charge e=1/3 and an upstream neutral mode only in ν=8/3-excluding the possibility of non-abelian Read-Rezayi states and supporting Laughlin-like states. The absence of an upstream neutral mode in the ν=7/3 state also proves that edge reconstruction was not present in the ν=7/3 state, suggesting its absence also in ν=5/2 state, and thus may provide further support for the non-abelian anti-pfaffian nature of the ν=5/2 state.

Plasma analogy and non-Abelian statistics for Ising-type quantum Hall states

We study the non-Abelian statistics of quasiparticles in the Ising-type quantum Hall states which are likely candidates to explain the observed Hall conductivity plateaus in the second Landau level, most notably the one at filling fraction nu=5/2. We complete the program started in Nucl. Phys. B 506, 685 (1997) and show that the degenerate four-quasihole and six-quasihole wavefunctions of the Moore-Read Pfaffian state are orthogonal with equal constant norms in the basis given by conformal blocks in a c=1+1/2 conformal field theory. As a consequence, this proves that the non-Abelian statistics of the excitations in this state are given by the explicit analytic continuation of these wavefunctions. Our proof is based on a plasma analogy derived from the Coulomb gas construction of Ising model correlation functions involving both order and (at most two) disorder operators. We show how this computation also determines the non-Abelian statistics of collections of more than six quasiholes and give an explicit expression for the corresponding conformal block-derived wavefunctions for an arbitrary number of quasiholes. Our method also applies to the anti-Pfaffian wavefunction and to Bonderson-Slingerland hierarchy states constructed over the Moore-Read and anti-Pfaffian states.

Identification of 331 quantum Hall states with Mach-Zehnder interferometry

It has been shown recently that non-Abelian states and the spin-polarized and unpolarized versions of the Abelian 331 state may have identical signatures in Fabry-P\'erot interferometry in the quantum Hall effect at filling factor 5/2. We calculate the Fano factor for the shot noise in a Mach-Zehnder interferometer in the 331 states and demonstrate that it differs from the Fano factor in the proposed non-Abelian states. The Fano factor depends periodically on the magnetic flux through the interferometer. Its maximal value is $2\ifmmode\times\else\texttimes\fi{}1.4e$ for the 331 states with a symmetry between two flavors of quasiparticles. In the absence of such symmetry the Fano factor can reach $2\ifmmode\times\else\texttimes\fi{}2.3e$. On the other hand, for the Pfaffian and anti-Pfaffian states the maximal Fano factor is $2\ifmmode\times\else\texttimes\fi{}3.2e$. The period of the flux dependence of the Fano factor is one flux quantum. If only quasiparticles of one flavor can tunnel through the interferometer then the period drops to one half of the flux quantum. We also discuss transport signatures of a general Halperin state with the filling factor $2+k/(k+2)$.

Exotic resonant level models in non-Abelian quantum Hall states coupled to quantum dots

In this paper, we study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state. We assume the dot is small enough that its level spacing is large compared to both the temperature and the coupling to the spatially proximate bulk non-Abelian fractional quantum Hall state. We focus on the physics of level degeneracy with electron number on the dot. The physics of such a resonant level is governed by a $k$-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction $\ensuremath{\nu}=2+k/(k+2)$ or its particle-hole conjugate at $\ensuremath{\nu}=2+2/(k+2)$. The $k$-channel Kondo model is channel symmetric even without fine tuning any couplings in the former state; in the latter, it is generically channel asymmetric. The two limits exhibit non-Fermi-liquid and Fermi-liquid properties, respectively, and therefore may be distinguished. By exploiting the mapping between the resonant level model and the multichannel Kondo model, we discuss the thermodynamic and transport properties of the system. In the special case of $k=2$, our results provide a distinct venue to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction $\ensuremath{\nu}=5/2$. We present numerical estimates for realizing this scenario in experiment.

Transport in line junctions ofν=52quantum Hall liquids

We calculate the tunneling current through long-line junctions of a $\ensuremath{\nu}=\frac{5}{2}$ quantum Hall liquid and (i) another $\ensuremath{\nu}=\frac{5}{2}$ liquid, (ii) an integer quantum Hall liquid, and (iii) a quantum wire. Momentum-resolved tunneling provides information about the number, propagation directions, and other features of the edge modes and thus helps distinguish several competing models of the 5/2 state. We investigate transport properties of two proposed Abelian states: $K=8$ and 331 state, and four possible non-Abelian states: Pfaffian, edge-reconstructed Pfaffian, and two versions of the anti-Pfaffian state. We also show that the nonequilibrated anti-Pfaffian state has a different resistance from other proposed states in the bar geometry.

Particle-hole symmetry breaking and theν=52fractional quantum Hall effect

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.