Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction $ν={5/2}$ fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from $g=1/4$ to $g=1/2$ for the edge reconstructed Pfaffian state, and changes from $g=1/2$ to $g=0.55-0.75$ for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed $ν={5/2}$ state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.
We present a low-energy effective field theory describing the universality class of the Pfaffian quantum Hall state. To arrive at this theory, we observe that the edge theory of the Pfaffian state of bosons at $ν=1$ is an $SU(2)_2$ Kac-Moody algebra. It follows that the corresponding bulk effective field theory is an SU(2) Chern-Simons theory with coupling constant $k=2$. The effective field theories for other Pfaffian states, such as the fermionic one at $ν=1/2$ are obtained by a flux-attachment procedure. We discuss the non-Abelian statistics of quasiparticles in the context of this effectiv…
We present a method for the exact construction of the fully particle-hole symmetric PH-Pfaffian ground state and its charged excitations on a sphere. We adopt the Moore-Read state, but with a nonholomorphic pairing component as in previous studies, and project it to the lowest Landau level. We study the energetics as well as other properties of these states and find that in a pure system interacting with the Coulomb forces the PH-Pfaffian cannot compete with either the Moore-Read state or its particle-hole conjugate, the anti-Pfaffian state, as an explanation for the 5/2-effect.
We discuss the implications of approximate particle-hole symmetry in a half-filled Landau level in which a paired quantum Hall state forms. We note that the Pfaffian state is not particle-hole symmetric. Therefore, in the limit of vanishing Landau level mixing, in which particle-hole transformation is an exact symmetry, the Pfaffian spontaneously breaks this symmetry. There is a particle-hole conjugate state, which we call the anti-Pfaffian, which is degenerate with the Pfaffian in this limit. We observe that strong Landau level mixing should favor the Pfaffian, but it is an open problem which…
Coupled atom-cavity arrays, such as those described by the Jaynes-Cummings Hubbard model, have the potential to emulate a wide range of condensed matter phenomena. In particular, the strongly correlated states of the fractional quantum Hall effect can be realised. At some filling fractions, the fraction quantum Hall effect has been shown to possess ground states with non-abelian excitations. The most well studied of these states is the Pfaffian state of Moore and Read, which is the groundstate of a Hall Liquid with a 3-body interaction. In this paper we show how an effective 3-body interaction…
The essence of the $ν=5/2$ fractional quantum Hall effect is believed to be well captured by the Moore-Read Pfaffian (or anti-Pfaffian) description. However, an important mystery regarding the formation of the Pfaffian state is the role of the three-body interaction Hamiltonian $H_3$ that produces the Pfaffian as an exact ground state and the concomitant particle-hole symmetry breaking. We show that a two-body interaction Hamiltonian $H_2$ constructed via particle-hole symmetrization of $H_3$ produces a ground state nearly exactly approximating the Pfaffian and anti-Pfaffian states, respective…
We study non-equilibrium noise in the tunnelling current between the edges of a quantum Hall liquid in the Pfaffian state, which is a strong candidate for the plateau at $ν=5/2$. To first non-vanishing order in perturbation theory (in the tunneling amplitude) we find that one can extract the value of the fractional charge of the tunnelling quasiparticles. We note however that no direct information about non-abelian statistics can be retrieved at this level. If we go to higher-order in the perturbative calculation of the non-equilibrium shot noise, we find effects due to non-Abelian statistics.…
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 ($p$-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 an…
The Pfaffian model has been proposed for the fractional quantum Hall effect (FQHE) at nu=5/2. We examine it for the quasihole excitations by comparison with exact diagonalization results. Specifically, we consider the structure of the low-energy spectrum, accuracy of the microscopic wave functions, particle-hole symmetry, splitting of the degeneracies, and off-diagonal long range order. We also review how the 5/2 FQHE can be understood without appealing to the Pfaffian model. Implications for nonabelian braiding statistics will be mentioned.
It is demonstrated that an understanding of the 5/2 fractional quantum Hall effect can be achieved within the composite fermion theory without appealing to the Pfaffian wave function. The residual interaction between composite fermions plays a crucial role in establishing incompressibility at this filling factor. This approach has the advantage of being amenable to systematic perturbative improvements, and produces ground as well as excited states. It, however, does not relate to non-Abelian statistics in any obvious manner.
We study tunneling phenomena at the edge of the anti-Pfaffian quantum Hall state at the filling factor $ν=5/2$. 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-termina…
We find the finite-width, i.e., the layer thickness, of experimental quasi-two dimensional systems produces a physical environment sufficient to stabilize the Moore-Read Pfaffian state thought to describe the fractional quantum Hall effect at filling factor $ν=5/2$. This conclusion is based on exact calculations performed in the spherical and torus geometries, studying wavefunction overlap and ground state degeneracy
The $ν=5/2$ 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 bac…
Inter-Landau-level transitions break particle hole symmetry and will choose either the Pfaffian or the anti-Pfaffian state as the absolute ground state at 5/2 filling of the fractional quantum Hall effect. An approach based on truncating the Hilbert space has favored the anti-Pfaffian. A second approach based on an effective Hamiltonian produced the Pfaffian. In this letter perturbation theory is applied to finite sizes without bias to any specific pseudo-potential component. This method also singles out the anti-Pfaffian. A critical piece of the effective Hamiltonian, which was absent in prev…
We consider the properties of the Moore-Read Pfaffian state under particle-hole conjugation. We show that the particle-hole conjugate of the Pfaffian state - or "anti-Pfaffian" state - is in a different universality class from the Pfaffian state, with different topological order. The two states can be distinguished by both their bulk and edge physics though the difference is most dramatic at the edge: the edge of the anti-Pfaffian state has a composite structure that leads to a different thermal Hall conductance and different tunneling exponents than the Pfaffian state. At the same time, the t…
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 dis…
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 the clear evidence of a direct sharp transition (possibly first-order) from the Pfaffian to the anti-Pfaffian state at the…
We calculate the electron spectral functions at the edges of the Moore-Read Pfaffian and anti-Pfaffian fractional quantum Hall states, in the clean limit. We show that their qualitative differences can be probed using momentum resolved tunneling, thus providing a method to unambiguously distinguish which one is realized in the fractional quantum Hall state observed at filling factor $ν=5/2$. We further argue that edge reconstruction, which may be less important in the first excited Landau level (LL) than in the lowest LL, can also be detected this way if present.
Several topological orders have been proposed to explain the quantum Hall plateau at $ν=5/2$. 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…
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 consi…
We extend the topological quantum computation scheme using the Pfaffian quantum Hall state, which has been recently proposed by Das Sarma et al., in a way that might potentially allow for the topologically protected construction of a universal set of quantum gates. We construct, for the first time, a topologically protected Controlled-NOT gate which is entirely based on quasihole braidings of Pfaffian qubits. All single-qubit gates, except for the pi/8 gate, are also explicitly implemented by quasihole braidings. Instead of the pi/8 gate we try to construct a topologically protected Toffoli ga…
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 PH-Pfaffian topological order by Son is consistent with all experiments. The absence of the 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.
Since the charged mode is much faster than the neutral modes on quantum Hall edges at large filling factors, the edge may remain out of equilibrium in thermal conductance experiments. This sheds light on the observed imperfect quantization of the thermal Hall conductance at $ν=8/3$ and can increase the observed thermal conductance by two quanta at $ν=8/5$. Under certain unlikely but not impossible assumptions, this might also reconcile the observed thermal conductance at $ν=5/2$ with not only the PH-Pfaffian order but also the anti-Pfaffian order.
We calculate the tunneling current through long line junctions of a $ν=5/2$ quantum Hall liquid and i) another $ν=5/2$ 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 state and 331 state, and four possible non-Abelian states: Pfaffian, edge-reconstructed Pfaffian, and two versions of the anti-Pfaffian state. …
Through exact numerical diagonalization, the von Neumann entropy is calculated for the Laughlin and Pfaffian quantum Hall states in rotating interacting Bose gases at zero temperature in the lowest Landau level limit. The particles comprising the states are indistinguishable, so the required spatial bipartitioning is effected by tracing over a subset of single-particle orbitals. The topological entropy is then extracted through a finite-size scaling analysis. The results for the Laughlin and the Pfaffian states agree with the expected values of $\ln\sqrt{2}$ and $\ln\sqrt{4}$, respectively.
The Pfaffian state, which may describe the quantized Hall plateau observed at Landau level filling fraction $ν= 5/2$, can support topologically-protected qubits with extremely low error rates. Braiding operations also allow perfect implementation of certain unitary transformations of these qubits. However, in the case of the Pfaffian state, this set of unitary operations is not quite sufficient for universal quantum computation (i.e. is not dense in the unitary group). If some topologically unprotected operations are also used, then the Pfaffian state supports universal quantum computation, al…
We compute the tunneling current in a double point contact geometry of a Quantum Hall system at filling fraction $ν=5/2$, as function of voltage and temeprature, in the weak tunneling regime. We quantitatively compare two possible candidates for the state at $ν=5/2$: the Moore-Read Pfaffian state, and its particle-hole conjugate, the anti-Pfaffian. We find that both possibilities exhibit the same qualitative behavior, and both have an even-odd effect that reflects their non-Abelian nature, but differ quantitatively in their voltage and temperature dependance.
We present an approach to the computation of the non-Abelian statistics of quasiholes in quantum Hall states, such as the Pfaffian state, whose wavefunctions are related to the conformal blocks of minimal model conformal field theories. We use the Coulomb gas construction of these conformal field theories to formulate a plasma analogy for the quantum Hall states. A number of properties of the Pfaffian state follow immediately, including the Berry phases which demonstrate the quasiholes' fractional charge, the abelian statistics of the two-quasihole state, and equal-time ground state correlatio…
We analyze the Moore-Read Pfaffian state on a thin torus. The known six-fold degeneracy is realized by two inequivalent crystalline states with a four- and two-fold degeneracy respectively. The fundamental quasihole and quasiparticle excitations are domain walls between these vacua, and simple counting arguments give a Hilbert space of dimension $2^{n-1}$ for $2n-k$ holes and $k$ particles at fixed positions and assign each a charge $\pm e/4$. This generalizes the known properties of the hole excitations in the Pfaffian state as deduced using conformal field theory techniques. Numerical calcul…
When a gas of electrons is confined to two dimensions, application of a strong magnetic field may lead to startling phenomena such as emergence of electron pairing. According to a theory this manifests itself as appearance of the fractional quantum Hall effect with a quantized conductivity at an unusual half-integer nu=5/2 Landau level filling. Here we show that similar electron pairing may occur in quantum dots where the gas of electrons is trapped by external electric potentials into small quantum Hall droplets. However, we also find theoretical and experimental evidence that, depending on t…
We present a pure Chern-Simons formulation of families of interesting Conformal Field Theories describing edge states of non-Abelian Quantum Hall states. These theories contain two Abelian Chern-Simons fields describing the electromagnetically charged and neutral sectors of these models, respectively. The charged sector is the usual Abelian Chern-Simons theory that successfully describes Laughlin-type incompressible fluids. The neutral sector is a 2+1-dimensional theory analogous to the 1+1-dimensional orbifold conformal field theories. It is based on the gauge group O(2) which contains a ${\b…
The recent measurement of a half-integer thermal conductance for the $ν=5/2$ fractional quantum Hall state has confirmed its non-Abelian nature, making the question of the underlying topological order highly intriguing. We analyze the shot noise at the edge of the three most prominent non-Abelian candidate states. We show that the noise scaling with respect to the edge length can, in combination with the thermal conductance, be used to experimentally distinguish between the Pfaffian, anti-Pfaffian, and particle-hole-Pfaffian edge structures.
An important development in the field of the fractional quantum Hall effect has been the proposal that the 5/2 state observed in the Landau level with orbital index $n = 1$ of two dimensional electrons in a GaAs quantum well originates from a chiral $p$-wave paired state of composite fermions which are topological bound states of electrons and quantized vortices. This state is theoretically described by a "Pfaffian" wave function or its hole partner called the anti-Pfaffian, whose excitations are neither fermions nor bosons but Majorana quasiparticles obeying non-Abelian braid statistics. This…
We study the tunneling current between edge states of quantum Hall liquids across a single long contact region, and predict a resonance at a bias voltage set by the scale of the edge velocity. For typical devices and edge velocities associated with charged modes, this resonance occurs outside the physically accessible bias domain. However, for edge states that are expected to support neutral modes, such as the $ν={2/3}$, and $ν={5/2}$ Pfaffian and anti-Pfaffian states, the neutral velocity can be orders of magnitude smaller than the charged mode and if so the resonance would be accessible. The…
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 [Banerjee et al., Nature 559, 205 (2018)] at the ever enigmatic $ν=5/2$ 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 s…
It was recently pointed out that Halperin's 113 topological order explains the transport experiments in the quantum Hall liquid at filling factor $ν=5/2$. The 113 order, however, cannot be easily distinguished from other likely topological orders at $ν=5/2$ 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 $ν=5/2$ states. Specifically, the $I$-$V$ curve for the tunneling current through the interferometer is more asy…
Nature of the fractional quantum Hall state at Landau level filling factor 5/2 remains elusive despite intensive experimental and theoretical work. While the leading theoretical candidates are Moore-Read Pfaffian (Pf) and its particle-hole conjugate anti-Pfaffian (APf), neither received unambiguous experimental support. We show that a state that is intermediate between them, made of alternating stripes of Pf and APf in the bulk, is a viable candidate. Such a state is shown to be incompressible and thus a charge insulator in the bulk, but a heat conductor due to the presence of gapless neutral …
Several states were proposed as candidates for the $ν=5/2$ quantum Hall plateau. We suggest an experiment which can determine the physical state. The proposal involves transport measurements in the geometry with three quantum Hall edges connected by two quantum point contacts. In contrast to interference experiments, this approach can distinguish the Pfaffian and anti-Pfaffian states as well as different states with identical Pfaffian or anti-Pfaffian statistics. The transport is not sensitive to the fluctuations of the number of the quasiparticles trapped in the system.
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 …
While the simplest quantum Hall plateaus, such as the $ν= 1/3$ 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 $ν= 5/2$, where scattering into higher Landau levels chooses between the competing non-Abelian Pfaffian and anti-Pfaffian states. We address…
In search of states with non-Abelian statistics, we explore the fractional quantum Hall effect in a system of two-dimensional charge carrier holes. We propose a new method of mapping states of holes confined to a finite width quantum well in a perpendicular magnetic field to states in a spherical shell geometry. This method provides single-particle hole states used in exact diagonalization of systems with a small number of holes in the presence of Coulomb interactions. An incompressible fractional quantum Hall state emerges in a hole liquid at the half-filling of the ground state in a magnetic…
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau level filling factor 1/2 bilayer problem and make new predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong tunneling regime. In th…
The PH-Pfaffian topological order has been proposed as a candidate order for the $ν=5/2$ quantum Hall effect. The PH-Pfaffian liquid is known to be the ground state in several coupled wire and coupled stripe constructions. No translationally and rotationally invariant models with the PH-Pfaffian ground state have been identified so far. By employing anyon condensation on top of a topological order, allowed in an isotropic system, we argue that the PH-Pfaffian order is possible in the presence of rotational and translational symmetries.
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. We introduce a general method to analyze the Moore-Read Pfaffian state and its particle-hole conjugate, the anti-Pfaffian, under periodic boundary conditions in a "quartered" Brillouin zone scheme containing both even and odd numbers of electrons. We examine the rotational quantum numbers on the torus, and 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 t…
We consider electrical and thermal equilibration of the edge modes of the Anti-Pfaffian quantum Hall state at $ν=5/2$ due to tunneling of the Majorana edge mode to trapped Majorana zero modes in the bulk. Such tunneling breaks translational invariance and allows scattering between Majorana and other edge modes in such a way that there is a parametric difference between the length scales for equilibration of charge and heat transport between integer and Bose mode on the one hand, and for thermal equilibration of the Majorana edge mode on the other hand. We discuss a parameter regime in which th…
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 conjug…
Results from exact diagonalization show that the spin-polarized Coulomb ground state at nu=5/2 is adiabatically connected with the Moore-Read wave function for systems with up to Nel = 16 electrons on the surface of a sphere. Varying the Haldane pseudopotentials v1 and v3, keeping all others at their value for Coulomb interaction, energy gap and (v1,v3)-dependent ground state overlap with the Moore-Read state form two ridges whose peak positions and extent in the (v1,v3)-plane essentially coincide. Thus, the physics of the Coulomb ground state at nu=5/2 is indeed captured by the Moore-Read sta…
The quantum Hall phase diagram of the half-filled bilayer system in the second Landau level is studied as a function of tunneling and layer separation using exact diagonalization. We make the striking prediction that bilayer structures would manifest two distinct branches of incompressible fractional quantum Hall effect (FQHE) corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the quantum phase transition between them w…
We numerically study a 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 states wave functions, which is consistent with the low-lying …
Although numerical studies modeling the quantum hall effect at filling fraction $5/2$ predict either the Pfaffian (Pf) or its particle hole conjugate, the anti-Pfaffian (aPf) state, recent experiments appear to favor a quantized thermal hall conductivity with quantum number $K=5/2$ , rather than the value $K=7/2$ or $K=3/2$ expected for the Pf or aPF state, respectively. While a particle hole symmetric topological order (the PH-Pfaffian) would be consistent with the experiments, this state is believed to be energetically unfavorable in a homogenous system. Here we study the effects of disorder…
The Pfaffian quantum Hall states, which can be viewed as involving pairing either of spin-polarized electrons or of composite fermions, are generalized by finding the exact ground states of certain Hamiltonians with k+1-body interactions, for all integers k > 0. The remarkably simple wavefunctions of these states involve clusters of k particles, and are related to correlators of parafermion currents in two-dimensional conformal field theory. The k=2 case is the Pfaffian. For k > 1, the quasiparticle excitations of these systems are expected to possess nonabelian statistics, like those of the P…
Topological states of matter, such as fractional quantum Hall states, are an active field of research due to their exotic excitations. In particular, ultracold atoms in optical lattices provide a highly controllable and adaptable platform to study such new types of quantum matter. However, finding a clear route to realize non-Abelian quantum Hall states in these systems remains challenging. Here we use the density-matrix renormalization-group (DMRG) method to study the Hofstadter-Bose-Hubbard model at filling factor $ν= 1$ and find strong indications that at $α=1/6$ magnetic flux quanta per pl…
Certain well known quantum Hall states -- including the Laughlin states, the Moore-Read Pfaffian, and the Read-Rezayi Parafermion states -- can be defined as the unique lowest degree symmetric analytic function that vanishes as at least p powers as some number (g+1) of particles approach the same point. Analogously, these same quantum Hall states can be generated as the exact highest density zero energy state of simple angular momentum projection operators. Following this theme we determine the highest density zero energy state for many other values of p and g.
Landau level mixing plays an important role in the Pfaffian (or anti-Pfaffian) states. In ZnO the Landau level gap is essentially an order of magnitude smaller than that in a GaAs quantum well. We introduce the screened Coulomb interaction in a single Landau level to tackle that situation. Here we study the overlap of the ground state and the Pfaffian (or anti-Pfaffian) state at evendenominator fractional quantum Hall (FQH) states present in ZnO. The overlap is strongly system size-dependent which suggests a newly proposed particle-hole symmetry Pfaffian ground state in the extreme Landau leve…
