Paired Quantum Hall States Research Articles

Paired parton quantum Hall states: A coupled wire construction

The Pfaffian fractional quantum Hall (FQH) states are incompressible non-Abelian topological fluids present in a half-filled electron Landau level, where there is a balanced population of electrons and holes. They give rise to half-integral quantum Hall plateaus that divide critical transitions between integer quantum Hall (IQH) states. On the other hand, there are Abelian FQH states, such as the Laughlin state, that can be understood using partons, which are fermionic divisions of the electron. In this paper, we propose a new family of incompressible paired parton FQH states at filling $\nu=1/6$ (modulo 1) that emerge from critical transitions between IQH states and Abelian FQH states at filling $\nu=1/3$ (modulo 1). These paired parton states are originated from a half-filled parton Landau level, where there is an equal amount of partons and holes. They generically support Ising-like anyonic quasiparticle excitations and carry non-Abelian Pfaffian topological orders (TO) for partons. We prove the principle existence of these paired parton states using exactly solvable interacting arrays of electronic wires under a magnetic field. Moreover, we establish a new notion of particle-hole (PH) symmetry for partons and relate the PH symmetric parton Pfaffian TO with the gapped symmetric surface TO of a fractional topological insulator in three dimension.

Bilayer mapping of the paired quantum Hall state: Instability toward anisotropic pairing

One of the most dominant candidates for the paired quantum Hall (QH) state at filling factor $\ensuremath{\nu}=5/2$ is the Moore-Read (MR) Pfaffian state. A salient problem, however, is that it does not occur exactly at the Coulomb interaction, but rather at a modified interaction, which favors particle-hole-symmetry breaking. In an effort to find a better state, in this work, we investigate the possible connection between the paired QH state and the antisymmetrized bilayer ground state, which is inspired by the intriguing identity that the MR Pfaffian state is entirely equivalent to the antisymmetrized projection of the bilayer QH state called the Halperin (331) state, which is valid at interlayer distance, $d$, roughly equal to the magnetic length, ${l}_{\mathrm{B}}$. Specifically, by using exact diagonalization in the torus geometry, we show that the exact $5/2$ state at a given Haldane pseudopotential variation is intimately connected with the antisymmetrized bilayer ground state at a corresponding $d/{l}_{\mathrm{B}}$ via one-to-one mapping, which we call the bilayer mapping. One of the most important discoveries in this work is that the paired QH state occurring at the Coulomb interaction is mapped onto the antisymmetrized bilayer ground state at $d\ensuremath{\gg}{l}_{\mathrm{B}}$, which is equivalent to the antisymmetrized product state of two composite fermion seas at quarter filling, not the MR Pfaffian state. While maintaining high overlap with the paired QH state, the antisymmetrized bilayer ground state at $d\ensuremath{\gg}{l}_{\mathrm{B}}$ exhibits an abrupt change under the influence of small anisotropy. This suggests that the paired QH state occurring at the Coulomb interaction might be susceptible to anisotropic instability, opening up the possibility of anisotropic ${p}_{x}$- or ${p}_{y}$-wave pairing instead of ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$-wave pairing in the MR Pfaffian/anti-Pfaffian state.

Dynamical mass generation of composite Dirac fermions and fractional quantum Hall effects near charge neutrality in graphene

We develop a composite Dirac fermion theory for the fractional quantum Hall effects (QHE) near charge neutrality in graphene. We show that the interactions between the composite Dirac fermions lead to a dynamical mass generation through exciton condensation. The four-fold spin–valley degeneracy is fully lifted due to the mass generation and exchange effects such that the odd-denominator fractional QHE observed in the vicinity of charge neutrality can be understood in terms of the integer QHE of composite Dirac fermions. At a filling factor ν = 1/2, we show that the massive composite Dirac fermion liquid is unstable against chiral p-wave pairing for weak Coulomb interactions and the ground state is a paired non-Abelian quantum Hall state described by the Moore–Read Pfaffian in the long wavelength limit.

Microscopic Model of Quasiparticle Wave Packets in Superfluids, Superconductors, and Paired Hall States

We study the structure of Bogoliubov quasiparticles, bogolons, the fermionic excitations of paired superfluids that arise from fermion (BCS) pairing, including neutral superfluids, superconductors, and paired quantum Hall states. The naive construction of a stationary quasiparticle in which the deformation of the pair field is neglected leads to a contradiction: it carries a net electrical current even though it does not move. However, treating the pair field self-consistently resolves this problem: in a neutral superfluid, a dipolar current pattern is associated with the quasiparticle for which the total current vanishes. When Maxwell electrodynamics is included, as appropriate to a superconductor, this pattern is confined over a penetration depth. For paired quantum Hall states of composite fermions, the Maxwell term is replaced by a Chern-Simons term, which leads to a dipolar charge distribution and consequently to a dipolar current pattern.

Quantum phase transitions and coherent tunneling in a bilayer of ultracold atoms with dipole interactions

We study the quantum phase transitions and coherent tunneling in bilayer ultracold bosonic atoms with dipolar interactions at total filling factor ν T = 1/2. We demonstrate that, due to the unique characters of the dipole interactions, the ground state of the system may undergo paired quantum Hall state (PQH), dimer quantum-Hall state (DQH), and super-counter-fluidity phase (SCF) by tuning the dipole strength. The SCF state which breaks the easy-plane U(1) symmetry is of particular interest because it can exhibit interlayer coherent tuneling in the incompressible quantum-Hall (QH) state. We suggest that this coherent tunneling current could be observed directly in future experiments.

Gapless excitations in the Haldane-Rezayi state: The thin-torus limit

We study the thin torus limit of the Haldane-Rezayi state. Eight of the ten ground states are found to assume a simple product form in this limit, as is known to be the case for many other quantum Hall trial wave functions. The two remaining states have a somewhat unusual thin torus limit, where a "broken" pair of defects forming a singlet is completely delocalized. We derive these limits from the wave functions on the cylinder, and deduce the dominant matrix elements of the thin torus hollow-core Hamiltonians. We find that there are gapless excitations in the thin torus limit. This is in agreement with the expectation that local Hamiltonians stabilizing wave functions associated with non-unitary conformal field theories are gapless. We also use the thin torus analysis to obtain explicit counting formulas for the zero modes of the hollow-core Hamiltonian on the torus, as well as for the parent Hamiltonians of several other paired and related quantum Hall states.

Correlation-Hole Induced Paired Quantum Hall States in the Lowest Landau Level

A theory is developed for the paired even-denominator fractional quantum Hall states in the lowest Landau level. We show that electrons bind to quantized vortices to form composite fermions, interacting through an exact instantaneous interaction that favors chiral p-wave pairing. There are two canonically dual pairing gap functions related by the bosonic Laughlin wave function (Jastrow factor) due to the correlation holes. We find that the ground state is the Moore-Read Pfaffian in the long-wavelength limit for weak Coulomb interactions, a new Pfaffian with an oscillatory pairing function for intermediate interactions, and a Read-Rezayi composite Fermi liquid beyond a critical interaction strength. Our findings are consistent with recent experimental observations of the 1/2 and 1/4 fractional quantum Hall effects in asymmetric wide quantum wells.

Paired quantum Hall states on noncommutative two-tori

By exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter θ (in appropriate units), a one-to-one correspondence between an Abelian noncommutative field theory (NCFT) and a non-Abelian theory of twisted fields on ordinary space can be established. Starting from this general result, we focus on the conformal field theory (CFT) describing a quantum Hall fluid (QHF) at paired states fillings ν = m p m + 2 Cristofano et al. (2000) [1], recently obtained by means of m-reduction procedure, and show that it is the Morita equivalent of a NCFT. In this way we extend the construction proposed in Marotta and Naddeo (2008) [2] for the Jain series ν = m 2 p m + 1 . The case m = 2 is explicitly discussed and the role of noncommutativity in the physics of quantum Hall bilayers is emphasized. Our results represent a step forward the construction of a new effective low energy description of certain condensed matter phenomena and help to clarify the relationship between noncommutativity and quantum Hall fluids.

Spin Order in Paired Quantum Hall States

We consider quantum Hall states at even-denominator filling fractions, especially nu=5/2, in the limit of small Zeeman energy. Assuming that a paired quantum Hall state forms, we study spin ordering and its interplay with pairing. We give numerical evidence that at nu=5/2 an incompressible ground state will exhibit spontaneous ferromagnetism. The Ginzburg-Landau (GL) theory for the spin degrees of freedom of paired Hall states is a perturbed CP2 model. We compute the coefficients in the GL theory by a BCS Stoner mean-field theory for coexisting order parameters, and show that even if repulsion is smaller than that required for a Stoner instability, ferromagnetic fluctuations can induce a partially or fully polarized superconducting state.

Particle-Hole Symmetry and theν=52Quantum Hall State

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 state is favored for the moderate Landau-level mixing which is present in experiments. We discuss the bulk and edge physics of the anti-Pfaffian. We analyze a simplified model in which transitions between analogs of the two states can be studied in detail. Finally, we discuss experimental implications.

Wavefunctions for topological quantum registers

We present explicit wavefunctions for quasi-hole excitations over a variety of non-abelian quantum Hall states: the Read–Rezayi states with k ⩾ 3 clustering properties and a paired spin-singlet quantum Hall state. Quasi-holes over these states constitute a topological quantum register, which can be addressed by braiding quasi-holes. We obtain the braid properties by direct inspection of the quasi-hole wavefunctions. We establish that the braid properties for the paired spin-singlet state are those of ‘Fibonacci anyons’, and thus suitable for universal quantum computation. Our derivations in this paper rely on explicit computations in the parafermionic conformal field theories that underly these particular quantum Hall states.

Parafermion statistics and application to non-Abelian quantum Hall states

The (exclusion) statistics of parafermions is used to study degeneracies of quasiholes over the paired (or in general clustered) quantum Hall states. Focus is on the Z_k and su(3)_k/u(1)^2 parafermions, which are used in the description of spin-polarized and spin-singled clustered quantum Hall states.

Separation of spin and charge in paired spin-singlet quantum Hall states

We propose a series of paired spin-singlet quantum Hall states, which exhibit a separation of spin and charge degrees of freedom. The fundamental excitations over these states, which have filling fraction $\ensuremath{\nu}=2/(2m+1)$ with m an odd integer, are spinons (spin-$\frac{1}{2}$ and charge zero) or fractional holons (charge $\ifmmode\pm\else\textpm\fi{}1/(2m+1)$ and spin zero). The braid statistics of these excitations are non-Abelian. The mechanism for the separation of spin and charge in these states is topological: spin and charge excitations are liberated by binding to a vortex in a p-wave pairing condensate. We briefly discuss related, Abelian spin-singlet states and possible transitions.

Strong Quasi-Particle Tunneling Study in the Paired Quantum Hall States

The quasi-particle tunneling phenomena in the paired fractional quantum Hall states are studied. A single point-contact system is first considered. Because of relevancy of the quasi-particle tunneling term, the strong tunneling regime should be investigated. Using the instanton method it is shown that the strong quasi-particle tunneling regime is described as the weak electron tunneling regime effectively. Expanding to the network model the paired quantum Hall liquid to insulator transition is discussed.

Pairing Symmetry Transitions in the Even-Denominator FQHE System

Transitions from a paired quantum Hall state to another quantum Hall state in bilayer systems are discussed in the framework of the edge theory. Starting from the edge theory for the Haldane-Rezayi state, it is shown that the charging effect of a bilayer system which breaks the SU(2) symmetry of the pseudo-spin shifts the central charge and the conformal dimensions of the fermionic fields which describe the pseudo-spin sector in the edge theory. This corresponds to the transition from Haldane-Rezayi state to Halperin's 331 state, or singlet d-wave to triplet p-wave ABM type paired state in the composite fermion picture. Considering interlayer tunneling, the tunneling rate-capacitance phase diagram for the $\nu=5/2$ paired bilayer system is discussed.

Absence of a metallic phase in random-bond Ising models in two dimensions: Applications to disordered superconductors and paired quantum Hall states

When the two-dimensional random-bond Ising model is represented as a noninteracting fermion problem, it has the same symmetries as an ensemble of random matrices known as class D. A nonlinear sigma model analysis of the latter in two dimensions has previously led to the prediction of a metallic phase, in which the fermion eigenstates at zero energy are extended. In this paper we argue that such behavior cannot occur in the random-bond Ising model, by showing that the Ising spin correlations in the metallic phase violate the bound on such correlations that results from the reality of the Ising couplings. Some types of disorder in spinless or spin-polarized p-wave superconductors and paired fractional quantum Hall states allow a mapping onto an Ising model with real but correlated bonds, and hence a metallic phase is not possible there either. It is further argued that vortex disorder, which is generic in the fractional quantum Hall applications, destroys the ordered or weak-pairing phase, in which nonabelian statistics is obtained in the pure case.

Persistent edge currents for paired quantum Hall states

We study the behavior of the persistent edge current for paired quantum Hall states on the cylinder. We show that the currents are periodic with the unit flux $\phi_0=hc/e$. At low temperatures, they exhibit anomalous oscillations in their flux dependence.The shape of the functions converges to the sawtooth function periodic with $\phi_0/2$.

Incompressible paired hall state, stripe order, and the composite fermion liquid phase in half-filled landau levels

We consider the two lowest Landau levels at half filling. In the higher Landau level (nu = 5/2), we find a first-order phase transition separating a compressible striped phase from a paired quantum Hall state, which is identified as the Moore-Read state. The critical point is very near the Coulomb potential and the transition can be driven by increasing the width of the electron layer. We find a much weaker transition (either second-order or a crossover) from pairing to the composite fermion Fermi-liquid behavior. A very similar picture is obtained for the lowest Landau level, but the transition point is not near the Coulomb potential.

Parafermion statistics and quasihole excitations for the generalizations of the paired quantum Hall states

We continue the program started in cond-mat/9809384 and explain the statistics of the excitations for the generalizations of the paired states in the quantum Hall effect in terms of the parafermion statistics. We show that these excitations behave as combinations of bosons and parafermions. That generalizes the prior treatment of the paired (Pfaffian) state where the excitations behave as combinations of bosons and fermions. We explain what it means, from a quantum mechanical point of view, for a particle to be a `parafermion' rather than a boson or a fermion and work through several explicit examples. The resulting multiplets coincide exactly with the angular momentum multiplets found numerically for the $k+1$ particle interaction Hamiltonian on a sphere. We also present a proof that the wave functions found in cond-mat/9809384 are indeed the correlation functions of the parafermion conformal field theory.

Coulomb drag as a signature of the paired quantum Hall state

Motivated by the recent Coulomb drag experiment of M. P. Lilly et. al, we study the Coulomb drag in a two-layer system with Landau level filling factor $\nu=1/2$. We find that the drag conductivity in the incompressible paired quantum Hall state at zero temperature can be finite. The drag conductivity is also greatly enhanced above $T_c$, at which the transition between the weakly coupled compressible liquids and the paired quantum Hall liquid takes place. We discuss the implications of our results for the recent experiment.