Read Rezayi Research Articles

Braiding properties of paired spin-singlet and non-Abelian hierarchy states

We study explicit model wave functions describing the fundamental quasiholes in a class of non-Abelian fractional quantum Hall states. This class is a family of paired spin-singlet states with internal degrees of freedom. We determine the braid statistics of the quasiholes by determining the monodromy of the explicit quasihole wave functions, that is how they transform under exchanges of quasihole coordinates. The statistics is shown to be the same as that of the quasiholes in the Read–Rezayi states, up to a phase. We also discuss the application of this result to a class of non-Abelian hierarchy wave functions.

Parton construction of particle-hole-conjugate Read-Rezayi parafermion fractional quantum Hall states and beyond

The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling $\nu = k/(k+2)$. Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted "$\bar{2}^{k}1^{k+1}$," and show that they lie in the same universality classes as the particle-hole-conjugate RR ("anti-RR") states. Our analytical results imply that a $(U(1)_{k+1} \times U(2k)_{-1})/(SU(k)_{-2} \times U(1)_{-1})$ coset conformal field theory describes the edge excitations of the $\bar{2}^{k}1^{k+1}$ state, suggesting non-trivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence "$\bar{n}\bar{2}^{2}1^{4}$," with $n=1,2,3$, to describe the FQH states observed at $\nu=2+1/2,~2+2/5$ and $2+3/8$.

Coupled wire model of Z4 orbifold quantum Hall states

We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors \nu = 2/(2m-p) for odd integers $m$ and $p$, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge $c=1$ at radius $R=\sqrt{p/2}$. When $p=1$ the state is Abelian. The state with $p=3$ is the $Z_4$ Read Rezayi state, and the series of $p\ge 3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge $e$ electrons coexist with charge $4e$ bosons and a strong clustered phase where the electrons are strongly bound in groups of 4. The transition between these two phases on a wire is mapped to the critical point of the 4 state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge $4e$ bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons. Coupled wires near the critical state form quantum Hall states with a gapless neutral mode described by the orbifold theory. The coupled wire approach allows us to employ the Abelian bosonization technique to fully analyze the physics of single wire, and then to extract most topological properties of the resulting non-Abelian quantum Hall states. These include the list of quasiparticles, their fusion rules, the correspondence between bulk quasiparticles and edge topological sectors, and most of the phases associated with quasiparticles winding one another.

Scaling analysis of the non-Abelian quasiparticle tunneling in FQH states

Quasiparticle tunneling between two counter propagating edges through point contacts could provide information on its statistics. Previous study of the short distance tunneling displays a scaling behavior, especially in the conformal limit with zero tunneling distance. The scaling exponents for the non-Abelian quasiparticle tunneling exhibit some non-trivial behaviors. In this work, we revisit the quasiparticle tunneling amplitudes and their scaling behavior in a full range of the tunneling distance by putting the electrons on the surface of a cylinder. The edge–edge distance can be smoothly tuned by varying the aspect ratio for a finite size cylinder. We analyze the scaling behavior of the quasiparticles for the Read–Rezayi states for and 4 both in the short and long tunneling distance region. The finite size scaling analysis automatically gives us a critical length scale where the anomalous correction appears. We demonstrate this length scale is related to the size of the quasiparticle at which the backscattering between two counter propagating edges starts to be significant.

Universal properties of the FQH state from the topological entanglement entropy and disorder effects

The topological entanglement entropy (TEE) is a robust measurement of the quantum many-body state with topological order. In fractional quantum Hall (FQH) state, it has a connection to the quantum dimension of the state itself and its quasihole excitations from the conformal field theory (CFT) description. We study the entanglement entropy (EE) in the Moore–Read (MR) and Read–Rezayi (RR) FQH states. The non-Abelian quasihole excitation induces an extra correction of the TEE which is related to its quantum dimension. With considering the effects of the disorder, the ground state TEE is stable before the spectral gap closing and the level statistics seems to have significant change with a stronger disorder, which indicates a many-body localization (MBL) transition.

Identification of the fractional quantum Hall edge modes by density oscillations

The neutral fermionic edge mode is essential to the non-Abelian topological property and its experimental detection in $Z_k$ fractional quantum Hall (FQH) state for $k > 1$. Usually, the identification of the edge modes in a finite size system is difficult, especially near the region of the edge reconstruction, due to mixing with the bosonic edge mode and the bulk states as well. We study the edge-mode excitations of the Moore-Read (MR) and Read-Rezayi (RR) states by using Jack polynomials in the truncated subspace. It is found that the electron density, as a detector, has marked different behaviors between the bosonic and fermionic edge modes. As an application, it helps us to identify them near the edge reconstruction and in the RR edge spectrum. On the other hand, we systematically study the edge excitations for the RR state, extrapolate the edge velocities and their related coherence length and temperature in the interferometer experiments.

Jack 3/5 State from Two-Body Interaction

We investigate the Read–Rezayi parafermion state of correlated electrons at the fractional Landau level filling ν = 3/5. It is a Jack polynomial generated by contact four-body repulsion. We show by exact diagonalization that it is also emerges from a suitable short-range two-body interaction. We find that it closely matches Coulomb ground state in the second Landau level of non-relativistic fermions, and thus possibly describes the ν = 13/5 (and, by conjugation, ν = 12/5) fractional quantum Hall effect in GaAs.

Stability of topological insulators with non-Abelian edge excitations

Chiral–antichiral pairs of non-Abelian Hall states, like the Pfaffian, Read–Rezayi and NASS states, can be used to model two-dimensional time-reversal invariant topological insulators. Their stability was shown to be associated to the presence of a anomaly and characterized by the same index introduced for free fermion and Abelian systems. In this work, we continue the stability analysis by providing the form of time-reversal invariant interactions that gap the non-Abelian edge excitations. Our approach is based on the description of non-Abelian states as projections of corresponding ‘parent’ Abelian states.

Mathematical Structure of Bosonic and Fermionic Jack States and Their Application in Fractional Quantum Hall Effect

Fractional quantum Hall e ect is a remarkable behaviour of correlated electrons, observed exclusively in two dimensions, at low temperatures, and in strong magnetic elds. The most prominent fractional quantum Hall state occurs at Landau level lling factor ν = 1/3 and it is described by the famous Laughlin wave function, which (apart from the trivial Gaussian factor) is an example of Jack polynomial. Fermionic Jack polynomials also describe another pair of candidate fractional quantum Hall states: Moore Read and Read Rezayi states, believed to form at the ν = 1/2 and 3/5 llings of the second Landau level, respectively. Bosonic Jacks on the other hand are candidates for certain correlated states of cold atoms. We examine here a continuous family of fermionic Jack polynomials whose special case is the Laughlin state as approximate wave functions for the 1/3 fractional quantum Hall e ect.

From Luttinger liquid to non-Abelian quantum Hall states

We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non Abelian states. In the present context, the coset construction arises when the original chiral modes are fractionalized into coset sectors, and the different sectors acquire energy gaps due to coupling in "different directions". The coupled wire construction can also can be used to describe anisotropic lattice systems, and provides a starting point for models of fractional and non-Abelian Chern insulators. This paper also includes an extended introduction to the coupled wire construction for Abelian quantum Hall states, which was introduced earlier.

Non-Abelian fractional Chern insulators from long-range interactions

The recent theoretical discovery of fractional Chern insulators (FCIs) has provided an important new way to realize topologically ordered states in lattice models. In earlier works, on-site and nearest neighbor Hubbard-like interactions have been used extensively to stabilize Abelian FCIs in systems with nearly flat, topologically nontrivial bands. However, attempts to use two-body interactions to stabilize non-Abelian FCIs, where the ground state in the presence of impurities can be massively degenerate and manipulated through anyon braiding, have proven very difficult in uniform lattice systems. Here, we study the remarkable effect of long-range interactions in a lattice model that possesses an exactly flat lowest band with a unit Chern number. When spinless bosons with two-body long-range interactions partially fill the lowest Chern band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases with non-Abelian anyon statistics. We characterize these states through studying topological degeneracies, the overlap between the ground states of two-body interactions and the exact RR ground states of three- and four-body interactions, and state counting in the particle-cut entanglement spectrum. Moreover, we demonstrate how an approximate lattice form of Haldane's pseudopotentials, analogous to that in the continuum, can be used as an efficient guiding principle in the search for lattice models with stable non-Abelian phases.

Composite fermion wavefunctions derived by conformal field theory

The Jain theory of hierarchical Hall states is reconsidered in the light of recent analyses that have found exact relations between projected Jain wavefunctions and conformal field theory correlators. We show that the underlying conformal theory is precisely given by the W-infinity minimal models introduced earlier. This theory involves a reduction of the multicomponent Abelian theory that is similar to the projection to the lowest Landau level in the Jain approach. The analysis closely parallels the bosonic conformal theory description of the Pfaffian and Read–Rezayi states.

Quasiparticle agglomerates in the Read–Rezayi and anti-Read–Rezayi states

We calculate the dominant excitations for the k-level () Read–Rezayi (RR) states and their particle–hole conjugates, the anti-Read–Rezayi (), proposed for quantum Hall states. These states are supposed to be built over the second Landau level with total filling factor ν = 2 + ν* with ν* = k/(k + 2) for RR and ν* = 2/(k + 2) for . In the k-level RR states, based on parafermions, the dominant excitations are the fundamental quasiparticles (qps) with fractional charge e*k = e/(k + 2), with e the electron charge, if k = 2,3. For k = 4 the single-qp and the 2-agglomerate, with charge 2e*k, have the same scaling and both dominate, while for k > 4 the 2-agglomerates are dominant. Anyway the dominance of the 2-agglomerates can be affected by the presence of environmental renormalizations. For all the k-level states, the single-qp and the 2-agglomerate have the same scaling and both dominate. In this case, only the presence of environmental renormalizations can make one dominant over the other. We determine the conditions in which the environmental renormalizations of the charged and neutral modes make the Abelian 2-agglomerates dominant over the non-Abelian single-qps in the two models and for any value of k. We conclude by observing that, according to these predictions, the dominance of 2-agglomerates, at very low energies for the ν = 5/2, can be an interesting indication supporting the validity of the anti-Pfaffian model in comparison with the Pfaffian.

Stability of the k = 3 Read–Rezayi state in chiral two-dimensional systems with tunable interactions

The k = 3 Read–Rezayi (RR) parafermion quantum Hall state hosts non-Abelian excitations which provide a platform for universal topological quantum computation. Although the RR state may be realized at the filling factor ν = 12/5 in GaAs-based two-dimensional electron systems, the corresponding quantum Hall state is weak and at present nearly impossible to study experimentally. Here we argue that the RR state can alternatively be realized in a class of chiral materials with massless and massive Dirac-like band structure. This family of materials encompasses monolayer and bilayer graphene, as well as topological insulators. We show that, compared to GaAs, these systems provide several important advantages in realizing and studying the RR state. Most importantly, the effective interactions can be tuned in situ by varying the external magnetic field, and by designing the dielectric environment of the sample. This tunability enables the realization of RR state with controllable energy gaps in different Landau levels. It also allows one to probe the quantum phase transitions to other compressible and incompressible phases.

Fractional quantum Hall effect in a U(1)×SU(2) gauge field

We consider the bosonic fractional quantum Hall (FQH) effect in the presence of a non-Abelian gauge field in addition to the usual Abelian magnetic field. The non-Abelian field breaks the twofold internal state degeneracy, but preserves the Landau level degeneracy. Using exact diagonalization, we find that for moderate non-Abelian field strengths the system's behaviour resembles a single internal state quantum Hall system, while for stronger fields there is a phase transition to either two internal state behaviour or the complete absence of FQH plateaus. Usually the energy gap is reduced by the presence of a non-Abelian field, but some non-Abelian fields appear to slightly increase the gap of the ν=1 and ν=3/2 Read–Rezayi states.

Scaling and non-Abelian signature in fractional quantum Hall quasiparticle tunneling amplitude

We study the scaling behavior in the tunneling amplitude when quasiparticles tunnel along a straight path between the two edges of a fractional quantum Hall annulus. Such scaling behavior originates from the propagation and tunneling of charged quasielectrons and quasiholes in an effective field analysis. In the limit when the annulus deforms continuously into a quasi-one-dimensional (1D) ring, we conjecture the exact functional form of the tunneling amplitude for several cases, which reproduces the numerical results in finite systems exactly. The results for Abelian quasiparticle tunneling is consistent with the scaling analysis; this allows for the extraction of the conformal dimensions of the quasiparticles. We analyze the scaling behavior of both Abelian and non-Abelian quasiparticles in the Read–Rezayi -parafermion states. Interestingly, the non-Abelian quasiparticle tunneling amplitudes exhibit non-trivial k-dependent corrections to the scaling exponent.

TOPOLOGICAL ENTANGLEMENT ENTROPY IN THE SECOND LANDAU LEVEL

The entanglement entropy of the incompressible states of a realistic quantum Hall system in the second Landau level is studied by direct diagonalization. The subdominant term of the area law, the topological entanglement entropy, which is believed to carry information about topological order in the ground state, was extracted for filling factors ν = 12/5 and ν = 7/3. While it is difficult to make strong conclusions about ν = 12/5, the ν = 7/3 state appears to be very consistent with the topological entanglement entropy for the k = 4 Read–Rezayi state. The effect of finite thickness corrections to the Coulomb potential used in the direct diagonalization is also systematically studied.

Search for electron liquids with non-Abelian quasiparticles

We use exact numerical diagonalization in the search of fractional quantum Hall states with non-Abelian quasiparticle statistics. For the (most promising) states in a partially filled second Landau level, the search is narrowed to the range of filling factors 7/3 < ve < 8/3. In this range, the analysis of energy spectra and correlation functions, calculated including finite width and Landau level mixing, supports the prominent non-Abelian candidates at ve = 5/2 (paired Moore–Read "pfafian" state) and 12/5 (clustered Read–Rezayi "parafermion" state). Outside of this range, the noninteracting composite fermion model with four attached flux quanta is validated, yielding the family of quantum liquids with fractional, but Abelian statistics. The borderline ve = 7/3 state is shown to be adiabatically connected to the Laughlin liquid, but its short-range correlations are significantly different.

Clustering properties, Jack polynomials and unitary conformal field theories

Recently, Jack polynomials have been proposed as natural generalizations of Z k Read–Rezayi states describing non-Abelian fractional quantum Hall systems. These polynomials are conjectured to be related to correlation functions of a class of W-conformal field theories based on the Lie algebra A k − 1 . These theories can be considered as non-unitary solutions of a more general series of CFTs with Z k symmetry, the parafermionic theories. Starting from the observation that some parafermionic theories admit unitary solutions as well, we show, by computing the corresponding correlation functions, that these theories provide trial wavefunctions which satisfy the same clustering properties as the non-unitary ones. We show explicitly that, although the wavefunctions constructed by unitary CFTs cannot be expressed as a single Jack polynomial, they still show a fine structure where the mathematical properties of the Jack polynomials play a major role.

Degeneracy of non-Abelian quantum Hall states on the torus: domain walls and conformalfield theory

We analyze the non-Abelian Read–Rezayi quantum Hall states on the torus, where it isnatural to employ a mapping of the many-body problem onto a one-dimensionallattice model. On the thin torus—the Tao–Thouless (TT) limit—the interactingmany-body problem is exactly solvable. The Read–Rezayi states at fillingν = k/(kM+2) are known to be exact ground states of a local repulsivek+1-bodyinteraction, and in the TT limit this is manifested in that all states in the ground state manifold have exactlyk particleson any kM+2 consecutive sites. For the two-body correlations of these states also imply that there is no more than one particle onM adjacent sites. The fractionally charged quasiparticles and quasiholes appear as domainwalls between the ground states, and we show that the number of distinct domain wallpatterns gives rise to the nontrivial degeneracies, required by the non-Abelianstatistics of these states. In the second part of the paper we consider the quasiholedegeneracies from a conformal field theory (CFT) perspective, and show that thecounting of the domain wall patterns maps one to one on the CFT counting via thefusion rules. Moreover we extend the CFT analysis to topologies of higher genus.