Composite Fermion States Research Articles

Composite fermion states on the torus

We extend the composite fermion construction to the torus geometry. We verify the validity of our construction by computing the overlap of the composite fermion state to the exact diagonalization ground state of both Coulomb interaction and Haldane-pseudopotential interaction V_0 (V_1) for bosonic (fermionic) states.

Fractional Chern insulators beyond Laughlin states

We report the first numerical observation of composite fermion (CF) states in fractional Chern insulators (FCI) using exact diagonalization. The ruby lattice Chern insulator model for both fermions and bosons exhibits a clear signature of CF states at filling factors $2/5$ and $3/7$ ($2/3$ and $3/4$ for bosons). The topological properties of these states are studied through several approaches. Quasihole and quasielectron excitations in FCI display similar features as their fractional quantum hall (FQH) counterparts. The entanglement spectrum of FCI ground states shows an identical fingerprint to its FQH partner. We show that the correspondence between FCI and FQH obeys the emergent symmetry already established, proving the validity of this approach beyond the clustered states. We investigate other Chern insulator models and find similar signatures of CF states. However, some of these systems exhibit strong finite size effects.

Composite fermion state of spin-orbit-coupled bosons

We consider spinor Bose gas with the isotropic Rashba spin-orbit coupling in two dimensions. We argue that at low density, its ground state is a composite fermion state with a Chern-Simons gauge field and filling factor of one. The chemical potential of such a state scales with the density as $\ensuremath{\mu}\ensuremath{\propto}{n}^{3/2}$. This is a lower energy per particle than $\ensuremath{\mu}\ensuremath{\propto}n$ for the earlier suggested ground-state candidates: a condensate with broken time-reversal symmetry and a spin-density wave state.

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 Hall states in rapidly rotating two-component Bose gases

We investigate strongly correlated phases of two-component (or pseudospin-$1/2$) Bose gases under rapid rotation through exact diagonalization on a torus geometry. In the case of pseudospin-independent contact interactions, we find the formation of gapped spin-singlet states at the filling factors $\ensuremath{\nu}=k/3+k/3$ ($k/3$ filling for each component) with integer $k$. We present numerical evidences that the gapped state with $k=2$ is well described as a non-Abelian spin-singlet (NASS) state, in which excitations feature non-Abelian statistics. Furthermore, we find the phase transition from the product of composite fermion states to the NASS state by changing the ratio of the intercomponent-to-intracomponent interactions.

Evaluation of Ranks of Real Space and Particle Entanglement Spectra for Large Systems

We devise a way to calculate the dimensions of symmetry sectors appearing in the particle entanglement spectrum (PES) and real space entanglement spectrum (RSES) of multiparticle systems from their real space wave functions. We first note that these ranks in the entanglement spectra equal the dimensions of spaces of wave functions with a number of particles fixed. This also yields equality of the multiplicities in the PES and the RSES. Our technique allows numerical calculations for much larger systems than were previously feasible. For somewhat smaller systems, we can find approximate entanglement energies as well as multiplicities. We illustrate the method with results on the RSES and PES multiplicities for integer quantum Hall states, Laughlin and Jain composite fermion states, and for the Moore-Read state at filling ν = 5/2 for system sizes up to 70 particles.

Spinful composite fermions in a negative effective field

In this paper we study fractional quantum Hall composite fermion wavefunctions at filling fractions \nu = 2/3, 3/5, and 4/7. At each of these filling fractions, there are several possible wavefunctions with different spin polarizations, depending on how many spin-up or spin-down composite fermion Landau levels are occupied. We calculate the energy of the possible composite fermion wavefunctions and we predict transitions between ground states of different spin polarizations as the ratio of Zeeman energy to Coulomb energy is varied. Previously, several experiments have observed such transitions between states of differing spin polarization and we make direct comparison of our predictions to these experiments. For more detailed comparison between theory and experiment, we also include finite-thickness effects in our calculations. We find reasonable qualitative agreement between the experiments and composite fermion theory. Finally, we consider composite fermion states at filling factors \nu = 2+2/3, 2+3/5, and 2+4/7. The latter two cases we predict to be spin polarized even at zero Zeeman energy.

Quasiparticles and excitons for the Pfaffian quantum Hall state

We propose trial wave functions for quasiparticle and exciton excitations of the Moore-Read Pfaffian fractional quantum Hall states, both for bosons and for fermions, and study these numerically. Our construction of trial wave functions employs a picture of the bosonic Moore-Read state as a symmetrized double layer composite fermion state. We obtain the number of independent angular momentum multiplets of quasiparticle and exciton trial states for systems of up to 20 electrons. We find that the counting for quasielectrons at large angular momentum on the sphere matches that expected from the conformal field theory (CFT) that describes the Moore-Read state's boundary theory. In particular, the counting for quasielectrons is the same as for quasiholes, in accordance with the idea that the CFT describing both sides of the Fractional Quantum Hall plateau should be the same. We also show that our trial wave functions have good overlaps with exact wave functions obtained using various interactions, including second Landau level Coulomb interactions and the three-body delta interaction for which the Pfaffian states and their quasiholes are exact ground states. We discuss how these results relate to recent work by Sreejith et al. on a similar set of trial wave functions for excitations over the Pfaffian state as well as to earlier work by Hansson et al., which has produced trial wave functions for quasiparticles based on conformal field theory methods and by Bernevig and Haldane, which produced trial wave functions based on clustering properties and ``squeezing.''

Gap Structure and Gapless Structure in Fractional Quantum Hall Effect

Higher-order composite fermion states are correlated with many quasiparticles. The energy calculations are very complicated. We develop the theory of Tao and Thouless to explain them. The total Hamiltonian is (𝐻𝐷

Unpaired Composite Fermion, Topological Exciton, and Zero Mode

The paired state of composite fermions is expected to support two kinds of excitations: vortices and unpaired composite fermions. We construct an explicit microscopic description of the unpaired composite fermions, which we demonstrate to be accurate for a 3-body model interaction and, possibly, adiabatically connected to the Coulomb solution. This understanding reveals that an unpaired composite fermion carries with it a charge-neutral "topological" exciton, which, in turn, helps provide microscopic insight into the origin of zero modes, fusion rules, and energetics.

Bipartite Composite Fermion States

We study a class of ansatz wave functions in which composite fermions form two correlated "partitions." These "bipartite" composite fermion states are demonstrated to be very accurate for electrons in a strong magnetic field interacting via a short-range 3-body interaction potential over a broad range of filling factors. Furthermore, this approach gives accurate approximations for the exact Coulomb ground state at 2+3/5 and 2+4/7 and is thus a promising candidate for the observed fractional quantum Hall states at the hole conjugate fractions at 2+2/5 and 2+3/7.

Extracting Excitations from Model State Entanglement

We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.

Activation energy gap of the layer-imbalanced bilayer [formula omitted] quantum Hall states

We investigate the bilayer fractional quantum Hall state (FQHS) in layer-imbalanced configurations at the total Landau level filling factor ν = 1 3 , which is interpreted as the bilayer ν = 1 integer quantum Hall state (IQHS) of composite fermions (CFs). We carried out detailed magnetotransport experiments, especially focused on measurements of the activation energy gap Δ in the bilayer ν = 1 3 FQHS under tilted magnetic fields. We observed Δ as a function of in-plane magnetic field for various layer density imbalance parameters σ , which is compared with the reported data of the bilayer ν = 1 IQHS. At small σ , the FQHSs collapse by applying large in-plane fields. This behavior is interpreted as the transition from the commensurate (C)-like phase to no-FQHS. At large σ , Δ stays finite even in large in-plane field, which means the incommensurate (IC)-like phase appears also in the ν = 1 3 layer-imbalanced bilayer FQHS. We discuss the interactions among CFs from the phase transition point between the C-like and the IC-like phases in the layer-imbalanced ν = 1 3 FQHS.

States of interacting composite fermions at the Landau level fillingν=2+3/8

There is increasing experimental evidence for fractional quantum Hall effect at filling factor $\nu=2+3/8$. Modeling it as a system of composite fermions, we study the problem of interacting composite fermions by a number of methods. In our variational study, we consider the Fermi sea, the Pfaffian paired state, and bubble and stripe phases of composite fermions, and find that the Fermi sea state is favored for a wide range of transverse thickness. However, when we incorporate interactions between composite fermions through composite-fermion diagonalization on systems with up to 25 composite fermions, we find that a gap opens at the Fermi level, suggesting that inter-composite fermion interaction can induce fractional quantum Hall effect at $\nu=2+3/8$. The resulting state is seen to be distinct from the Pfaffian wave function.

Activation study of the bilayer [formula omitted] quantum Hall state

We investigated the bilayer fractional quantum Hall state (FQHS) at total Landau level filling factor ν = 1 3 , which is interpreted as the bilayer ν = 1 integer quantum Hall state (IQHS) of composite fermions. We carried out detailed magnetotransport experiments of the bilayer ν = 1 3 FQHS in tilted magnetic field. The data are compared with reported data of the bilayer ν = 1 IQHS. We found that the commensurate - incommensurate phase transition, which occurs in the bilayer ν = 1 IQHS, is missing in the bilayer ν = 1 3 FQHS.

Theoretical study of even denominator fractions in graphene: Fermi sea versus paired states of composite fermions

The physics of the state at even denominator fractional fillings of Landau levels depends on the Coulomb pseudopotentials, and produces, in different GaAs Landau levels, a composite fermion Fermi sea, a stripe phase, or, possibly, a paired composite fermion state. We consider here even denominator fractions in graphene, which has different pseudopotentials as well as a possible fourfold degeneracy of each Landau level. We test various composite fermion (CF) Fermi sea wave functions [fully polarized, SU(2) singlet, SU(4) singlet] as well as the paired composite fermion states in the $n=0$ and 1 Landau levels and predict that (i) paired states are not favorable, (ii) CF Fermi seas occur in both Landau levels, and (iii) an SU(4) singlet composite fermion Fermi sea is stabilized in the appropriate limit. The results from detailed microscopic calculations are generally consistent with the predictions of the mean field model of composite fermions.

Non-Abelian States with Negative Flux: A New Series of Quantum Hall States

By applying the idea of parafermionic clustering to composite bosons with positive as well as negative flux, a new series of trial wave functions to describe fractional quantum Hall states is proposed. These states compete at filling factors nu=k/(3k+/-2) with other ground states like stripes or composite-fermion states. These series contain all the states recently discovered by Pan et al. [Phys. Rev. Lett. 90, 016801 (2003)10.1103/Phys. Rev. Lett. 90, 016801(2003)] including the even-denominator cases. Exact diagonalization studies on the sphere and torus point to their relevance for nu=3/7, 3/11, and 3/8.

Condensation of Nonabelian Moore-Read Quasiholes in Correlated Composite Fermion Liquids

Twoand three-body correlation functions (number of pairs or triplets vs. relative angular momentum) of electrons or Laughlin quasielectrons (i.e., composite fermions in their first excited Landau level) are studied numerically in several fractional quantum Hall liquids. It is shown directly that the νe = 4/11 liquid (corresponding to a ν = 1/3 filling of composite fermions in their first excited Landau level) is a paired state of quasielectrons, hence interpreted as a condensate of “second-generation” quasiholes of Moore–Read ν = 1/2 state of composite fermions.

Composite fermion theory of rapidly rotating two-dimensional bosons

Ultracold neutral bosons in a rapidly rotating atomic trap have been predicted to exhibit fractional quantum Hall-like states. We describe how the composite fermion theory, used in the description of the fractional quantum Hall effect for electrons, can be applied to interacting bosons. Numerical evidence supporting the formation of composite fermions, each being the bound state of a boson and one flux quantum, is shown for filling fractions of the type ν = p/(p + 1), both by spectral analysis and by direct comparison with trial wavefunctions. The rapidly rotating system of two-dimensional bosons thus constitutes an interesting example of ‘statistical transmutation’, with bosons behaving like composite fermions. We also describe the difference between the electronic and the bosonic cases when p approaches infinity. Residual interactions between composite fermions are attractive in this limit, resulting in a paired composite fermion state described by the Moore–Read wavefunction.

Angular magnetoresistance oscillations in bilayers in tilted magnetic fields

Angular magnetoresistance oscillations (AMRO) were originally discovered in organic conductors and then found in many other layered metals. It should be possible to observe AMRO to semiconducting bilayers as well. Here we present an intuitive geometrical interpretation of AMRO as the Aharonov–Bohm interference effect, both in real and momentum spaces, for balanced and imbalanced bilayers. Applications to the experiments with bilayers in tilted magnetic fields in the metallic state are discussed. We speculate that AMRO may be also observed when each layer of the bilayer is in the composite-fermion state.