This chapter appears in "Fractional Quantum Hall Effects: New Development," edited by B. I. Halperin and J. K. Jain (World Scientific, 2020). The chapter begins with a primer on composite fermions, and then reviews three directions that have recently been pursued. It reports on theoretical calculations making detailed quantitative predictions for two sets of phenomena, namely spin polarization transitions and the phase diagram of the crystal. This is followed by the Kohn-Sham density functional theory of the fractional quantum Hall effect. The chapter concludes with recent applications of the parton theory of the fractional quantum Hall effect to certain delicate states.
Heavy-Fermions provide an extreme example of the utility of the idea of continuity and analyticity in physics. Their discovery and study in the past thirty years has added a fascinating chapter to condensed matter physics. I briefly review the origins of the heavy-fermion problem out of the study of magnetic moments in metals and the study of mixed-valent rare-earth compounds. I also review the principal ideas underlying the features understood in their fermi-liquid phase as well as in their anisotropic superconductivity. The unsolved issues are also briefly mentioned.
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.
We present a mapping of elementary fermion operators onto a quadratic form of composite fermionic and bosonic operators. The mapping is an exact isomorphism as long as the physical constraint of one composite particle per cluster is satisfied. This condition is treated on average in a composite particle mean-field approach, which consists of an ansatz that decouples the composite fermionic and bosonic sectors. The theory is tested on the one- and two-dimensional Hubbard models. Using a Bogoliubov determinant for the composite fermions and either a coherent or Bogoliubov state for the bosons, w…
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.
The particle-hole (PH) symmetry of {\em electrons} is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. This article investigates an emergent symmetry in the fractional quantum Hall effect, namely the PH symmetry of {\em composite fermions}, which relates states at composite fermion filling factors $ν^*=n+\barν$ and $ν^*=n+1-\barν$, where the integer $n$ is the $Λ$ level index and $0\leq \barν\leq 1$. Detailed calculations using the microscopic theory of composite…
We present a generalization of the composite Fermion picture for a muticomponent quantum Hall plasma which contains particle with different effective charges. The model predicts very well the low-lying states of a $ν=5/2$ quantum Hall state found in numerical diagonalization.
A recent experimental study [Pan et al., arXiv: 1902.10262] has shown that fractional quantum Hall effect gaps are essentially consistent with particle-hole symmetry in the lowest Landau level. Motivated by this result, we consider a clean two dimensional electron system (2DES) from the viewpoint of composite fermion mean-field theory. In this short note, we show that while the experiment is manifestly consistent with a Dirac composite fermion theory proposed recently by Son, it can equally well be explained within the framework of non-relativistic composite fermions, first put forward by Halp…
The Wigner crystal of composite fermions is a strongly correlated state of complex emergent particles, and therefore its unambiguous detection would be of significant importance. Recent observation of optical resonances in the vicinity of filling factor ν = 1/3 has been interpreted as evidence for a pinned Wigner crystal of composite fermions [Zhu et al., Phys. Rev. Lett. 105, 126803 (2010)]. We evaluate in a microscopic theory the shear modulus and the magnetophonon and magnetoplasmon dispersions of the composite fermion Wigner crystal in the vicinity of filling factors 1/3, 2/5, and 3/7. We …
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 dynamical mass generation through exciton condensation. The four-fold spin-valley degeneracy is fully lifted due to the mass generation and the 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 the composite Dirac fermions. At the filling factor $ν=1/2$, we show that the massive composite Di…
We predict that an incompressible fractional quantum Hall state is likely to form at $ν=3/8$ as a result of a chiral p-wave pairing of fully spin polarized composite fermions carrying four quantized vortices, and that the pairing is of the Anti-Pfaffian kind. Possible experimental ramifications are discussed.
We formulate a Chern-Simons composite fermion theory for Fractional Chern Insulators (FCIs), whereby bare fermions are mapped into composite fermions coupled to a lattice Chern-Simons gauge theory. We apply this construction to a Chern insulator model on the kagome lattice and identify a rich structure of gapped topological phases characterized by fractionalized excitations including states with unequal filling and Hall conductance. Gapped states with the same Hall conductance at different filling fractions are characterized as realizing distinct symmetry fractionalization classes.
We study the effect of the Chern-Simons gauge fields on the possible transition from two decoupled composite fermion metals to the interlayer coherent composite fermion state proposed by Alicea et al. [Phys. Rev. Lett. 103, 256403 (2009)] in a symmetrically doped quantum Hall bilayer with total Landau level filling fraction $ν_{tot} = 1$. In this transition, interlayer Coulomb repulsion leads to excitonic condensation of composite fermions which are then free to tunnel coherently between layers. We find that this coherent tunneling is strongly suppressed by the layer-dependent Aharonov-Bohm ph…
We theoretically investigate the nature of the state at quarter filled lowest Landau level and predict that, as the quantum well width is increased, a transition occurs from the composite fermion Fermi sea into a novel non-Abelian fractional quantum Hall state that is topologically equivalent to $f$-wave pairing of composite fermions. This state is topologically distinct from the familiar $p$-wave paired Pfaffian state. We compare our calculated phase diagram with experiments and make predictions for many observable quantities.
Haldane's geometrical description of fractional quantum Hall states is generalized to compressible states. It is shown that anisotropy in the composite fermion Fermi surface is a direct reflection of this intrinsic geometry. A simple model is introduced in which the geometric parameter can be obtained exactly from other parameters including electron mass anisotropy. Our results compare favorably with recent measurements of anisotropy in composite fermion Fermi surface [D. Kamburov, Y. Liu, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W. Baldwin, Phys. Rev. Lett. 110, 206801 (2013)]. Broader…
The particle-hole (PH) symmetry at half-filled Landau level requires the relationship between the flux number N_phi and the particle number N on a sphere to be exactly N_phi - 2(N-1) = 1. The wave functions of composite fermions with 1/2 "orbital spin", which contributes to the shift "1" in the N_phi and N relationship, are proposed, shown to be PH symmetric, and validated with exact finite system results. It is shown the many-body composite electron and composite hole wave functions at half-filling can be formed from the two components of the same spinor wave function of a massless Dirac ferm…
We investigate the problem of intertwined orders in fractional Chern insulators by considering lattice fractional quantum Hall (FQH) states arising from pairing of composite fermions in the square-lattice Hofstadter model. At certain filling fractions, magnetic translation symmetry ensures the composite fermions form Fermi surfaces with multiple pockets, leading to the formation of finite-momentum Cooper pairs in the presence of attractive interactions. We obtain mean-field phase diagrams exhibiting a rich array of striped and topological phases, establishing paired lattice FQH states as an id…
We construct a density functional theory for two-dimension electron (hole) gases subjected to both strong magnetic fields and external potentials. In particular, we are focused on regimes near even-denominator filling factors, in which the systems form composite fermion liquids. Our theory provides a systematic and rigorous approach to determine the properties of ground states in fractional quantum Hall regime modified by artificial structures. We also propose a practical way to construct approximated functional.
While the integer quantum Hall effect of composite fermions manifests as the prominent fractional quantum Hall effect (FQHE) of electrons, the FQHE of composite fermions produces further, more delicate states, arising from a weak residual interaction between composite fermions. We study the spin phase diagram of these states, motivated by the recent experimental observation by Liu {\em et al.} \cite{Liu14a,Liu14b} of several spin-polarization transitions at 4/5, 5/7, 6/5, 9/7, 7/9, 8/11 and 10/13 in GaAs systems. We show that the FQHE of composite fermions is much more prevalent in multicompon…
We report on exact-diagonalization studies of correlated many-electron states in the half-filled Landau levels of graphene, including pseudospin (valley) degeneracy. We demonstrate that the polarized Fermi sea of non-interacting composite fermions remains stable against a pairing transition in the lowest two Landau levels. However, it undergoes spontaneous depolarization, which is unprotected owing to the lack of single-particle pseudospin splitting. These results suggest the absence of the Pfaffian phase in graphene.
There is increasing experimental evidence for fractional quantum Hall effect at filling factor $ν=2+3/8$. 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 fer…
The relation between the conductivity tensors of Composite Fermions and electrons is extended to second generation Composite Fermions. It is shown that it crucially depends on the coupling matrix for the Chern-Simons gauge field. The results are applied to a model of interacting Composite Fermions that can explain both the anomalous plateaus in spin polarization and the corresponding maxima in the resistivity observed in recent transport experiments.
We argue that the naively expected singularities of the Fermi surface, in the mixed composite boson - composite fermion states proposed [S.H. Simon et al., PRL 91, 046803(2003)] for the evolution of ν= 1 bilayer quantum Hall system with distance, are obliterated. Our conclusion is based on a careful analysis of the momentum distribution in ν= 1/2 single-layer composite-fermion state. We point out to a possibility of the phenomenon hitherto unknown outside Kondo lattice systems when, in a translationally invariant system, Fermi-liquid-like portion of electrons enlarges its volume.
A rich pattern of fractional quantum Hall states in graphene double layers can be naturally explained in terms of two-component composite fermions carrying both intra- and inter-layer vortices.
The classic composite fermion field theory (Ref. 1) builds up an excellent framework to uniformly study important physical objects and globally explain anomalous experimental phenomena in fractional quantum Hall physics while there are also inherent weaknesses. We present a nonperturbative emergent Dirac fermion theory from this strongly correlated composite fermion field theory, which overcomes these serious long-standing shortcomings. The particle-hole symmetry of Dirac equation resolves this particle-hole symmetry enigma in the composite fermion field theory. With the help of presented nume…
We investigate the properties of composite fermions with four attached flux quanta through tilted field experiments near Landau level filling factor nu = 3/4. The observed collapse of fractional quantum Hall gaps in the vicinity of this quarter-filling state can be comprehensively understood in terms of composite fermions with mass and spin. Remarkably, the effective mass and g-factor of these four flux quanta composite fermions around nu = 3/4 are very similar to those of two flux quanta composite fermions around nu = 3/2.
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 four fold degeneracy of each Landau level. We test various composite fermion 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 $n=1$ L…
In two-dimensional electron systems confined to wide AlAs quantum wells, composite fermions around the filling factor $ν$ = 3/2 are fully spin polarized but possess a valley degree of freedom. Here we measure the energy needed to completely valley polarize these composite fermions as a function of electron density. Comparing our results to the existing theory, we find overall good quantitative agreement, but there is an unexpected trend: The measured composite fermion valley polarization energy, normalized to the Coulomb energy, decreases with decreasing density.
Fractional edge states can be viewed as integer edge states of composite fermions. We exploit this to discuss the conductance of the fractional quantized Hall states and the velocity of edge magnetoplasmons.
A recently developed model of interacting composite fermions, is used to investigate different composite-fermion phases. Their interaction potential allows for the formation of both solid and new quantum-liquid phases, which are interpreted in terms of second-generation composite fermions and which may be responsible for the fractional quantum Hall states observed at unusual filling factors, such as nu=4/11. Projection of the composite-fermion dynamics to a single level, involved in the derivation of the Hamiltonian of interacting composite fermions, reveals the underlying self-similarity of t…
We construct a class of composite fermion states for bilayer electron systems in a strong transverse magnetic field, and determine quantitatively the phase diagram as a function of the layer separation, layer thickness, and electron density. We find, in general, that there are several transitions, and that the incompressible phases are separated by compressible ones. The paired states of composite fermions, described by Pfaffian wave functions, are also considered.
We consider a model of Fermi-Bose mixture with strong hard-core repulsion between particles of the same sort and attraction between particles of different sorts. In this case, besides the standard anomalous averages of the type $<b>$; $<bb>$ and $<cc>$, a pairing between fermion and boson of the type $bc$ is possible. This pairing corresponds to a creation of composite fermions in the system. At low temperatures and equal densities of fermions and bosons composite fermions are further paired in quartets. Our investigations are important for high-$T_c$ superconductors and in connection with rec…
Strong interactions and topology drive a wide variety of correlated ground states. Some of the most interesting of these ground states, such as fractional quantum Hall states and fractional Chern insulators, have fractionally charged quasiparticles. Correlations in these phases are captured by the binding of electrons and vortices into emergent particles called composite fermions. Composite fermion quasiparticles are randomly localized at high levels of disorder and may exhibit charge order when there is not too much disorder in the system. However, more complex correlations were predicted whe…
In the framework of a recently developed model of interacting composite fermions restricted to a single level, we calculate the activation gaps of a second generation of spin-polarized composite fermions. These composite particles consist each of a composite fermion of the first generation and a vortex-like excitation and may be responsible for the recently observed fractional quantum Hall states at unusual filling factors such as nu=4/11,5/13,5/17, and 6/17. Because the gaps of composite fermions of the second generation are found to be more than one order of magnitude smaller than those of t…
We discuss the composite fermion pairing state in bilayer quantum Hall systems. After the evaluation of the range of the inter-layer separation in which the quantum Hall state is stabilized, we discuss the effect of inter-layer tunneling on the composite fermion pairing state at ν=1/2. We show that there is a cusp at the transition point between the Halperin (3,3,1) state and the Pfaffian state.
Spin excitations from a partially populated composite fermion level are studied above and below $ν=1/3$. In the range $2/7<ν<2/5$ the experiments uncover significant departures from the non-interacting composite fermion picture that demonstrate the increasing impact of interactions as quasiparticle Landau levels are filled. The observed onset of a transition from free to interacting composite fermions could be linked to condensation into the higher order states suggested by transport experiments and numerical evaluations performed in the same filling factor range.
Charged excitons of composite fermions (CFs) are considered in the fractional quantum Hall effect. Energies of the charged CF excitons are computed at $ν=1/3$ as a function of the size of exciton. We show that the charged CF exciton with size of roton is lower in energy than the unbound state of a neutral roton and a lone charge. Therefore we propose that the lowest-energy excitation of the fractional quantum Hall effect is in fact due to the charged excitons of composite fermions composed of two CF-particles and one CF-hole. We believe that charged excitons of composite fermions shed new ligh…
We study the integer quantum Hall plateau transition using composite fermion mean-field theory. We show that the topological $θ= π$ term in the associated nonlinear sigma model [P. Kumar et al., Phys. Rev. B 100, 235124 (2019)] is stable against a certain particle-hole symmetry violating perturbation, parameterized by the composite fermion effective mass. This result, which applies to both the Halperin, Lee, and Read and Dirac composite fermion theories, represents an emergent particle-hole symmetry. For a disorder ensemble without particle-hole symmetry, we find that $θ$ can vary continuously…
We show that the entanglement spectrum associated with a certain class of strongly correlated many-body states --- the wave functions proposed by Laughlin and Jain to describe the fractional quantum Hall effect --- can be very well described in terms of a simple model of non-interacting (or weakly interacting) composite fermions.
Composite fermion metal states emerge in quantum Hall bilayers at total Landau level filling factor $ν_T$=1 when the tunneling gap collapses by application of in-plane components of the external magnetic field. Evidence of this transformation is found in the continua of spin excitations observed by inelastic light scattering below the spin-wave mode at the Zeeman energy. The low-lying spin modes are interpreted as quasiparticle excitations with simultaneous changes in spin orientation and composite fermion Landau level index.
We present a continuum operator map that inverts the role of particles and antiparticles in three dimensional QED. This is accomplished by the attachment of specific holomorphic Wilson lines to Dirac fermions. We show that this nonlocal map provides a continuum realization of Son's composite fermions at $ν= \frac{1}{2}$. The inversion of Landau levels and the Dirac-cone--composite-Fermion duality is explicitly demonstrated for the case of slowly varying magnetic fields. The role of Maxwell-terms as well as the connection of this construction to a gauge-invariant formulation of 3D gauge theorie…
We propose that a composite fermion operator $f_{iσ}(2n_{i{\bar σ}}-1)$ could have coherent excitations, where $f_{iσ}$ is the fermion operator for interacting electrons and $n_{i{\bar σ}}$ is the number operator of the opposite spin. In the two-impurity Anderson model, it is found that the excitation of this composite fermion has a pseudogap in the Kondo regime, and has a finite spectral weight in the regime where the excitation of the regular fermion $f_{iσ}$ has a pseudogap. In the latter regime, the self-energy of $f_{iσ}$ is found to be singular near Fermi energy. We argue that this compo…
In several two-dimensional films that exhibit a magnetic field-tuned superconductor to insulator transition (SIT), stable metallic phases have been observed. Building on the `dirty boson' description of the SIT, we suggest that the metallic region is analogous to the composite Fermi liquid observed about half-filled Landau levels of the two-dimensional electron gas. The composite fermions here are mobile vortices attached to one flux quantum of an emergent gauge field. The composite vortex liquid is a 2D non-Fermi liquid metal, which we argue is stable to weak quenched disorder. We describe se…
We investigate the response of a two-dimensional electron gas, in the fractional quantum Hall regime, to the sudden appearance of a localised charged probe using the Chern-Simons theory of composite fermions. The dynamic structure factor of the electron gas is found to have a major influence on the spectral function of the probe. In particular, there is an orthogonality catastrophe when the filling factor is an even-denominator filling fraction due to the compressibility of the state, but there is no catastrophe at odd-denominator filling factors because these states have a gap to excitations.…
We propose a (4+1) dimensional Chern-Simons field theoretical description of the fractional quantum Hall effect. It suggests that composite fermions reside on a momentum manifold with a nonzero Chern number. Based on derivations from microscopic wave functions, we further show that the momentum manifold has a uniformly distributed Berry curvature. As a result, composite fermions do not follow the ordinary Newtonian dynamics as commonly believed, but the more general symplectic one. For a Landau level with the particle-hole symmetry, the theory correctly predicts its Hall conductance at half-fi…
The response of composite Fermions to large wavevector scattering has been studied through phonon drag measurements. While the response retains qualitative features of the electron system at zero magnetic field, notable discrepancies develop as the system is varied from a half-filled Landau level by changing density or field. These deviations, which appear to be inconsistent with the current picture of composite Fermions, are absent if half-filling is maintained while changing density. There remains, however, a clear deviation from the temperature dependence anticipated for 2k_F scattering.
We propose a $\mathbb{Z}_{2}$ classification of Abelian time-reversal fractional topological insulators in terms of the composite fermions picture. We consider the standard toy model where spin up and down electrons are subjected to opposite magnetic fields and only electrons of the same spin interact via a repulsive force. By applying the composite fermions approach to this time-reversal symmetric system, we are able to obtain a hierarchy of topological insulators with spin Hall conductance $σ_{s}=\frac{e}{2π}\frac{p}{2mp+1} $, being $p,m \in\mathbb{N}$. They show stable edge states only for …
We review the recently proposed Dirac composite fermion theory of the half-filled Landau level. This paper is based on a talk given at the Nambu Symposium at University of Chicago, March 11-13, 2016.
Composite fermion wavefuctions have been used to describe electrons in a strong magnetic field. We show that the polynomial part of these wavefunctions can be obtained by applying a normal ordered product of suitably defined annihilation and creation operators to an even power of the Vandermonde determinant, which can been considered as a kind of a non-trivial Fermi sea. In the case of the harmonic interaction we solve the system exactly in the lowest Landau level. The solution makes explicit the boson-fermion correspondence proposed recently.
We discuss the emergence of an orthogonality catastrophe in the response of a composite fermion liquid as the filling factor νapproaches 1/2m, where m=1,2,3.... A tunneling experiment is proposed in which dramatic changes in the I-V characteristic should be observable as νis varied. Explicit I-V characteristics calculated within the so-called Modified Random Phase Approximation, are provided for ν=1/3 -> ν=1/2.
We construct an action for the composite Dirac fermion consistent with symmetries of electrons projected to the lowest Landau level. First we construct a generalization of the $g=2$ electron that gives a smooth massless limit on any curved background. Using the symmetries of the microscopic electron theory in this massless limit we find a number of constraints on any low-energy effective theory. We find that any low-energy description must couple to a geometry which exhibits nontrivial curvature even on flat space-times. Any composite fermion must have an electric dipole moment proportional an…
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-…
We show that the QCD Dirac spectrum at finite chemical potential using a matrix model in the spontaneously broken phase, is amenable to a generic 2-dimensional effective action. The eigenvalues form a droplet with strong screening and plasmon oscillation. The droplet is threaded by a magnetic vortex which is at the origin of a Berry phase. For quarks in the complex or Dirac representation, the anomalous transport in the droplet of eigenvalues bear some similarities with that in droplets of composite fermions at half filling suggesting that the latters maybe Dirac fermions.
Exact diagonalization of a two-dimensional electron gas in a strong magnetic field in the disk geometry shows that there exists a filling factor range in the second Landau level where the states significantly differ from those in the lowest Landau level. We show that the difference arises because the interaction between composite fermions is not negligible in higher Landau levels, as indicated by a substantial mixing between composite-fermion Landau-like levels, or lambda levels. We find that the exact ground state is well reproduced by composite fermion theory with lambda level mixing incorpo…
Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a $p$-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling facto…
