Composite Fermions Research Articles

Particle-hole mirror symmetries around the half-filled shell: Quantum numbers and algebraic structure of composite fermions

Composite fermions (CFs) of the fractional quantum Hall effect are described as spherical products of electron and vortex spinors, built from underlying L=1/2 ladder operators aligned so that the spinor angular momenta Le and Lv are maximal. We identify the CF's quantum numbers as the angular momentum L in (L_e L_v)L, its magnetic projection m_L, the electron number N, with L_v={N-1)/2, and magnetic \nu-spin, m_\nu=L_e-L_v. Translationally invariant FQHE states are formed by filling p subshells with their respective CFs, in order of ascending L for fixed L_e and L_v, beginning with the lowest allowed value, L=|m_\nu|. We show that this wave function has an exactly equivalent hierarchical form. FQHE states can be grouped into \nu-spin multiplets mirror symmetric around m_\nu=0, with N held constant. Electron particle-hole conjugation with respect to this vacuum is identified as the mirror symmetry relating FQHE states of the same N but distinct fillings \nu = p/(2p+1} and p/( 2p-1). Alternatively, mirror symmetric \nu-spin multiplets can be constructed in which the magnetic field strength is held fixed: the valence states are electron particle-vortex hole excitations. Particle-hole symmetry -- relating the N-particle FQHE state of filling \nu=p/(2p+1} to the $\bar{N}$-particle state of filling {p+1)/(2p+1} -- is shown to be equivalent to electron-vortex exchange. In this construction $\bar{N}$-N CFs of the higher density state occupy an extra zero-mode subshell. We link this structure, familiar from supersymmetric quantum mechanics, to the CF Pauli Hamiltonian, which we show is isospectral, quadratic in the \nu-spin raising and lowering operators, and four-fold degenerate. On linearization, it takes a Dirac form similar to that found in the integer quantum Hall effect (IQHE).

Possible higher temperature superconductivity in the modulation-doped superlattice structure of cuprate superconductors

We design a new structure for a cuprate superconductor indicating the possibility of higher temperature superconductivity using our recently proposed composite fermions theory. It is constructed with modulation-doped superlattice structures, which are often used in the design of semiconductor superlattice devices. The superconductive critical temperature (Tc) was calculated in the superlattice structures of the superconductor in which the optimal doped CuO2 layer was sandwiched between two less-doped CuO2 layers. We find that if these structures could be realized in a cuprate superconductor such as Bi2Sr2Ca2Cu3O10 or HgBa2Ca2Cu3O9, the highest Tc could attain the level of 300 K at atmospheric pressure.

Generalization of Laughlin’s theory for the fractional quantum Hall effect

Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin’s quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions , respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their ‘conjugates’, formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.

Fractional Excitonic Insulator.

We argue that a correlated fluid of electrons and holes can exhibit a fractional quantum Hall effect at zero magnetic field analogous to the Laughlin state at filling 1/m. We introduce a variant of the Laughlin wave function for electrons and holes and show that for m=1 it is the exact ground state of a free fermion model that describes p_{x}+ip_{y} excitonic pairing. For m>1 we develop a simple composite fermion mean field theory, and we present evidence that our wave function correctly describes this phase. We derive an interacting Hamiltonian for which our wave function is the exact ground state, and we present physical arguments that the m=3 state can be realized in a system in which energy bands with angular momentum that differ by 3 cross at the Fermi energy. This leads to a gapless state with (p_{x}+ip_{y})^{3} excitonic pairing, which we argue is conducive to forming the fractional excitonic insulator in the presence of interactions. Prospects for numerics on model systems and band structure engineering to realize this phase in real materials are discussed.

Crystallization in the Fractional Quantum Hall Regime Induced by Landau-Level Mixing.

The interplay between strongly correlated liquid and crystal phases for two-dimensional electrons exposed to a high transverse magnetic field is of fundamental interest. Through the nonperturbative fixed-phase diffusion MonteCarlo method, we determine the phase diagram of the Wigner crystal in the ν-κ plane, where ν is the filling factor and κ is the strength of Landau-level (LL) mixing. The phase boundary is seen to exhibit a striking ν dependence, with the states away from the magic filling factors ν=n/(2pn+1) being much more susceptible to crystallization due to Landau-level mixing than those at ν=n/(2pn+1). Our results explain the qualitative difference between the experimental behaviors observed in n- and p-doped gallium arsenide quantum wells and, in particular, the existence of an insulating state for ν<1/3 and also for 1/3<ν<2/5 in low-density p-doped systems. We predict that, in the vicinity of ν=1/5 and ν=2/9, increasing LL mixing causes a transition not into an ordinary electron Wigner crystal, but rather into a strongly correlated crystal of composite fermions carrying two vortices.

Paired states at 5/2: Particle-hole Pfaffian and particle-hole symmetry breaking

We study Cooper pairing in the Dirac composite fermion (CF) system. The presence of the mass term in the Dirac CF description (which may simulate Landau level mixing) i.e. breaking of particle-hole (PH) symmetry in this system, is a necessary condition for the existence of a PH Pfaffian-like topological state. In the scope of RPA approximation and hydrodynamic approach, we find some signatures of pairing at finite frequencies. Motivated by this insight, we extend our analysis to the case of a different but still Dirac quasi-particle (CF) representation, appropriate in the presence of a mass term, and discuss the likelihood of PH Pfaffian pairing and Pfaffian pairing in general. On the basis of gauge field effects, we find for small Dirac mass anti-Pfaffian or Pfaffian instability depending on the sign of mass, while for large mass (Landau level mixing), irrespective of its sign, PH Pfaffian-like instability.

Topological phase transition underpinning particle-hole symmetry in the Halperin-Lee-Read theory

Long wavelength descriptions of a half-filled lowest Landau level ($\nu = 1/2$) must be consistent with the experimental observation of particle-hole (PH) symmetry. The traditional description of the $\nu=1/2$ state pioneered by Halperin, Lee and Read (HLR) naively appears to break PH symmetry. However, recent studies have shown that the HLR theory with weak quenched disorder can exhibit an emergent PH symmetry. We find that such inhomogeneous configurations of the $\nu=1/2$ fluid, when described by HLR mean-field theory, are tuned to a topological phase transition between an integer quantum Hall state and an insulator of composite fermions with a dc Hall conductivity $\sigma_{xy}^{\rm (cf)} = - {1 \over 2} {e^2 \over h}$. Our observations help explain why the HLR theory exhibits PH symmetric dc response.

Roton minimum at ν = 1∕2 filled fractional quantum Hall effect of Bose particles

We have studied the collective excitation of fractional quantum Hall effect (FQHE) in the rotating Bose-Einstein condensate (BEC) using CF theory at the filling fraction ν = 1∕2. The roton type of excitation in the FQHE of electron system is established over the years for all the filling fraction, whereas the collective excitation at ν = 1∕2 filling fraction in the rotating BEC shows no roton minimum. We have investigated this using composite fermion theory with the Pöschl-Teller interaction potential between the particles. We have seen that the long-range interaction will give the roton, whereas short-range interaction gives no roton minimum.

Numerical exploration of trial wave functions for the particle-hole-symmetric Pfaffian

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 $\nu=5/2$ 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 state performs reasonably well energetically in the half-filled lowest Landau level, but is likely not a good starting point for understanding the $\nu=5/2$ ground state. Our results suggest that the PH-Pfaffian model wave function either encodes anomalously weak $p$-wave pairing of composite fermions or fails to represent a gapped, incompressible phase altogether.

Study of partially polarized fractional quantum Hall states

We have studied the partially spin polarized fractional quantum Hall states using Chern Simon's theory and plasma picture proposed by Halperin. Using these theoretical techniques we have tried to find the stable polarized states of different filling fractions observed in experiments. We have calculated the ground state energies of those states and also pair correlation function. We have described the nature of the states by the behaviour of this quantity. In our study, we have seen that the partially polarized states, which do not fit with Jain's composite fermion description are basically the mixed state of up-spin liquid phase and down-spin solid phase.

Berry phase of the composite-fermion Fermi sea: Effect of Landau-level mixing

We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low energy excitations following a recently developed approach [Pu, Wu and Jain, Phys. Rev. B 96, 195302 (2018)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $\pi$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau level mixing. For this purpose, we consider a simple model wherein we determine the variational ground state as a function of Landau level mixing within the space spanned by two basis functions: the lowest-Landau-level projected and the unprojected composite-fermion Fermi sea wave functions. We evaluate Berry phase for a path around the Fermi circle within this model following a recent prescription, and find that it rotates rapidly as a function of Landau level mixing. We also consider the effect of a particle-hole symmetry breaking three-body interaction on the Berry phase while confining the Hilbert space to the lowest Landau level. Our study deepens the connection between the $\pi$ Berry phase and the exact particle-hole symmetry in the lowest Landau level.

Parity Anomaly and Duality Web

AbstractWe review the parity anomaly and a duality web in 2+1 dimensions. An odd dimensional non‐interacting Dirac fermion theory is not parity invariant at quantum level. We demonstrate the parity anomaly in a three dimensional non‐interacting Dirac fermion theory and a one dimensional non‐interacting Dirac fermion theory. These theories can generate non‐gauge invariant Abelian Chern‐Simons terms at a finite temperature through an effective action. The parity anomaly also leads us to study the duality web in 2+1 dimensions, in which the 2+1 dimensional duality web begins from the conjecture of a duality between a three dimensional Dirac fermion theory and a three dimensional interacting scalar field theory at the Wilson‐Fisher fixed point. We first review the duality web for the flat background, then we discuss its extension to the spinc manifold to avoid inconsistency from the spin structure. This also leads a global effect. We discuss that the composite fermions approach of the quantum Hall system also suffers from the same issue of the global description. Finally, we use perspective of the electric‐magnetic duality of the Abelian gauge theory in 3+1 dimensions to study the 2+1 dimensional duality web.

Unconventional fractional quantum Hall states and Wigner crystallization in suspended Corbino graphene

Competition between liquid and solid states in two-dimensional electron systems is an intriguing problem in condensed matter physics. We have investigated competing Wigner crystal and fractional quantum Hall (FQH) liquid phases in atomically thin suspended graphene devices in Corbino geometry. Low-temperature magnetoconductance and transconductance measurements along with IV characteristics all indicate strong charge density dependent modulation of electron transport. Our results show unconventional FQH phases which do not fit the standard Jain’s series for conventional FQH states, instead they appear to originate from residual interactions of composite fermions in partially filled Landau levels. Also at very low charge density with filling factors nu ,lesssim, 1/5, electrons crystallize into an ordered Wigner solid which eventually transforms into an incompressible Hall liquid at filling factors around ν ≤ 1/7. Building on the unique Corbino sample structure, our experiments pave the way for enhanced understanding of the ordered phases of interacting electrons.

Even-denominator fractional quantum Hall states at an isospin transition in monolayer graphene

Magnetic fields quench the kinetic energy of two dimensional electrons, confining them to highly degenerate Landau levels. In the absence of disorder, the ground state at partial Landau level filling is determined only by Coulomb interactions, leading to a variety of correlation-driven phenomena. Here, we realize a quantum Hall analog of the Ne\'el-to-valence bond solid transition within the spin- and sublattice- degenerate monolayer graphene zero energy Landau level by experimentally controlling substrate-induced sublattice symmetry breaking. The transition is marked by unusual isospin transitions in odd denominator fractional quantum Hall states for filling factors $\nu$ near charge neutrality, and the unexpected appearance of incompressible even denominator fractional quantum Hall states at $\nu=\pm1/2$ and $\pm1/4$ associated with pairing between composite fermions on different carbon sublattices.

Effect of Magnetization on the Tunneling Anomaly in Compressible Quantum Hall States.

Tunneling of electrons into a two-dimensional electron system is known to exhibit an anomaly at low bias, in which the tunneling conductance vanishes due to a many-body interaction effect. Recent experiments have measured this anomaly between two copies of the half-filled Landau level as a function of in-plane magnetic field, and they suggest that increasing spin polarization drives a deeper suppression of tunneling. Here, we present a theory of the tunneling anomaly between two copies of the partially spin-polarized Halperin-Lee-Read state, and we show that the conventional description of the tunneling anomaly, based on the Coulomb self-energy of the injected charge packet, is inconsistent with the experimental observation. We propose that the experiment is operating in a different regime, not previously considered, in which the charge-spreading action is determined by the compressibility of the composite fermions.

Thermopower and Nernst measurements in a half-filled lowest Landau level

Motivated by recent proposal by Potter et al. [Phys. Rev. X 6, 031026 (2016)] concerning possible thermoelectric signatures of Dirac composite fermions, we perform a systematic experimental study of thermoelectric transport of an ultrahigh-mobility GaAs/AlxGa1-xAs two dimensional electron system at filling factor v = 1/2. We demonstrate that the thermopower Sxx and Nernst Sxy are symmetric and anti-symmetric with respect to B = 0 T, respectively. The measured properties of thermopower Sxx at v = 1/2 are consistent with previous experimental results. The Nernst signals Sxy of v = 1/2, which have not been reported previously, are non-zero and show a power law relation with temperature in the phonon-drag dominant region. In the electron-diffusion dominant region, the Nernst signals Sxy of v = 1/2 are found to be significantly smaller than the linear temperature dependent values predicted by Potter et al., and decreasing with temperature faster than linear dependence.

Exotic bilayer crystals in a strong magnetic field

Electron bilayers in a strong magnetic field exhibit insulating behavior for a wide range of interlayer separation $d$ for total Landau level fillings $\nu\leq 1/2$, which has been interpreted in terms of a pinned crystal. We study theoretically the competition between many strongly correlated liquid and crystal states and obtain the phase diagram as a function of quantum well width and $d$ for several filling factors of interest. We predict that three crystal structures can be realized: (a) At small $d$, the Triangular Ising AntiFerromagnetic (TIAF) crystal is stabilized in which the particles overall form a single-layer like triangular crystal while satisfying the condition that no nearest-neighbor triangle has all three particles in the same layer. (b) At intermediate $d$, a Correlated Square (CS) crystal is stabilized, in which particles in each layer form a square lattice, with the particles in one layer located directly across the centers of the squares of the other. (c) At large $d$, we find a Bilayer Graphene (BG) crystal in which the A and B sites of the graphene lattice lie in different layers. All crystals that we predict are strongly correlated crystals of composite fermions; a theory incorporating only electron Hartree-Fock crystals does not find any crystals besides the `trivial' ones occurring at large interlayer separations for total filling factor $\nu\leq1/3$ (when layers are uncorrelated and each layer is in the long familiar single-layer crystal phase). The TIAF, CS and BG crystals come in several varieties, with different flavors of composite fermions and different interlayer correlations. The appearance of these exotic crystal phases adds to the richness of the physics of electron bilayers in a strong magnetic field, and also provides insight into experimentally observed bilayer insulator as well as transitions within the insulating part of the phase diagram.

Direct Observation of Composite Fermions and Their Fully-Spin-Polarized Fermi Sea near ν=5/2.

The enigmatic even-denominator fractional quantum Hall state at Landau level filling factor ν=5/2 is arguably the most promising candidate for harboring Majorana quasiparticles with non-Abelian statistics and, thus, of potential use for topological quantum computing. The theoretical description of the ν=5/2 state is generally believed to involve a topological p-wave pairing of fully-spin-polarized composite fermions through their condensation into a non-Abelian Moore-Read Pfaffian state. There is, however, no direct and conclusive experimental evidence for the existence of composite fermions near ν=5/2 or for an underlying fully-spin-polarized Fermi sea. Here, we report the observation of composite fermions very near ν=5/2 through geometric resonance measurements and find that the measured Fermi wave vector provides direct demonstration of a Fermi sea with full spin polarization. This lends crucial credence to the model of 5/2 fractional quantum Hall effect as a topological p-wave paired state of composite fermions.

Fractional Quantum Hall Effect in SiGe/Si/SiGe Quantum Wells in Weak Quantizing Magnetic Fields

We have experimentally studied the fractional quantum Hall effect (FQHE) in SiGe/Si/SiGe quantum wells in relatively weak magnetic fields, where the Coulomb interaction between electrons exceeds the cyclotron splitting by a factor of a few XX. Minima of the longitudinal resistance have been observed corresponding to the quantum Hall effect of composite fermions with quantum numbers $p$ = 1, 2, 3, and 4. Minima with $p$ = 3 disappear in magnetic fields below 7 Tesla, which may be a consequence of the intersection or even merging of the quantum levels of the composite fermions with different orientations of the pseudo-spin, i.e., those belonging to different valleys. We have also observed minima of the longitudinal resistance at filling factors $\nu$ = 4/5 and 4/11, which may be due to the formation of the second generation of the composite fermions.

Particle-hole symmetry and composite fermions in fractional quantum Hall states

We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write down an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi liquid description, we demonstrate the appearance of the Girvin--Macdonlald--Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the HLR theory is particle-hole symmetric, correlation functions obtained from HLR are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the HLR theory violate it.