Composite Fermion Picture Research Articles

Dualities of Paired Quantum Hall Bilayer States at ν_{T}=1/2+1/2.

Density-balanced, widely separated quantum Hall bilayers at ν_{T}=1 can be described as two copies of composite Fermi liquids (CFLs). The two CFLs have interlayer weak-coupling Bardeen-Cooper-Schrieffer instabilities mediated by gauge fluctuations, the resulting pairing symmetry of which depends on the CFL hypothesis used. If both layers are described by the conventional Halperin-Lee-Read (HLR) theory-based composite electron liquid (CEL), the dominant pairing instability is in the p+ip channel; whereas if one layer is described by CEL and the other by a composite hole liquid (CHL, in the sense of anti-HLR), the dominant pairing instability occurs in the s-wave channel. Using the Dirac composite fermion (CF) picture, we show that these two pairing channels can be mapped onto each other by particle-hole (PH) transformation. Furthermore, we derive the CHL theory as the nonrelativistic limit of the PH-transformed massive Dirac CF theory. Finally, we prove that an effective topological field theory for the paired CEL-CHL in the weak-coupling limit is equivalent to the exciton condensate phase in the strong-coupling limit.

Floquet engineering of a dynamical Z2 lattice gauge field with ultracold atoms

Abstract Gauge field theory is a fundamental concept in modern physics, attracting many theoretical and experimental efforts towards its simulation. In this paper we propose that a simple model, in which fermions coupled to a dynamical lattice gauge field, can be engineered via the Floquet approach. The model possesses both an independent Maxwell term and local Z 2 gauge symmetry. Our proposal relies on a species-dependent optical lattice, and can be achieved in one, two or three dimensions. By a unitary transformation, this model can be mapped into a non-interacting composite fermion system with fluctuating background charge. With the help of this composite fermion picture, two characteristic observations are predicted. One is radio-frequency spectroscopy, which exhibits no dispersion in all parameter regimes. The second is dynamical localization, which depends on the structure of the initial states.

Violation of Luttinger's theorem in the simplest doped Mott insulator: Falicov-Kimball model in the strong-correlation limit

The Luttinger's theorem has long been taken as the key feature of Landau's Fermi liquid, which signals the presence of quasiparticles. Here, by the unbiased Monte Carlo method, violation of Luttinger's theorem is clearly revealed in the Falicov-Kimball (FK) model, indicating the robust correlation-driven non-Fermi liquid characteristic under any electron density. Introducing hole carriers to the half-filled FK leads to Mott insulator-metal transition, where the Mott quantum criticality manifests unconventional scaling behavior in transport properties. Further insight on the violation of the Luttinger's theorem is examined by combining Hubbard-I approximation with a composite fermion picture, which emphasizes the importance of a mixed excitation of the itinerant electron and the composite fermion. Interestingly, when compared FK model with a binary disorder system, it suggests that the two-peak band structure discovered by Monte Carlo and Hubbard-I approaches is underlying the violation of Luttinger's theorem.

Properties of Strange Quark Matter in the Context of Diquark Correlation

AbstractProperties of strange quark matter (SQM) have been studied as diquark matter with diquark correlation. The dense SQM is under strong magnetic field due to electroweak interaction of quarks. Here it is suggested that u, d, s quarks get correlated to form diquark matter of different flavors. The diquarks are suggested to behave like quasiparticles under the influence of strong magnetic field inside the dense quark matter. In the current work diquarks are described as the composite fermions which behave like quasiparticle under the strong magneticfield in an analogy with the electrons behaving in strong magnetic field. The thermodynamic potential for SQM has been estimated and behavior is studied with respect to chemical potential. The equation of state and the velocity of sound through SQM are also studied. Some interesting observations are made. It is suggested that the composite fermion picture of diquark correlation lowers the binding energy of the system making the SQM a more stable state.

Adiabatic construction of hierarchical quantum Hall states

Intuitive arguments and numerical simulations have long been used to argue for the incompressibility of fractionally quantized Hall states at the most prominent odd-denominator Landau-level filling fractions $\ensuremath{\nu}$ = $p$/($p$ ($m$ - 1) + 1), where $p$ is an integer and $m$ an odd integer. Here, the authors strengthen the case by deriving these states more rigorously via adiabatic localization of magnetic flux onto the particles. They arrive at wave functions suggested by Jain's composite-fermion picture.

Asymmetry of the geometrical resonances of composite fermions

We propose an experiment to test the uniform-Berry-curvature picture of composite fermions. We show that the asymmetry of geometrical resonances observed in a periodically modulated composite fermion system can be explained with the uniform-Berry-curvature picture. Moreover, we show that an alternative way of modulating the system, i.e., modulating the external magnetic field, will induce an asymmetry opposite to that of the usual periodic grating modulation which effectively modulates the Chern-Simons field. The experiment can serve as a critical test of the uniform-Berry-curvature picture, and probe the dipole structure of composite fermions proposed by Read.

Quantum Hall hierarchy from coupled wires

The coupled-wire construction provides a useful way to obtain microscopic Hamiltonians for various two-dimensional topological phases, among which fractional quantum Hall states are paradigmatic examples. Using the recently introduced flux attachment and vortex duality transformations for coupled wires, we show that this construction is remarkably versatile to encapsulate phenomenologies of hierarchical quantum Hall states: the Jain-type hierarchy states of composite fermions filling Landau levels and the Haldane-Halperin hierarchy states of quasiparticle condensation. The particle-hole conjugate transformation for coupled-wire models is also given as a special case of the hierarchy construction. We also propose coupled-wire models for the composite Fermi liquid, which turn out to be compatible with a sort of the particle-hole symmetry implemented in a nonlocal way at $\nu=1/2$. Furthermore, our approach shows explicitly the connection between the Moore-Read Pfaffian state and a chiral $p$-wave pairing of the composite fermions. This composite fermion picture is also generalized to a family of the Pfaffian state, including the anti-Pfaffian state and Bonderson-Slingerland hierarchy states.

Matrix product state representation of quasielectron wave functions

Matrix product state techniques provide a very efficient way to numerically evaluate certain classes of quantum Hall wave functions that can be written as correlators in two-dimensional conformal field theories. Important examples are the Laughlin and Moore-Read ground states and their quasihole excitations. In this paper, we extend the matrix product state techniques to evaluate quasielectron wave functions, a more complex task because the corresponding conformal field theory operator is not local. We use our method to obtain density profiles for states with multiple quasielectrons and quasiholes, and to calculate the (mutual) statistical phases of the excitations with high precision. The wave functions we study are subject to a known difficulty: the position of a quasielectron depends on the presence of other quasiparticles, even when their separation is large compared to the magnetic length. Quasielectron wave functions constructed using the composite fermion picture, which are topologically equivalent to the quasielectrons we study, have the same problem. This flaw is serious in that it gives wrong results for the statistical phases obtained by braiding distant quasiparticles. We analyze this problem in detail and show that it originates from an incomplete screening of the topological charges, which invalidates the plasma analogy. We demonstrate that this can be remedied in the case when the separation between the quasiparticles is large, which allows us to obtain the correct statistical phases. Finally, we propose that a modification of the Laughlin state, that allows for local quasielectron operators, should have good topological properties for arbitrary configurations of excitations.

Realizing and adiabatically preparing bosonic integer and fractional quantum Hall states in optical lattices

We study the ground states of 2D lattice bosons in an artificial gauge field. Using state of the art DMRG simulations we obtain the zero temperature phase diagram for hardcore bosons at densities $n_b$ with flux $n_\phi$ per unit cell, which determines a filling $\nu=n_b/n_\phi$. We find several robust quantum Hall phases, including (i) a bosonic integer quantum Hall phase (BIQH) at $\nu=2$, that realizes an interacting symmetry protected topological phase in 2D (ii) bosonic fractional quantum Hall phases including robust states at $\nu=2/3$ and a Laughlin state at $\nu=1/2$. The observed states correspond to the bosonic Jain sequence ($\nu=p/(p+1)$) pointing towards an underlying composite fermion picture. In addition to identifying Hamiltonians whose ground states realize these phases, we discuss their preparation beginning from independent chains, and ramping up interchain couplings. Using time dependent DMRG simulations, these are shown to reliably produce states close to the ground state for experimentally relevant system sizes. Besides the wave-function overlap, we utilize a simple physical signature of these phases, the non-monotonic behavior of a two-point correlation, a direct consequence of edge states in a finite system, to numerically assess the effectiveness of the preparation scheme. Our proposal only utilizes existing experimental capabilities.

Many-Body Chern Numbers of ν = 1/3 and 1/2 States on Various Lattices

For various two dimensional lattices such as honeycomb, kagome, and square-octagon, gauge conventions (string gauge) realizing minimum magnetic fluxes that are consistent with the lattice periodicity are explicitly given. Then many-body interactions of lattice fermions are projected into the Hofstadter bands to form pseudopotentials. By using these pseudopotentials, degenerate many-body ground states are numerically obtained. We further formulate a scheme to calculate the Chern number of the ground state multiplet by the pseudopotentials. For the filling factor of the lowest Landau level, $\nu=1/3$, a simple scaling form of the energy gap are numerially obtained and the ground state is unique except the three-fold topological degeneracy. This is a quantum liquid, which can be lattice analogue of the Laughlin state. For the $\nu=1/2$ case, validity of the composite fermion picture is discussed in relation to the existence of the Fermi surface. Effects of disorder are also described.

Robust fractional quantum Hall effect in the N=2 Landau level in bilayer graphene.

The fractional quantum Hall effect is a canonical example of electron–electron interactions producing new ground states in many-body systems. Most fractional quantum Hall studies have focussed on the lowest Landau level, whose fractional states are successfully explained by the composite fermion model. In the widely studied GaAs-based system, the composite fermion picture is thought to become unstable for the N≥2 Landau level, where competing many-body phases have been observed. Here we report magneto-resistance measurements of fractional quantum Hall states in the N=2 Landau level (filling factors 4<|ν|<8) in bilayer graphene. In contrast with recent observations of particle–hole asymmetry in the N=0/N=1 Landau levels of bilayer graphene, the fractional quantum Hall states we observe in the N=2 Landau level obey particle–hole symmetry within the fully symmetry-broken Landau level. Possible alternative ground states other than the composite fermions are discussed.

On the absence of higher generations of incompressible daughter states of composite Fermion quasiparticles

Jain [1] introduced a simple mean-field (MF) composite Fermion (CF) picture by attaching to each electron in a quantum Hall system a flux tube producing a Chern- Simons magnetic field b(r) = 2pϕ0 Σi δ(r - ri)ẑ. Here ϕ0 = hc/e is the quantum of flux, and the sum is over all electron coordinates ri. He then averaged the total flux and the total charge (electronic plus positive background) over the entire sample. This MF picture gave a system of noninteracting CFs in an effective magnetic field B*0 = νB0. It predicted incompressible quantum liquid (IQL) states at filling factors ν = n(1 + 2pn)-1 for integral values of n. Chen and Quinn [2] demonstrated that Jain’s MF CF picture predicted the total angular momentum values of the lowest energy band of states for any value of the applied magnetic field B0. Justification of when the MFCF picture was valid was given by Wojs and Quinn [3], who extended the CF hierarchy scheme of Sitko et al. [4]. The CF hierarchy gave the Jain states for integrally filled CF Landau levels (CF LLs), and the Haldane hierarchy of all odd denominator fractions when quasiparticles in the highest (partially filled) CF angular momentum shell had interactions sufficiently similar to the Coulomb interactions of electrons in the lowest Landau level. Sitko et al. showed that the predictions of the CF hierarchy scheme were not always correct. By using a simple pair angular momentum identity and the concept of fractional grandparentage, Wojs and Quinn showed that higher generations of CFs could result from the interactions of the original CF quasiparticles only if their interaction energy VQP(L2) as a function of their pair angular momentum L2 increased with increasing L2 faster than L2(L2 + 1). For Laughlin quasielectrons of the ν = 1/3 IQL state this condition was not satisfied. Therefore, no second generation of CFs could occur. The observed IQL at electron filling factor ν = 4/11 can not be attributed to a daughter IQL state at νQE = 1/3 in a totally spin polarized system. Rather, it is an IQL state of electron pairs with pair angular momentum ιp = 2ι – 1. A description of the IQL state of electron pairs and its low energy excitations will be presented, and the possibility of a spin flip quantum phase transition as a function of well width and applied magnetic field will be discussed.

Is the Composite Fermion a Dirac Particle?

We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the $\nu=\frac12$ quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of $\pi$ around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, $\mathcal C\mathcal P$. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors $\frac n{2n+1}$ and $\frac{n+1}{2n+1}$, which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, $n$ and $n+1$, is now mapped to the same half-integer filling factor $n+\frac12$ of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as $d$-wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while $s$-wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a $\mathcal C\mathcal P$-breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.

Microwave pinning modes near Landau fillingν=1in two-dimensional electron systems with alloy disorder

We report measurements of microwave spectra of two-dimensional electron systems hosted in dilute Al alloy ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ for a range of Landau level fillings $\ensuremath{\nu}$ around 1. For $\ensuremath{\nu}g0.8$ or $\ensuremath{\nu}l1.2$, the samples exhibit a microwave resonance whose frequency decreases as $\ensuremath{\nu}$ moves away from 1. A resonance with this behavior is the signature of solids of quasiparticles or quasiholes in the partially occupied Landau level, and was previously seen only in ultralow-disorder samples. For $\ensuremath{\nu}l0.8$ down to as low as $\ensuremath{\nu}=0.54$, a resonance in the spectra is still present in the Al alloy-disordered samples, although it is partially or completely suppressed at $\ensuremath{\nu}=\frac{3}{5}$ and $\frac{1}{2}$, and is strongly damped over much of this $\ensuremath{\nu}$ range. The resonance also shows a striking enhancement in peak frequency for $\ensuremath{\nu}$ just below $\frac{3}{4}$. We discuss possible explanations of the resonance behavior for $\ensuremath{\nu}l0.8$ in terms of the composite fermion picture.

Integer and fractional quantum Hall effect in a strip of stripes

We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the one-dimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe with a reciprocal wavevector 8k_F. This wavevector gives rise to Umklapp scattering and resonant scattering that results in gaps and chiral edge states at all known integer and fractional filling factors \nu. The integer and odd denominator filling factors arise for a uniform distribution of stripes, whereas the even denominator filling factors arise for a non-uniform stripe distribution. We calculate the Hall conductance via the Streda formula and show that it is given by \sigma_H=\nu e^2/h for all filling factors. We show that the composite fermion picture follows directly from the condition of the resonant Umklapp scattering.

Coherent states on the Grassmannian U(4)/U(2)2: oscillator realization and bilayer fractional quantum Hall systems

Bilayer quantum Hall (BLQH) systems, which underlie a U(4) symmetry, display unique quantum coherence effects. We study coherent states (CS) on the complex Grassmannian , orthonormal basis, U(4) generators and their matrix elements in the reproducing kernel Hilbert space of analytic square-integrable holomorphic functions on , which carries a unitary irreducible representation of U(4) with index . A many-body representation of the previous construction is introduced through an oscillator realization of the U(4) Lie algebra generators in terms of eight boson operators. This particle picture allows us to make a physical interpretation of our abstract mathematical construction in the BLQH jargon. In particular, the index λ is related to the number of flux quanta bound to a bi-fermion in the composite fermion picture of Jain for fractions of the filling factor ν = 2. The simpler, and better known, case of spin-s CS on the Riemann–Bloch sphere is also treated in parallel, of which Grassmannian -CS can be regarded as a generalized (matrix) version.

Thickness-induced violation of de Haas-van Alphen effect through exact analytical solutions at a one-electron and a one-composite fermion level

A systematic study of the energetics of electrons in an interface in a magnetic field is reported with exact analytical calculations based on a Landau level (LL) picture, by serious consideration of the finite thickness of the quantum well (QW). The approach is physically transparent and subtly different in its line of reasoning from standard methods avoiding any semi-classical approximation. We find “internal” phase transitions (at partial LL filling) for magnetisation and susceptibility that are not captured by other approaches and that give rise to nontrivial violations of the standard de Haas-van Alphen periods, in a manner that reproduces the exact quantal astrophysical behaviours in the limit of full three-dimensional (3D) space. Upon inclusion of Zeeman splitting, additional features are also found, such as global energy minima originating from the interplay of QW, Zeeman and LL Physics, while a corresponding calculation in a composite fermion picture with Λ-levels, leads to new predictions on magnetic properties of an interacting electron liquid. By pursuing the same line of reasoning for a topologically nontrivial system with a relativistic spectrum, we find evidence that similar effects might be operative in the dimensionality crossover of 3D strong topological insulators to 2D topological insulator quantum wells.

Spin-chain description of fractional quantum Hall states in the Jain series

We discuss the relationship between fractional quantum Hall (FQH) states at filling factor $\nu=p/(2p+1)$ and quantum spin chains. This series corresponds to the Jain series $\nu=p/(2mp+1)$ with $m=1$ where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin S=1 chains with $p$ spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as $p$ is increased which is similar to $S=p$ integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.

Excitation properties ofν=2/3bilayer quantum Hall phases investigated by magnetotransport methods

We report on novel excitation properties in the \ensuremath{\nu} $=$ 2/3 bilayer quantum Hall (QH) phases by carrying out magnetotransport measurements using a GaAs$/$AlGaAs double-quantum-well sample. We refer to the areas with different particular excitation properties in a single phase as phase internal areas (pi-areas). The differences of the excited states between the pi-areas in the high- and low-magnetic-field region of the same phase lead to different behavior of thermal activation energy \ensuremath{\Delta} under a tilted magnetic field. A new pi-area appears between the balanced point and the single-layer region when \ensuremath{\Delta}${}_{\mathrm{SAS}}$ is small. A simple theoretical model based on the composite-fermion picture is advanced for bilayer fractional QH systems to interpret the characteristics of these pi-areas. Finite-size exact-diagonalization calculations are also executed to verify the validity of the theoretical model.

Dynamic Nuclear Polarization in the Fractional Quantum Hall Regime

We investigate dynamic nuclear polarization in quantum point contacts (QPCs) in the integer and fractional quantum Hall regimes. Following the application of a dc bias, fractional plateaus in the QPC shift symmetrically about half filling of the lowest Landau level, ν=1/2, suggesting an interpretation in terms of composite fermions. Polarizing and detecting at different filling factors indicates that Zeeman energy is reduced by the induced nuclear polarization. Mapping effects from integer to fractional regimes extends the composite fermion picture to include hyperfine coupling.