Motivated by the discovery of extremely low energy collective modes in the fractional quantum Hall effect (Kang, Pinczuk {\em et al.}), with energies below the Zeeman energy, we study theoretically the spin reversed excitations for fractional quantum Hall states at $ν=2/5$ and 3/7 and find qualitatively different behavior than for $ν=1/3$. We find that a low-energy, charge-neutral "spin roton," associated with spin reversed excitations that involve a change in the composite-fermion Landau level index, has energy in reasonable agreement with experiment.
The true nature of the lowest-energy, long-wavelength neutral excitation of the fractional quantum Hall effect has been a long outstanding problem. In this Letter, we establish that it is a two-roton bound state.
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 P$\ddot{o}$schl-Teller interaction potential between the particles. We have seen that the long range interaction will give the ro…
Composite Fermion approach using extensive and parallalized numerical analysis has recently established a two-roton bound state as the lowest energy long wavelength neutral excitation of FQHE for finite particle (N~30) system. By focussing on the "oriented dipole" character of magneto roton, we model the two roton problem and solve it variationally (analytically) to find a two-roton bound state with binding energy which is in good agreement with the composite fermion numerical results.
A major obstacle toward a {\em quantitative} verification, by comparison to experiment, of the theory of the excitations of the fractional quantum Hall effect has been the lack of a proper understanding of disorder. We circumvent this problem by studying the {\em neutral} magneto-roton excitations, whose energy is expected to be largely insensitive to disorder. The calculated energies of the roton at 1/3, 2/5 and 3/7 fillings of the lowest Landau level are in good agreement with those measured experimentally.
We apply the magneto-roton theory of the fractional quantum Hall effect to study the collective excitation spectrum of rotating dipolar Fermi gases. The predicted spectrum has a finite energy gap in the long wavelength limit and a roton minimum at finite wave vector. The roton minimum being deepened from filling factor 1/3 to filling factor 1/5 is a signature of incipient crystallization near filling factor 1/7. We also demonstrate that there are no low-lying single-particle excitations below the roton mode. The fractional-quantum-Hall fluid rotating dipolar fermions behaves as an incompressib…
We consider the effect of interactions on electrons confined to two dimensions at Landau level filling $ν=2$, with the specific aim to determine the range of parameters where the fully polarized state is stable. We calculate the charge and the spin density collective modes in random phase approximation (RPA) including vertex corrections (also known as time dependent Hartree Fock), and treating the Landau level mixing accurately within the subspace of a single particle hole pair. It is found that the spin wave excitation mode of the fully polarized state has a roton minimum which deepens as a r…
Exciton-roton mode in moiré fractional Chern insulators
Tsinghua University（清华大学）；Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；Frontier Science Center for Quantum Information, Beijing（北京量子前沿科学中心）
We construct model wavefunctions for the collective modes of fractional quantum Hall systems. The wavefunctions are expressed in terms of symmetric polynomials characterized by a root partition and a "squeezed" basis, and show excellent agreement with exact diagonalization results for finite systems. In the long wavelength limit, the model wavefunctions reduce to those predicted by the single-mode approximation, and remain accurate at energies above the continuum of roton pairs.
According to Wen's theory, a universal behavior of the fractional quantum Hall edge is expected at sufficiently low energies, where the dispersion of the elementary edge excitation is linear. A microscopic calculation shows that the actual dispersion is indeed linear at low energies, but deviates from linearity beyond certain energy, and also exhibits an "edge roton minimum." We determine the edge exponent from a microscopic approach, and find that the nonlinearity of the dispersion makes a surprisingly small correction to the edge exponent even at energies higher than the roton energy. We exp…
Relaxation of collective plasmon inter-Landau-level excitation is determined by emission of LO-phonons or by Auger-like processes when this emission is suppressed off the magneto-phonon resonance conditions. The decay of "one-cyclotron" magneto-plasmons with wave-vectors near the roton minimum is studied under the condition of filling $ν=1$. Some features of this relaxation should be helpful for the experimental detection of the magneto-rotons in a strongly correlated 2D electron gas.
We consider gapped fractional quantum Hall states on the lowest Landau level when the Coulomb energy is much smaller than the cyclotron energy. We introduce two spectral densities, ρ_T(ω) and \bar ρ_T(ω), which are proportional to the probabilities of absorption of circularly polarized gravitons by the quantum Hall system. We prove three sum rules relating these spectral densities with the shift S, the q^4 coefficient of the static structure factor S_4, and the high-frequency shear modulus of the ground state μ_\infty, which is precisely defined. We confirm an inequality, first suggested by Ha…
We calculate the dynamical structure factor of the ν=5/2 non-abelian quantum Hall state in the dipole approximation, valid for large momenta. Due to the fact that both quasi-particles (qps) and quasi-holes (qhs) have an internal Majorana degree of freedom, a qp-qh pair has a fermionic degree of freedom which can be either empty or occupied, and leads to a splitting of the roton mode. Observation of this splitting by means of finite wavelength optical spectroscopy could provide evidence for Majorana modes in the ν=5/2 quantum Hall state.
We determine the lowest and higher order collective modes in both spin-conserving and spin-reversed sectors by calculating energy differences of the appropriate linear combinations of different levels of composite-fermion-excitons and the fully spin-polarized ground states at filling factors $ν=1/3$, 2/5, 3/7, and 4/9. Apart from providing the detailed study of previously reported modes that have also been observed in the experiments, we predict additional higher energy modes at different filling factors. The lowest and the next higher spin-conserving modes have equal number of "magneto-rotons…
We have calculated the energy spectra for almost all the filling fractions in the Jain series low energy fundamental mode as well as higher energy modes using composite fermion theory. The nature of the low energy mode and higher energy mode is nearly identical to the roton mode. We have observed that, these series of filling fractions $ν=\frac{n}{2pn+1}$ possess $n$ number of roton minima in their higher energy mode.
The Ultra Quantum Crystal phases observed in quasi-one-dimensional conductors of the Bechgaard salts family under magnetic field exhibit both Spin Density Wave order and a Quantized Hall Effect. As a result, they also possess a Magneto-Roton mode within the single particle gap, and the usual Goldstone modes. The sign reversals of the Quantum Hall Effect have recently been explained within the Quantized Nesting Model. I show here that the sign reversed phases of the Ultra Quantum Crystal (the "Ribault Phases") have novel properties compared to the majority sign phases. They exhibit at least two…
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…
The Field Induced Spin Density Wave phases observed in quasi-one-dimensional conductors of the Bechgaard salts family under magnetic field exhibit both Spin Density Wave order and a Quantized Hall Effect, which may exhibit sign reversals. The original nature of the condensed phases is evidenced by the collective mode spectrum. Besides the Goldstone modes, a quasi periodic structure of Magneto-Roton modes, predicted to exist for a monotonic sequence of Hall Quantum numbers, is confirmed, and a second mode is shown to exist within the single particle gap. We present numerical estimates of the Ma…
We present here the results of polarized magneto-photoluminescence measurements on a high mobility single-heterojunction. The presence of a doublet structure over a large magnetic field range (2>nu>1/6) is interpreted as possible evidence for the existence of a magneto-roton minima of the charged density waves. This is understood as an indication of strong electronic correlation even in the case of the IQHE limit.
We show model wavefunctions for neutral collective modes in fractional quantum Hall (FQH) states have simple analytic forms obtained from judicially reducing the powers of selected pairs in the ground state Jastrow factor. This scheme of "pair excitations" works for the magneto-roton modes of single-component Abelian and non-Abeliean FQH states, as well as neutral fermion mode for the Moore-Read (MR) state. The analytic wavefunctions enable computations in the thermodynamic limit previous inaccessible to numerics, and the long wavelength limit of the neutral energy gap of the magneto-roton mod…
Inelastic light scattering measurements of low-lying collective excitations of electron double layers in the quantum Hall state at total filling nu_T=1 reveal a deep magnetoroton in the dispersion of charge-density excitations across the tunneling gap. The roton softens and sharpens markedly when the phase boundary for transitions to highly correlated compressible states is approached. The findings are interpreted with Hartree-Fock evaluations that link soft magnetorotons to enhanced excitonic Coulomb interactions and to quantum phase transitions in the ferromagnetic bilayers.
We report inelastic light scattering measurements of dispersive spin and charge density excitations in dilute 2D electron systems reaching densities less than 10^{10} cm^{-2}. In the quantum Hall state at nu=2, roton critical points in the spin inter--Landau level mode show a pronounced softening as r_s is increased. Instead of a soft mode instability predicted by Hartree--Fock calculations for r_s ~ 3.3, we find evidence of multiple rotons in the dispersion of the softening spin excitations. Extrapolation of the data indicates the possibility of an instability for r_s >~ 11.
Quasiparticles with fractional charge and fractional statistics are key features of the fractional quantum Hall effect. We discuss in detail the definitions of fractional charge and statistics and the ways in which these properties may be observed. In addition to theoretical foundations, we review the present status of the experiments in the area. We also discuss the notions of non-Abelian statistics and attempts to find experimental evidence for the existence of non-Abelian quasiparticles in certain quantum Hall systems.
We give a brief introduction to the phenomenon of the Fractional Quantum Hall effect, whose discovery was awarded the Nobel prize in 1998. We also explain the composite fermion picture which describes the fractional quantum Hall effect as the integer quantum Hall effect of composite fermions.
A fractional quantization in a two dimensional space is proposed. The angular momenta of the two dimensional electrons are quantized in fractional numbers by the boundary conditions on a multi-layered Riemann surface. Extended wave functions for the incompressible quantum fluid states are presented and the cohesive and the excitation energies are given.
Two kinds of topological excitations, vortices and skyrmions, are studied in the frame of Ginzburg-Landau theory. We obtain the rigorous relation between the topological excitation and the order parameters in the fractional quantum Hall systems. We also discuss the evolution of the vortices in fractional quantum Hall systems.
We investigate the time dependent thermal relaxation of a two-dimensional electron system in the fractional quantum Hall regime where ballistic phonons are used to heat up the system to a non-equilibrium temperature. The thermal relaxation of a 2DES at $ν=1/2$ can be described in terms of a broad band emission of phonons, with a temperature dependence proportional to $T^4$. In contrast, the relaxation at fractional filling $ν=2/3$ is characterized by phonon emission around a single energy, the magneto-roton gap. This leads to a strongly reduced energy relaxation rate compared to $ν=1/2$ with o…
We obtain a ``mean field'' scaling flow of the longitudinal and the Hall conductivities in the fractional quantum Hall regime. Using the composite fermion picture and assuming that the composite fermions follow the Khmelnitskii-Pruisken scaling flow for the integer quantum Hall effect, the unstable fixed points which govern the transitions between different fractional quantum Hall states are identified. Distributions of the critical longitudinal and Hall conductivities at the unstable fixed points are obtained and implications of the results for the experiments on mesoscopic quantum Hall syste…
We constructed a Hall resistance formula for the fractional quantum Hall effect by analyzing the experimental data reported in [J. P. Eisenstein and H. L. Stormer, {Science} {\bf 248}, 1510 (1990)]. The formula is given as a function of magnetic induction, chemical potential and temperature. The Hall resistance function contains a single-electron energy spectrum, which has phenomenological perturbation terms with three tunable parameters. The formula yields 12 plateaus that are consistent with the experiment. The perturbations can be interpreted as precession or nutation of a Landau orbital in…
The quantum magnetotransport properties of a monolayer of molybdenum disulfide are derived using linear response theory. Especially, the effect of topological terms on longitudinal and Hall conductivity is analyzed. The Hall conductivity exhibits fractional steps in the integer quantum Hall regime. Further complete spin and valley polarization of the longitudinal conductivity is seen in presence of these topological terms. Finally, the Shubnikov-de Hass oscillations are suppressed or enhanced contingent on the sign of these topological terms.
Quasielectrons and quasiholes in the fractional quantum Hall liquids obey fractional (including nontrivial mutual) exclusion statistics. Their statistics matrix can be determined from several possible state-counting scheme, involving different assumptions on statistical correlations. Thermal activation of quasiparticle pairs and thermodynamic properties of the fractional quantum Hall liquids near fillings $1/m$ ($m$ odd) at low temperature are studied in the approximation of generalized ideal gas. The existence of hierarchical states in the fractional quantum Hall effect is shown to be a manif…
In the striped Hall state, a magnetic translation in one direction is spontaneously broken to the discrete translation. The spectrum of the neutral collective excitation is obtained in the single mode approximation at half-filled third and fourth Landau levels. The spectrum is anisotropic and has a multiple line node structure. In one direction, the spectrum resembles the liquid Helium spectrum with the phonon and roton minimum.
We argue that dynamics of gapless Fractional Quantum Hall Edge states is essentially non-linear and that it features fractionally quantized solitons propagating along the edge. Observation of solitons would be a direct evidence of fractional charges. We show that the non-linear dynamics of the Laughlin's FQH state is governed by the quantum Benjamin-Ono equation. Non-linear dynamics of gapless edge states is determined by gapped modes in the bulk of FQH liquid. The dispersion of edge modes is traced to the double boundary layer of FQH states.
We consider interacting bosonic atoms in an optical lattice subject to a large simulated magnetic field. We develop a model similar to a bilayer fractional quantum Hall system valid near simple rational numbers of magnetic flux quanta per lattice cell. Then we calculate its ground state, magnetic lengths, fractional fillings, and find unexpected sign changes in the Hall current. Finally we study methods for detecting these novel features via shot noise and Hall current measurements.
The activation energy measured in transport experiments in the fractional Hall regime corresponds to the energy required to create a far separated particle hole pair. We calculate, for several fractional quantum Hall states, the energy gaps for the excitation in which the excited composite fermion reverses its spin quantum number. Our results indicate that such spin reversed excitations may be relevant in experimentally accessible regimes.
In the fractional quantum Hall effect regime we measure diagonal ($ρ_{xx}$) and Hall ($ρ_{xy}$) magnetoresistivity tensor components of two-dimensional electron system (2DES) in gated GaAs/Al$_{x}$Ga$_{1-x}$As heterojunctions, together with capacitance between 2DES and the gate. We observe 1/3- and 2/3-fractional quantum Hall effect at rather low magnetic fields where corresponding fractional minima in the thermodynamical density of states have already disappeared manifesting complete suppression of the quasiparticle energy gaps.
With the recent observation of graphene-like Landau levels at the surface of topological insulators, the possibility of fractional quantum Hall effect, which is a fundamental signature of strong correlations, has become of interest. Some experiments have reported intra-Landau level structure that is suggestive of fractional quantum Hall effect. This paper discusses the feasibility of fractional quantum Hall effect from a theoretical perspective, and argues that while this effect should occur, ideally, in the $n=0$ and $|n|=1$ Landau levels, it is ruled out in higher $|n|$ Landau levels. Unlike…
It is shown that the description of the FQHE as condensation of the electron system into a ground state with fractionally charged excitations is incompatible with experimental observations. The nature of the Hall voltage in fractional plateaux as a chemical potential difference eliminates the condensate as current carrier and the temperature dependence of longitudinal conductance eliminates the excitation picture of charge carriers.
Two dimensional electron systems exhibiting the fractional quantum Hall effects are characterized by a quantized Hall conductance and a dissipationless bulk. The transport in these systems occurs only at the edges where gapless excitations are present. We present a {\it microscopic} calculation of the edge states in the fractional quantum Hall systems at various filling factors using the extended Hamiltonian theory of the fractional quantum Hall effect. We find that at $ν=1/3$ the quantum Hall edge undergoes a reconstruction as the background potential softens, whereas quantum Hall edges at hi…
We generalize the fractional quantum Hall hierarchy picture to apply to arbitrary, possibly non-Abelian, fractional quantum Hall states. Applying this to the nu = 5/2 Moore-Read state, we construct new explicit trial wavefunctions to describe the fractional quantum Hall effect in the second Landau level. The resulting hierarchy of states, which reproduces the filling fractions of all observed Hall conductance plateaus in the second Landau level, is characterized by electron pairing in the ground state and an excitation spectrum that includes non-Abelian anyons of the Ising type. We propose thi…
Previously we have demonstrated that, on a torus, the abelian quantum hall liquid is adiabatically connected to a charge density wave as the smaller dimension of the torus is varied. In this work we extend this result to the non-abelian bosonic Hall state. The outcome of these works is the realization that the paradigms of quantum number fractionalization in one dimension (polyacetylene) and two dimensions (fractional quantum Hall effect) are in fact equivalent.
We have experimentally identified fractional quasiparticle creation in a tunneling process through a local fractional quantum Hall (FQH) state. The local FQH state is prepared in a low-density region near a quantum point contact (QPC) in an integer quantum Hall (IQH) system. Shot-noise measurements reveal a clear transition from elementary-charge tunneling at low bias to fractional-charge tunneling at high bias. The fractional shot noise is proportional to T1(1 ? T1) over a wide range of T1, where T1 is the transmission probability of the IQH edge channel. This binomial distribution indicates …
Most of the fractions observed to date belong to the sequences $ν=n/(2pn\pm 1)$ and $ν=1-n/(2pn\pm 1)$, $n$ and $p$ integers, understood as the familiar {\em integral} quantum Hall effect of composite fermions. These sequences fail to accommodate, however, many fractions such as $ν=4/11$ and 5/13, discovered recently in ultra-high mobility samples at very low temperatures. We show that these "next generation" fractional quantum Hall states are accurately described as the {\em fractional} quantum Hall effect of composite fermions.
Edge states of the quantum Hall fluid provide an opportunity to study mesoscopic effects in a highly correlated electron system that is both experimentally accessible and theoretically tractable. In this paper we review recent work on the persistent current and Aharonov-Bohm magnetoconductance oscillations in the fractional quantum Hall effect regime.
We derive the effective field theory from the microscopic Hamiltonian of interacting two-dimensional (pseudo) Dirac electrons by performing a statistic gauge transformation. The quantized Hall conductance are expected to be $σ_{xy}=\frac{e^2}{h}(2k-1)$ with $k$ is arbitrary integer. There are also topological excitations which have fractional charge and obey fractional statistics.
Experimental data available in the literature for peak values of the diagonal resistivity in the transitions between fractional quantum Hall states are compared with the theoretical predictions. It is found that the majority of the peak values are close to the theoretical values for two-dimensional systems with moderate mobilities.
We study the behavior of two-dimensional electron gas in the fractional quantum Hall regime in the presence of finite layer thickness and correlated disordered potential. Generalizing the Chern number calculation to many-body systems, we determine the mobility gaps of fractional quantum Hall states based on the distribution of Chern numbers in a microscopic model. We find excellent agreement between experimentally measured activation gaps and our calculated mobility gaps, when combining the effects of both disordered potential and layer thickness. We clarify the difference between mobility gap…
Using a Hamiltonian formulation of Composite Fermions that I recently developed with R. Shankar, I compute the dependence of the spin polarization on the temperature for the translationally invariant fractional quantum Hall states at $ν=1/3$ and $ν=2/5$. I compare my results to experiments at $ν=1/3$ and find reasonably good agreement.
We report a simple route to generate magnetotransport data that results in fractional quantum Hall plateaus in the conductance. Ingredients to the generating model are conducting tiles with integer quantum Hall effect and metallic linkers, further Kirchhoff rules. When connecting few identical tiles in a mosaic, fractional steps occur in the conductance values. Richer spectra representing several fractions occur when the tiles are parametrically varied. Parts of the simulation data are supported with purposefully designed graphene mosaics in high magnetic fields. The findings emphasize that th…
Intra-Landau level excitations in the fractional quantum Hall regime are not accessible via optical absorption measurements. We point out that optical probes are enabled by the periodic potentials produced by a moiré pattern. Our observation is motivated by the recent observations of fractional quantum Hall incompressible states in moiré-patterned graphene on a hexagonal boron nitride substrate, and is theoretically based on $f-$sum rule considerations supplemented by a perturbative analysis of the influence of the moiré potential on many-body states.
Stationary solutions of the Chern-Simons effective field theory for the fractional quantum Hall systems with edges are presented for Hall bar, disk and annulus. In the infinitely long Hall bar geometry (non compact case), the charge density is shown to be monotonic inside the sample. In sharp contrast, spatial oscillatory modes of charge density are found for the two circular geometries, which indicate that in systems with compact geometry, charge and current exist also far from the edges.
The temperature driven flow lines of the Hall and dissipative magnetoconductance data (σ_{xy},σ_{xx}) are studied in the fractional quantum Hall regime for a 2D electron system in GaAs/Al_{x}Ga_{1-x}As heterostructures. The flow lines are rather well described by a recent unified scaling theory developed for both the integer and the fractional quantum Hall effect in a totally spin-polarized 2D electron system which predicts that one (σ_{xy},σ_{xx}) point determines a complete flow line.
We study theoretically the dispersion of a single quasiparticle or quasihole of the fractional quantum Hall effect, obtained by injecting or removing a composite fermion. By comparing to a free fermion system, we estimate the regime of validity of the Landau-Fermi-liquid-type description and deduce the Landau mass.
We report localization of fractional quantum Hall (QH) quasiparticles on graphene antidots. By studying coherent tunneling through the localized QH edge modes on the antidot, we measured the QH quasiparticle charges to be approximately $\pm e/3$ at fractional fillings of $ν= \pm 1/3$. The Dirac spectrum in graphene allows large energy scales and robust quasiparticle localization against thermal excitation. The capability of localizing fractional quasiparticles on QH antidots brings promising opportunities for realizing anyon braiding and novel quantum electronics.
The fractional quantum hall effect (FQHE) is a milestone of modern day physics, its disovery paved the way for the study of fractional charges which do not obey abelian physics. However, all FQHE require an external magnetic field in order for there to be fractionally charged electrons. This work shows a theoretical study of strongly interaction electrons in the Kagome lattice with magnetism. Under optimal conditions, a band gap opens on the surface of the material which hosts fractional Fermions. These fractional Fermions form composite quasi-electrons without the need for external magnetic f…
