Session G54: Fractional Quantum Hall Effect and Composite Fermions

Bloch Ferromagnetism of Composite Fermions

The magnetic properties of the ground state of a low-density, 2D electron system (2DES) have been a topic of intense theoretical and experimental speculation and controversy, because the physics here is governed by strong correlations. Bloch predicted a fully spin-polarized ground state for a dilute fermionic system in 1929. However, such a state has eluded experimental realization for the last nine decades. Here we present an experimental realization of the elusive interaction-driven spin polarization in a dilute, 2D fermionic system, namely composite fermions (CFs). CFs are exotic quasiparticles, each composed of an electron and two flux quanta, formed in the half-filled Landau level of a 2DES. We determine the spin-polarization of these CFs via direct measurements of the CFs’ Fermi wavevector. We find that at high electron densities (n_e), the CFs are fully spin-polarized, consistent with previous experiments. As we lower n_e, the CFs lose their magnetization, also as expected. Remarkably, however, as n_e is further reduced, the CFs make a sudden transition and become fully spin-polarized. This spontaneous magnetization of CFs closely resembles the Bloch ferromagnetism. We also performed theoretical calculations that provide a semi-quantitative understanding of the phenomenon.   

Hall viscosity of composite fermions

Hall viscosity, also known as the Lorentz shear modulus, has been proposed as a topological property of a quantum Hall fluid. Using a recent formulation of the composite fermion theory on the torus [1], we evaluate the Hall viscosities for a large number of fractional quantum Hall states at filling factors of the form ν=n/(2pn±1) [2], where n and p are integers. The calculated Hall viscosities η^A agree with the expression η^A =hSρ/8π, where ρ is the density and S is the ``shift'' in the spherical geometry [3]. We show that the Hall viscosity for ν=n/(2pn+1) may be derived analytically from the microscopic wave functions providing some assumption. This derivation is applicable to a class of states in the parton construction, which are products of integer quantum Hall states with magnetic fields pointing in the same direction [4].
[1] S. Pu, Y.-H. Wu, and J. K. Jain, Phys. Rev. B 96, 195302 （2017）
[2] S. Pu, M. Fremling and J. K. Jain, “Hall Viscosity of Composite Fermions”, arXiv: 1910.06496, (2019)
[3] N. Read, Phys. Rev. B 79, 045308 (2009)
[4] J. K. Jain, Phys. Rev. B 40, 8079(R), (1989)   

Fractional Chern insulators in magic-angle twisted bilayer graphene in the Hofstadter regime

We apply a perpendicular magnetic field to the minimal effective two-orbital Fermi-Hubbard
model describing the low-energy physics of twisted bilayer graphene at the first magic angle. Through
the use of a Peierl’s substitution, we determine the Landau level splitting and study the structure of
the resulting Chern sub-bands for a range of magnetic flux per plaquette. We identify the minimal examples
of low-energy topological flat sub-bands in this framework. We show that, with the inclusion of a
density-density interaction, fractional Chern insulator states can be realized solely within these flat bands.
Specifically, we characterize the ν = 1/3 Laughlin state through the use of change pumping, spectral flow,
entanglement scaling, and CFT edge state counting; and we analyze its dependence on band flatness.
Ultimately, we comment on the requirements for the observation of fractional Chern insulators for this
system in experiment.   

Quantum Well Width Dependence of the Fractional Quantum Hall Energy Gaps

The fractional quantum Hall effect (FQHE) stems from the strong short-range Coulomb interaction in a two-dimensional (2D) electron system at high magnetic fields. However, the Coulomb interaction is softened in realistic samples which contain quasi-2D systems with a finite (non-zero) electron layer thickness. This softening leads to a weakening of the FQHE states, as manifested in a lowering of their energy gaps in both experiments [1] and calculations [2,3]. Here we present measurements of the energy gaps, from the temperature dependence of the magnetoresistance minima, of several fractional states in the lowest Landau level as a function of the width of the confining square quantum well. Our samples have a fixed density (n ≈ 1x10¹¹ cm^-2) while the width of the confining GaAs well varies between 20 and 80 nm. This is in contrast to measurements reported in [1] where the electrons were confined to parabolic Al_xGa_1-xAs quantum wells. The gaps in our measurements change in magnitude when plotted versus the GaAs well width. We contrast our results with numerical calculations that include the role of finite layer thickness [2,3].

[1] M. Shayegan et al., PRL 65, 2916 (1990).
[2] S. He et al., PRB 42, 11376 (1990).
[3] K. Park et al., J. Phys.: Condens. Matter 11, 7283 (1999).   

Achievement of Systematically Higher Transport Mobilities for a Large Range of Two-dimensional Electron Densities in GaAs Quantum Wells

For the range of two-dimensional (2D) electron densities between 1.5 x 10¹⁰ cm^-2 and 1.4 x 10¹¹ cm^-2, we report measurements that confirm a ~60 to 70 % increase in the transport mobility of remotely-doped GaAs quantum wells compared to all previous GaAs material. A characteristic example of this development is the 2D transport mobility at an electron density of 1 x 10¹¹ cm^-2 in a 50 nm quantum well. Previously, in this type of sample the electron mobilities were limited to ~ 1.8 x 10⁷ cm²V^-1s^-1; now we have grown several with transport mobilities reaching ~3.0 x 10⁷ cm²V^-1s^-1. We attribute these results to our program of high-temperature in-situ bakes that have cleaned our AlGaAs barrier material of oxygen impurities, and to the use of large-area cryogenic cold-plates operating at 17 K in the vacuum sump of our molecular beam epitaxy chamber. When cooled to temperatures below 100 mK, the magnetotransport data of the 1 x 10¹¹ cm^-2 samples described above are remarkable, displaying clear quantum Hall features at the Landau level filling factors ν = 11/23, 12/25, and even 13/27.   

Renormalization Group Flow of Dirac Composite Fermions at Half Filling

We compare the effects of gauge field fluctuations between different composite fermion descriptions of the half-filled Landau level: the Halperin-Lee-Read (HLR) theory of non-relativistic composite fermions and the theory of Dirac composite fermions recently proposed by Son. While the Dirac theory is explicitly particle-hole symmetric, the HLR theory contains a fluctuating statistical gauge field which appears to break particle-hole symmetry. To better characterize this distinction, we present a renormalization group study of these theories in the presence of Coulomb interactions, using a controlled expansion both in the number of fermion species and the width of the Coulomb potential. In particular, we pay close attention to the effects in the HLR theory that break particle-hole symmetry. We find that both theories are described by marginal Fermi liquid fixed points and compare their universal features.   

Heterostructure Design for High Two-dimensional Electron Density in GaAs/AlGaAs Quantum Wells at Large Hydrostatic Pressures

The application of hydrostatic pressure to GaAs/AlGaAs heterostructures has been a topic of great interest due to the possibility of achieving a g-factor ≈ 0 in high-quality, two-dimensional electron systems (2DESs). Despite the exciting prospect of studying electron-electron interactions in the absence of spin splitting, there are very few experimental reports on this subject. This is because in standard modulation-doped GaAs/AlGaAs heterostructures, the 2DES density decreases significantly as the magnitude of hydrostatic pressure applied to the system is increased. Here, we present a heterostructure design that circumvents such density degradation, so that high-density, high-quality 2DESs can be obtained at large hydrostatic pressures. We show that by implementing this scheme, the 2DES density only decreases by ~ 4 x 10¹⁰ cm^-2 even at a hydrostatic pressure of 8.2 kbar. This is a factor of ~4 improvement compared to conventional structures. We will also present low-temperature magnetotransport data that demonstrate the high quality of our samples at large hydrostatic pressures.
  

Diagnosis of universal geometric responses of fractional quantum Hall liquids

Geometric response of topological quantum liquids is expected to reveal rich phenomenon. However, explicit demonstration or identification of the geometric response in a microscopic model is very challenging. Here, we demonstrate that Dehn-twist deformation is able to reveal both the universal modular properties and the microscopic features. We provide numerical evidences for various fractional quantum Hall (FQH) states, including fermionic and bosonic Laughlin states, Hierarchy states, Halperin states and Moore-Read states, by means of exact diagonalization calculations. We conclusively show, geometric transformation applied on torus geometry gives rise to a viscosity related Berry phase, which reflects the geometric metric of elementary FQH droplets. It also captures intrinsic modular information like topological spin and chiral central charge. These findings not only provide a unified description of Berry phase induced by geometric deformation, but also provide a systematical way to inspect the incompressibility of a gapped topological order in the projected Landau level.   

Particle-hole symmetry of the fractional quantum Hall effect in the lowest Landau level

Electrons confined to two-dimensions experience the fractional quantum Hall effect (FQHE) at low electron densities, high magnetic fields, and low temperatures. FQHE states are topologically ordered phases characterized by the electron filling factor ν which is the electron number divided by the Landau level degeneracy. Alternatively, under particle-hole conjugation of a spin-polarized system confined to a single Landau level one can consider the system in terms of holes (the absence of an electron) with a hole filling factor of νh = 1 - ν. Naively, if the system maintains particle-hole symmetry, then the FQHE at filling factor ν will also occur at 1- ν with all the same properties. However, realistic effects such as Landau level mixing can break particle-hole symmetry at the level of the Hamiltonian through the inclusion of three-body terms. We study the nature of particle-hole symmetry on the FQHE in the lowest Landau level under realistic conditions numerically using exact diagonalization.   

Identification of topological orders in fractional quantum Hall state at ν=1/4

The nature of the fractional quantum Hall state at quarter filling in a wide quantum well is still under debate. Both one-component non-Abelian and two-component Abelian orders have been proposed to describe the system. Interestingly, these candidates received support from different experiments under disparate conditions. In this article, we focus on non-Abelian orders from Cooper pairing between composite fermions and the Abelian Halperin-(5,5,3) order. We discuss and predict systematically different experimental signatures to identify them in future experiment. In particular, we address the Mach-Zehnder interferometry experiment and show that it can identify the recently proposed 22111 parton order.   

Anomalous nematic states in half-filled high Landau levels

It is well established that the ground states of a two-dimensional electron gas with half-filled high (N ≥ 2) Landau levels are compressible charge-ordered states, known as quantum Hall stripe (QHS) phases. The generic features of QHSs are a maximum (minimum) in a longitudinal resistance R_xx (R_yy) and a non-quantized Hall resistance R_H. This talk will report on emergent minima (maxima) in R_xx (R_yy) and plateau-like features in R_H in half-filled N ≥ 3 Landau levels. Remarkably, these unexpected features emerge at temperatures considerably lower than the onset temperature of QHSs, suggestive of a new ground state.   

Strain-induced resistance anisotropy near the FQHE v=5/2 in two-dimensional GaAs single quantum wells

We report strain-dependent low temperature magnetotransport measurements of a two-dimensional electron gas (2DEG) confined in GaAs single quantum wells. The samples are mounted to a piezoelectric-based strain device with which we can, in situ, apply and vary tensile strain. With this apparatus, we have achieved strain as large as ~0.5% in GaAs quantum wells at cryogenic temperatures. We find that increasing strain causes the magnetoresistance of the 2DEG confined in the quantum well to become anisotropic near the filling factor v = 5/2. Additionally, we find that the effects of strain are highly dependent on the magnitude of the magnetic field applied. These anisotropic states are metastable and resume an isotopic state after a temperature dependent decay.
  

Density Functional Theory of the Abelian Anyon Gas

A Kohn-Sham density functional theory of composite fermions takes into account the flux attached to composite fermions in a self-consistent fashion [1]. We extend this method to treat a system of many anyons. We construct a series of states that do not involve the center of mass excitations, and explore behavior in the vicinity of the fermionic statistics. We compare the energy obtained from our DFT calculation with exact energies known for small systems and find a qualitative and semi-quantitative agreement. Our method provides an understanding of the existing exact result and also a self-consistent way to calculate the ground state properties of the anyon gas in the thermodynamic limit.

[1] Yayun Hu and J. K. Jain, Phys. Rev. Lett. 123, 176802(2019)   

From CFT matrix product states to parent Hamiltonians

We study frustration free (FF) Hamiltonians of fractional quantum Hall (FQH) states from the point of view of the Matrix-product-state (MPS) representation of their ground/excited states. There is a wealth of solvable models relating to FQH physics, which, however, is mostly derived and analyzed from vantage point of first quantized "analytic clustering properties". In contrast, one obtains long-ranged FF lattice models when these Hamiltonians are studied in an orbital basis, which is the natural basis for the MPS representation of FQH state that has been of much interest lately. The connection between MPS-like states and frustration free parent Hamiltonians is central guiding principle in the construction of solvable lattice models, but thus far, only for short range Hamiltonians and MPS of finite bond dimension. The situation in the FQH context is fundamentally different. Here we expose the direct link between the infinite-bond-dimension MPS structure of Laughlin and Moore-Read-type CFT states and their parent Hamiltonians. Possible uses for states that lack a representation with nice analytic clustering properties are discussed.   

Manifestations of the Uniform Berry Curvature in the Physics of Composite Fermions

It was proposed that the composite fermion is subject to a uniform Berry curvature in the momentum space [1]. Based on the picture, we theoretically consider a number of possible manifestations of the uniform Berry curvature. We show that the presence of the Berry curvature induces a shift in the commensurability condition of the geometrical resonance experiments. We argue that the shift had actually been observed in experiments. We further show that an electrostatic modulation, which is equivalent to a modulation of the Chern-Simons field of composite fermions, and the modulation of an externally applied magnetic field, yield different commensurability conditions. Moreover, we determine the low-frequency and long wavelength electromagnetic responses of a composite fermion system, and find results different from both the Halperin-Lee-Read theory and Son's Dirac theory.
[1] J. Shi and W. Ji, Phys. Rev. B 97, 125133 (2018); G. Ji and J. Shi, preprint, arXiv:1901.00321 (2019).