Fractional quantum Hall effect in higher dimensions

Quantum hall effect in wide parabolic GaAs/Al xGa 1−xAs wells

We investigate the integer quantum Hall effect (IQHE) and the fractional quantum Hall effect (FQHE). We derive the quantized Hall resistance of IQHE in the presence of the high magnetic field using the scheme of standing waves by de Broglie matter wave of electron gas confined within a two-dimensional square-type quantum well. Without any modification of electrons and holes, it is shown that FQHE is only a decoupling mode of the Hall resistance by two-band-type of electrons and holes, which are governed by IQHE respectively.

The quantum Hall effect (QHE) has captivated the attention of theorists and experimentalists following its discovery. First came the astounding integer quantum Hall effect (IQHE) discovered by von Klitzing, Dorda, and Pepper in 1980 [1]. Then came the even more mysterious discovery of the fractional quantum Hall effect (FQHE) by Tsui, Stormer, and Gossard in 1982 [2]. Obviously I cannot provide even an overview of this vast subject. Instead, I will select two techniques that come into play in the theoretical description of the FQHE. Along the way I will cover some aspects of IQHE. However, of necessity, I will be forced to leave out many related developments, too numerous to mention. The books in [3–7] and online notes in [8] may help you with further reading. The first technique is due to Bohm and Pines (BP) [9], and was used to describe an excitation of the electron gas called the plasmon . Since the introduction by BP of this technique in first quantization, it has been refined and reformulated in the diagrammatic framework. I will stick to the wavefunction-based approach because it is very beautiful, and because two of the great problems in recent times – the theory of superconductivity and the theory of the FQHE – were first cracked open by ingenious trial wavefunctions that captured all the essentials. I will introduce the BP approach in terms of the electron gas. The second technique is Chern–Simons field theory. Originally a product of the imaginations of the mathematicians S. S. Chern and J. Simons, it first entered particle physics in the work of Deser, Jackiw, and Templeton [10], and then condensed matter [11–14]. I will describe its role in the FQHE after introducing the problem to you. The Bohm–Pines Theory of Plasmons: The Goal Consider a system of N spinless fermions experiencing the Coulomb interaction Invoking the Fourier transformation (in unit spatial volume) we find that is the density operator (in first quantization), and the q =0 component is presumed to have been neutralized by some background charge.

The quantum Hall effect and associated quantum transport phenomena in low-dimensional systems have been the focus of much attention for more than a decade. Recent theoretical development of interesting quasiparticles - `composite fermions' - has led to significant advances in understanding and predicting the behaviour of two-dimensional electron systems under high transverse magnetic fields. Composite fermions may be viewed as fermions carrying attached (fictitious) magnetic flux. Here we review models of the integer and fractional quantum Hall effects, including the development of a unified picture of the integer and fractional effects based upon composite fermions. The composite fermion picture predicts remarkable new physics: the formation of a Fermi surface at high magnetic fields, and anomalous ballistic transport, thermopower, and surface acoustic wave behaviour. The specific theoretical predictions of the model, as well as the body of experimental evidence for these phenomena are reviewed. We also review recent edge-state models for magnetotransport in low-dimensional devices based on the composite fermion picture. These models explain the fractional quantum Hall effect and transport phenomena in nanoscale devices in a unified framework that also includes edge state models of the integer quantum Hall effect. The features of the composite fermion edge-state model are compared and contrasted with those of other recent edge-state models of the fractional quantum Hall effect.

The last three years have witnessed a growing interest in optical research of the two-dimensional (2D) electron gas in the regimes of the integer and fractional quantum Hall effects. The goal of such studies is the discovery of new behaviors that are not accessible in magnetotransport experiments. The seminal photoluminescence measurements were reported in Si mosfet devices [1] and in multiple GaAs-AlGaAs quantum wells [2]. The optical recombination of the 2D electron system with acceptors in the Si devices showed anomalies at the Landau level filling factors v = p/q of the fractional quantum Hall effect [1]. The results of intrinsic optical emission from the modulation doped quantum wells revealed temperature-dependent intensity anomalies at v = 1 and v = 2/3 [2].

Area non-preserving transformations in the non-commutative plane are introduced with the aim to map the ν=1 integer quantum Hall effect (IQHE) state on the \(\nu=\frac{1}{2p+1}\) fractional quantum Hall effect (FQHE) states. Using the hydrodynamical description of the quantum Hall fluid, it is shown that these transformations are generated by vector fields satisfying the Gauss law in the interacting non-commutative Chern–Simons gauge theory, and the corresponding field-theory Lagrangian is reconstructed. It is demonstrated that the geometric transformations induce quantum-mechanical non-unitary similarity transformations, establishing the interplay between integral and fractional QHEs.

The interplay between interaction and disorder-induced localization is of fundamental interest. This article addresses localization physics in the fractional quantum Hall state, where both interaction and disorder have nonperturbative consequences. We provide compelling theoretical evidence that the localization of a single quasiparticle of the fractional quantum Hall state at filling factor ν=n/(2n+1) has a striking quantitative correspondence to the localization of a single electron in the (n+1)th Landau level. By analogy to the dramatic experimental manifestations of Anderson localization in integer quantum Hall effect, this leads to predictions in the fractional quantum Hall regime regarding the existence of extended states at a critical energy, and the nature of the divergence of the localization length as this energy is approached. Within a mean field approximation, these results can be extended to situations where a finite density of quasiparticles is present.

The main phenomenological features of Integer Quantum Hall Effect (IQHE) and Fractional Quantum Hall Effect (FQHE) are reviewed. A theory is proposed based on a new basis for the single particle states, given by a representation of the Magnetic Translation Group (MTG).

The energy of the activation gaps in the quantum Hall effect has been thought to be reduced by the broadening of the Landau levels due to disorder. The isotopic mass can affect the electron-phonon interactions and the 0 K renormalization of the conduction and valence bands, leading to appreciable energy offsets between lattice sites with different nuclear masses. The isotopic disorder originating from the natural occurrence of nuclear masses could be the dominant broadening mechanism remaining, leading to the experimentally observed quantum Hall effect gaps in high-mobility devices. The Landau level broadening due to the isotopic disorder has been calculated microscopically without fitting parameters.

The two dimensional electron gas subjected to a perpendicular magnetic field is a model system that supports a variety of electronic phases. Perhaps the most well-known are the fractional quantum Hall states, but in recent years there has been an upsurge of interest in the charge ordered phases commonly referred to as electron solids. These solids are a consequence of electron-electron interactions in a magnetic field. While some solid phases form in the lowest Landau level, the charged ordered phases are most abundant in the higher Landau levels. Examples of such phases include the Wigner solids, electronic bubble phases and stripe or nematic phases. Open questions surround the exact role of disorder, confinement potential, temperature and the Landau level index in determining the stability and competition of these phases with other ground states. The interface of GaAs/AlGaAs remains the cleanest host for the two-dimensional electron gas due to the extremely high quality of materials available and the advancement in molecular beam epitaxy growth techniques. As a result, exceptionally high electron mobilities in this system have been instrumental in the discovery of numerous electron solids. In this Thesis, I discuss the discovery and properties of several electron solids that develop in such state-of-the-art two dimensional electron gases. These electron solids often develop at ultra low temperatures, in the milliKelvin temperature range. After an introduction to the physics of the quantum Hall effect in two dimensions, in chapter 3, I discuss electron solids developing in the N=1 Landau level. While these solids have been known for some time, details of the competition of these phases xiii with the nearby fractional quantum Hall states remains elusive. A number of reports observe new fractional quantum Hall states at filling factors where electron solids are found in other experiments. We undertook a systematic study to answer some of these unsettled questions. We see evidence for incipient fractional quantum Hall states at 2+2/7 and 2+5/7 at intermediate temperatures which are overtaken by the electronic bubble phases at lower temperatures. Several missing fractional states including those at filling factors 2+3/5, 2+3/7, 2+4/9 highlight the relative stability of the electronic solids called the bubble phases in the vicinity in our sample. In chapter 4, I discuss a newly seen electron crystal which manifests itself in transport measurements as a reentrant integer quantum Hall state. Reentrant integer behavior is common in high Landau levels, but so far it was not observed in the lowest Landau level in narrow quantum well samples. In contrast to high Landau levels, where such reentrant integer behavior was associated with electronic bubbles, we believe that the same signature in the N=0 Landau level is due to an electronic Wigner crystal. The filling factors at which we observe such reentrance reveal that it is a crystal of holes, rather than electrons. The discovery of this reentrant integer state paints a complex picture of the interplay of the Wigner crystal and fractional quantum Hall states. Finally, in chapter 5, I discuss the observation of a novel phenomenon, that of reentrant fractional quantum Hall effect. In the lowest Landau level, we observe a fractional quantum Hall state, but as the field is increased, we see a deviation and then a return to quantization in the Hall resistance. Such a behavior indicates a novel electron solid. In contrast to the collective localization of electrons evidenced by the reentrant integer quantum Hall effect, such reentrance to a fractional Hall resistance clearly points to the involvement of composite fermion quasiparticles. This property thus distinguishes the ground state we observed as a solid formed of composite fermions. Such a solid phase is evidence for exotic electron-electron correlations at play which are clearly different from those in the traditional Wigner solid of electrons.

Duality and the fractional quantum Hall effect

Chapter 1 Laser Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields

The fractional quantum Hall effect (FQHE) arises from strong correlations between electrons when they are confined to two dimensions and exposed to a strong magnetic field. The underlying physics is the formation of topological particles called composite fermions (CFs), electron-vortex bound states whose integer quantum Hall effect explains a large majority of the observed FQHE states. In recent years, the focus has shifted to the more exotic states that originate from a weak residual interaction between composite fermions. These include chiral p-wave paired states of composite fermions at certain even denominator fractions, unconventional FQHE of composite fermions, and a series of CF crystals at low fillings. Aside from these states, we also review the FQHE in multicomponent systems, which has attracted renewed attention because of the observation of well-developed FQHE in several multivalley systems, such as graphene and AlAs quantum wells.

We demonstrate that formulating the composite-fermion theory of the fractional quantum Hall (FQH) effect in terms of quaternions greatly expands its reach and opens the door into many interesting issues that were previously not amenable to quantitative theoretical investigation. As an illustration, we explore the possibility of a nematic or a charge-density wave instability of the composite-fermion Fermi sea at half-filled Landau level and of the nearby FQH states by looking for a gap closing instability of the neutral magneto-roton excitation. Our quaternion formulation of the FQH effect has been inspired by mathematical developments in the theoretical analyses of gravitational wave modes and cosmic microwave background radiation, where an important role is played by spin-weighted spherical harmonics that are nothing but monopole harmonics appearing in the spherical geometry for the FQH effect.

The Quantum Hall Effect (QHE) in low-dimensional electron systems represents one of the most significant discoveries in condensed matter physics, revealing topologically protected quantization of Hall conductance under strong magnetic fields and low temperatures. This paper presents a comprehensive study of integer and fractional quantum Hall effects with emphasis on low-dimensional electron gases, quantum wells, and two-dimensional materials, highlighting major theoretical and experimental contributions from Indian research groups. The study synthesizes Indian work on GaAs/AlGaAs heterostructures, graphene-based systems, and quantum transport measurements, and proposes an experimental methodology framework consistent with laboratory practices in Indian nanoscience institutes. Simulated analytical results are presented to illustrate quantized conductance behavior and carrier density dependence. The discussion connects Indian advances in low-dimensional transport physics with emerging directions in topological electronics and quantum metrology. The study concludes that Indian contributions have significantly strengthened experimental and theoretical understanding of quantum Hall physics and continue to influence next-generation low-dimensional electronic systems.