Duality and the fractional quantum Hall effect

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

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.

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 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].

It was recently discovered that for a boundary system in the presence of a background magnetic field, the quantum fluctuation of the vacuum would create a non-uniform magnetization density for the vacuum and a magnetization current is induced in the vacuum [1]. It was also shown that this “magnetic Casimir effect” of the vacuum is closely related to another quantum effect of the vacuum, the Weyl anomaly. Furthermore, the phenomena can be understood in terms of the holography of the boundary system [2]. In this paper, we generalize this four dimensional effect to six dimensions. We use the AdS/BCFT holography to show that in the presence of a 3-form magnetic field strength H, a string current is induced in a six dimensional boundary conformal field theory. This allows us to determine the gauge field contribution to the Weyl anomaly in six dimensional conformal field theory in a H-flux background. For the (2,0) superconformal field theory of N M5-branes, the current has a magnitude proportional to N3 for large N. This suggests that the degree of freedoms scales as N3 in the (2,0) superconformal theory of N multiple M5-branes. The prediction we have for the Weyl anomaly is a new criteria that the (2,0) theory should satisfy.

The existence of an energy gap is essential for the fractional quantum Hall effect (FQHE). However, in the case of the FQHE, the origin of the gap is different from that in the case of the IQHE. In the latter, the gap already exists in the single-electron spectrum. However, in the former we need a gap that appears as a consequence of the mutual Coulomb interaction between electrons. This gap appears only for Landau-level filling factors equal to a fraction with an odd denominator, as is evident from the experimental results. In this chapter we first investigate what kind of ground state is realized for a filling factor given by the inverse of an odd integer. Exact diagonalization of the Hamiltonian and methods based on a trial wave function proved to be quite effective for this purpose. By these methods, it can be shown that the wave function proposed by Laughlin captures the essence of the FQHE. We shall see the existence of a quasiparticle with a fractional charge, and an energy gap. Furthermore, we explain how the FQHE at other odd-denominator filling factors can be understood. Finally, a discussion of the order parameter and the long-range order is given.

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.

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.

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

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 study of the electronic properties of quasi two dimensional systems has been a very exciting area of condensed matter physics during the last quarter of the 20th century. Among the most interesting discoveries in this area are the incompressible states showing integral and fractional quantum Hall effects. Incompressible quantum liquid states of the integral quantum Hall effect result from an energy gap in the single particle spectrum. The incompressibility of the fractional quantum Hall effect is completely the result of electron–electron interactions in a highly degenerate fractionally filled Landau level. Since the quantum Hall effect involves electrons moving on a two dimensional surface in the presence of a perpendicular magnetic field, we begin with a description of this problem.

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.

Generalizing from previous work on the integer quantum Hall effect, we construct the effective action for the analog of Laughlin states for the fractional quantum Hall effect in higher dimensions. The formalism is a generalization of the parton picture used in two spatial dimensions, the crucial ingredient being the cancellation of anomalies for the gauge fields binding the partons together. Some subtleties which exist even in two dimensions are pointed out. The effective action is obtained from a combination of the Dolbeault and Dirac index theorems. We also present expressions for some transport coefficients such as Hall conductivity and Hall viscosity for the fractional states. Published by the American Physical Society 2025