Composite fermions in the quantum Hall effect

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.

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

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.

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 particle-hole (PH) symmetry of {\em electrons} is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. This article investigates an emergent symmetry in the fractional quantum Hall effect, namely the PH symmetry of {\em composite fermions}, which relates states at composite fermion filling factors $\nu^*=n+\bar{\nu}$ and $\nu^*=n+1-\bar{\nu}$, where the integer $n$ is the $\Lambda$ level index and $0\leq \bar{\nu}\leq 1$. Detailed calculations using the microscopic theory of composite fermions demonstrate that for low lying $\Lambda$ levels (small $n$): (i) the 2-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\Lambda$ level; and (ii) the 3-body interaction for composite fermions is an order of magnitude smaller than the 2-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\Lambda$ levels, which is also supported by exact diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\nu={n+\bar{\nu}\over 2(n+\bar{\nu})\pm 1}$ and $\nu={n+1-\bar{\nu}\over 2(n+1-\bar{\nu})\pm 1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\Lambda$ level index, the 2-body and 3-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\nu=1/2$.

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

In two-dimensional (2D) electron systems under strong magnetic fields, interactions can cause fractional quantum Hall (FQH) effects. Bringing two 2D conductors to proximity, a new set of correlated states can emerge due to interactions between electrons in the same and opposite layers. Here we report interlayer correlated FQH states in a system of two parallel graphene layers separated by a thin insulator. Current flow in one layer generates different quantized Hall signals in the two layers. This result is interpreted by composite fermion (CF) theory with different intralayer and interlayer Chern-Simons gauge-field coupling. We observe FQH states corresponding to integer values of CF Landau level (LL) filling in both layers, as well as "semi-quantized" states, where a full CF LL couples to a continuously varying partially filled CF LL. Remarkably, we also recognize a quantized state between two coupled half-filled CF LLs, attributable to an interlayer CF exciton condensate.

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.

Experimental signatures of a Berry phase for composite fermions in the fractional quantum Hall effect provide support for the predictions that these composite fermions are Dirac particles. The fractional quantum Hall effect (FQHE)1,2 in two-dimensional electron systems is an exotic, superfluid-like matter with an emergent topological order. From the consideration of the Aharonov–Bohm interaction between electrons and magnetic field, the ground state of a half-filled lowest Landau level is mathematically transformed to a Fermi sea of composite objects of electrons bound to two flux quanta, termed composite fermions (CFs)3,4,5. A strong support for the CF theories comes from experimental confirmation of the predicted Fermi surface at ν = 1/2 (where ν is the Landau level filling factor) from the detection of the Fermi wavevector in semi-classical geometrical resonance experiments2,6,7,8,9. Recent developments in the theory of CFs10,11,12,13,14,15,16,17,18,19,20,21 have led to the prediction of a π Berry phase for the CF circling around the Fermi surface at half-filling10,14,17,18,19,20. In this paper we provide experimental evidence for the detection of the Berry phase of CFs in the fractional quantum Hall effect. Our measurements of the Shubnikov–de Haas oscillations of CFs as a function carrier density at a fixed magnetic field provide strong support for the existence of a π Berry phase at ν = 1/2. We also discover that the conductivity of composite fermions at ν = 1/2 displays an anomalous linear density dependence, whose origin remains mysterious yet tantalizing.

The origin of the fractional quantum Hall effect (FQHE) at 4/11 and 5/13 has remained controversial. We make a compelling case that the FQHE is possible here for fully spin polarized composite fermions, but with an unconventional underlying physics. Thanks to a rather unusual interaction between composite fermions, the FQHE here results from the suppression of pairs with a relative angular momentum of three rather than one, confirming the exotic mechanism proposed by Wójs, Yi, and Quinn [Phys. Rev. B 69, 205322 (2004)]. We predict that the 4/11 state reported a decade ago by Pan etal. [Phys. Rev. Lett. 90, 016801 (2003)] is a conventional partially spin polarized FQHE of composite fermions, and we estimate the Zeeman energy where a phase transition into the unconventional fully spin polarized state will occur.

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].

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.

Thermopower of composite fermions in the fractional quantum Hall effect regime

Single particle basis functions for composite fermions are obtained from which many-composite fermion states confined to the lowest electronic Landau level can be constructed in the standard manner, i.e. by building Slater determinants. This representation enables a Monte Carlo study of systems containing a large number of composite fermions, yielding new quantitative and qualitative information. The ground state energy and the gaps to charged and neutral excitations are computed for a number of fractional quantum Hall effect (FQHE) states, earlier off-limits to a quantitative investigation. The ground state energies are estimated to be accurate to ~0.1% and the gaps at the level of a few percent. It is also shown that at Landau level fillings smaller than or equal to 1/9 the FQHE is unstable to a spontaneous creation of excitons of composite fermions. In addition, this approach provides new conceptual insight into the structure of the composite fermion wave functions, resolving in the affirmative the question of whether it is possible to motivate the composite fermion theory entirely within the lowest Landau level, without appealing to higher Landau levels.

Quasiparticles of the fractional quantum Hall systems obeyfractional (including mutual) exclusion statistics. In this letterwe study the effects of exclusion statistics on thermal activationof quasiparticle pairs in the approximation of generalizedideal gas. The distribution function for thermally activatedquasiparticles is found explicitly for the statistics matrixgiven by the composite fermion picture and the thermodynamicconsequences are discussed analytically.In particular, at lowtemperatures, there is a quasielectron-quasihole duality in thedescription of the system between two adjacent magic fillings.