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The fractional quantum Hall effect (FQHE) was discovered by Tsui, Stormer and Gossard in 1982 at Bell Labs. They observed that at very high magnetic fields, a 2DEG shows fractional quantization of the Hall conductance. In particular, they got a quantized Hall plateau of magnitude ρxy = 3h/e2, which is accompanied by the vanishing of the longitudinal conductivity, ρxx, at low temperature (T < 5 K) in GaAs and AlGaAs samples. As opposed to the integer quantum Hall effect (IQHE), where an integer number of Landau levels (LLs) are occupied, here in FQHE the LLs are partially occupied. If onemakes themagnetic field large enough, the lowest Landau level (LLL) will be partially filled. Whatwe can expect is that the system will form some kind of a lattice, for example, a Wigner crystal or a charge density wave. Thus, it naively seems to be reasonable that the system would like to minimize its potential energy, since there is no kinetic energy left in the system corresponding to the LLL, and only a trivial zero point energy is present in the system. Thus, the ions tend to stay away from each other and form something similar to a crystal lattice. However, surprisingly that does not happen, and instead the system becomes an incompressible quantum liquid, which has gaps in the energy spectrum at filling 1/m (m: odd, or a rational fraction of the form n/m). So it is inevitable that the systemminimizes its energy by having gaps at fractional values of filling. The reason is that, owing to the presence of a large number of electrons (macroscopically degenerate in any of the LLs), a many-body interaction is induced, which in fact makes the excitations above this incompressible ground state to be fractional. So in essence, the Hall current carries a fractional charge.
Chapter 4 aims at establishing that the fractional charge calculated in Chapter 3 is sharp. To this end, the calculation of the mean value and second cumulant of the electronic charge localized in one of two wells of a double-well potential in quantum mechanics is contrasted to that of the mean value and second cumulant of the fractional charge localized around a soliton in a dimerization profile of polyacetylene support a pair of soliton and anti-soliton defects far apart from each other.
The date of discovery of the quantum Hall effect (QHE) is known pretty accurately. It occurred at 2:00 a.m. on 5 February 1980 at the high magnetic lab in Grenoble, France (see Fig. 1.1). There was an ongoing research on the transport properties of silicon field-effect transistors (FETs). The main motive was to improve the mobility of these FET devices. The devices that were provided by Dorda and Pepper allowed direct measurement of the resistivity tensor. The system is a highly degenerate two-dimensional (2D) electron gas contained in the inversion layer of a metal oxide semiconductor field effect transistor (MOSFET) operated at low temperatures and strong magnetic fields. The original notes appear in Fig. 1.1, where it is clearly stated that the Hall resistivity involves universal constants and hence signals towards the involvement of a very fundamental phenomenon.
In the classical version of the phenomenon discovered by E. Hall in 1879, just over a hundred years before the discovery of its quantum analogue, one may consider a sample with a planar geometry so as to restrict the carriers to move in a 2D plane. Next, turn on a bias voltage so that a current flows in one of the longitudinal directions and a strong magnetic field perpendicular to the plane of the gas (see Fig. 1.2). Because of the Lorentz force, the carriers drift towards a direction transverse to the direction of the current flowing in the sample. At equilibrium, a voltage develops in the transverse direction, which is known as the Hall voltage. The Hall resistivity, R, defined as the Hall voltage divided by the longitudinal current, is found to linearly depend on the magnetic field, B, and inversely on the carrier density, n, through R = B/nq (q is the charge). A related and possibly more familiar quantity is the Hall coefficient, denoted by RH = R/B, which via its sign yields information on the type of the majority carriers, that is, whether they are electrons or holes.
At very low temperature or at very high values of the magnetic field (or at both), the resistivity of the sample assumes quantized values of the form rxy = h/ne2. Initially, n was found to be an integer with extraordinary precession (one part in ∼ 108). This is shown in Fig. 1.3.
Chapter 8 extends the 10-fold way of gapped phases from one to any dimension of space. This is done by presenting the homotopy groups of the classifying spaces of normalized Dirac masses. There follows two applications. First, there is the interplay between Anderson localization and the topology of classifying spaces for disordered quantum wires. Second, it is possible to derive the breakdown of the 10-fold way due to short-range interactions in any dimension. The chapter closes with the relationship between invertible topological phases and invertible topological field theories.
Having studied a prototype model Hamiltonian in one-dimensional (1D), we turn our focus towards two-dimensional (2D), now with the lens on graphene. Particularly, we shall explore whether graphene possesses the credibility of becoming a topological insulator. That may happen, provided by some means, we are able to open a spectral gap at the Dirac cones. Since a non-zero Berry phase can be a smoking gun for non-trivial properties, let us first look at the Berry phase of graphene.
Broadband frequency-tripling pulses with high energy are attractive for scientific research, such as inertial confinement fusion, but are difficult to scale up. Third-harmonic generation via nonlinear frequency conversion, however, remains a trade-off between bandwidth and conversion efficiency. Based on gradient deuterium deuterated potassium dihydrogen phosphate (KDxH2-xPO4, DKDP) crystal, here we report the generation of frequency-tripling pulses by rapid adiabatic passage with a low-coherence laser driver facility. The efficiency dependence on the phase-matching angle in a Type-II configuration is studied. We attained an output at 352 nm with a bandwidth of 4.4 THz and an efficiency of 36%. These results, to the best of our knowledge, represent the first experimental demonstration of gradient deuterium DKDP crystal in obtaining frequency-tripling pulses. Our research paves a new way for developing high-efficiency, large-bandwidth frequency-tripling technology.
The tangential drift of the trapped alpha particles in bounce or transit averaged kinetic treatments of stellarators reverses direction on each flux surface at a particular value of pitch angle. The vanishing of the tangential drift corresponds to a resonance that allows a narrow collisional boundary layer to form due to the presence of pitch angle scattering by the background ions. The alphas in and adjacent to this drift reversal layer are particularly sensitive to collisions because they are in or very close to resonance. As a result, enhanced collisional transport occurs due to the existence of this drift reversal resonance in a nearly quasisymmetric stellarator with a single helicity imperfection. Moreover, the value of the resonant pitch angle for drift reversal on neighbouring flux surfaces varies continuously, with the inner flux surfaces having a larger resonant pitch angle than the outer ones. This pitch angle dependence means phase space ‘tubes’ or ‘pods’ exist that connect the inner flux surfaces to the outer ones. These pods allow collisional radial transport of the alphas to extend over the entire radial cross section. When collisions are finite, but weak, and the single helicity departure from quasisymmetry large enough, the collisionless alpha particle motion remains constrained by collisions as they complete their drift trajectories in phase. In particular, the small radial scales introduced by the radial extent or width of the phase space pods require the retention of the nonlinear radial drift term in the kinetic equation. The associated collisional radial transport is evaluated and found to be significant, but is shown to preferentially remove slower speed alphas without substantially affecting birth alphas.
In laser systems requiring a flat-top distribution of beam intensity, beam smoothing is a critical technology for enhancing laser energy deposition onto the focal spot. The continuous phase modulator (CPM) is a key component in beam smoothing, as it introduces high-frequency continuous phase modulation across the laser beam profile. However, the presence of the CPM makes it challenging to measure and correct the wavefront aberration of the input laser beam effectively, leading to unwanted beam intensity distribution and bringing difficulty to the design of the CPM. To address this issue, we propose a deep learning enabled robust wavefront sensing (DLWS) method to achieve effective wavefront measurement and active aberration correction, thereby facilitating active beam smoothing using the CPM. The experimental results show that the average wavefront reconstruction error of the DLWS method is 0.04 μm in the root mean square, while the Shack–Hartmann wavefront sensor reconstruction error is 0.17 μm.
In this work, we propose a method for optimising stellarator devices to favour the presence of an electron root solution of the radial electric field. Such a solution can help avoid heavy impurity accumulation, improve neoclassical thermal ion confinement and helium ash exhaust, and possibly reduce turbulence. This study shows that an optimisation for such a root is possible in quasi-isodynamic stellarators. Examples are shown for both vacuum and finite plasma pressure configurations.
With the escalating laser peak power, modulating and detecting the intensity, duration, phase and polarization of ultra-intense laser pulses progressively becomes increasingly arduous due to the limited damage thresholds of conventional optical components. In particular, the generation and detection of ultra-intense vortex lasers pose great challenges for current laser technologies, which has limited the widely potential applications of relativistic vortex lasers in various domains. In this study, we propose to reconstruct the vortex phase and generate and amplify the relativistic vortex lasers via surface plasma holograms (SPHs). By interfering with the object laser and reference laser, SPHs are formed on the target and the phase of the interfering laser is imprinted through the modulation of surface plasma density. In particular, using the quadrature phase-shift interference, the vortex phase of the object laser can be well reconstructed. The generated vortex lasers can be focused and enhanced further by one order of magnitude, up to $1.7\times {10}^{21}$ W/cm${}^2$, which has been demonstrated by full three-dimensional particle-in-cell simulations. For the first time, we provide a practical way to detect the phase of relativistic vortex lasers, which can be applied in large 1–10 PW laser facilities. This will promote future experimental research of vortex-laser–plasma interaction and open a new avenue of plasma optics in the ultra-relativistic regime.
The Hele-Shaw–Cahn–Hilliard model, coupled with phase separation, is numerically simulated to demonstrate the formation of anomalous fingering patterns in a radial displacement of a partially miscible binary-fluid system. The composition of injected fluid is set to be less viscous than the displaced fluid and within the spinodal or metastable phase-separated region, in which the second derivative of the free energy is negative or positive, respectively. Because of phase separation, concentration evolves non-monotonically between the injected and displaced fluids. The simulations reveal four areas of the concentration distribution between the fluids: the inner core; the low-concentration grooves/high-concentration ridges; the isolated fluid fragments or droplets; the mixing zone. The grooves/ridges and the fragments/droplets, which are the unique features of phase separation, form in the spinodal and metastable regions. Four typical types of patterns are categorized: core separation (CS); fingering separation (FS); separation fingering (SF); lollipop fingering, in the order of the dominance of phase separation, respectively. For the patterns of CS and FS, isolated fluid fragments or droplets around the inner core are the main features. Fingering formation is better maintained with droplets in the SF pattern if the phase separation is relatively weaker than viscous fingering (VF). Even continuous fingers are well preserved in the case of dominant VF; phase separation results in lollipop-shaped fingers. The evolving trend of the patterns is in line with the experiments. These patterns are summarized in a pattern diagram, mainly by the magnitude of the second derivative of the free energy profile.
Polarization smoothing can effectively improve the uniformity of focal spots. In this study, we theoretically and experimentally investigated the polarization synthesis of the focal spot under a birefringent wedge (BW) and speckle under the coupling of the BW and continuous phase plate. Polarization distribution was experimentally obtained using rotating quarter-wave plate measurement under a specific wedge angle. The simulated and experimental results are consistent, demonstrating that the focal spot is in a state of coexistence of elliptical and linear polarizations. In addition, the polarization state is determined by the ratio of the amplitudes and the phase difference between the sub-beams. The simulation results showed that the proportion of linear polarization increased with the separation angle of the sub-beam. In contrast, it decreased with the incident light aperture. This research is crucial for accurately describing the polarization distribution and further understanding the laser–plasma interactions.
The continental plates of Earth are known to drift over a geophysical time scale, and their interactions have led to some of the most spectacular geoformations of our planet while also causing natural disasters such as earthquakes and volcanic activity. Understanding the dynamics of interacting continental plates is thus significant. In this work, we present a fluid mechanical investigation of the plate motion, interaction and dynamics. Through numerical experiments, we examine the coupling between a convective fluid and plates floating on top of it. With physical modelling, we show the coupling is both mechanical and thermal, leading to the thermal blanket effect: the floating plate is not only transported by the fluid flow beneath, it also prevents the heat from leaving the fluid, leading to a convective flow that further affects the plate motion. By adding several plates to such a coupled fluid–structure interaction, we also investigate how floating plates interact with each other, and show that under proper conditions, small plates can converge into a supercontinent.
Large-eddy simulations (LES) of a hypersonic boundary layer on a $7^\circ$-half-angle cone are performed to investigate the effects of highly cooled walls (wall-to-recovery temperature ratio of $T_w / T_r \sim 0.1$) on fully developed turbulence and to validate a newly developed rescaling method based on volumetric flow extraction. Two Reynolds numbers are considered, $Re_m = 4.1 \times 10^6\ \text {m}^{-1}$ and $6.4 \times 10^6\ \text {m}^{-1}$, at free-stream Mach numbers of $M_\infty = 7.4$. A comparison with a reference laminar-to-turbulent simulation, capturing the full history of the transitional flow dynamics, reveals that the volumetric rescaling method can generate a synthetic turbulent inflow that preserves the structure of the fluctuations. Equilibrium conditions are recovered after approximately 40 inlet boundary layer thicknesses. Numerical trials show that a longer streamwise extent of the rescaling box increases numerical stability. Analyses of turbulent statistics and flow visualizations reveal strong pressure oscillations, up to $50\,\%$ of local mean pressure near the wall, and two-dimensional longitudinal wave structures resembling second-mode waves, with wavelengths up to 50 % of the boundary layer thickness, and convective Mach numbers of $M_c \simeq 4.5$. It is shown that their quasi-periodic recurrence in the flow is not an artefact of the rescaling method. Strong and localized temperature fluctuations and spikes in the wall-heat flux are associated with such waves. Very high values of temperature variance near the wall result in oscillations of the wall-heat flux exceeding its average. Instances of near-wall temperature falling below the imposed wall temperature of $T_w=300$ K result in pockets of instantaneous heat flux oriented against the statistical mean direction.