The quantum Hall＇s effect： A quantum electrodynamic phenomenon

<正>We have applied Maxwell’s equations to study the physics of quantum Hall’s effect.The electromagnetic properties of this system are obtained.The Hall’s voltage,V_H = 2πh²n_s/em,where n_s is the electron number density,for a 2- dimensional system,and h = 2πh is the Planck’s constant,is found to coincide with the voltage drop across the quantum capacitor.Consideration of the cyclotronic motion of electrons is found to give rise to Hall’s resistance. Ohmic resistances in the horizontal and vertical directions have been found to exist before equilibrium state is reached. At a fundamental level,the Hall’s effect is found to be equivalent to a resonant LCR circuit with L_H = 2πm/e²n_s and C_H = me²/2πh²n_s satisfying the resonance condition with resonant frequency equal to the inverse of the scattering （relaxation） time,τ_s.The Hall’s resistance is found to be R_H =（（L_H）/C_H）^1/2.The Hall’s resistance may be connected with the impedance that the electron wave experiences when it propagates in the 2-dimensional gas.     

Effect of Magnetic-Impurity Scattering in Disordered Two-Dimensional Square Lattices Around Half Filling

Weak-localization effect in the presence of magnetic impurities is studied in disordered two-dimensional tight-binding square lattices around half filling. Both the magnetic and nonmagnetic impurities are assumed to be randomly distributed on small fractions of the sites, while the nonmagnetic impurities have a strong potential yielding a unitary-limit scattering. We derive in details the expressions of diffusive π modes in the retarded-retarded (or advancedadvanced) channel, which result from the existence of particle-hole symmetry. The quantum interference correction to the density of states is calculated. While the magnetic-impurity scattering suppresses the quantum correction from π-mode cooperon, it does not affect the contribution of π-mode diffuson.     

Finite size effects on helical edge states in HgTe quantum wells with the spin orbit coupling due to bulk- and structure-inversion asymmetries

There is a quantum spin Hall state in the inverted HgTe quantum well, characterized by the topologically protected gapless helical edge states lying within the bulk gap. It has been found that for a strip of finite width, the edge states on the two sides can couple together to produce a gap in the spectrum. The phenomenon is called the finite size effect in quantum spin Hall systems. In this paper, we investigate the effects of the spin-orbit coupling due to bulk- and structure-inversion asymmetries on the finite size effect in the HgTe quantum well by means of the numerical diagonalization method. When the bulk-inversion asymmetry is taken into account, it is shown that the energy gap Eg of the edge states due to the finite size effect features an oscillating exponential decay as a function of the strip width of the HgTe quantum well. The origin of this oscillatory pattern on the exponential decay is explained. Furthermore, if the bulk- and structure-inversion asymmetries are considered simultaneously, the structure-inversion asymmetry will induce a shift of the energy gap Eg closing point. Finally, based on the roles of the bulk- and structure-inversion asymmetries on the finite size effects, a way to realize the quantum spin Hall field effect transistor is proposed.     

Intrinsic Hall effect and separation of Rashba and Dresselhaus spin splittings in semiconductor quantum wells

We have proposed a method to separate Rashba and Dresselhaus spin splittings in semiconductor quantum wells by using the intrinsic Hall effect. It is shown that the interference between Rashba and Dresselhaus terms can deflect the electrons in opposite transverse directions with a change of sign in the macroscopic Hall current, thus providing an alternative way to determine the different contributions to the spin--orbit coupling.     

Spin Chern numbers and time-reversal-symmetry-broken quantum spin Hall effect

The quantum spin Hall （QSH） effect is considered to be unstable to perturbations violating the time-reversal （TR） symmetry. We review some recent developments in the search of the QSH effect in the absence of the TR symmetry. The possibility to realize a robust QSH effect by artificial removal of the TR symmetry of the edge states is explored. As a useful tool to characterize topological phases without the TR symmetry, the spin-Chern number theory is introduced.     

Entangled multi-knot lattice model of anyon current

We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyons. Long-range coupling interaction is a fundamental character of this knot lattice model. The short-range coupling models, such as the Ising model,Hamiltonian model of quantum Hall effect, fermion pairing model, Kitaev honeycomb lattice model, and so on, are the short-range coupling cases of this knot lattice model. The long-range coupling knot lattice model bears Abelian and nonAbelian anyons, and shows integral and fractional filling states like the quantum Hall system. The fusion rules of anyons are explicitly demonstrated by braiding crossing states. The eigenstates of quantum models can be represented by a multilayer link lattice pattern whose topology is characterized by the linking number. This topological linking number offers a new quantity to explain and predict physical phenomena in conventional quantum models. For example, a convection flow loop is introduced into the well-known Bardeen–Cooper–Schrieffer fermion pairing model to form a vortex dimer state that offers an explanation of the pseudogap state of unconventional superconductors, and predicts a fractionally filled vortex dimer state. The integrally and fractionally quantized Hall conductance in the conventional quantum Hall system has an exact correspondence with the linking number in this multi-knot lattice model. The real-space knot pattern in the topological insulator model has an equivalent correspondence with the Lissajous knot in momentum space. The quantum phase transition between different quantum states of the quantum spin model is also directly quantified by the change of topological linking number, which revealed the topological character of phase transition. Circularized photons in an optical fiber network are a promising physical implementation of this multi-knot lattice, and provide a different path to topological quantum computation.     

Entropy of the rotating and charged black string to all orders in the Planck length

<正>By using the entanglement entropy method,this paper calculates the statistical entropy of the Bose and Fermi fields in thin films,and derives the Bekenstein-Hawking entropy and its correction term on the background of a rotating and charged black string.Here,the quantum field is entangled with quantum states in the black string and thin film to the event horizon from outside the rotating and charged black string.Taking into account the effect of the generalized uncertainty principle on quantum state density,it removes the difficulty of the divergence of state density near the event horizon in the brick-wall model.These calculations and discussions imply that high density quantum states near the event horizon of a black string are strongly correlated with the quantum states in a black string and that black string entropy is a quantum effect.The ultraviolet cut-off in the brick-wall model is not reasonable.The generalized uncertainty principle should be considered in the high energy quantum field near the event horizon.From the viewpoint of quantum statistical mechanics,the correction value of Bekenstein-Hawking entropy is obtained.This allows the fundamental recognition of the correction value of black string entropy at nonspherical coordinates.     

Anomalous Hall effect of heavy holes in III-V semiconductor quantum wells

The anomalous Hall effect of heavy holes in semiconductor quantum wells is studied in the intrinsic transport regime, where the Berry curvature governs the Hall current properties. Based on the first--order perturbation of wave function the expression of the Hall conductivity the same as that from the semiclassical equation of motion of the Bloch particles is derived. The dependence of Hall conductivity on the system parameters is shown. The amplitude of Hall conductivity is found to be balanced by a competition between the Zeeman splitting and the spin--orbit splitting.     

From magnetically doped topological insulator to the quantum anomalous Hall effect

Quantum Hall effect （QHE）, as a class of quantum phenomena that occur in macroscopic scale, is one of the most important topics in condensed matter physics. It has long been expected that QHE may occur without Landau levels so that neither external magnetic field nor high sample mobility is required for its study and application, Such a QHE free of Landau levels, can appear in topological insulators （TIs） with ferromagnetism as the quantized version of the anomalous Hall effect, i.e., quantum anomalous Hall （QAH） effect. Here we review our recent work on experimental realization of the QAH effect in magnetically doped TIs. With molecular beam epitaxy, we prepare thin films of Cr-doped （Bi,Sb）2Te3 TIs with well- controlled chemical potential and long-range ferromagnetic order that can survive the insulating phase. In such thin films, we eventually observed the quantization of the Hall resistance at h/e2 at zero field, accompanied by a considerable drop in the longitudinal resistance. Under a strong magnetic field, the longitudinal resistance vanishes, whereas the Hall resistance remains at the quantized value. The realization of the QAH effect provides a foundation for many other novel quantum phenomena predicted in TIs, and opens a route to practical applications of quantum Hall physics in low-power-consumption electronics.     

Weak-Localization Effect on the Density of States in Disordered d-wave Superconductors

The weak-localization effect on the quasiparticle density of states (DOS) is studied with the diagrammatic technique in the binary-alloy model of disordered two-dimensional d-wave superconductors both in the Born and the unitary limits. We derive in details the expressions of the Goldstone modes (cooperon and diffuson) for quasiparticle diffuson. For generic Fermi surfaces, the DOS is shown to be subject to a quantum interference correction of logarithmic suppression. In the combined limit of unitarity and nested Fermi surface (the UN limit), it is found that the self-energy diagrams with two π-mode diffusons make additional contributions to the weak-localization effect, which has not been considered in the previous diagrammatic analysis. Due to the contributions of these new diagrams, the DOS in the UN limit is shown to have also a negative logarithmic correction, which is qualitatively different from the previous prediction.     

Abelian and non-abelian Hall liquids and charge-density wave: quantum number fractionalization in one and two dimensions

Previously, we have demonstrated that, on a torus, the Abelian quantum Hall liquid is adiabatically connected to a charge-density wave as the smaller dimension of the torus is varied. In this work, we extend this result to the non-Abelian bosonic Hall state. The outcome of these works is the realization that the paradigms of quantum number fractionalization in one dimension (polyacetylene) and two dimensions (fractional quantum Hall effect) are, in fact, equivalent.     

Quantized anomalous Hall effect in two-dimensional ferromagnets: quantum Hall effect in metals

We study the effect of disorder on the anomalous Hall effect (AHE) in two-dimensional ferromagnets. The topological nature of the AHE leads to the integer quantum Hall effect from a metal, i.e., the quantization of sigma(xy) induced by the localization except for the few extended states carrying Chern numbers. Extensive numerical study on a model reveals that Pruisken's two-parameter scaling theory holds even when the system has no gap with the overlapping multibands and without the uniform magnetic field. Therefore, the condition for the quantized AHE is given only by the Hall conductivity sigma(xy) without the quantum correction, i.e., /sigma(xy)/>e(2)/(2h).     

量子霍尔效应的发展历程

量子霍尔效应的发现是新兴的低维凝聚态物理发展中的一件大事,分数量子霍尔效应的发现更是开创了一个研究多体现象的新时代,并将影响到物理学的很多分支,这个领域两次被授予诺贝尔物理奖,引起了人们很大的兴趣,文章介绍了量子霍尔效应发展的历程,主要内容包括１８９７年霍尔发现霍尔效应、１９８０年Ｋｌａｕｓ ｖｏｎ Ｋｌｉｔｚｉｎｇ发现整数量子霍尔效应、１９８２年崔琦和Ｈｏｒｓｔ Ｌ．Ｓｔｏｒｍｅｒ发现分数量子霍     

Exotic quantum order in low-dimensional systems

Strongly correlated quantum systems in low dimensions often exhibit novel quantum ordering. This ordering is sometimes hidden and can be revealed only by examining new “dual” types of correlations. Such ordering leads to novel collection modes and fractional quantum numbers. Examples will be presented from quantum spin chains and the quantum Hall effect.     

Weak localization, hole-hole interactions, and the "Metal"-insulator transition in two dimensions

A detailed investigation of the metallic behavior in high-quality GaAs-AlGaAs two-dimensional hole systems reveals the presence of quantum corrections to the resistivity at low temperatures. Despite the low density ( r(s)>10) and high quality of these systems, both weak localization (observed via negative magnetoresistance) and weak hole-hole interactions (giving a correction to the Hall constant) are present in the so-called metallic phase where the resistivity decreases with decreasing temperature. If these quantum corrections persist down to T = 0, the results suggest that even at high r(s) there is no metallic phase in two dimensions.     

Current breakdown of the integer and fractional quantum Hall effects detected by torque magnetometry

We have developed a novel technique that enables measurements of the breakdown of both the integer and fractional quantum Hall effects in a two-dimensional electron system without the need to contact the sample. The critical Hall electric fields that we measure are significantly higher than those reported by other workers, and support the quasi-elastic inter-Landau-level tunnelling model of breakdown. Comparison of the fractional quantum Hall effect results with those obtained on the integer quantum Hall effect allows the fractional quantum Hall effect energy gap to be determined and provides a test of the composite-fermion theory. The temperature dependence of the critical current gives an insight into the mechanism by which momentum may be conserved during the breakdown process.     

Insulator, semiclassical oscillations and quantum Hall liquids at low magnetic fields

Magneto-transport measurements are performed on two-dimensional GaAs electron systems to probe the quantum Hall (QH) effect at low magnetic fields. Oscillations following the Shubnikov-de Haas (SdH) formula are observed in the transition from the insulator to QH liquid when the observed almost temperature-independent Hall slope indicates insignificant interaction correction. Our study shows that the existence of SdH oscillations in such a transition can be understood based on the non-interacting model.     

Integer quantum Hall effect in a PbTe quantum well

Transport measurements have been carried out on a 10 nm n-type PbTe/Pb_0.9Eu_0.1Te quantum well at millikelvin temperatures. The Hall and longitudinal resistances are measured in a Van der Pauw geometry under high magnetic fields up to 23 T. A robust signature of the integer quantum Hall effect is observed without any sign of parasitic parallel conduction. The unconventional sequence of filling factors associated with the integer quantum Hall effect is discussed in terms of the occupancy of multiple valleys.     

Transverse electric field-induced quantum valley Hall effects in zigzag-edge graphene nanoribbons

We have investigated gapless edge states in zigzag-edge graphene nanoribbons under a transverse electric field across the opposite edges by using a tight-binding model and the density functional theory calculations. The tight-binding model predicted that a quantum valley Hall effect occurs at the vacuum-nanoribbon interface under a transverse electric field and, in the presence of edge potentials with opposite signs on opposite edges, an additional quantum valley Hall effect occurs under a much lower field. Dangling bonds inevitable at the edges of real nanoribbons, functional groups terminating the edge dangling bonds, and spin polarizations at the edges result in the edge potentials. The density functional theory calculations confirmed that asymmetric edge terminations, such as one having hydrogen at an edge and fluorine at the other edge, lead to the quantum valley Hall effect even in the absence of a transverse electric field. The electric field-induced half-metallicity in the antiferromagnetic phase, which has been intensively investigated in the last decade, was revealed to originate from a half-metallic quantum valley Hall effect.