We propose a generalized Dirac fermion description for the electronic state of graphene terminated by a zigzag edge. This description admits a spin-orbit coupling needed to preserve time-reversal invariance of the zigzag confinement, otherwise, for spinless particles, showing the parity anomaly typical of quantum electrodynamics in (2+1) dimensions. At a certain critical strength the spin-orbit coupling induces a phase transition of the quantum-spin-Hall type. It is manifested by a novel type of the edge states consisting of a Kramers' pair of counter-propagating modes with opposite spin orientations. Such edge states are capable of accumulating an integer spin in response to a transverse electric field in the absence of a magnetic one. They exist without any excitation gap in the bulk, due to which our system stands out among other quantum spin Hall systems studied earlier. We show that at the transition the local density of states is discontinuous and its energy dependence reflects the phase diagram of the system.
We review some elementary aspects of the critical properties of the series of metal-insulator transitions that constitute the integer quantum Hall effect. Numerical work has proven essential in charting out this phenomenon. Without being complete, we review network models that seem to capture the essentials of this critical phenomenon.
We investigate numerically whether the chiral symmetry is the sole factor dominating the criticality of the quantum Hall transitions in disordered graphene. When the disorder respects the chiral symmetry, the plateau-to-plateau transition at the $n=0$ Landau level is shown to become anomalous (i.e., step-function-like). Surprisingly, however, the anomaly is robust against the inclusion of the uniform next-nearest neighbor hopping, which degrades the chiral symmetry of the lattice models. We have also shown that the ac (optical) Hall conductivity exhibits a robust plateau structure when the dis…
We argue that by inducing superconductivity in graphene via the proximity effect, it is possible to observes the "quantum valley Hall effect". In the presence of magnetic field, supercurrent causes "valley pseudospin" to accumulate at the edges of the superconducting strip. This, and the structure of the superconducting vortex core, provide possibilities to experimentally observe aspects of the "deconfined quantum criticality".
We report the results of a microscopic theory, based on the topological concept of a $θ$ vacuum, which show that the Coulomb potential, unlike any finite ranged interaction potential, renders the longstanding problem of the plateau transitions in the quantum Hall regime non-Fermi liquid like. Our present results, which are of outstanding significance for quantum phase transitions in general and composite fermion ideas in particular, provide a novel understanding of the critical exponent values that have recently been (re)taken from a series of state-of-the-art quantum Hall samples.
Characterized by zigzag and armchair boundaries, the narrow ribbons display the very different characteristics in Hall conductivities. It is shown that the multi-band-crossings occur in the energy spectrum for armchair ribbons, and the number of them depends on the width of ribbons. Theoretically, it is predicated that the conductivities exhibit drastic sign reversals for narrow ribbons as the Fermi energy sweep over the band-crossings. A new classification of armchair ribbons is suggested based on the emergence of a flat band in the energy spectrum only for odd armchair ribbons. The evolution…
Based on differences of generalized Rényi entropies nontrivial constraints on the shape of the distribution function of broadly distributed observables are derived introducing a new parameter in order to quantify the deviation from lognormality. As a test example the properties of the two--measure random Cantor set are calculated exactly and finally using the results of numerical simulations the distribution of the eigenvector components calculated in the critical region of the lowest Landau--band is analyzed.
Mott Criticality as the Confinement Transition of a Pseudogap-Mott Metal
Indian Institute of Science Education and Research Kolkata（印度科学教育研究所加尔各答分校）；The Institute of Mathematical Sciences, HBNI（数学科学研究所，HBNI）；National Institute of Science, Education and Research, HBNI（国家科学、教育和研究学院，HBNI）；Jawaharlal Nehru Center for Advanced Scientific Research（贾瓦哈拉尔·尼赫鲁高级科学研究中心）；Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）
We find that the confinement potential of a few electron quantum dot can be tuned to significantly increase the overlap with certain quantum Hall trial wave functions. Besides manipulating inter-electron interaction, this approach may prove useful in quantum point contact experiments, which involve narrow constrictions.
A model consisting of a mixture of superconducting and quantum links is proposed to describe the integer quantum Hall transition. The quantum links correspond to tunneling of electrons between trajectories trapped in adjacent potential valleys, while the superconducting links correspond to merging of these trajectories once the Fermi energy crosses the saddle point energy separating the two valleys. The quantum Hall transition in this model corresponds to percolation of the superconducting links. Numerical calculations and scaling analysis using two different approaches yield the critical expo…
Multifractal scaling of critical wave functions at a disorder-driven (Anderson) localization transition is modified near boundaries of a sample. Here this effect is studied for the example of the spin quantum Hall plateau transition using the supersymmetry technique for disorder averaging. Upon mapping of the spin quantum Hall transition to the classical percolation problem with reflecting boundaries, a number of multifractal exponents governing wave function scaling near a boundary are obtained exactly. Moreover, additional exact boundary scaling exponents of the localization problem are extr…
The long-thought charge Kondo effects have recently been experimentally realized in the quantum Hall regime. This experiment, supported by numerics, exemplifies the realization of two-channel Kondo state, a non-Fermi Liquid, and its crossover to the one-channel counterpart, a Fermi liquid. Scaling up such a platform, we find a hierarchy of non-Fermi Liquids and their tunable crossovers based on a renormalization group analysis. Utilizing results from a conformal field theory, we further examine the universal conductances of this strongly correlated system and their finite temperature scaling, …
We develop a theory of the non-local transport of two counter-propagating $ν= 1$ quantum Hall edges coupled via a narrow disordered superconductor. The system is self-tuned to the critical point between trivial and topological phases by the competition between tunneling processes with or without particle-hole conversion. The critical conductance is a random, sample-specific quantity with a zero average and unusual bias dependence. The negative values of conductance are relatively stable against variations of the carrier density, which may make the critical state to appear as a topological one.
In quantum anomalous Hall (QAH) insulators, the interior is insulating but electrons can travel with zero resistance along one-dimensional conducting paths known as chiral edge channels (CECs). These CECs have been predicted to be confined to the one-dimensional (1D) edges and exponentially decay in the two-dimensional (2D) bulk. In this work, we present the results of a systematic study of QAH devices fashioned in a Hall bar geometry of different widths. At the charge neutral point, the QAH effect persists in a Hall bar device with a width of only ~72 nm, implying the intrinsic decaying lengt…
Confinement of small-gapped fractional quantum Hall states facilitates quasiparticle manipulation and is an important step towards quasiparticle interference measurements. Demonstrated here is conduction through top gate defined, narrow channels in high density, ultra-high mobility heterostructures. Transport evidence for the persistence of a correlated state at filling fraction 5/3 is shown in channels of 2micron length but gated to near 0.3micron in width. The methods employed to achieve this confinement hold promise for interference devices proposed for studying potential non-Abelian statis…
We investigate the integer quantum Hall effect in Cd3As2 thin films under conditions of strong to moderate quantum confinement (thicknesses of 10 nm, 12 nm, 15 nm). In all the films, we observe the integer quantum Hall effect in the spin-polarized lowest Landau level (filling factor ν = 1) and at spin-degenerate higher index Landau levels with even filling factors (ν = 2,4,6). With increasing quantum confinement, we also observe a lifting of the Landau level spin degeneracy at ν = 3, manifest as the emergence of an anomaly in the longitudinal and Hall resistivity. Tight-binding calculations sh…
We report new experimental data on the plateau-insulator transition in the quantum Hall regime, taken from a low mobility InGaAs/InP heterostructure. By employing the fundamental symmetries of the quantum transport problem we are able to disentangle the universal quantum critical aspects of the magnetoresistance data (critical indices and scaling functions) and the sample dependent aspects due to macroscopic inhomogeneities. Our new results and methodology indicate that the previously established experimental value for the critical index (kappa = 0.42) resulted from an admixture of both univer…
Interaction driven symmetry breaking in a metallic (doped) Dirac system can manifest in the spontaneous gap generation at the nodal point buried below the Fermi level. Across this transition linear conductivity remains finite making its direct observation difficult in linear transport. We propose the nonlinear Hall effect as a direct probe of this transition when inversion symmetry is broken. Specifically, for a two-dimensional Dirac material with a tilted low-energy dispersion, we first predict a transformation of the characteristic inter-band resonance peak into a non-Lorentzian form in the …
Static disorder in a noninteracting gas of electrons confined to two dimensions can drive a continuous quantum (Anderson) transition between a metallic and an insulating state when time-reversal symmetry is preserved but spin-rotation symmetry is broken. The critical exponent $ν$ that characterizes the diverging localization length and the bulk multifractal scaling exponents that characterize the amplitudes of the critical wave functions at the metal-insulator transition do not depend on the topological nature of the insulating state, i.e., whether it is topologically trivial (ordinary insulat…
We have observed an anomalous shift of the high field reentrant insulating phases in a two-dimensional electron system (2DES) tightly confined within a narrow GaAs/AlGaAs quantum well. Instead of the well-known transitions into the high field insulating states centered around $ν= 1/5$, the 2DES confined within an 80Å-wide quantum well exhibits the transition at $ν= 1/3$. Comparably large quantum lifetime of the 2DES in narrow well discounts the effect of disorder and points to confinement as the primary driving force behind the evolution of the reentrant transition.
Band-inverted electron-hole bilayers support quantum spin Hall insulator and exciton condensate phases. We investigate such a bilayer in an external magnetic field. We show that the interlayer correlations lead to formation of a helical quantum Hall exciton condensate state. In contrast to the chiral edge states of the quantum Hall exciton condensate in electron-electron bilayers, existence of the counterpropagating edge modes results in formation of a ground state spin-texture not supporting gapless single-particle excitations. This feature has deep consequences for the low energy behavior of…
The anomalous Floquet Anderson insulator (AFAI) is a two dimensional periodically driven system in which static disorder stabilizes two topologically distinct phases in the thermodynamic limit. The presence of a unit-conducting chiral edge mode and the essential role of disorder induced localization are reminiscent of the integer quantum Hall (IQH) effect. At the same time, chirality in the AFAI is introduced via an orchestrated driving protocol, there is no magnetic field, no energy conservation, and no (Landau level) band structure. In this paper we show that in spite of these differences th…
We explore the longitudinal conductivity of graphene at the Dirac point in a strong magnetic field with two types of short-range scatterers: adatoms that mix the valleys and "scalar" impurities that do not mix them. A scattering theory for the Dirac equation is employed to express the conductance of a graphene sample as a function of impurity coordinates; an averaging over impurity positions is then performed numerically. The conductivity $σ$ is equal to the ballistic value $4e^2/πh$ for each disorder realization provided the number of flux quanta considerably exceeds the number of impurities.…
Two-dimensional (2D) Dirac fermions are a central paradigm of modern condensed matter physics, describing low-energy excitations in graphene, in certain classes of superconductors, and on surfaces of 3D topological insulators. At zero energy E=0, Dirac fermions with mass m are band insulators, with the Chern number jumping by unity at m=0. This observation lead Ludwig et al [Phys. Rev. B 50, 7526 (1994)] to conjecture that the transition in 2D disordered Dirac fermions (DDF) and the integer quantum Hall transition (IQHT) are controlled by the same fixed point and possess the same universal cri…
We present an analytic theory of quantum criticality in the quasi one-dimensional topological Anderson insulators of class AIII and BDI. We describe the systems in terms of two parameters $(g,χ)$ representing localization and topological properties, respectively. Surfaces of half-integer valued $χ$ define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow describing the class A quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entir…
In the absence of spin-orbit coupling, the conventional dogma of Anderson localization asserts that all states localize in two dimensions, with a glaring exception: the quantum Hall plateau transition (QHPT). In that case, the localization length diverges and interference-induced quantum-critical spatial fluctuations appear at all length scales. Normally QHPT states occur only at isolated energies; accessing them therefore requires fine-tuning of the electron density or magnetic field. In this paper we show that QHPT states can be realized throughout an energy continuum, i.e. as an "energy sta…
Quantum anomalous Hall (QAH) effect in the films with nontrivial band structure accompanies the ferromagnetic transition in the system of magnetic dopants. Experimentally, the QAH transition manifests itself as a jump in the dependence of longitudinal resistivity on a weak external magnetic field. Microscopically, this jump originates from the emergence of a chiral edge mode on one side of the ferromagnetic transition. We study analytically the effect of an extended confinement on the structure of the edge modes. We employ the simplest model of the extended confinement in the form of potential…
Under hydrostatic pressure, the ground state of a two-dimensional electron gas at $ν=5/2$ changes from a fractional quantum Hall state to the stripe phase. By measuring the energy gap of the fractional quantum Hall state and of the onset temperature of the stripe phase we mapped out a phase diagram of these competing phases in the pressure-temperature plane. Our data highlight the dichotomy of two descriptions of the half-filled Landau level near the quantum critical point: one based on electrons and another on composite fermions.
An asymmetric break-down of the integer quantized Hall effect is investigated. This rectification effect is observed as a function of the current value and its direction in conjunction with an asymmetric lateral confinement potential defining the Hall-bar. Our electrostatic definition of the Hall-bar via Schottky-gates allows a systematic control of the steepness of the confinement potential at the edges of the Hall-bar. A softer edge (flatter confinement potential) results in more stable Hall-plateaus, i.e. a break-down at a larger current density. For one soft and one hard edge the break-dow…
We investigate how the criticality of the quantum Hall plateau transition in disordered graphene differs from those in the ordinary quantum Hall systems, based on the honeycomb lattice with ripples modeled as random hoppings. The criticality of the graphene-specific n=0 Landau level is found to change dramatically to an anomalous, almost exact fixed point as soon as we make the random hopping spatially correlated over a few bond lengths. We attribute this to the preserved chiral symmetry and suppressed scattering between K and K' points in the Brillouin zone. The results suggest that a fixed p…
Screening-controlled dynamical criticality in the quantum Hall regime
Indian Institute of Science（印度科学学院）；Indian Institute of Science Education and Research, Pune（印度科学教育与研究学院，浦那）；National Institute for Materials Science（国立材料研究所）；Weizmann Institute of Science（魏茨曼科学研究所）
We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibit an unusual mechanism of topological protection against Anderson localization. These describe surface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological superconductors (TSCs), subject to the effects of quenched disorder. Numerics reveal a surprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall effects in classes A, C, and D. Conventional arguments derived from the non-linear $σ$-model picture imply that most TSC surface states should A…
We describe magnetotransport experiments performed on Hall crosses made from quantum wires at LaAlO3/SrTiO3 interfaces. Shubnikov-de Haas oscillations measured in a 14-nm wide structure exhibit modulations that are consistent with spin-orbit coupling or valley degeneracies. Hall measurements performed on the 6 nm-wide Hall cross reveal dissipative coupling to magnetic phases. Hall plateaus are observed that deviate significantly from two-dimensional quantum Hall counterparts. Non-monotonic dips in the Hall resistance are attributed to one-dimensional confinement and spin-orbit coupling.
We incorporate quantum size effect to investigate the extrinsic spin-Hall effect in ultrathin metal films. A Lippmann-Schwinger formalism based theoretical method, accounting for quantum confinement and surface roughness scattering, is developed to calculate both spin-Hall and longitudinal resistivities and spin-Hall angle. The presence of quantum confinement gives rise to a linear relation $ρ_{sH}=αρ+β$ between the extrinsic spin-Hall resistivity $ρ_{sH}$ and longitudinal charge resistivity $ρ$. The linear term $αρ$ originates from side jump, and the constant $β$ is due to skew scattering. Th…
Using the non-commutative Kubo formula for aperiodic solids [1-3] and a recently developed numerical implementation [4], we study the conductivity $σ$ and resistivity $ρ$ tensors as functions of Fermi level and temperature, for models of strongly disordered Chern insulators. The formalism enabled us to converge the transport coefficients at temperatures low enough to enter the quantum critical regime at the Chern-to-trivial insulator transition. We find that the $ρ_{xx}$-curves at different temperatures intersect each other at one single critical point, and that they obey a single-parameter sc…
Motivated by recent transport experiments, we theoretically study the quantum Hall effect in topological semimetal films. Owing to the confinement effect, the bulk subbands originating from the chiral Landau levels establish energy gaps that have quantized Hall conductance and can be observed in relatively thick films. We find that the quantum Hall state is strongly anisotropic for different confinement directions not only due to the presence of the surface states but also because of the bulk chiral Landau levels. As a result, we re-examine the quantum Hall effect from the surface Fermi arcs a…
Even-denominator fractional quantum Hall (FQH) states, such as 5/2 and 7/2, have been well known in a two-dimensional electron gas (2DEG) for decades and are still investigated as candidates of non-Abelian statistics. In this paper, we present the observation of a 3/2 FQH plateau in a single-layer 2DEG with lateral confinement at a bulk filling factor of 5/3. The 3/2 FQH plateau is quantized at (h/e^2 )/(3/2) within 0.02%, and can survive up to 300 mK. This even-denominator FQH plateau may imply intriguing edge structure and excitation in FQH system with lateral confinement. The observations i…
Universal chiral Luttinger liquid behavior has been predicted for fractional quantum Hall edge states, but so far has not been observed experimentally in semiconductor-based two-dimensional electron gases. One likely cause of this absence of universality is the generic occurrence of edge reconstruction in such systems, which is the result of a competition between confinement potential and Coulomb repulsion. We show that due to a completely different mechanism of confinement, edge reconstruction can be avoided in graphene, which allows for the observation of the predicted universality.
We study the effect of electron-electron interactions on the charge and spin structures of a Quantum Hall strip in a triangularly confined potential. We find that the strip undergoes a spin-unpolarized to spin-polarized transition as a function of magnetic field perpendicular to the strip. For sharp confinements the spin-polarization transition is spontaneous and first develops at the softer side of the triangular potential which shows up as an "eye-structure" in the electron dispersion. For sufficiently weak confinements this spin-polarization transition is preceded by a charge reconstruction…
We consider the conductivity $σ_{xx}$ of graphene with negligible intervalley scattering at half filling. We derive the effective field theory, which, for the case of a potential disorder, is a symplectic-class $σ$-model including a topological term with $θ=π$. As a consequence, the system is at a quantum critical point with a universal value of the conductivity of the order of $e^2/h$. When the effective time reversal symmetry is broken, the symmetry class becomes unitary, and $σ_{xx}$ acquires the value characteristic for the quantum Hall transition.
We investigate the spectral statistics of a network model for a three dimensional layered quantum Hall system numerically. The scaling of the quantity $J_0={1/2}< s^2>$ is used to determine the critical exponent $ν$ for several interlayer coupling strengths. Furthermore, we determine the level spacing distribution $P(s)$ as well as the spectral compressibility $χ$ at criticality. We show that the tail of $P(s)$ decays as $\exp(-κs)$ with $κ=1/(2χ)$ and also numerically verify the equation $χ=(d-D_2)/(2d)$, where $D_2$ is the correlation dimension and $d=3$ the spatial dimension.
Scaling of various local observables with a system size at Anderson transition criticality is characterized by a generalized multifractality. We study the generalized multifractality in the spin quantum Hall symmetry class (class C) in the presence of interaction. We employ Finkel'stein nonlinear sigma model and construct the pure scaling derivativeless operators for class C in the presence of interaction. Within the two-loop renormalization group analysis we compute the anomalous dimensions of the pure scaling operators and demonstrate that they are affected by the interaction. We find that t…
Spin-orbit coupling in semiconductors relates the spin of an electron to its momentum and provides a pathway for electrically initializing and manipulating electron spins for applications in spintronics and spin-based quantum information processing. This coupling can be regulated with quantum confinement in semiconductor heterostructures through band structure engineering. Here we investigate the spin Hall effect and current-induced spin polarization in a two-dimensional electron gas confined in (110) AlGaAs quantum wells using Kerr rotation microscopy. In contrast to previous measurements, th…
The sharp ν=1 quantum Hall edge present for hard confinement is shown to have two modes that go soft as the confining potential softens. This signals a second order transition to a reconstructed edge that is either a depolarized spin-texture edge or a polarized charge density wave edge.
We combine electrostatic and magnetic confinement to define a quantum dot in bilayer graphene. The employed geometry couples $n$-doped reservoirs to a $p$-doped dot. At magnetic field values around $B = 2~$T, Coulomb blockade is observed. This demonstrates that the coupling of the co-propagating modes at the $p$-$n$ interface is weak enough to form a tunnel barrier, facilitating transport of single charge carriers onto the dot. This result may be of use for quantum Hall interferometry experiments.
The lowest Landau level of graphene is studied numerically by considering a tight-binding Hamiltonian with disorder. The Hall conductance $σ_\mathrm{xy}$ and the longitudinal conductance $σ_\mathrm{xx}$ are computed. We demonstrate that bond disorder can produce a plateau-like feature centered at $ν=0$, while the longitudinal conductance is nonzero in the same region, reflecting a band of extended states between $\pm E_{c}$, whose magnitude depends on the disorder strength. The critical exponent corresponding to the localization length at the edges of this band is found to be $2.47\pm 0.04$. W…
We propose a realization of Kondo-lattice physics in moiré superlattices at the interface between a WX$_2$ homobilayer and MoX$_2$ monolayer (where X=S,Se). Under appropriate gating conditions, the interface-WX$_2$-layer forms a triangular lattice of local moments that couple to itinerant electrons in the other WX$_2$-layer via a gate-tunable Kondo exchange interaction. Using a parton mean-field approach we identify a range of twist-angles which support a gate-tuned quantum phase transition between a heavy-fermion liquid with large anomalous Hall conductance and a fractionalized chiral spin-li…
The interplay between the confinement potential and electron-electron interactions causes reconstructions of Quantum Hall edges. We study the consequences of this edge reconstruction for the relaxation of hot electrons injected into integer quantum Hall edge states. In translationally invariant edges, the relaxation of hot electrons is governed by three-body collisions which are sensitive to the electron dispersion and thus to reconstruction effects. We show that the relaxation rates are significantly altered in different reconstruction scenarios.
We study mesoscopic spin Hall effect on the surface of topological insulator with a step-function potential. The giant spin polarization induced by a transverse electric current is derived analytically by using McMillan method in the ballistic transport limit, which oscillates across the potential boundary with no confinement from the potential barrier due to the Klein paradox, and should be observable in spin resolved scanning tunneling microscope.
In a Laughlin fractional quantum Hall state, one- and two-quasihole states can be obtained by diagonalizing the many-body Hamiltonian with a trapping potential or, for larger systems, from the linear combination of the edge Jack polynomials. The quasihole states live entirely in the subspace of the lowest-energy branch in the energy spectrum with a fixed number of orbits, or a hard-wall confinement. The reduction in the Hilbert space dimension facilitates the study of time evolution of the quasihole states after, say, the removal of the trapping potential. We explore the quench dynamics under …
We study the spin relaxation (SR) of a two-dimensional electron gas (2DEG) in the quantized Hall regime and discuss the role of spatial inhomogeneity effects on the relaxation. The results are obtained for small filling factors ($ν\ll 1$) or when the filling factor is close to an integer. In either case SR times are essentially determined by a smooth random potential. For small $ν$ we predict a "magneto-confinement" resonance manifested in the enhancement of the SR rate when the Zeeman energy is close to the spacing of confinement sublevels in the low-energy wing of the disorder-broadened Land…
