We argue that the static non-linear Hall conductivity can always be represented as a vector in two-dimensions and as a pseudo-tensor in three-dimensions independent of its microscopic origin. In a single band model with a constant relaxation rate this vector or tensor is proportional to the Berry curvature dipole \cite{Sodemann_2015}. Here, we develop a quantum Boltzmann formalism to second order in electric fields. We find that in addition to the Berry Curvature Dipole term, there exist additional disorder mediated corrections to the non-linear Hall tensor that have the same scaling in impurity scattering rate. These can be thought of as the non-linear counterparts to the side-jump and skew-scattering corrections to the Hall conductivity in the linear regime. We illustrate our formalism by computing the different contributions to the non-linear Hall conductivity of two-dimensional tilted Dirac fermions.
We propose a modified Boltzmann nonlinear electric-transport framework which differs from the nonlinear generalization of the linear Boltzmann formalism by a contribution that has no counterpart in linear response. This contribution follows from the interband-coherence effect of dc electric-fields during scattering and is related to the interband Berry connection. As an application, we demonstrate it in the second-order nonlinear Hall effect of the tilted massive Dirac model. The intuitive Boltzmann constructions are confirmed by a quantum kinetic theory, which shows that arbitrary $n$th-order…
We study theoretically transverse photoconductivity induced by circularly polarized radiation, i.e. the photovoltaic Hall effect, and linearly polarized radiation causing intraband optical transitions in two-dimensional electron gas (2DEG). We develop a microscopic theory of these effects based on analytical solution of the Boltzmann equation for arbitrary electron spectrum and scattering mechanism. We calculate the transverse photoconductivity of 2DEG with parabolic and linear dispersion for short-range and Coulomb scatterers at different temperatures. We show that the transverse electric cur…
The kinetics of the extrinsic spin Hall conductivity induced by the skew scattering is performed from the fully microscopic kinetic spin Bloch equation approach in $(001)$ GaAs symmetric quantum well. In the steady state, the extrinsic spin Hall current/conductivity vanishes for the linear-$\mathbf k$ dependent spin-orbit coupling and is very small for the cubic-$\mathbf k$ dependent spin-orbit coupling. The spin precession induced by the Dresselhaus/Rashba spin-orbit coupling plays a very important role in the vanishment of the extrinsic spin Hall conductivity in the steady state. An in-plane…
We present a direct numerical evidence for reflection symmetry of longitudinal resistivity $ρ_{xx}$ and quantized Hall resistivity $ρ_{xy}$ near the transition between $ν=1$ quantum Hall state and insulator, in accord with the recent experiments. Our results show that a universal scaling behavior of conductances, $σ_{xx}$ and $σ_{xy}$, in the transition regime decide the reflection symmetry of $ρ_{xx}$ and quantization of $ρ_{xy}$, independent of particle-hole symmetry. We also find that in insulating phase away from the transition region $ρ_{xy}$ deviates from the quantization and diverges wi…
Based on quantum field theory, we formulate the Wigner function and quantum kinetic theory for polarized photons in curved spacetimes which admit a covariantly constant timelike vector. From this framework, the photonic chiral/zilch vortical effects are reproduced in a rigidly rotating coordinate. In a spatially inhomogeneous coordinate, we derive the spin Hall effect for the photon helicity current and energy current in equilibrium. Our derivation reveals that such photonic Hall effect are related to the photonic vortical effect via the Lorentz invariance and their transport coefficients matc…
Inversion symmetry breaking and spin-orbit coupling lead to valley and spin Hall effect in MoS2. Because of the large valley separation in momentum space, the valley index is expected to be robust. In this paper, quantum Hall effect in MoS2 originated from Berry curvature is analyzed after review of symmetry structure and spin-orbit coupling Hamiltonian of MoS2. Finally an expression and rough calculation is given for valley and spin Hall effect.
We present a two-band kinetic equation method to investigate the anomalous Hall effect in a Rashba two-dimensional electron system subjected to a homogeneous magnetization. The electron-impurity scattering is taken into account in the self-consistent Born approximation. It is demonstrated that the impurity-density-free anomalous Hall conductivity arises from an intrinsic and a disorder-mediated mechanisms, associated respectively with the electron states under and near the Fermi surface. The intrinsic mechanism relates to a dc-field-induced transition process, or in other words, a linear stati…
We numerically investigate the second-order nonlinear Hall transport properties of a four-terminal system with time-reversal symmetry and broken inversion symmetry. Within the nonequilibrium Green's function formalism, the second-order nonlinear conductances are derived, where the internal Coulomb potential in response to external voltages is explicitly included to guarantee the gauge invariance. For the system with single mirror symmetry Mx, nonlinear Hall properties are only observable in the y direction and contributed solely from the second-order nonlinear effect. From the symmetry point o…
We investigate the topological phase derived by time-reversal breaking fields in a nonsymmorphic symmetry-protected two-dimensional Dirac semimetal. When the nonsymmorphic symmetry is preserved even in the presence of the field, the two-dimensional electronic states change into two distinct topological phases with the insulating gap. One phase is well-known as quantum Hall states with chiral edge modes accompanying the Hall current, but the other one is an unconventional topological phase with helical edge modes in the absence of time-reversal symmetry.
In this article we briefly review recent experimental and theoretical work on quantum Hall effect in graphene, and argue that some of the quantum Hall states exhibit spontaneous symmetry breaking that is driven by electron-electron interaction. We will also discuss how to experimentally determine the actual manner in which symmetry breaking occurs, and detect the collective charge and neutral excitations associated with symmetry breaking. Other issues will also be briefly mentioned.
In a four terminal sample microscopic time-reversibility leads to symmetry relations between resistance measurements where the role of current and voltage leads are exchanged. These reciprocity relations are a manifestation of general Onsager-Casimir symmetries in equilibrium systems. We investigate experimentally the validity of time reversal symmetry in a $GaAs/Ga_{1-x}Al_xAs$ Hall bar irradiated by an external AC field, at zero magnetic field. For inhomogeneous AC fields we find strong deviations from reciprocity relations and show that their origin can be understood from the the billiard m…
Intrinsic Hall conductivity, emerging when chiral symmetry is broken, is at the heart of future low energy consumption devices because it can generate non-dissipative charge neutral current. A symmetry breaking state is also induced by electronic correlation even for the centro-symmetric crystalline materials. However, generation of non-dissipative charge neutral current by intrinsic Hall conductivity induced by such spontaneous symmetry breaking is experimentally elusive. Here we report intrinsic Hall conductivity and generation of a non-dissipative charge neutral current in a spontaneous ant…
The nonlinear Hall effect has opened the door towards deeper understanding of topological states of matter. It can be observed as the double-frequency Hall voltage response to an ac longitudinal current in the presence of time-reversal symmetry. Disorder plays indispensable roles in various linear Hall effects, such as the localization in the quantized Hall effects and the extrinsic mechanisms of the anomalous, spin, and valley Hall effects. Unlike in the linear Hall effects, disorder enters the nonlinear Hall effect even in the leading order. However, the disorder-induced contribution to the …
The coupled kinetic equation for density and spin Wigner functions are derived including spin-orbit coupling, electric and magnetic field as well as selfconsistent Hartree meanfields suited for SU(2) transport. The interactions are assumed to be with scalar and magnetic impurities as well as scalar and spin-flip potentials among the particles. The spin-orbit interaction is used in a form suitable to solid state physics with Rashba or Dresselhaus coupling, graphene, extrinsic spin-orbit coupling as well as effective nuclear matter coupling. The deficiencies of the two-fluid model are worked out…
We examine the magnetic field driven insulator-quantum Hall-insulator transitions of the two dimensional hole gas in a Ge/SiGe quantum well. We observe direct transitions between low and high magnetic field insulators and the $ν=1$ quantum Hall state. With increasing magnetic field, the transitions from insulating to quantum Hall and quantum Hall to insulating are very similar with respect to their transport properties. We address the temperature dependence around the transitions and show that the characteristic energy scale for the high field transition is larger.
Following a previous work [Shen, Ma, Xie and Zhang, Phys. Rev. Lett. 92, 256603 (2004)] on the resonant spin Hall effect, we present detailed calculations of the spin Hall conductance in two-dimensional quantum wells in a strong perpendicular magnetic field. The Rashba coupling, generated by spin-orbit interaction in wells lacking bulk inversion symmetry, introduces a degeneracy of Zeeman-split Landau levels at certain magnetic fields. This degeneracy, if occuring at the Fermi energy, will induce a resonance in the spin Hall conductance below a characteristic temperature of order of the Zeeman…
Based on the construction of generalized Halperin wave functions, we predict the possible existence of a large class of broken spin symmetry states in bilayer quantum Hall structures, generalizing the recently suggested canted antiferromgnetic phase to many fractional fillings. We develop the appropriate Chern-Simons theory, and establish explicitly that the low-lying neutral excitation is a Goldstone mode and that the charged excitations are bimerons with continuously tunable (through the canted antiferromagnetic order parameter) electric charge on the individual merons.
I review recent novel experimental and theoretical advances in the physics of quantum Hall effect bilayers. Of particular interest is a broken symmetry state which optimizes correlations by putting the electrons into a coherent superposition of the two different layers.
We calculate in an analyical fashion the energies and net spins of skyrmions in fractional quantum Hall systems, based on the suggestion that skyrmion states are spontaneously $L_Z$ and $S_Z$ symmetry-breaking states. The quasihole-skyrmion state with a charge $-e/3$ around $ν$ = 1/3, where the ground state is known as a spin-polarized ferromagnetic state, is found to exist even in high magnetic fields up to about 7 T for GaAs samples.
We construct a family of holomorphic $β$-functions whose RG flow preserves the $Γ(2)$ modular symmetry and reproduces the observed stability of the Hall plateaus. The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram. The generic scale dependance of the conductivities is found to agree qualitatively with the present experimental data. The existence of a crossing point occuring in the …
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…
A monolayer of WTe$_2$ has been shown to display quantum spin Hall (QSH) edge modes persisting up to 100~K in transport experiments. Based on density-functional theory calculations and symmetry-based model building including the role of correlations and substrate support, we develop an effective electronic model for WTe$_2$ which fundamentally differs from other prototypical QSH settings: we find that the extraordinary robustness of quantum spin Hall edge modes in WTe$_2$ roots in a glide symmetry due to which the topological gap opens away from high-symmetry points in momentum space. While th…
In this paper we discuss the implication of the existence of a sliding symmetry, equivalent to the absence of a shear modulus, on the low-energy theory of the quantum hall smectic (QHS) state. We show, through renormalization group calculations, that such a symmetry causes the naive continuum approximation in the direction perpendicular to the stripes to break down through infrared divergent contributions originating from naively irrelevant operators. In particular, we show that the correct fixed point has the form of an array of sliding Luttinger liquids which is free from superficially "irre…
In this Letter, a tunable valley polarization is investigated for honeycomb systems with broken inversion symmetry such as transition-metal dichalcogenide MX2 (M = Mo, W; X = S, Se) monolayers through elliptical pumping. As compared to circular pumping, elliptical pumping is more universal and effective method to create coherent valley polarization. When two valleys of MX2 monolayers are doped or polarized, a novel anomalous valley orbital magnetic moment Hall effect driven by opposite Berry curvatures at different valleys is predicted and can generate orbital magnetic moment current without t…
The quantum spin Hall (QSH) effect is known to be unstable to perturbations violating time-reversal symmetry. We show that creating a narrow ferromagnetic (FM) region near the edge of a QSH sample can push one of the counterpropagating edge states to the inner boundary of the FM region, and leave the other at the outer boundary, without changing their spin polarizations and propagation directions. Since the two edge states are spatially separated into different "lanes", the QSH effect becomes robust against symmetry-breaking perturbations.
We present a systematic microscopic derivation of the semiclassical Boltzmann equation for band structures with the finite Berry curvature based on Keldysh technique of nonequilibrium systems. In the analysis, an ac electrical driving field is kept up to quadratic order, and both cases of small and large frequencies corresponding to intra- and interband transitions are considered. In particular, this formulation is suitable for the study of nonlinear Hall effect and photogalvanic phenomena. The role of impurity scattering is carefully addressed. Specifically, in addition to previously studied …
The third-order nonlinear Hall effect induced by Berry-connection polarizability tensor has been observed in Weyl semimetals T$_d$-MoTe$_2$ as well as T$_d$-TaIrTe$_4$. The experiments were performed on bulk samples, and the results were interpreted with the semiclassical Boltzmann approach. Beyond the bulk limit, we develop a quantum nonlinear transport theory to investigate the third-order Hall response of a four-terminal setup with time-reversal symmetry in quantum regime. The quantum nonlinear theory is verified on a model system of monolayer MoTe$_2$, and numerical results on the angle-re…
We study the coupled dynamics of spin and charge currents in a two-dimensional electron gas in the transport diffusive regime. For systems with inversion symmetry there are established relations between the spin Hall effect, the anomalous Hall effect and the inverse spin Hall effect. However, in two-dimensional electron gases of semiconductors like GaAs, inversion symmetry is broken so that the standard arguments do not apply. We demonstrate that in the presence of a Rashba type of spin-orbit coupling (broken structural inversion symmetry) the anomalous Hall effect, the spin Hall and inverse s…
The appearance of a Hall conductance necessarily requires breaking of time-reversal symmetry, either by an external magnetic field or by the internal magnetization of a material. However, as a second response, Hall dissipationless transverse currents can appear even in time-reversal symmetric conditions in non-centrosymmetric materials. Moreover, this non-linear effect has a quantum origin: it is related to the geometric properties of the electronic wavefunctions and encoded in the dipole moment of the Berry curvature. Here we review the general theory underpinning this effect and discuss vari…
We study a delocalization transition for non-interacting quasiparticles moving in two dimensions, which belongs to a new symmetry class. This symmetry class can be realised in a dirty, gapless superconductor in which time reversal symmetry for orbital motion is broken, but spin rotation symmetry is intact. We find a direct transition between two insulating phases with quantized Hall conductances of zero and two for the conserved quasiparticles. The energy of quasiparticles acts as a relevant, symmetry-breaking field at the critical point, which splits the direct transition into two conventiona…
We study the spin current induced spin-Hall effect that a longitudinal spin dependent chemical potential $qV_{s=x,y,z}$ induces a transverse spin conductances $G^{ss'}$. A four terminal system with Rashba and Dresselhaus spin-orbit interaction (SOI) in the scattering region is considered. By using Landauer-B$\ddot u$ttiker formula with the aid of the Green function, various spin current induced spin-Hall conductances $G^{ss'}$ are calculated. With the charge chemical potential $qV_c$ or spin chemical potential $qV_{s=x,y,z}$, there are 16 elements for the transverse conductances $G^{μν}_p=J_{p…
The band-inverted electron-hole bilayers, such as InAs/GaSb, are an interesting playground for the interplay of quantum spin Hall effect and correlation effects because of the small density of electrons and holes and the relatively small hybridization between the electron and hole bands. It has been proposed that Coulomb interactions lead to a time-reversal symmetry broken phase when the electron and hole densities are tuned from the trivial to the quantum spin Hall insulator regime. We show that the transport properties of the system in the time-reversal symmetry broken phase are consistent w…
In this paper, we present a phenomenological picture based on the composite fermion theory, in responding to the recent discovery by Shahar et al. of a new transport regime near the transition from a $ν=1$ quantum Hall liquid to a Hall insulator(ref[8]). In this picture, the seemingly unexpected reflection symmetry in the longitudinal resistivity $ρ_{xx}$ can be understood clearly as due to the symmetry of the gapful excitations which dominate $σ_{xx}$ across the transition, and the abrupt change in $σ_{xy}$ at the transition. The parameter $α$ in the linear fit of $ν_0(T)$ in ref[8] is also g…
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…
Recently, a zero Hall conductance plateau with random domains is experimentally observed in quantum anomalous Hall (QAH) effect. We study the effects of random domains on the zero Hall plateau in QAH insulators. We find the structure inversion symmetry determines the scaling property of the zero Hall plateau transition in the QAH systems. In the presence of structure inversion symmetry, the zero Hall plateau state shows a quantum-Hall-type critical point, originating from the two decoupled subsystems with opposite Chern numbers. However, the absence of structure inversion symmetry leads to mix…
The quantum anomalous Hall effect (QAHE), the last member of Hall family, was predicted to exhibit quantized Hall conductivity e2/h without any external magnetic field. The QAHE shares a similar physical phenomenon with the integer quantum Hall effect (QHE), whereas its physical origin relies on the intrinsic topological inverted band structure and ferromagnetism. Since the QAHE does not require external energy input in the form of magnetic field, it is believed that this effect has unique potential for applications in future electronic devices with low-power consumption. More recently, the QA…
Magnetic-field-induced phase transitions in the integer quantum Hall effect are studied under the formation of paired Landau bands arising from Zeeman spin splitting. By investigating features of modular symmetry, we showed that modifications to the particle-hole transformation should be considered under the coupling between the paired Landau bands. Our study indicates that such a transformation should be modified either when the Zeeman gap is much smaller than the cyclotron gap, or when these two gaps are comparable.
Instanton analysis for the spin quantum Hall symmetry class: Non-perturbative corrections to physical observables and generalized multifractal spectrum
We investigate effects of the filled Dirac sea on the $SU(4)$ symmetry breaking in the integer quantum Hall states of graphene with long-ranged Coulomb interactions. Our model also includes Hubbard and nearest neighbour repulsive interactions with strengths $U$ and $V$ respectively. We find that the symmetry breaking of the $n=0$ Landau levels induces an $SU(4)$ polarization of the Dirac sea. This results in several phases which are absent when the effects of the Dirac sea are neglected. We compute the phase diagram in the $U$-$V$ space. We also calculate the excitation gaps in tilted magnetic…
Two dimensional inversion symmetry ($180^{\circ}$ rotations in the ``Hall plane'' that hosts the incompressible electron fluid that exhibits the quantized Hall effect) is identified as its fundamental unbroken symmetry. A consequence is that the integers $p$ and $q$ which define both the Landau level filling factor $ν$ = $p/q$ and the elementary fractional charge $\pm e/q$ of topological excitations, cannot have a common divisor greater than 2.
We report on results of systematic numerical studies of two-dimensional electron gas systems subject to a perpendicular magnetic field, with a high Landau level partially filled by electrons. Our results are strongly suggestive of a breakdown of translational symmetry and the presence of crystalline order in the ground state. This is in sharp contrast with the physics of the lowest and first excited Landau levels, and in good qualitative agreement with earlier Hartree-Fock studies. Experimental implications of our results are discussed.
It has been proposed that band-inverted electron-hole bilayers support a phase transition from an insulating phase with spontaneously broken time-reversal symmetry to a quantum spin Hall insulator phase as a function of increasing electron and hole densities. Here, we show that in the presence of proximity-induced superconductivity it is possible to realize Majorana zero modes in the time-reversal symmetry broken phase in the absence of magnetic field. We develop an effective low-energy theory for the system in the presence of time-reversal symmetry breaking order parameter to obtain analytica…
We discuss the implications of approximate particle-hole symmetry in a half-filled Landau level in which a paired quantum Hall state forms. We note that the Pfaffian state is not particle-hole symmetric. Therefore, in the limit of vanishing Landau level mixing, in which particle-hole transformation is an exact symmetry, the Pfaffian spontaneously breaks this symmetry. There is a particle-hole conjugate state, which we call the anti-Pfaffian, which is degenerate with the Pfaffian in this limit. We observe that strong Landau level mixing should favor the Pfaffian, but it is an open problem which…
Magnetic-field-induced phase transitions were studied with a two-dimensional electron AlGaAs/GaAs system. The temperature-driven flow diagram shows the features of the $Γ$(2) modular symmetry, which includes distorted flowlines and shiftted critical point. The deviation of the critical conductivities is attributed to a small but resolved spin splitting, which reduces the symmetry in Landau quantization. [B. P. Dolan, Phys. Rev. B 62, 10278.] Universal scaling is found under the reduction of the modular symmetry. It is also shown that the Hall conductivity could still be governed by the scaling…
Symmetry breaking in a quantum system often leads to complex emergent behavior. In bilayer graphene (BLG), an electric field applied perpendicular to the basal plane breaks the inversion symmetry of the lattice, opening a band gap at the charge neutrality point. In a quantizing magnetic field electron interactions can cause spontaneous symmetry breaking within the spin and valley degrees of freedom, resulting in quantum Hall states (QHS) with complex order. Here we report fractional quantum Hall states (FQHS) in bilayer graphene which show phase transitions that can be tuned by a transverse el…
Unconventional responses upon breaking discrete or crystal symmetries open avenues for exploring emergent physical systems and materials. By breaking inversion symmetry, a nonlinear Hall signal can be observed, even in the presence of time-reversal symmetry, quite different from the conventional Hall effects. Low-symmetry two-dimensional materials are promising candidates for the nonlinear Hall effect, but it is less known when a strong nonlinear Hall signal can be measured, in particular, its connections with the band-structure properties. By using model analysis, we find prominent nonlinear …
The flow of momentum and energy in a fluid is typically associated with dissipative transport coefficients: viscosity and thermal conductivity. Fluids that break certain symmetries such as mirror symmetry and time-reversal invariance can display non-dissipative transport coefficients called odd (or Hall) viscosities and thermal conductivities. The goal of this paper is to elucidate the microscopic origin of these dissipationless transport coefficients using kinetic theory. We show that odd viscosity and odd thermal conductivity arise when the linearized collision operator is not Hermitian. Thi…
Nonequilibrium quantum states can be controlled via the driving field in periodically driven systems. Such control, which is called Floquet engineering, has opened various phenomena, such as the light-induced anomalous Hall effect. There are expected to be some essential differences between the anomalous Hall and spin Hall effects of periodically driven systems because of the difference in time-reversal symmetry. However, these differences remain unclear due to the lack of Floquet engineering of the spin Hall effect. Here we show that when the helicity of circularly polarized light is changed …
Numerical results suggest that the quantum Hall effect at ν = 5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau level mixing are strong. We show that the recent proposal of a PH-Pfaffian topological order by Son is consistent with all experiments. The absence of the particle-hole symmetry at ν = 5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.
