It is well known that a nonvanishing Hall conductivity requires broken time-reversal symmetry. However, in this work, we demonstrate that Hall-like currents can occur in second-order response to external electric fields in a wide class of time-reversal invariant and inversion breaking materials, at both zero and twice the driving frequency. This nonlinear Hall effect has a quantum origin arising from the dipole moment of the Berry curvature in momentum space, which generates a net anomalous velocity when the system is in a current-carrying state. The nonlinear Hall coefficient is a rank-two pseudotensor, whose form is determined by point group symmetry. We discus optimal conditions to observe this effect and propose candidate two-and three-dimensional materials, including topological crystalline insulators, transition metal dichalcogenides, and Weyl semimetals. DOI: 10.1103/PhysRevLett.115.216806 PACS numbers: 73.43.-f, 03.65.Vf, 72.15.-v, 72.20.My Introduction.-The Hall conductivity of an electron system whose Hamiltonian is invariant under time-reversal symmetry is forced to vanish. Crystals with sufficiently low symmetry can have resistivity tensors which are anisotropic, but Onsager's reciprocity relations [1] force the conductivity to be a symmetric tensor in the presence of time-reversal symmetry. Hence, when the electric field is along its principal axes the current and the electric field are collinear, at least to the first order in electric fields. However, this constraint is only about the linear response and does not necessarily enforce the full current to flow collinearly with the local electric field.In this Letter we study a special type of such nonlinear Hall-like currents. We will demonstrate that metals without inversion symmetry can have a nonlinear Hall-like current arising from the Berry curvature in momentum space. The conventional Hall conductivity can be viewed as the zeroorder moment of the Berry curvature over occupied states, namely, as an integral of the Berry curvature within the metal's Fermi surface. The effect we discuss here is determined by a pseudotensorial quantity that measures a first-order moment of the Berry curvature over the occupied states, and hence we call it the Berry curvature dipole. This nonlinear Hall effect has a quantum origin arising from the anomalous velocity of Bloch electrons generated by the Berry curvature [2], but it is not expected to be quantized.In a time-reversal invariant system, the Berry curvature is odd in momentum space, Ω a ðkÞ ¼ −Ω a ð−kÞ, and hence its integral weighed by the equilibrium Fermi distribution is forced to vanish, because Kramers pair states at k and −k are equally occupied. However, the second-order response is determined by the integral of the Berry curvature evaluated in the nonequilibrium distribution of electrons computed to first order in the electric field. Since the

We estimate the strength of interaction-enhanced coherence between two graphene or topological insulator surface-state layers by solving imaginary-axis gap equations in the random phase approximation. Using a self-consistent treatment of dynamic screening of Coulomb interactions in the gapped phase, we show that the excitonic gap can reach values on the order of the Fermi energy at strong interactions. The gap is discontinuous as a function of interlayer separation and effective fine structure constant, revealing a first order phase transition between effectively incoherent and interlayer coherent phases. To achieve the regime of strong coherence the interlayer separation must be smaller than the Fermi wavelength, and the extrinsic screening of the medium embedding the Dirac layers must be negligible. In the case of a graphene double-layer we comment on the supportive role of the remote π-bands neglected in the two-band Dirac model.

The first-order interaction correction to the irreducible polarization function of pristine graphene is studied at arbitrary relation between momentum and frequency. The results are used to calculate the dielectric function and the dynamical conductivity of graphene beyond the standard random-phase approximation. The computed static dielectric constant compares favorably with recent experiments.

Transitions between topologically distinct electronic states have been predicted in different classes of materials and observed in some. A major goal is the identification of measurable properties that directly expose the topological nature of such transitions. Here we focus on the giant-Rashba material bismuth tellurium iodine (BiTeI) which exhibits a pressure-driven phase transition between topological and trivial insulators in threedimensions. We demonstrate that this transition, which proceeds through an intermediate Weyl semi-metallic state, is accompanied by a giant enhancement of the Berry curvature dipole which can be probed in transport and optoelectronic experiments. From first-principles calculations, we show that the Berrry-dipole -a vector along the polar axis of this material-has opposite orientations in the trivial and topological insulating phases and peaks at the insulator-to-Weyl critical points, at which the nonlinear Hall conductivity can increase by over two orders of magnitude.

We derive effective Hamiltonians for the fractional quantum Hall effect in n = 0 and n = 1 Landau levels that account perturbatively for Landau level mixing by electron-electron interactions. To second order in the ratio of electron-electron interaction to cyclotron energy, Landau level mixing is accounted for by constructing effective interaction Hamiltonians that include two-body and three-body contributions characterized by Haldane pseudopotentials. Our study builds upon previous treatments, using as a stepping stone the observation that the effective Hamiltonian is fully determined by the few-body problem with N = 2 and N = 3 electrons in the partially filled Landau level. For the n = 0 case we use a first quantization approach to provide a compact and transparent derivation of the effective Hamiltonian which captures a class of virtual processes omitted in earlier derivations of Landau-level-mixing corrected Haldane pseudopotentials.

Samarium hexaboride is a classic three-dimensional mixed valence system with a high-temperature metallic phase that evolves into a paramagnetic charge insulator below 40 K. A number of recent experiments have suggested the possibility that the low-temperature insulating bulk hosts electrically neutral gapless fermionic excitations. Here we show that a possible ground state of strongly correlated mixed valence insulators—a composite exciton Fermi liquid—hosts a three dimensional Fermi surface of a neutral fermion, that we name the “composite exciton.” We describe the mechanism responsible for the formation of such excitons, discuss the phenomenology of the composite exciton Fermi liquids and make comparison to experiments in SmB6.

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 5 . 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 sidejump 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. PACS numbers: 73.43.-f, 03.65.Vf, 72.15.-v, 72.20.My Introduction-Two independent experimental studies have recently reported the discovery 1,2 of the time-reversalinvariant non-linear Hall effect (NLHE) in layered transition metal dichalcogenides. Unlike the ordinary Hall effect, the NLHE can occur in time-reversal-invariant metals lacking inversion symmetry 3-6 . Building upon previous studies 3,4 a simple semiclassical theory of this effect was developed in Ref. 5 based on the notion of the Berry curvature dipole (BCD): a tensorial object measuring the average gradient of the Berry curvature over the occupied states. In a single band model with a constant relaxation rate the non-linear conductivity of a time reversal invariant metal was found to be proportional to the BCD. Several subsequent studies have addressed the NLHE and related effects in a variety of contexts and material platforms 7-17 .

Electrons moving in a Bloch band are known to acquire an anomalous Hall velocity proportional to the Berry curvature of the band [1] which is responsible for the intrinsic linear Hall effect in materials with broken time-reversal symmetry [2]. Here, we demonstrate that there is also an anomalous correction to the electron acceleration which is proportional to the Berry curvature dipole [3] and is responsible for the Non-linear Hall effect recently discovered in materials with broken inversion symmetry [4,5]. This allows us to uncover a deeper meaning of the Berry curvature dipole as a nonlinear version of the Drude weight that serves as a measurable order parameter for broken inversion symmetry in metals. We also derive a "Quantum Rectification Sum Rule" in time reversal invariant materials by showing that the integral over frequency of the rectification conductivity depends solely on the Berry connection and not on the band energies. The intraband spectral weight of this sum rule is exhausted by the Berry curvature dipole Drude-like peak, and the interband weight is also entirely controlled by the Berry connection. This sum rule opens a door to search for alternative photovoltaic technologies based on the Berry geometry of bands. We also describe the rectification properties of Weyl semimetals which are a promising platform to investigate these effects.