Intrinsic nonlinear conductivity induced by the quantum metric dipole Shibalik Lahiri, 1, ∗ Kamal Das, 1, † Dimitrie Culcer, 2, 3, ‡ and Amit Agarwal 1, § 1 Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016 2 School of Physics, The University of New South Wales, Sydney 2052, Australia 3 Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies The nonlinear current to second-order in the electric field is known to originate from three different physical mechanisms: extrinsic i) nonlinear Drude and ii) Berry curvature dipole, and intrinsic iii) Berry connection polarizabilty. Here, we predict a new intrinsic contribution to the nonlinear current related to the quantum metric, a quantum geometric property of the electronic wave-function. In contrast to the other three contributions, the new quantum metric dipole induced conductivity is a Fermi sea response. More interestingly, it manifests in both longitudinal as well as Hall current in systems which simultaneously break the time reversal and the inversion symmetry. This is the first instance of an intrinsic quantum coherence effect in longitudinal transport that can be observed in doped electron system. The quantum metric dipole induced NL Hall current dominates transport in parity-time reversal symmetric systems in the bandgap and near the band edges, something we show explicitly for topological antiferromagnets. Introduction:— The non-linear conductivity provides new physical insight into the quantum geometry of the electronic wave-function [1–7]. Additionally, it plays a fundamental role in the identification of different topo- logical and magnetic states [8, 9]. For instance, the non- linear (NL) anomalous Hall conductivity [3] which deter- mines the Hall response in time-reversal symmetric sys- tems, provides information on the Berry curvature dipole. It also acts as a sensor for topological phase transitions of the valley-Chern type [8, 10]. On the other hand, the intrinsic NL Hall conductivity [11] provides information on the Berry connection polarizability (BCP). Interest- ingly, it can sense the orientation of the Neel vector in parity-time reversal symmetric systems [9]. Most of the transport coefficients are extrinsic in nature. In these extrinsic conductivities, the information about the elec- tronic state of the system is entangled with the effect of disorder. This has motivated an energetic search for in- trinsic (scattering-independent) transport coefficients. In the linear response regime, several intrinsic Hall conduc- tivities such as the anomalous Hall [12–14], spin Hall [15– 17], and quantum anomalous Hall conductivity [18, 19] are known. In the NL response regime, however, this list is short and only the intrinsic NL BCP Hall conductivity has been discovered recently. However, there is no known intrinsic longitudinal conductivity induced by the band geometric quantities. Here, we predict a new intrinsic second-order NL re- sponse, which gives rise to both a Hall and a longitudinal current. This new second order NL conductivity can be expressed as, σ QMD abc = e 3   m,p  [dk] ∂ a G bc mp ǫ mp f m . (1) Here, f m is the Fermi function, the electronic charge is −e (with e> 0), ǫ mp = ǫ m − ǫ p is the energy difference between bands, ∂ a ≡ ∂/∂k a , and [dk]= d d k/(2π) d is the integration measure for a d-dimensional system. The FIG. 1. A schematic of all the four different second-order NL transport responses at zero frequency. The two scatter- ing time dependent extrinsic contributions are presented in the top row. The Drude conductivity arises from the second order correction to the distribution function and the band gra- dient velocity, while the anomalous Hall conductivity arises from the first order correction to the distribution function and the anomalous Hall velocity. The two intrinsic contribu- tions are shown in the bottom row. The left panel represents the nonlinear BCP conductivity which is a Fermi surface phe- nomenon. The right panel shows the NL QMD conductivity which is a Fermi sea phenomena. quantity G bc mp is the non-Abelian quantum metric which is gauge invariant. The quantum metric is the real part of the quantum geometric tensor Q bc mp = R b pm R c mp with R mp = i〈u m (k) |∇ k u p (k)〉. We denote the susceptibil- ity in Eq. (1) as the quantum metric dipole induced NL conductivity. Note that ∂ a G bc mp represents the quantum metric dipole (QMD) density, analogous to the Berry cur- vature dipole density (BCD) introduced in Ref. [3]. We emphasize that the longitudinal part of the QMD con- ductivity predicted here, is the only second order intrinsic arXiv:2207.02178v1 [cond-mat.mes-hall] 5 Jul 2022