Quantum nonlinear Hall effect induced by Berry curvature dipole in time-reversal invariant materials (1508.00571v1)

Abstract: It is well-known that a non-vanishing Hall conductivity requires time-reversal symmetry breaking. However, in this work, we demonstrate that a Hall-like transverse current can occur in second-order response to an external electric field in a wide class of time-reversal invariant and inversion breaking materials, at both zero and twice the optical 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. We show that the nonlinear Hall coefficient is a rank-two pseudo-tensor, 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.

• The paper introduces a novel mechanism where a Berry curvature dipole induces a nonlinear Hall effect in time-reversal invariant materials.
• It develops a pseudo-tensor framework linking crystal symmetry to nonlinear conductivity through rectified and second harmonic responses.
• The findings suggest observable effects in materials like transition metal dichalcogenides and Weyl semimetals, paving the way for future experimental studies.

Quantum Nonlinear Hall Effect Induced by Berry Curvature Dipole in Time-Reversal Invariant Materials

This paper by Sodemann and Fu addresses a notable deviation from conventional Hall effect theory by proposing a nonlinear Hall effect in materials that are time-reversal symmetric but lack inversion symmetry. Traditionally, a non-zero Hall conductivity is associated with breaking time-reversal symmetry. However, this research explores how higher-order responses facilitate a Hall-like current within certain materials that maintain time-reversal invariance.

Overview and Key Concepts

The central mechanism at play here is the Berry curvature dipole, which arises in momentum space when examining the quantum properties of electrons in specific crystalline materials. While Berry curvature is generally odd in momentum, leading to cancellation in time-reversal invariant systems, the authors point out that a second-order response to an electrical field allows for non-zero contributions due to asymmetric distribution of Berry curvature. This asymmetry results in an anomalous velocity component that can contribute to a transverse current, akin to the Hall effect, albeit non-quantized.

Mathematical Framework and Results

Sodemann and Fu develop a formalism that describes how the nonlinear Hall coefficient can be derived as a pseudo-tensor, whose properties are governed by the point group symmetries of the crystals in question. Mathematically, the paper presents expressions for the nonlinear current response terms, which include contributions from both rectified current and second harmonic generation. Notably, the expression for the conductivity tensor is directly linked to the Berry curvature dipole, a central concept of this work. The results suggest that the dipole moment of Berry curvature, an integral of the Berry curvature weighted by the non-equilibrium electron distribution, dictates the strength and presence of the nonlinear Hall effect in materials.

Material Implications and Experimental Considerations

The authors propose several material candidates where this nonlinear Hall effect could be observable, such as transition metal dichalcogenides and three-dimensional Weyl semimetals, each possessing suitable symmetry conditions. The phenomenon is particularly linked to materials with low crystal symmetry, where sizable Berry curvature effects emerge around band-crossing points such as Dirac or Weyl points. The theory suggests that the nonlinear Hall effect can be observed in situations where the electronic band structure is modified by structural distortions or inherent material properties, offering both theoretical intrigue and practical utility.

Theoretical Implications and Future Directions

From a theoretical perspective, this work provides insights into the quantum geometric aspects of electronic states in solids and enriches our understanding of response functions beyond the linear regime. The suggested manifestations of Berry curvature in nonlinear phenomena encourage further exploration of non-linear transport properties and their relationship with symmetry and topology in condensed matter. Future research could extend to explore the diverse applications of such effects in electronic and optoelectronic devices, particularly as materials with pronounced Berry curvature effects become technologically relevant.

The investigation of these effects implies a broader canvas for future studies into the role of quantum geometry in macroscopic material properties. The proposed nonlinear Hall effect in such systems can potentially inspire experimental verification and push the envelope of understanding synergies between quantum mechanics and material science functionalities.