Quantum third-order nonlinear Hall effect of a four-terminal device with time-reversal symmetry

The third-order nonlinear Hall effect induced by the Berry-connection polarizability tensor has been observed in Weyl semimetals ${\mathrm{T}}_d\text{\ensuremath{-}}{\mathrm{MoTe}}₂$ as well as ${\mathrm{T}}_d\text{\ensuremath{-}}{\mathrm{TaIrTe}}₄$. 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 the quantum regime. The quantum nonlinear theory is verified on a model system of monolayer ${\mathrm{MoTe}}₂$, and numerical results on the angle-resolved Hall currents are qualitatively consistent with the experiment. More importantly, quantum signatures of the third-order Hall effect are revealed, which are independent of the system symmetry. The first quantum signature is quantum enhancement of the third-order Hall current, which is characterized by sharp current peaks whose magnitudes are three orders larger than those of the first-order Hall current. Such quantum enhancement originates from quantum interference in coherent transport, and it can be easily destroyed by the dephasing effect. The second quantum signature is disorder-induced enhancement of the third-order Hall current for weak disorders. Our findings reveal quantum characteristics of the third-order Hall effect, and we propose feasible ways to enhance it in nanoscale systems. The quantum third-order theory developed in this paper provides a general formalism for describing nonlinear coherent transport properties in multiterminal devices, regardless of the system symmetry.