A Clarification on Quantum-Metric-Induced Nonlinear Transport

TL;DR

This work addresses inconsistencies in the form of the quantum-metric-induced second-order nonlinear conductivity by performing a unified analysis across standard perturbation theory, wave packet dynamics, and the Luttinger-Kohn approach. It shows that the intrinsic, relaxation-time–independent contribution is governed by the quantum metric dipole via $\sigma_{ijk}^{\text{qm}} = -\frac{e^3}{\hbar} \sum_n \int [d\mathbf k]\big[\partial_i \mathcal G_n^{jk} - \frac{1}{2}(\partial_j \mathcal G_n^{ik} + \partial_k \mathcal G_n^{ij})\big] f_0$, and that electric-field–induced corrections to the Berry connection and band energy satisfy $\tilde{\mathbfcal A}_n(\mathbf k) = \mathbfcal A_n(\mathbf k) + \mathbf G_n(\mathbf k) \cdot \mathbf E$ and $\tilde{\varepsilon}_{n\mathbf k} = \varepsilon_{n\mathbf k} + \frac{e}{2}\mathbf E \cdot \mathbf G_n(\mathbf k) \cdot \mathbf E$. The authors identify the source of discrepancies with prior works and show that consistent results emerge when the bare-band renormalization and the Fermi-sea correction are properly accounted for; a Schrieffer-Wolff–based Luttinger-Kohn treatment reproduces the same corrections, unifying the three formalisms. To enable targeted studies of quantum-metric–driven transport free from Berry-curvature contamination, they introduce a PT-symmetric toy model in which the quantum-metric contribution can be isolated and its dependence on symmetry breaking explored. This provides a clear theoretical foundation and practical platform for future experiments and extensions to disorder and higher-order geometric effects.

Abstract

Over the years, Berry curvature, which is associated with the imaginary part of the quantum geometric tensor, has profoundly impacted many branches of physics. Recently, quantum metric, the real part of the quantum geometric tensor, has been recognized as indispensable in comprehensively characterizing the intrinsic properties of condensed matter systems. The intrinsic second-order nonlinear conductivity induced by the quantum metric has attracted significant recent interest. However, its expression varies across the literature. Here, we reconcile this discrepancy by systematically examining the nonlinear conductivity using the standard perturbation theory, the wave packet dynamics, and the Luttinger-Kohn approach. Moreover, inspired by the Dirac model, we propose a toy model that suppresses the Berry-curvature-induced nonlinear transport, making it suitable for studying the quantum-metric-induced nonlinear conductivity. This work provides a clearer and more unified understanding of the quantum-metric contributions to nonlinear transport. It also establishes a solid foundation for future theoretical developments and experimental explorations in this highly active and rapidly evolving field.

Figures (2)

• Figure 1: Distinct physical origins of the second-order nonlinear conductivity. The contributions from the nonlinear Drude term, the Berry curvature dipole, and the quantum metric dipole are indicated in red, blue, and green, respectively. Note that $\mathbf E$ and $\mathbf E^2$ are used to demonstrate the order of electric-field-induced perturbation, and the exact $\mathbf E$ dependences of the relevant quantities are given in Eqs. \ref{['Eq: An_tot']}, \ref{['Eq: en_tot']}, and \ref{['Eq: f']}.
• Figure 2: Theoretical results from the toy model [Eq. \ref{['Eq: model']}]. (a) Band structure at $t/v=0.9$ with the Fermi surface placed at $\varepsilon_F=0.25$ eV. (b) Fermi surfaces at $\varepsilon_F=0.25$ eV for $t/v=0, 0.5, 0.9$. The color scale indicates the magnitude of the quantum metric dipole $\Lambda_n^{xyy}$. (c) Calculated nonlinear Hall conductivity $\sigma_{yxx}^{\text{qm}}$ as a function of the Fermi energy $\varepsilon_F$. (d) Momentum-resolved distribution of $\Lambda_n^{xyy}$ for $t/v=0.9$. The model parameters used are $v=1$ eV$\cdot$nm, $m=0.1$ eV, and $b=1$ eV$\cdot$nm$^2$.