Nonlinear Layer Hall Effect and Detection of the Hidden Berry Curvature Dipole in $\mathcal{PT}$-Symmetric Antiferromagnetic Insulators

Abstract

Recent experimental and theoretical studies have revealed the emergence of a linear layer Hall effect (LHE) induced by hidden Berry curvature in \textrm{MnBi}$_{2}$\textrm{Te}$_{4}$ thin films. This phenomenon underscores the layer degree of freedom as a novel mechanism for generating Hall transport in layered materials, providing a new pathway to probe and manipulate the internal structure of fully compensated topological antiferromagnets (AFMs). In this work, we predict a nonlinear LHE in $\mathcal{PT}$-symmetric layered AFMs, which manifests as a detectable nonlinear Hall conductivity even with respect to the AFM order and odd with respect to the vertical electric field, in contrast to the linear LHE. Furthermore, we demonstrate that the nonlinear Hall currents induced by the hidden BCD and quantum metric dipole (QMD) obey distinct symmetries and flow in different directions. Our proposed nonlinear LHE establishes an experimentally advantageous framework for exclusively probing the hidden BCD quantum geometry.

Figures (4)

• Figure 1: Schematics of the (a) hidden-BCD-induced nonlinear LHE, (b) QMD-induced nonlinear anomalous Hall effect, and (c)-(d) BCD-induced layer-polarized nonlinear Hall currents in the presence of a vertical electric field $E_{z}$ (black arrows). The blue and red filled hexagons indicate the distribution of the BC $\Omega_{n}^{xy}(\boldsymbol{k},z)$ and quantum metric $\mathcal{R}_{n}^{yx}(\boldsymbol{k},z)$ on the Fermi surfaces, with their dipoles $\boldsymbol{D}(z)=D_{x}^{xy}(z)\hat{x}$ and $\boldsymbol{Q}(z)=Q_{y}^{yx}(z)\hat{y}$ denoted by the red arrows inside the Fermi surfaces. The resulting nonlinear Hall currents $\boldsymbol{j}_{2}(z)$ are represented by the blue arrowed curves. In the nonlinear LHE, electrons on different layers are deflected spontaneously to opposite directions, due to the layer-locked BCD, while no such locking exists for the QMD.
• Figure 2: Numerical results for the layer-resolved BCD induced by (a)-(b) the AFM order with $m=0.5$, and (c)-(d) the tilt with $\lambda=0.1$, where $n_{z}=2$ and $n_{z}=4$ are chosen in the left and right panels, respectively. The rest parameters for the calculations are taken as $\theta=0$, $\phi=0$, $w=0.5$, $M_{0}=0.1$, $A_{1}=A_{2}=0.55$, $B_{1}=B_{2}=0.25$, and $E_{z}=0$.
• Figure 3: (a) The layer-resolved nonlinear Hall conductivity $\sigma_{y;xx}^{BCD}(Top)=-\sigma _{y;xx}^{BCD}(Bottom)$ as a function of $m$, with $\boldsymbol{N}=(0,0,1)$, $|M_{0}/B_{1}|=0.4$, and $n_{z}=2$. (b) The energy band for $m=0$ and $n_{z}=20$, with $M_{0}/B_{1}<0$ (right) and $M_{0}/B_{1}>0$ (left), where topological surface states emerge in the bulk band gap in the phase of topological AFM insulator. (c)-(d) $\sigma_{y;xx}^{BCD}\left( Top\right)$ as a function of $w$ and the direction of the Néel vector, respectively. The rest parameters are the same as in Fig. \ref{['Fig2']}(a).
• Figure 4: (a) The net nonlinear Hall conductivity $\sigma_{y;xx}^{BCD}$ as a function of the vertical electric field. (b) The underlying physics of the hidden-BCD-induced nonlinear Hall conductivity, where the vertical electric field induces a potential difference between the top and bottom surfaces, leading to an imbalance of their Hall conductivity, as a result of the uncompensated hidden BCD of different layers when the global $\mathcal{PT}$ symmetry is broken. (c) The $E_{z}$-dependence of the QMD component $\sigma_{x;yy}^{QMD}$ and (d) the nonlinear Hall conductivities when the driving electric field is misaligned from the crystal axis.