Intrinsic nonlinear Hall effect in two-dimensional honeycomb topological antiferromagnets

The quantum geometry of wave functions has a fundamental and deep connection with the behavior of electrons. Two basic quantum geometric quantities are the quantum metric and Berry curvature, which correspond to the real part and imaginary part of the so-called quantum geometric tensor Provost and Vallee (1980), respectively. The Berry curvature has been extensively studied over the past few decades and revealed to be an indispensable factor to understand many important phenomena in materials, with the most celebrated example being its application in understanding the quantized (anomalous) Hall effect Thouless et al. (1982); Chang et al. (2023) and the generic non-quantized anomalous Hall effect in magnetic metals Nagaosa et al. (2010) or topological semimetals Yang et al. (2011). Compared to the Berry curvature, the quantum metric started to attract wide interest in the condensed-matter field much more lately. The reason is partly due to that the quantum metric influences the electrons relatively more subtly, unlike the Berry curvature that gives a transparent contribution to the velocity operator Xiao et al. (2010). Nevertheless, recent studies have shown that the quantum metric is also fundamentally important for the understanding of many important phenomena, such as the superconductivity in flat bands Peotta and Törmä (2015); Julku et al. (2016); Liang et al. (2017); Hu et al. (2019); Xie et al. (2020); Huhtinen et al. (2022), optical responses Chen and Huang (2021); Ahn et al. (2022); Onishi and Fu (2023), etc Törmä (2023).

In the past few years, the generalized higher-order moments of Berry curvature and quantum metric have further generated considerable interest as they can induce Hall-type effects in the nonlinear response regime Moore and Orenstein (2010); Gao et al. (2014); Sodemann and Fu (2015); Xu et al. (2018); Zhang et al. (2018a); You et al. (2018); Zhang et al. (2018b); Du et al. (2018); Facio et al. (2018); Ma et al. (2019); Kang et al. (2019); Battilomo et al. (2019); Wang and Qian (2019); Xiao et al. (2019); Rostami and Juričić (2020); Singh et al. (2020); Zeng et al. (2021); Pantaleón et al. (2021); Samal et al. (2021); Kumar et al. (2021); Liao et al. (2021); Zhang et al. (2022); Bandyopadhyay et al. (2022); Roy and Narayan (2022); Okyay et al. (2022); Sinha et al. (2022); Chakraborty et al. (2022); Das et al. (2023); Wang et al. (2023a); Zhang et al. (2023); Atencia et al. (2023); Zhuang and Yan (2023); Huang et al. (2023); Saha and Narayan (2023); Kirikoshi and Hayami (2023); Hetényi and Lévay (2023); Huang et al. (2023); Kaplan et al. (2024); Mandal et al. (2023); Ma et al. (2021); Du et al. (2021); Ortix (2021); Bandyopadhyay et al. (2024). The two nonlinear Hall effects (NLHEs) that have attracted particular interest are the time-reversal-even NLHE of Berry-curvature-dipole origin Sodemann and Fu (2015) (for the convenience of discussion, we dub it as Berry-curvature-dipole NLHE) and the time-reversal-odd NLHE of quantum-metric-dipole (or say Berry connection polarizability) origin Gao et al. (2014) (known as intrinsic NLHE) in inversion-asymmetric systems. Because of the fundamental difference under time reversal, the Berry-curvature-dipole NLHE can appear in a time-reversal invariant system, whereas the intrinsic NLHE can only show up in systems without time-reversal symmetry. The adjective “intrinsic” refers to the fact that the effect does not depend on the scattering time and only depends on the band property. Being a time-reversal odd effect, the intrinsic NLHE has been shown in theory that it holds promise for applications in antiferromagnetic spintronics as it has the power to detect one key property of the antiferromagnets, the Néel vector Liu et al. (2021); Wang et al. (2021a). Besides the prospect of applications in spintronics, the detection of the direction of Néel vector is also of significant importance in its own right, since many properties of an antiferromagnet, such as band topology Šmejkal et al. (2017); Xu et al. (2019), sensitively depend on it. Remarkably, the intrinsic NLHE and its sign change upon reversing the Néel vector have recently been experimentally observed in even-layered topological antiferromagnets Gao et al. (2023); Wang et al. (2023b), MnBi₂Te₄. This breakthrough has paved the way to explore the interplay of antiferromagnetism and other factors of a system through the intrinsic NLHE in experiment.

The lattice structure, spin-orbit coupling and magnetism are three factors that strongly influence the band structure and the quantum geometry of the Bloch wave functions. When the band structure sensitively depends on their interplay, it is natural to expect that the intrinsic NLHE would exhibit characteristic features. Among various lattice structures, the two-dimensional (2D) honeycomb lattice is known to be a paradigmatic platform where the spin-orbit coupling and magnetism can influence the band topology in a nontrivial way Niu et al. (2017); Zou et al. (2022); Li et al. (2022). Therefore, a honeycomb topological antiferromagnet is expected to be an ideal platform to explore the intrinsic NLHE Wang et al. (2023a). With this picture in mind, in this work we consider honeycomb topological antiferromagets effectively described by a $\mathcal{PT}$-symmetric antiferromagnetic Kane-Mele model and explore the evolution of the intrinsic NLHE with respect to the change of lattice anisotropy, band topology, chemical potential, and the direction of the Néel vector. Our main findings include: (i) the lattice anisotropy breaking the $\mathcal{C}_{3z}$ rotation symmetry is crucial for having a nonzero intrinsic NLHE; (ii) when the band topology changes from a quantum spin Hall insulator to a trivial insulator or a boundary-obstructed atomic insulator, the intrinsic NLHE preserves its direction, which is distinct from the Berry-curvature-dipole NLHE; (iii) the intrinsic NLHE is valley-polarized in the weakly-doped regime and exhibit nonanalyticity when the Fermi surface undergoes Lifshitz transitions; (iv) the existence of spin-orbit coupling to lift the spin-rotation symmetry is decisive for detecting the direction of the Néel vector. These results suggest that the intrinsic NLHE provides an effective tool to measure basic properties of 2D honeycomb topological antiferromagnets.

The paper is organized as follows. In Sec.II, we give the effective tight-binding Hamiltonian and discuss the important symmetries and possible topological phases associated with the Hamiltonian. In Sec.III, we investigate the behavior of the intrinsic NLHE across two types of topological phase transitions. In Sec.IV, we study the dependence of the intrinsic NLHE on the chemical potential and the direction of the Néel vector. In Sec.V, we discuss our findings and conclude the paper.

A $\mathcal{PT}$-symmetric honeycomb collinear antiferromagnet with finite intrinsic spin-orbit coupling can be effectively described by the tight-binding Kane-Mele model. The Hamiltonian is given by $H=\sum_{\boldsymbol{k}}\Psi_{\boldsymbol{k}}^{\dagger}\mathcal{H}(\boldsymbol{k})\Psi_{\boldsymbol{k}}$, where the basis is chosen as $\Psi_{\boldsymbol{k}}^{\dagger}=(c_{A,\uparrow,\boldsymbol{k}}^{\dagger},c_{B,\uparrow,\boldsymbol{k}}^{\dagger},c_{A,\downarrow,\boldsymbol{k}}^{\dagger},c_{B,\downarrow,\boldsymbol{k}}^{\dagger})$ and the momentum-space Hamiltonian reads Kane and Mele (2005a, b)

	$\displaystyle\mathcal{H}(\boldsymbol{k})$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{3}t_{i}\left[\cos(\boldsymbol{k}\cdot\boldsymbol{a}_{i})s_{0}\sigma_{x}+\sin(\boldsymbol{k}\cdot\boldsymbol{a}_{i})s_{0}\sigma_{y}\right]$		(1)
			$\displaystyle+2\lambda_{\rm so}\sum_{i}\sin(\boldsymbol{k}\cdot\boldsymbol{b}_{i})s_{z}\sigma_{z}+(\boldsymbol{M}\cdot\boldsymbol{s})\sigma_{z}.$

Above the first line describes the nearest-neighbor hoppings, the first term in the second line represents the intrinsic spin-orbit coupling involving next-nearest-neighbor hoppings, and the last term denotes the exchange field associated with the antiferromagnetic order. $(s_{0};s_{x},s_{y},s_{z})$ and $(\sigma_{0};\sigma_{x},\sigma_{y},\sigma_{z})$ are the identity matrix and Pauli matrices in the spin and sublattice subspaces, respectively. The three nearest-neighbor lattice vectors are given by $\boldsymbol{a}_{1}=a(0,1)$, $\boldsymbol{a}_{2}=\frac{a}{2}(\sqrt{3},-1)$, $\boldsymbol{a}_{3}=\frac{a}{2}(-\sqrt{3},-1)$, and the three next-nearest-neighbor lattice vectors are determined by $\boldsymbol{a}_{i}$ through the relation: $\boldsymbol{b}_{1}=\boldsymbol{a}_{2}-\boldsymbol{a}_{3}$, $\boldsymbol{b}_{2}=\boldsymbol{a}_{3}-\boldsymbol{a}_{1}$ and $\boldsymbol{b}_{3}=\boldsymbol{a}_{1}-\boldsymbol{a}_{2}$. For notational simplicity, we set the lattice constant $a=1$ throughout.

Due to the antiferromagnetic exchange field, the above Hamiltonian does not have the spinful time-reversal symmetry (symmetry operator is $\mathcal{T}=-is_{y}\sigma_{0}\mathcal{K}$ with $\mathcal{K}$ the complex conjugate operator) and inversion symmetry ($\mathcal{P}=s_{0}\sigma_{x}$). However, the Hamiltonian has their combination, the spinful $\mathcal{PT}$ symmetry. The symmetry operator is $\mathcal{PT}=-is_{y}\sigma_{x}\mathcal{K}$, where $\mathcal{K}$ denotes the complex conjugate operator and the symmetry operator satisfies $(\mathcal{PT})^{2}=-1$. This combinational symmetry on one hand enforces the band to be doubly degenerate, and on the other hand makes the Berry curvature identically vanishing. As a result, any Hall-type effect with a Berry-curvature origin is expected to vanish. When the Néel vector is aligned in the $z$ direction and the hopping constants are isotropic, the Hamiltonian also contains several important crystallographic symmetries that could have a strong impact on the band topology and intrinsic NLHE, including the $\mathcal{C}_{3z}$ rotation symmetry ($\mathcal{C}_{3z}=e^{i\frac{\pi}{3}s_{z}\sigma_{0}}$), and the mirror symmetries about the $xy$ ($\mathcal{M}_{z}=is_{z}\sigma_{0}$) and $xz$ ($\mathcal{M}_{y}=is_{y}\sigma_{x}$) planes. To be general, we incorporate lattice anisotropy that can be caused by intrinsic lattice corrugation or extrinsic strain, and assume that the Néel vector can point to any direction. To be specific, for the lattice anisotropy, we set $t_{3}=t_{2}$, but allow these two hopping constants to be different from $t_{1}$, as illustrated in Fig. 1. For this type of lattice anisotropy, the $\mathcal{C}_{3z}$ rotation symmetry is broken once $t_{1}\neq t_{2}$, but the mirror symmetry about the $xz$ plane remains. For the convenience of discussion, we introduce the ratio $\eta=t_{1}/t_{2}$ to characterize the extent of lattice anisotropy. The more $\eta$ deviates from $1$, the stronger the lattice anisotropy is.

The band topology of the Hamiltonian in Eq.(1) sensitively depends on the spin-orbit coupling and antiferromagnetic exchange field. Without the spin-orbit coupling and antiferromagnetic exchange field, it is known that the Hamiltonian realizes a Dirac semimetal with two Dirac points at the two valleys, $\mathbf{K}$ and $\mathbf{K}^{\prime}$, for the case without lattice anisotropy Castro Neto et al. (2009). Weak lattice anisotropy shifts the locations of the two Dirac points, but cannot annihilate them due to the protection of a spinless $\mathcal{PT}$ symmetry (the corresponding symmetry operator is $\mathcal{PT}=s_{0}\sigma_{x}\mathcal{K}$, satisfying $(\mathcal{PT})^{2}=1$). When the lattice anisotropy reaches a critical condition ($\eta=\pm 2$), the two Dirac points converge and annihilate, and a further increase of the lattice anisotropy opens an energy gap and drives the system to an insulator Pereira et al. (2009), as illustrated in Fig. 2.

As long as the lattice anisotropy does not annihilate the two Dirac points, the presence of spin-orbit coupling will immediately gap out the Dirac points due to a lifting of the spinless $\mathcal{PT}$ symmetry, accompanying with a direct transition from the Dirac semimetal to a quantum spin Hall insulator with helical edge states Kane and Mele (2005a, b), as illustrated in Fig. 3(a). Interestingly, recent works have shown that if the lattice anisotropy is strong, the quantum spin Hall insulator does not become a featureless trivial insulator, instead, it will evolve to a boundary-obstructed atomic insulator which supports boundary floating bands or corner states for appropriate geometry Wang et al. (2021b); Lahiri and Basu (2023), as illustrated in Fig. 3(b).

In the quantum spin Hall regime, when the antiferromagnetic exchange field is also brought in, the time-reversal symmetry protecting the helical edge states is broken. Nevertheless, the helical edge states can remain stable if the Néel vector is in the $z$ direction and the strength of the exchange field is lower than a critical value ($M_{c}=3\sqrt{3}\lambda_{\rm so}$ for the isotropic-hopping case), as illustrated in Figs. 3(c) and 3(d). There are two ways to understand the robustness of the helical edge states. The first one is that the spin remains a good quantum number for this special case, therefore, the Hamiltonian remains to be characterized by spin Chern number Sheng et al. (2006). To be concrete, as $[s_{z},\mathcal{H}(\boldsymbol{k})]=0$, the Hamiltonian (1) can be decomposed as the direct sum of two independent parts in accordance with the two eigenvalues of $s_{z}$, i.e., $\mathcal{H}(\boldsymbol{k})=\mathcal{H}_{s_{z}=1}(\boldsymbol{k})\oplus\mathcal{H}_{s_{z}=-1}(\boldsymbol{k})$, where

	$\displaystyle\mathcal{H}_{s_{z}=s}(\boldsymbol{k})$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{3}t_{i}\left[\cos(\boldsymbol{k}\cdot\boldsymbol{a}_{i})\sigma_{x}+\sin(\boldsymbol{k}\cdot\boldsymbol{a}_{i})\sigma_{y}\right]$		(2)
			$\displaystyle+s\left[2\lambda_{\rm so}\sum_{i=1}^{3}\sin(\boldsymbol{k}\cdot\boldsymbol{b}_{i})+M_{z}\right]\sigma_{z}$

with $s=\pm 1$. The two-band Hamiltonians can be expressed as the form: $\boldsymbol{d}^{s}(\boldsymbol{k})\cdot\boldsymbol{\sigma}$, with the three components of $\boldsymbol{d}^{s}(\boldsymbol{k})$ being the coefficients before the corresponding Pauli matrices. For each spin sector, the Hamiltonian lacks time-reversal symmetry and is hence characterized by the Chern number,

$\displaystyle C_{s}=\frac{1}{4\pi}\int_{\rm BZ}dk_{x}dk_{y}\frac{\boldsymbol{d}^{s}(\boldsymbol{k})\cdot\left[\partial_{k_{x}}\boldsymbol{d}^{s}(\boldsymbol{k})\times\partial_{k_{y}}\boldsymbol{d}^{s}(\boldsymbol{k})\right]}{|\boldsymbol{d}^{s}(\boldsymbol{k})|^{3}}.$

(3)

As the two spin sectors are related by time-reversal symmetry, one has $C_{+1}=-C_{-1}$. While the total Chern number defined as $C_{T}=C_{+1}+C_{-1}$ is forced to be zero, the spin Chern number, which is defined as $C^{\text{spin}}=(C_{+1}-C_{-1})/2$ Sheng et al. (2006), can be nonzero. The spin Chern number counts the number of pairs of helical edge states. It will not change its nontrivial value unless the bulk energy gap closes when the antiferromagnetic exchange field reaches its critical value, as illustrated by the phase diagram shown in Fig.4. The second one is that the the Hamiltonian has the mirror symmetry about the $xy$ plane (recall $\mathcal{M}_{z}=is_{z}\sigma_{0}$) for this special case. This mirror symmetry can also protect the helical edge states as a mirror Chern number can be defined to characterize the Hamiltonian Teo et al. (2008). For this Hamiltonian, the mirror Chern number is just equal to the spin Chern number, as the two different mirror-sector Hamiltonians obtained by decomposing the Hamiltonian in accordance with the two opposite eigenvalues of $\mathcal{M}_{z}$ are exactly the same as the two spin-sector Hamiltonians given in Eq.(2). In other words, the topological gapped phase can be equivalently interpreted as either a quantum spin Hall insulator or a topological mirror insulator.

Once the direction of the Néel vector deviates away from the $z$ direction, the $\mathcal{M}_{z}$ mirror symmetry is broken and the spin is also no longer a conserved quantity. As a result, the helical edge states will immediately be gapped due to the lack of any symmetry protection. Commonly, the opening of a gap to the edge states suggests that the resulting phase becomes a trivial insulator. However, the trivialness is only strict in the first-order topology. The resulting phase without gapless edge states is in fact not completely topologically trivial. In Ref.Miao et al. (2023), the authors showed that, if the honeycomb lattice consists of two parts with opposite in-plane Néel vectors, despite the absence of gapless edge states in each part, 0D topological bound states will emerge at the ends of their interface. All of these results reflect that the quantum geometric properties of honeycomb-lattice materials have a sensitive dependence on the interplay of lattice anisotropy, spin-orbit coupling and antiferromagnetism.

As discussed above, when the Néel vector is aligned in $z$ direction, the system will undergo a topological phase transition from a quantum spin Hall insulator to a trivial insulator with the increase of the antiferromagnetic exchange field, or to a boundary-obstructed atomic insulator with the increase of lattice anisotropy. In the following, we explore the behavior of the intrinsic NLHE across these two types of topological phase transitions.

Before start, we first give a brief review of the intrinsic NLHE. In 2014, Gao, Yang and Niu showed that the electric field can induce a first-order correction to the Berry curvature Gao et al. (2014). As a result, a second-harmonic Hall-type current can arise. The Hall-type current is of the form $j_{\alpha}^{\rm int}=\chi_{\alpha\beta\gamma}^{\rm int}\mathcal{E}^{\beta}\mathcal{E}^{\gamma}$, where $\mathcal{E}^{\beta}$ represents the electric-field component in the $\beta$ direction, and $\chi_{\alpha\beta\gamma}^{\rm int}$ is a conductivity tensor independent of scattering time. The explicit expression of $\chi_{\alpha\beta\gamma}^{\rm int}$ is given by Gao et al. (2014); Liu et al. (2021); Wang et al. (2021a)

$\displaystyle\chi_{\alpha\beta\gamma}^{\rm int}=e^{3}\sum_{n}\int\frac{d^{D}k}{(2\pi)^{D}}\Lambda_{\alpha\beta\gamma}(\boldsymbol{k})\frac{\partial f(E_{n})}{\partial E_{n}},$

(4)

where $D$ is the dimension, $n$ is the band index, and $f(E_{n})$ is the equilibrium Fermi-Dirac distribution function of the $n$th band. The tensor $\Lambda_{\alpha\beta\gamma}(\boldsymbol{k})$ is given by

$\displaystyle\Lambda_{\alpha\beta\gamma}^{(n)}(\boldsymbol{k})$

$\displaystyle=$

$\displaystyle v_{\alpha}^{(n)}(\boldsymbol{k})G_{\beta\gamma}^{(n)}(\boldsymbol{k})-v_{\beta}^{(n)}(\boldsymbol{k})G_{\alpha\gamma}^{(n)}(\boldsymbol{k}),$

(5)

where $v_{\alpha}^{(n)}=\partial E_{n}/\partial k_{\alpha}$ is the group velocity of the $n$th band, and $G_{\beta\gamma}^{(n)}$ is of the formGao et al. (2014); Liu et al. (2021); Wang et al. (2021a)

$\displaystyle G_{\beta\gamma}^{(n)}(\boldsymbol{k})=2{\rm Re}\sum_{m\neq n}\frac{A_{nm,\beta}(\boldsymbol{k})A_{mn,\gamma}(\boldsymbol{k})}{E_{n}(\boldsymbol{k})-E_{m}(\boldsymbol{k})}.$

(6)

Above $A_{nm,\beta}(\boldsymbol{k})=i\langle u_{n}(\boldsymbol{k})|\partial_{k_{\beta}}u_{m}(\boldsymbol{k})\rangle$ with $n\neq m$ is the interband Berry connection. It is noteworthy that ${\rm Re}\sum_{m\neq n}A_{nm,\beta}(\boldsymbol{k})A_{mn,\gamma}(\boldsymbol{k})$ corresponds to the quantum metric of the $n$th band, suggesting the quantum metric origin of this second-order response. In Eq.(4), the derivative of the Fermi-Dirac distribution function indicates that this effect is a Fermi-surface property. From Eq.(5), it is easy to see that the tensor $\Lambda_{\alpha\beta\gamma}^{(n)}$, and so the conductivity tensor $\chi_{\alpha\beta\gamma}^{\rm int}$, is antisymmetric about the first two subscripts, i.e., $\chi_{\alpha\beta\gamma}^{\rm int}=-\chi_{\beta\alpha\gamma}^{\rm int}$, suggesting that the resulting current is a Hall-like current. Because of this property, the conductivity tensor only have two independent components in 2D, including $\chi_{xyx}^{\rm int}$ and $\chi_{xyy}^{\rm int}$. On the other hand, the energy difference between bands in the denominator of Eq.(6) implies that $G_{\beta\gamma}^{(n)}$ should be prominent near the band edge, and a decrease in the band energy gap could benefit the enhancement of this effect. Therefore, when the system is close to a topological phase transition, the intrinsic NLHE is expected to be prominent.

Let us first focus on the topological phase transition driven by the $z$-directional antiferromagnetic exchange field. To simplify the discussion, we consider the lattice anisotropy to be weak for this case. Accordingly, the band edge will be located near one of the two valleys when the system is close to the topological phase transition. By an expansion of the bulk Hamiltonian around the two valleys and only keeping the leading-order terms, we find that the corresponding low-energy Hamiltonians are given by

	$\displaystyle\mathcal{H}_{\chi}(\boldsymbol{q})$	$\displaystyle=$	$\displaystyle-\chi\frac{3t_{2}}{2}q_{x}\sigma_{x}+\left(t_{1}+\frac{t_{2}}{2}\right)q_{y}\sigma_{y}+M_{z}s_{z}\sigma_{z}$		(7)
			$\displaystyle-3\sqrt{3}\chi\lambda_{\rm so}s_{z}\sigma_{z},$

where $\chi=1$ for the valley $\mathbf{K}$ and $-1$ for the valley $\mathbf{K}^{{}^{\prime}}$. It is easy to see that the energy gap gets closed at $\mathbf{K}$ if $M_{z}=M_{c}\equiv 3\sqrt{3}\lambda_{\rm so}$, or at $\mathbf{K}^{\prime}$ if $M_{z}=-M_{c}$ (a more accurate analysis finds out that the critical value of the exchange field is $M_{c}^{\prime}=\lambda_{\rm so}(2+\eta)\sqrt{4-\eta^{2}}$).

Although the low-energy Hamiltonians above can capture the topological phase transition, it has higher symmetry than the full lattice Hamiltonian in Eq.(1), and the symmetry emergent from the leading-order approximation will force the intrinsic nonlinear Hall conductivity tensor (INLHCT) to vanish identically. To correctly obtain the INLHCT, we always adopt the full lattice Hamiltonian for calculations. It is noteworthy that if the lattice anisotropy is absent, the $\mathcal{C}_{3z}$ rotation symmetry will force all components of the INLHCT to vanish identically even for the full lattice Hamiltonian. When the lattice anisotropy is present, the remaining mirror symmetry about the $xz$ plane forces the component $\chi_{xyx}^{\rm int}$ to vanish identically. Therefore, only the component $\chi_{xyy}^{\rm int}$ needs to be considered. To intuitively see that $\chi_{xyx}^{\rm int}$ is forced to vanish while $\chi_{xyy}^{\rm int}$ is not, we plot the distribution of the geometric quantity $\Lambda_{xyx}(\boldsymbol{k})$ and $\Lambda_{xyy}(\boldsymbol{k})$ in the Brillouin zone, as shown in Fig. 5. From Figs. 5(a) and 5(c), one sees that $\Lambda_{xyx}(\boldsymbol{k})$ is odd about $k_{y}$, leading to the vanishing of $\chi_{xyx}^{\rm int}$ after the integration over the Fermi surface. In contrast, Figs. 5(b) and 4(d) show that $\Lambda_{xyy}(\boldsymbol{k})$ is even about $k_{y}$, hence a nonzero $\Lambda_{xyy}(\boldsymbol{k})$ is permitted.

The numerical results for the INLHCT across the topological phase transition are shown in Fig. 6. Several prominent conclusions can be read from Fig. 6(a). First, the INLHCT does not change sign in the weakly-doped regime when the system transits from a quantum spin Hall insulator to a trivial insulator. This is quite different from the Berry-curvature-dipole NLHE which shows a sign change when an inversion asymmetric topological insulator transits to a trivial insulator Du et al. (2018); Sinha et al. (2022); Chakraborty et al. (2022). The sign change of the Berry-curvature-dipole NLHE is simply due to the sign change of the Berry curvature across the topological phase transition. However, for topological phase transitions characterized by a Dirac Hamiltonian of the form in Eq.(7), it is easy to find that all components of the quantum metric tensor do not change sign when the Dirac mass changes sign. As the intrinsic NLHE is connected to the quantum metric, this explains why the INLHCT preserves its sign across this class of topological phase transitions. In a previous work, we have shown in the context of Hopf insulators that the INLHCT will change sign across a topological phase transition with the change of Hopf invariant Zhuang and Yan (2023). Therein, the critical point is different from a Dirac point in many aspects, and it turns out that the quantum metric and Berry curvature are closely connected. These results suggest that whether the INLHCT changes sign or not is not a universal property, but depends on the type of the topological phase transition. A second conclusion can be obtained from Fig. 6(a) is that the INLHCT switches sign when the antiferromagnetic exchange field reverses its direction, which is expected as the effect is time-reversal-odd. Last but not the least, under the condition of the same bulk energy gap, the INLHCT can be enhanced by increasing the antiferromagnetic exchange field to cross the topological phase transition.

Next we consider the topological phase transition from a quantum spin Hall insulator to a boundary-obstructed atomic insulator driven by the lattice anisotropy. In Fig. 2, we have shown that, in the absence of spin-orbit coupling and antiferromagnetic exchange field, the two Dirac points will merge together at the $\mathbf{M}$ point when $\eta=2$ or at the $\boldsymbol{\Gamma}$ point when $\eta=-2$ and form a critical semi-Dirac point. Since $\eta=-2$ means that the hopping constants $t_{1}$ and $t_{2}$ take opposite signs, which is not very realistic for a quantum material, below we will focus on the critical region at the neighborhood of $\eta=2$, and explore the behavior of the INLHCT across the topological phase transition driven by lattice anisotropy.

As the spin-orbit coupling vanishes at $\mathbf{M}$, a time-reversal invariant momentum, the mergence of Dirac points at this point when $\eta=2$ indicates that the topological phase transition is associated with the close of energy gap at $\mathbf{M}$. Therefore, we can do a low-energy expansion of the lattice Hamiltonian around this point. To capture the mergence of Dirac points, we keep the momentum up to the second order. The low-energy Hamiltonian is given by

	$\displaystyle\mathcal{H}_{\mathbf{M}}(\boldsymbol{q})$	$\displaystyle=$	$\displaystyle t_{2}\left\{\left[\left(1-\frac{\eta}{2}\right)-\frac{3}{8}q_{x}^{2}\right]-\frac{\sqrt{3}(1+\eta)}{2}q_{y}\right\}s_{0}\sigma_{x}$		(8)
			$\displaystyle-t_{2}\left\{\sqrt{3}\left[\left(1-\frac{\eta}{2}\right)-\frac{3}{8}q_{x}^{2}\right]+\frac{(1+\eta)}{2}q_{y}\right\}s_{0}\sigma_{y}$
			$\displaystyle+(4\sqrt{3}\lambda_{\rm so}q_{x}+M_{z})s_{z}\sigma_{z}.$

Without the last term, the energy spectrum is given by

$\displaystyle E_{\pm}(\boldsymbol{q})=\pm t_{2}\sqrt{4\left[\left(1-\frac{\eta}{2}\right)-\frac{3}{8}q_{x}^{2}\right]^{2}+(1+\eta)^{2}q_{y}^{2}},$

(9)

which displays the characteristic feature of a semi-Dirac point when $\eta=2$, namely, the energy spectrum is quadratic in one direction and linear in the other direction Samal et al. (2021); Dietl et al. (2008); Banerjee et al. (2009). Because the quantum metric and the density of states, two factors determining the conductivity tensor, are quite different between the Dirac point and the semi-Dirac point, different features are expected to show up in the INLHCT when the system undergoes this topological phase transition.

In Fig. 6(b), we show the INLHCT under different strength of lattice anisotropy and fixed spin-orbit coupling and exchange field. Fixing the band gap by adjusting the strength of lattice anisotropy, we find that the INLHCT still does not change sign across the topological phase transition. However, the INLHCT is considerably enhanced near this topological phase transition even for a weak exchange field. The result suggests that the lattice anisotropy can be applied as an effective factor to engineer strong intrinsic NLHE.

We have explored the intrinsic NLHE in 2D honeycomb antiferromagnets with $\mathcal{P}\mathcal{T}$ symmetry. As a class of systems supporting a number of interesting topological phases, we investigated the behavior of the intrinsic NLHE across two types of topological phase transitions. We found that, unlike the Berry-curvature-dipole NLHE, the intrinsic NLHE does not switch direction when the system undergoes a Dirac-type topological phase transition, suggesting that the intrinsic NLHE cannot be applied to detect such topological phase transitions. Nevertheless, the intrinsic NLHE could become prominent near these topological phase transitions, owing to that the quantum metric is inversely proportional to the band gap. We found that the lattice anisotropy breaking the $\mathcal{C}_{3z}$ rotation symmetry is not only necessary for the presence of intrinsic NLHE, but also serves as an effective factor to tune its magnitude and direction. As a Fermi-surface property, we found that the intrinsic NLHE can manifest Lifshitz transitions. Furthermore, we found that the existence of spin-orbit coupling to lift the spin-rotation symmetry is crucial for the intrinsic NLHE to detect the Néel vector. Our findings show that the 2D honeycomb antiferromagnets could serve as fertile ground to study the intrinsic NLHE.

This work is supported by the National Natural Science Foundation of China (Grant No. 12174455), the Natural Science Foundation of Guangdong Province (Grant No. 2021B1515020026) and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023B1515040023).