Quantum metric–induced giant and reversible nonreciprocal transport phenomena in chiral loop-current phases of kagome metals

Significance

Rich symmetry-breaking phenomena driven by nontrivial quantum geometry are central topics in condensed matter physics. In particular, the chiral loop-current order observed in kagome metals and cuprate superconductors has attracted significant attention. Notably, the giant electrical magnetochiral anisotropy (eMChA) observed in kagome metals provides compelling evidence for the breaking of time-reversal and inversion symmetries. Here, we demonstrate that the eMChA is proportional to the loop-current-induced orbital magnetization, which is reversible by the magnetic field. Remarkably, the quantum metric exhibits significant momentum dependence in the loop-current phase, leading to the giant eMChA. This work not only sheds light on the fundamental symmetry-breaking mechanisms in kagome metals but also opens broad avenues for exploring quantum metric–induced phenomena resulting from exotic quantum phases.

Abstract

Emergence of quantum orders with nontrivial quantum geometric properties in metals represent central issues in condensed matter physics. In this context, recently discovered chiral loop-current order in kagome metals has garnered significant attention. Particularly noteworthy is the giant electrical magnetochiral anisotropy (eMChA) observed in CsV₃Sb₅, which provides compelling evidence for the simultaneous breaking of time-reversal and inversion symmetries. However, the origin of the eMChA and its fundamental connection to the loop-current remain highly elusive, as the loop-current itself preserves inversion symmetry. Here, we demonstrate that the loop-current phase breaks inversion symmetry in the presence of the experimentally observed stripe charge-density wave, leading to finite eMChA coefficient ${\gamma }_{\mathrm{eM}}$. In this mechanism, ${\gamma }_{\mathrm{eM}}$ is proportional to the product of the loop-current-induced orbital magnetization, $M_{\mathrm{orb}}^0$, and the lifetime of conduction electrons, τ. Therefore, ${\gamma }_{\mathrm{eM}}$ is reversible by the magnetic fields, and it takes large value in kagome metals with $\tau v_{\mathrm{Fermi}}\gg a_0$ (=lattice constant). Surprisingly, the quantum metric, which defines a fundamental geometric aspect of Bloch wavefunctions, acquires significant momentum dependence in the loop-current phase, resulting in a dramatic enhancement of eMChA by ∼100 times. This research not only clarifies the fundamental symmetry-breaking states in kagome metals but also opens a path for exploring quantum metric–induced phenomena arising from exotic quantum phase transitions in various metals.

In the realm of condensed matter physics, the investigation of quantum phase transitions and spontaneous symmetry breaking provides fundamental insights into the essential behaviors of electrons within solids. The exploration of rotational symmetry, time-reversal symmetry (TRS), inversion symmetry (IS), and $U(1)$ gauge symmetry-breaking quantum states has been actively pursued in various strongly correlated metals (1–5). The discovery of kagome-lattice metals AV₃Sb₅ ($A =$K, Rb, Cs) has paved the way for investigating novel correlation-driven quantum states and superconductivity (6–13). Unconventional multiple-phase transitions, both with and without TRS, arise due to the suppression of conventional spin or charge density waves (DWs) by geometrical frustration. The emergence of the $2\times 2$ Star-of-David or Tri-Hexagonal DW is attributed to the bond order (BO). Interestingly, the BO phase hosts exotic chiral charge loop-current (LC) orders that break TRS (8–10, 14), which is brought by the “pure imaginary” modulation in the hopping integral (15–18). Under the BO+LC phase in kagome metals, s-wave superconducting (SC) state emerges with finite impurities (12, 13). In contrast, unconventional SC states with chirality and nematicity have been observed using various experimental techniques (9, 19–21). Therefore, many unresolved issues remain regarding the SC state of kagome metals.

Fig. 1A illustrates the $2\times 2$ loop-current (LC) order expected to emerge in kagome metals. This LC state has been theoretically obtained by considering intersite repulsive interactions, as shown in refs. 22–24. Unconventional SC states have also been studied actively (22, 25–27). More recently, a mechanism in which LC state and s-wave SC state are mediated by BO fluctuations was proposed (28, 29).

Fig. 1.

Lattice structure, Fermi surface, and BO form factor in kagome metal. (A) 2D kagome lattice model with $2\times 2$ triple-$\mathit{q}$ LC $\mathit{\eta }=({\eta }_1,{\eta }_2,{\eta }_3)$ on $d_{\mathit{xz}}$ orbital. It is equal to $({\eta }_1,-{\eta }_2,-{\eta }_3)$ after shifting $\mathit{R}={\mathit{a}}_{\mathrm{BC}}+{\mathit{a}}_{\mathrm{CA}}$. The extended unit cell contains 12 sublattices $l=1\sim 12$. (B) Triple-$\mathit{q}$ Tri-Hexagonal BO $\mathit{\varphi }=\varphi {\mathit{e}}_0$$(\varphi >0)$ on $d_{\mathit{xz}}$ orbital. Two orbitals ($d_{\mathit{xz}}$, $d_{\mathit{yz}}$) on the V-ion are depicted. (C) Original Fermi surfaces without density waves. (D) Folded Fermi surfaces under the $2\times 2$ density waves with the folded BZ. (E) Nematic LC+BO coexisting state $({\mathit{\eta }}_{\mathrm{I}},{\mathit{\varphi }}_0)$. The $C_6$ symmetry centers of the LC and BO are expressed as O and ${\mathrm{H}}_{\mathrm{I}}$, respectively. The director is parallel to O-${\mathrm{H}}_{\mathrm{I}}$ line. Here, IS exists with respect to the inversion center H_I (${\mathrm{H}}_{\mathrm{II}(\mathrm{III})}$ is the center of ${\mathit{\eta }}_{\mathrm{II}(\mathrm{III})}$).

Recently, significant efforts have been made to detect the LC order. Several μSR studies have observed staggered local magnetic fields consistent with LC order (30–32). From a symmetry perspective, a finite $M_{\mathrm{orb}}^0$ is a consequence of the global TRS breaking in the triple-$\mathit{q}$ LC state. That is, the LC state is not restored under time reversal followed by a translation (23, 33). Therefore, the LC order is magnified under $B_z$ (33, 34), and its chirality is switched by tiny $B_z$ as observed by the anomalous Hall effect (AHE) (35, 36), nonreciprocal transport (37), STM (8–10), and Kerr rotation (38) measurements. TRS breaking was not observed in another Kerr rotation measurement (39). Interestingly, recent magnetic torque measurements (40) reveal the emergence of the single-$\mathit{q}$ LC order above $T_{\mathrm{BO}}$, and the transition from single-$\mathit{q}$ LC to triple-$\mathit{q}$ LC occurs under the conical magnetic field. Because $M_{\mathrm{orb}}^0=0$ in the single-$\mathit{q}$ LC state due to its global TRS, this scenario does not conflict with previous studies (8–10, 30–32, 37, 38). Notably, the LC will also have significant impact on the superconductivity, such as leading to the pair-density-wave and the $4e,6e$ pairing states (41–44).

Nonlinear effects in bulk materials provide a probe of the symmetry and geometry of Bloch electrons in solids and also useful functionalities including rectification and frequency conversion, which makes nonlinear effects important in both fundamental physics and application purposes (45–47). Among them, nonreciprocal transport is a current rectification effect in IS broken systems. In particular, nonreciprocal current under the applied magnetic field is called electrical magnetochiral anisotropy (eMChA) and is intensively studied (48–51). Recently, giant eMChA coefficient ${\gamma }_{\mathrm{eM}}$ defined as

$$\begin{array}{cc}& {\rho }_{\mathit{zz}}={\rho }_{\mathit{zz}}^0(1+{\gamma }_{\mathrm{eM}}{B}_x{J}_z)\hfill \end{array}$$ [1]

has been discovered in CsV₃Sb₅ in ref. 37. The observation clearly indicates the breaking of IS, considering the inversion operation that transforms $\mathit{j}\to -\mathit{j}$ and $\mathit{B}\to \mathit{B}$. The observed ${\gamma }_{\mathrm{eM}}$ in CsV₃Sb₅ is one or two orders of magnitude larger than the chiral magnets, such as CrNb₃S₆ (52) and MnSi (53), in which the chiral spin texture gives the eMChA. Unfortunately, the fundamental electronic states of the LC phase indicated by this important experiment are still unclear because of the lack of the underlying microscopic mechanism for the giant and switchable eMChA in CsV₃Sb₅.

The complex band structure reconstructed by the $2\times 2$ LC order in kagome metals—accompanied by broken TRS and IS—gives rise to exotic geometric properties of Bloch electrons. In this regard, kagome metals serve as an ideal platform for exploring quantum geometric effects, which underlie a variety of unconventional physical phenomena (54–59). The quantum geometric tensor (QGT) $\mathcal{T}=G-i\mathrm{\Omega }/2$ encapsulates essential information about the Bloch electron geometry (55, 56). The imaginary part Ω is the Berry curvature that gives the anomalous (or spin) Hall effects (59–62) and nonlinear Hall effect (63). Large AHE (29, 64) and orbital magnetization (33) are caused by Ω in kagome metals. In contrast, the real part G is the quantum metric (QM), which quantifies the “distance” between Bloch wave functions at different momenta in the Hilbert space. The QM essentially describes the spread of the wave packet made of Bloch electrons and plays a crucial role in flat band superconductivity (65), nonreciprocal transport phenomena (66), and other emergent effects. However, its impact in kagome metals, especially in their nonreciprocal transport phenomena, remains largely unexplored.

Importantly, in kagome metals, prominent QGT $\mathcal{T}$ originates from the BO and LC: their microscopic mechanisms due to strong electron correlations and geometrical frustrations have been developed in refs. 22, 23, 25–29, 67, and 68. Beyond-mean-field electron correlations give rise to various BO and LC orders in cuprates and Fe-based superconductors (1–5, 15, 69–78) and twisted bilayer graphene (79, 80).

In this study, we investigate the mechanism of the field-tunable giant eMChA arising from chiral LC in kagome metals, by focusing on the $2a_0$ or $4a_0$ ($a_0$: the unit cell length) stripe charge DW (CDW) that is universally observed in Cs-based kagome metals (81–83). The realized TRS and IS breaking state lead to a giant eMChA coefficient ${\gamma }_{\mathrm{eM}}\propto M_{\mathrm{orb}}^0\tau I_{\mathrm{st}}$, where $I_{\mathrm{st}}$ is the stripe CDW potential and τ is the lifetime of electrons. The derived eMChA becomes substantial due to large $\tau (\gg a_0/v_{\mathrm{Fermi}})$, and exhibits a sharp change in sign upon variation of the small magnetic field $B_z$. Unexpectedly, the QM acquires significant momentum dependence in the loop-current phase, resulting in a dramatic enhancement of the eMChA by ∼100 times. This mechanism dominates the Berry-curvature dipole mechanism (${\gamma }_{\mathrm{eM}}\propto {\tau }^0$) and the QM dipole mechanism (${\gamma }_{\mathrm{eM}}\propto {\tau }^{-1}$) in kagome metals with large τ. The present theory not only explains the switchable giant eMChA in the LC phase but also reveals the emergence of resonantly enlarged QM that will cause various exotic phenomena.

Kagome Lattice Model with LC+BO Order

The unit cell of the original kagome lattice model is composed of the three sublattices A, B, C shown in Fig. 1A. Under the triple-$\mathit{q}$ density-waves, the unit cell is enlarged to twelve sublattices ($l=1-12$), as denoted in Fig. 1 A and B. The Fermi surfaces are mainly composed of $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ orbitals denoted in Fig. 1B. The $d_{\mathit{xz}}$ orbital mainly composes the “pure-type” Fermi surface depicted in Fig. 1C, where each van-Hove singular (vHS) point $\mathit{k}\approx {\mathit{k}}_X$ ($X=$A, B, C) is composed of single sublattice X, called as “sublattice interference” (25). Both BO and LC orders emerge on $d_{\mathit{xz}}$-orbitals due to strong electron correlations (28, 29).

The three nesting vectors ${\mathit{q}}_n$ ($n=1,2,3$) are expressed in Fig. 1C. To describe the LC order at the wavevector ${\mathit{q}}_n$, we introduce the form factor $f_{i,j}^{(n)}=-f_{\mathit{ji}}^{(n)}=+i$ or $-i$ for the nearest-neighbor bonds ($i,j$), as expressed in see Fig. 1A. A detailed explanation is given in Materials and Methods. Note that $f_{\mathit{ij}}^{(1)}$ describes the LC on the AB-bonds. Using the notation ${\mathit{f}}_{\mathit{ij}}=(f_{\mathit{ij}}^{(1)},f_{\mathit{ij}}^{(2)},f_{\mathit{ij}}^{(3)})$, the LC order shown in Fig. 1A is given as

$$\begin{array}{cc}\hfill \delta t_{\mathit{ij}}^{\mathrm{c}}& =\mathit{\eta }·{\mathit{f}}_{\mathit{ij}},\hfill \end{array}$$ [2]

where $\mathit{\eta }\equiv ({\eta }_1,{\eta }_2,{\eta }_3)$ is the set of the LC order parameters at $\mathit{q}={\mathit{q}}_n$ ($n=1,2,3$).

Also, the BO form factor $g_{\mathit{ij}}^{(n)}=g_{\mathit{ji}}^{(n)}=+1$ or $-1$ are expressed in Fig. 1B; see Materials and Methods for details. Using the notation ${\mathit{g}}_{\mathit{ij}}=(g_{\mathit{ij}}^{(1)},g_{\mathit{ij}}^{(2)},g_{\mathit{ij}}^{(3)})$, the BO shown in Fig. 1B is given as

$$\begin{array}{cc}\hfill \delta t_{\mathit{ij}}^{\mathrm{b}}& =\mathit{\varphi }·{\mathit{g}}_{\mathit{ij}},\hfill \end{array}$$ [3]

where $\mathit{\varphi }\equiv ({\varphi }_1,{\varphi }_2,{\varphi }_3)$ is the set of the BO parameters at $\mathit{q}={\mathit{q}}_n$.

Here, we introduce the triple-$\mathit{q}$ BO, ${\mathit{\varphi }}_0\equiv \varphi {\mathit{e}}_0$, and the triple-$\mathit{q}$ LC, ${\mathit{\eta }}_{\alpha }\equiv \eta {\mathit{e}}_{\alpha }$ ($\alpha =\mathrm{I}, \mathrm{II}, \mathrm{III})$, by using the notation ${\mathit{e}}_0=(1,1,1)/\sqrt{3}$, ${\mathit{e}}_{\mathrm{I}}=(1,-1,-1)/\sqrt{3}$, ${\mathit{e}}_{\mathrm{II}}=(-1,1,-1)/\sqrt{3}$, ${\mathit{e}}_{\mathrm{III}}=(-1,-1,1)/\sqrt{3}$. Then, the $Z_3$ nematic LC+BO coexisting state $({\mathit{\eta }}_{\alpha },{\mathit{\varphi }}_0)$ with $\alpha =\mathrm{I}, \mathrm{II}, \mathrm{III}$ is realized when $T<T_{\mathrm{LC}}<T_{\mathrm{BO}}$ according to the Ginzburg–Landau (GL) free-energy theory (29, 33); see also SI Appendix, section A (84). In this state, the $C_6$ symmetry center of the LC, denoted as ${\mathrm{H}}_{\alpha }$, differs from that of the BO, denoted as O. Therefore, the state $({\mathit{\eta }}_{\alpha },{\mathit{\varphi }}_0)$ exhibits the nematic director along the O-${\mathrm{H}}_{\alpha }$ line, as depicted in Fig. 1E.

In the $Z_3$ LC+BO state, the orbital magnetization is finite, so the LC order parameter can be switched by the magnetic field (33). However, the eMChA is absent because of the global IS with respect to the inversion center H_I. In the next section, we explain that the eMChA occurs when the LC and the stripe CDW coexist, even in the absence of the BO. The stripe CDW has been widely reported in experiments (81–83).

Symmetry Analysis of NLH and eMChA Conductivities.

In this section, we study the second-order response to the electric field $\mathit{E}$ in the LC state in kagome metals. The nonlinear Hall conductivity ${\sigma }_{\alpha \beta {\beta }^{\prime }}$ is defined as

$$\begin{array}{c}\hfill \mathrm{\Delta }{j}_{\alpha }={\sigma }_{\alpha \beta {\beta }^{\prime }}{E}_{\beta }{E}_{{\beta }^{\prime }},\end{array}$$ [4]

where $j_{\alpha }$ is the charge current along the α-axis of order $O(E^2)$. Here, $\alpha ,\beta ,{\beta }^{\prime }=x$, y, or z. The eMChA conductivity ${\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }$ is defined as

$$\begin{array}{c}\hfill \mathrm{\Delta }{j}_{\alpha }={\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }{E}_{\beta }{E}_{{\beta }^{\prime }}{B}_{\gamma },\end{array}$$ [5]

where $B_{\gamma }$ is the magnetic field along the γ axis. Note that ${\gamma }_{\mathrm{eM}}$ in Eq. 1 is given by $-{\sigma }_{zzz,x}/{({\sigma }_{\mathit{zz}})}^2$ (37). That is, the relationship ${\sigma }_{zzz,x}\propto {\gamma }_{\mathrm{eM}}{\tau }^2$ holds. Under the inversion operation, $\mathrm{\Delta }\mathit{j}$ and $\mathit{E}$ are reversed while $\mathit{B}$ is unchanged. In addition, ${\sigma }_{\alpha \beta {\beta }^{\prime }}$ and ${\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }$ are unchanged if the system is centrosymmetric. Therefore, ${\sigma }_{\alpha \beta {\beta }^{\prime }}={\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }=0$ in any centrosymmetric system. Thus, the absence of IS is a necessary condition for the finite ${\sigma }_{\alpha \beta {\beta }^{\prime }}$ and ${\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }$. Under the time-reversal operation, the signs of $\mathrm{\Delta }\mathit{j}$ and $\mathit{B}$ are reversed. For this reason, the eMChA (NLH) conductivity is T-even (T-odd).

In the following, we discuss the conditions for the realization of eMChA in kagome metals in greater detail from the perspective of symmetry. For this purpose, we examine how the elementary symmetry operations [i.e., inversion (I), mirror ($M_{\mu }$), and time-reversal (T)] affect the model parameters (η, ϕ, $I_{\mathrm{st}}$) and external fields ($\mathit{E}$, $\mathit{B}$). Based on the resulting transformation rules, we derive the necessary conditions for nonzero components of ${\sigma }_{\alpha \beta {\beta }^{\prime }}$ and ${\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }$. In the original 2D kagome lattice shown in Fig. 2A, the IS centers are located at A, B, C, and H, represented by filled square symbols. Note that H is the center of the $C_6$ rotational symmetry. When the triple-$\mathit{q}$ LC order is introduced, only the inversion centers H and H₂ survive as shown in Fig. 2B. Here, Point H (H₂) is a $C_6$ ($C_2$) rotational symmetry point. Importantly, IS is broken by introducing the stripe CDW, as in Fig. 2C. Here, the yellow circles represent the stripe CDW potentials $I_{\mathrm{st}}$, which is consistent with the energy-independent (vertical) QPI signal with period $2a_0$ observed in refs. 81–83. Hereafter, we analyze only the $2a_0$ stripe CDW in the numerical study. (IS is also violated by introducing the period $4a_0$ CDW, while stripe CDW alone does not break IS.)

Fig. 2.

Inversion symmetry breaking in the LC+stripe state without BO. (A) Original 2D crystal structure. The IS centers are described by A, B, C, H. (B) $2\times 2$ LC order, in which the IS centers H and H₂ survive. Point H (H₂) is a $C_6$ ($C_2$) rotational symmetry point. (C) Coexisting phase of $2\times 2$ LC and $2a_0$ stripe CDW. Thus, IS is broken when the LC and stripe CDW coexist.

In Fig. 2C, $M_x$ ($M_y$) represents the mirror operation $x\to -x$ ($y\to -y$) and the inversion operation is $I=M_x{M}_y{M}_z$. Below, we assume $2\times 2\times 1$ LC to simplify the discussion. We denote the transformation of a physical quantity A under the symmetry operation X as $A^X$.

For $X=I$, real coordinate $\mathit{r}$, charge current $\mathit{j}$, and magnetic field $\mathit{B}$ are respectively transformed as ${\mathit{r}}^I=-\mathit{r}$, ${\mathit{j}}^I=-\mathit{j}$, and ${\mathit{B}}^I=\mathit{B}$. Note that $\mathit{r}$ and $\mathit{j}$ are polar vectors, while $\mathit{B}$ is an axial vector. For $X=M_x$, we obtain ${\mathit{r}}^{M_x}=(-x,y,z)$, ${\mathit{j}}^{M_x}=(-j_x,j_y,j_z)$, and ${\mathit{B}}^{M_x}=(B_x,-B_y,-B_z)$. Next, we consider the transformation of the model order parameters shown in Fig. 2C, where the LC order is $\mathit{\eta }=\eta {\mathit{e}}_{\mathrm{I}}$ and the stripe CDW potential denoted by the yellow circles is $I_{\mathrm{st}}$. ($-I_{\mathrm{st}}$ on the other A, B sites.) Then, the following relations are derived from Fig. 2C: ${(\eta ,I_{\mathrm{st}})}^{M_x}=(-\eta ,-I_{\mathrm{st}})$, ${(\eta ,I_{\mathrm{st}})}^{M_y}=(-\eta ,I_{\mathrm{st}})$, and ${(\eta ,I_{\mathrm{st}})}^T=(-\eta ,I_{\mathrm{st}})$, where T is the time-reversal operation. The symmetries of model order parameters are summarized in Table 1.

Table 1.

Parity of each model parameter ($\eta ,\varphi ,I_{\mathrm{st}}$) under the operation $X=M_x$, $M_y$, I, and T

	η	ϕ	I st
M x	−1	1	−1
M y	−1	1	1
I	1	1	−1
T	−1	1	1

The model parameters and the x-y coordinates are shown in Fig. 2C.

When $\eta I_{\mathrm{st}}\ne 0$, the spin-independent band dispersion of the present model loses its inversion symmetry, resulting in ${ϵ}_{b,\mathit{k}}\ne {ϵ}_{b,-\mathit{k}}$. In more detail, the relations ${ϵ}_{b,k_x,k_y}={ϵ}_{b,-k_x,k_y}\ne {ϵ}_{b,k_x,-k_y}$ is satisfied because ${[\eta I_{\mathrm{st}}]}^{M_x}=\eta I_{\mathrm{st}}$ and ${[\eta I_{\mathrm{st}}]}^{M_y}=-\eta I_{\mathrm{st}}$. Therefore, the Fermi surfaces shift along the $k_y$-axis due to the dipole potential ${ϵ}_{b,\mathit{k}}^{-}\equiv \frac{1}{2}({ϵ}_{b,\mathit{k}}-{ϵ}_{b,-\mathit{k}})$ (Fig. 3B).

Fig. 3.

QM-driven giant eMChA conductivity due to the dipole potential. (A) Fermi surfaces of the $d_{\mathit{xz}}+d_{\mathit{yz}}$ orbital 3D kagome lattice model in the original BZ. The $d_{\mathit{xz}}$ orbital weight is expressed as red color. These Fermi surfaces qualitatively fit the main Fermi surfaces of the first-principles DFT calculation (brown dotted lines). (B) Dipole potential ${ϵ}_{b,\mathit{k}}^{-}$ ($b=9$) in the folded BZ for $\eta =0.05$ and $I_{\mathrm{st}}=0.01$. (C) Schematic picture of the dipole potential. It gives no macroscopic current ($⟨v⟩=\mathbf{0}$) because of the Bloch’s theorem. However, it gives finite $⟨v^{\mathit{yzz}}⟩$. Then, the NLH ${\sigma }_{\mathit{yzz}}$ and the eMChA ${\sigma }_{zzz,x}$ becomes finite. (D) ${\sigma }_{zzz,x}^{-}(\mathit{k})$ in $k_z=\pi /2$ plane and $k_x=0.08\pi$ plane for $\eta =0.024$ and $I_{\mathrm{st}}=0.01$ at $T=0.01$. It is an even function of $\mathit{k}$, so its $\mathit{k}$ summation remains finite. (E) η-odd part of the QM at the Fermi level, $G_{\mathit{k}}^{zz,-}$, in the ${\mathit{k}}_z=\pi /2$ plane. Its resonant positions are the same as those of ${\sigma }_{zzz,x}^{-}(\mathit{k})$ in (E). Thus, giant eMChA originates from the resonantly enhanced η-odd QM. (F) eMChA and (G) NLH conductivities in the v-LC state with AB direction $2a_0$ stripe CDW ($I_{\mathrm{st}}=0.01$) as functions of n for $\eta =0.008-0.024$. We put $\gamma =1$ for simplicity. The numerical result is almost unchanged for the s-LC state; see SI Appendix, section C (84).

Here, we discuss the symmetry of the eMChA conductivity given by Eq. 5. Under the symmetry operation X, it is transformed as

$$\begin{array}{c}\hfill \mathrm{\Delta }{j}_{\alpha }^X={[{\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }]}^X{E}_{\beta }^X{E}_{{\beta }^{\prime }}^X{B}_{\gamma }^X,\end{array}$$ [6]

where ${[{\sigma }_{\alpha \beta \beta ,\gamma }]}^X$ obeys the same symmetry transformation as that for the Hamiltonian. When $X=I$, Eq. 6 gives $-\mathrm{\Delta }{j}_{\alpha }={[{\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }]}^I{E}_{\beta }{E}_{{\beta }^{\prime }}{B}_{\gamma }$. Therefore, IS breaking (i.e., ${[{\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }]}^I\ne {\sigma }_{\alpha \beta {\beta }^{\prime },\gamma }$) is a necessary condition for the emergence of the eMChA. Now, we set $\alpha =\beta ={\beta }^{\prime }=z$ and $\gamma =x$ to understand the experimentally observed eMChA in kagome metals. When $X=M_y$, Eq. 6 gives $\mathrm{\Delta }{j}_z={[{\sigma }_{zzz,x}]}^{M_y}{E}_z^2(-B_x)$. Therefore, ${\sigma }_{zzz,x}$ can appear when $M_y$ symmetry is broken, as observed in the present model from Table 1; ${[\eta I_{\mathrm{st}}]}^{M_y}=-\eta I_{\mathrm{st}}$.

When $X=T$, Eq. 6 gives $-\mathrm{\Delta }{j}_z={[{\sigma }_{zzz,x}]}^T{E}_z^2(-B_x)$, where ${[{\sigma }_{zzz,x}]}^T={(-1)}^{m+m^{\prime }}{\sigma }_{zzz,x}$ when ${\sigma }_{zzz,x}\propto {\tau }^m{\eta }^{m^{\prime }}$ with the relaxation time τ. Therefore, ${\sigma }_{zzz,x}$ can appear when $m+m^{\prime }$ is even. In Boltzmann theory, the most divergent eMChA conductivity with respect to $\tau (\gg a_0/v_{\mathrm{Fermi}})$ is $O({\tau }^3)$, as we will explain later and in the SI Appendix, section B (84). Thus, giant ${\tau }^3$-eMChA conductivity can appear for odd $m^{\prime }$, especially for $m^{\prime }=1$ in the least order.

In Materials and Methods, we present the three-dimensional (3D) LC state, focusing on the widely discussed $2\times 2\times 2$ order: 1) The vertical LC (v-LC) state, where the same 2D orders are stacked vertically (Fig. 5B). 2) The shift-stack LC (s-LC)” state, where the 2D order on even-number layers and that on the odd-number layers are shifted by the vector ${\mathit{a}}_{\mathrm{AB}}$ (Fig. 5D). Once IS is broken, the resulting eMChA conductivities for the v-LC and s-LC become nearly identical.

Fig. 5.

Inversion symmetry breaking in LC+stripe state. (A) Original 3D crystal structure. The IS centers are described by A, B, C, H, A^′, B^′, C^′, H^′. (B) v-LC with remaining IS centers H and H^′. (C) Coexisting phase of v-LC and $2a_0$ stripe CDW, resulting in IS breaking (D) s-LC with remaining IS center C^′. (E) Coexisting phase of s-LC and $2a_0$ stripe CDW, where the shift-stacking direction and the stripe direction are parallel, leading to IS breaking.

To summarize, the giant and reversible eMChA conductivity ${\sigma }_{zzz,x}\propto {\tau }^3\eta$ in the LC phase of kagome metals is understandable from the perspective of symmetry. Below, we conduct a numerical study of eMChA conductivity to gain a quantitative understanding.

TRS Breaking Dipole Potential due to Finite $\eta I_{\mathrm{st}}$

To perform a quantitative numerical analysis of the nonlinear conductivities, we introduce a 3D kagome lattice model with $d_{\mathit{xz}}$ and $d_{\mathit{yz}}$ orbitals. The Fermi surfaces of the present model are shown in Fig. 3A for the electron filling at $n_{\exp }=4.6$. [See SI Appendix, section D (84) for detailed explanation.] Here, both pure-type and mix-type Fermi surfaces are very close in energy. In addition, the DFT calculation shows that the pure/mix-type Fermi surface contains $10\sim 30$% $d_{\mathit{yz}}$/$d_{\mathit{xz}}$-orbital component (33). Therefore, the $2\times 2$ LC order on the $d_{\mathit{xz}}$ orbitals, ${\mathit{\eta }}_{\alpha }$, causes prominent pure-mix band hybridization. This LC-induced band reconstruction causes not only large $M_{\mathrm{orb}}^0$ (33) but also giant eMChA as we will explain below.

As explained above, the IS of the triple-$\mathit{q}$ LC state is broken by introducing a $2a_0$ or $4a_0$ CDW potential. Here, we specifically introduce the $2a_0$ stripe CDW potential $I_{\mathrm{st}}$, as schematically illustrated in Fig. 2C. In this paper, we denote the η-odd and η-even parts of a general function A by $A^{-}$ and $A^{+}$, respectively. They are explicitly defined as

$$\begin{array}{c}\hfill A_{\mathit{k},\eta }^{\pm }\equiv \frac{1}{2}(A_{\mathit{k},\eta }\pm A_{\mathit{k},-\eta }).\end{array}$$ [7]

In the following, we frequently refer to ${ϵ}_{b,\mathit{k},\eta }^{\pm }\equiv \frac{1}{2}({ϵ}_{b,\mathit{k},\eta }\pm {ϵ}_{b,\mathit{k},-\eta })$. In the present model, the relation ${ϵ}_{b,\mathit{k}}^{\pm }=\frac{1}{2}({ϵ}_{b,\mathit{k}}\pm {ϵ}_{b,-\mathit{k}})$ is satisfied for a fixed η, because time-reversal operation gives ${ϵ}_{b,\mathit{k},\eta }={ϵ}_{b,-\mathit{k},-\eta }$ for a fixed $I_{\mathrm{st}}$. Therefore, ${ϵ}_{b,\mathit{k}}^{-}$ is both the “η-odd part” and the “$\mathit{k}$-odd part” of ${ϵ}_{b,\mathit{k}}$.

Fig. 3B shows the numerical result of a conduction band ${ϵ}_{b,\mathit{k}}^{-}$ ($b=9$) under the AB stripe CDW for $\eta =0.05$. ${ϵ}_{b,\mathit{k}}^{-}$ give a “dipole potential in $\mathit{k}$-space” because it is proportional to $k_y$. It has the symmetry-protected nodes on the $k_x$-axis line and other two nodes on $k_y\approx \pm \sqrt{3}{k}_x$ line. Its schematic picture is given in Fig. 3C. To understand the origin of the dipole potential, we consider the symmetry of the triple-$\mathit{q}$ LC order depicted in Figs. 1A and 2C. The kagome lattice belongs to the $D_{6h}$ point group. Around point H, the LC pattern preserve IS, while all the parities of the six mirror operations along the $\theta =(\pi /6)m$ axis ($m=0,1,\cdots 5$ for $3{\sigma }_h$ and $3{\sigma }_v$) are $-1$. Thus, the LC order belongs to the $A_{2g}$ irreducible representation, whose basis function is $b_{A_{2g}}=x(x^2-3y^2)y(3x^2-y^2)$. Also, the AB direction stripe CDW changes its sign under $M_x$ mirror operation, while it is unchanged under $M_y$ mirror operation. Based on the second-order perturbation theory, the derived ${ϵ}_{\mathit{k}}^{-}$ of order $O(\eta I_{\mathrm{st}})$ is proportional to $x\times b_{A_{2g}}\sim k_y(k_x^2-3k_y^2)(3k_x^2-k_y^2)$, which reproduces the $k_y$-dipole with complex sign reversals shown in Fig. 3B. Thus, the dipole potential is simply expressed as ${ϵ}_{\mathit{k}}^{-}\propto k_y\eta I_{\mathrm{st}}$ with accidental sign reversals as shown in Fig. 3 B and C.

Due to the dipole potential ${ϵ}_{\mathit{k}}^{-}$, one may consider that the bulk current $⟨v^y⟩\propto \sum _{\mathit{k}}{v}^{y}f$ is finite, where $f_0$ is the Fermi distribution function, which is expanded as $f_0({ϵ}_{\mathit{k}})\approx f_0({ϵ}_{\mathit{k}}^{+})+f_0^{\prime }({ϵ}_{\mathit{k}}^{+}){ϵ}_{\mathit{k}}^{-}$. However, $⟨v^y⟩$ is exactly zero because the LC order is periodic in the $2\times 2$ unit cell, known as the Bloch’s theory based on the Galilean transformation argument (85). This is also simply proved as $⟨\mathit{v}⟩\propto \sum _{\mathit{k}}\mathit{v}{f}_0({ϵ}_{\mathit{k}})=\sum _{\mathit{k}}{\mathbf{\nabla }}_{\mu }({ϵ}_{\mathit{k}}-\mu )f_0({ϵ}_{\mathit{k}})=-\sum _{\mathit{k}}({ϵ}_{\mathit{k}}-\mu )f_0^{\prime }({ϵ}_{\mathit{k}})\mathit{v}=\mathbf{0}$ at $T=0$.

Boltzmann Transport Theory, Impact of Quantum Metric

In general multiband systems, the expressions of both linear and nonlinear conductivities are derived from the Boltzmann equation method. The Boltzmann equation provides a closed integral equation for the nonequilibrium distribution function f under the external electric field E and magnetic field B. It is drastically simplified by applying the relaxation time approximation for the collision integral term: ${(\partial f/\partial t)}_{\mathrm{coll}}=-(f-f_0)/\tau$, where $f_0$ is the equilibrium Fermi distribution function and τ is the relaxation time of the conduction electrons. In this approximation, the solution for f can be obtained by expanding f in powers of τ as $f=\sum _{n=0}^{\infty }{f}_n$, where $f_n\propto {\tau }^n$. The detailed analysis is performed in SI Appendix, section B (84).

Here, we denote the f of order $E^i{B}^j$ as $f^{(i,j)}$. Then, the linear conductivity ${\sigma }_{\alpha \alpha }$, the nonlinear Hall conductivity ${\sigma }_{\alpha \beta \beta }$, and the eMChA conductivity ${\sigma }_{\alpha \beta \beta ,\gamma }$ are derived from $f^{(1,0)}$, $f^{(2,0)}$, and $f^{(2,1)}$, respectively. $f^{(i,j)}$ is expressed as $\sum _{n=0}^{n^{\ast }}{f}_n^{(i,j)}$ with $n^{\ast }\equiv i+j$, because $f_n^{(i,j)}=0$ for $n>n^{\ast }$. In good metals with $\tau v_{\mathrm{F}}\gg a_0$, where $v_{\mathrm{F}}$ is the Fermi velocity and $a_0$ is the unit cell length, $f_{n^{\ast }}^{(i,j)}$ gives the most significant contribution. Therefore, the leading divergent terms of the conductivities with respect to τ are given as ${\sigma }_{\alpha \alpha }\propto {\tau }^1$, ${\sigma }_{\alpha \beta \beta }\propto {\tau }^2$, and ${\sigma }_{\alpha \beta \beta ,\gamma }\propto {\tau }^3$, respectively. When TRS exists, the most divergence terms of ${\sigma }_{\alpha \beta \beta }$, and ${\sigma }_{\alpha \beta \beta ,\gamma }$ vanish according to the abovementioned symmetry argument. However, they can remain finite in the LC phase, where TRS is broken.

The expressions of ${\sigma }_{\alpha \alpha }$, ${\sigma }_{\alpha \beta \beta }$, and ${\sigma }_{\alpha \beta \beta ,\gamma }$ corresponding to the leading divergent terms are derived in SI Appendix, section B (84). They are expressed as

$$\begin{array}{cc}\hfill {\sigma }_{\alpha \alpha }& =-e^2\frac{g_s}{N}{\displaystyle \sum _{a,\mathit{k}}}{(f_0^{\prime }{v}^{\alpha }{v}^{\alpha })}_{a,\mathit{k}}\frac{1}{2\gamma },\hfill \end{array}$$ [8]

$$\begin{array}{cc}\hfill {\sigma }_{\alpha \beta \beta }& =e^3\frac{g_s}{N}{\displaystyle \sum _{a,\mathit{k}}}{(f_0^{\prime }{v}^{\beta }{v}^{\alpha \beta })}_{a,\mathit{k}}\frac{1}{4{\gamma }^2},\hfill \end{array}$$ [9]

$$\begin{array}{cc}\hfill {\sigma }_{\alpha \beta \beta ,\gamma }& =e^4\frac{g_s}{N}{\displaystyle \sum _{a,\mathit{k}}}{ϵ}^{\gamma \varphi \eta }{(-f_0^{\prime }{v}^{\alpha \beta }{v}^{\beta \eta }{v}^{\varphi })}_{a,\mathit{k}}\frac{1}{4{\gamma }^3}\hfill \\ \hfill & +\mathrm{\Delta }{\sigma }_{\alpha \beta \beta ,\gamma },\hfill \end{array}$$ [10]

where $v_{a,\mathit{k}}^{\alpha }\equiv {\partial }_{\alpha }{ϵ}_{a,\mathit{k}}$ and $v_{a,\mathit{k}}^{\alpha \beta }\equiv {\partial }_{\alpha }{\partial }_{\beta }{ϵ}_{a,\mathit{k}}$, and a is the band index. $\gamma =1/2\tau$ is the quasiparticle (QP) damping rate, $f_0^{\prime }$ is the ϵ-derivative of Fermi distribution function, $g_s=2$ is the spin degeneracy, and ${ϵ}^{\alpha \beta \gamma }$ is Levi–Civita symbol. Here, $\mathrm{\Delta }{\sigma }_{\alpha \beta \beta ,\gamma }\equiv e^4\frac{g_s}{N}\sum _{a,\mathit{k}}{ϵ}^{\gamma \varphi \eta }{(f_0^{\prime }{v}^{\alpha \eta }{v}^{\beta \varphi }{v}^{\beta })}_{a,\mathit{k}}\frac{1}{8{\gamma }^3}$, which vanishes when $\alpha =\beta$.

In the present model with $\eta I_{\mathrm{st}}\ne 0$, the symmetries of $v_{\mathit{k}}^{\mu ,\pm }$ and $v_{\mathit{k}}^{\mu \nu ,\pm }$ are summarized in Table 2. Importantly, $v_{b,\mathit{k}}^{\mu ,-}$ is even in $\mathit{k}$ and $v_{b,\mathit{k}}^{\mu \nu ,-}$ is odd in $\mathit{k}$. For example, $v_{\mathit{k}}^{y,-}\propto k_{\mu }^0$ and $v_{\mathit{k}}^{zz,-}\propto k_y$. This table is useful to understand what kinds of nonreciprocal conductivities become finite. Based on Table 2, we understand that the NLH is finite for ${\sigma }_{y\mu \mu }$ ($\mu =x,y,z$). Also, the eMChA is finite for ${\sigma }_{z\mu \mu ,x}$ and ${\sigma }_{x\mu \mu ,z}$.

Table 2.

(A) $\mathit{k}$-dependences of $v^{\mathit{\mu }\mathbf{,}\mathbf{\pm }}$ and ${\mathrm{\Omega }}^{\mathit{\mu }\mathbf{,}\mathbf{\pm }}$, and (B) $\mathit{k}$-dependences of $v^{\mathit{\mu }\mathit{\nu }\mathbf{,}\mathbf{\pm }}$ in the LC state with AB-stripe CDW

$\mu =$	x		y		z
$v^{\mu ,-}$	$k_x{k}_y$		1		$k_y{k}_z$
$v^{\mu ,+}$	$k_x$		$k_y$		$k_z$
${\mathrm{\Omega }}^{\mu ,-}$	$k_x{k}_z$		$k_y{k}_z$		1
${\mathrm{\Omega }}^{\mu ,+}$	$k_x{k}_y{k}_z$		$k_z$		$k_y$
$\mu \nu =$	xx, yy	zz	xz	yz	xy
$v^{\mu \nu ,-}$	$k_y$	$k_y$	$k_x{k}_y{k}_z$	$k_z$	$k_x$
$v^{\mu \nu ,+}$	1	1	$k_x{k}_z$	$k_y{k}_z$	$k_x{k}_y$

Note that each function in the table should be multiplied by an $A_{1g}$ (totally symmetric) function in taking the derivative.

In the numerical study, we derive $v_{a,\mathit{k}}^{\alpha \beta }\equiv {\partial }_{\alpha }{\partial }_{\beta }{ϵ}_{a,\mathit{k}}$ from the sum rule $v_{a,\mathit{k}}^{\alpha \beta }={\tilde{v}}_{a,\mathit{k}}^{\alpha \beta }+2g_{a,\mathit{k}}^{\alpha \beta }$ to improve numerical accuracy. Here, ${\tilde{v}}_{a,\mathit{k}}^{\alpha \beta }\equiv ⟨u_{a,\mathit{k}}|({\partial }_{\alpha }{\partial }_{\beta }{\widehat{H}}_{\mathit{k}}^0)|u_{a,\mathit{k}}⟩$, where ${\widehat{H}}_{\mathit{k}}^0$ is the Hamiltonian for the kinetic term in the orbital basis. Also, $g_{a,\mathit{k}}^{\alpha \beta }$ is given by

$$\begin{array}{c}\hfill g_a^{\alpha \beta }=\frac{1}{2}{\displaystyle \sum _{b (\ne a)}}\frac{\mathrm{Re}[v_{\mathit{ab}}^{\alpha }{v}_{\mathit{ba}}^{\beta }+v_{\mathit{ab}}^{\beta }{v}_{\mathit{ba}}^{\alpha }] {ϵ}_{\mathit{ab}}}{{ϵ}_{\mathit{ab}}^2+{\delta }^2},\end{array}$$ [11]

where $\delta =0$ corresponds to the sum rule for the free-electron, while we introduce $\delta =0.001$ to obtain reliable numerical results. (δ would correspond to Im${\mathrm{\Sigma }}^{\mathrm{A}}$.) Note that Eq. 11 is derived from the covariant differential of the operator $v^z={\partial }_z{H}^0$: $D_z{v}^z={\partial }_z{v}^z-i[A^z,v^z]$, where $A_{a,b}^z=i⟨u_a|{\partial }_z{u}_b⟩$ is the Berry connection. We call $g_a^{\alpha \beta }$ as the quasi-quantum metric (qQM) of the Bloch wavefunction because of its close relationship to the quantum metric (QM) given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill G_a^{\alpha \beta }& \hfill ={\displaystyle \sum _{b (\ne a)}}\frac{\mathrm{Re}[v_{\mathit{ab}}^{\alpha }{v}_{\mathit{ba}}^{\beta }+v_{\mathit{ab}}^{\beta }{v}_{\mathit{ba}}^{\alpha }]}{{ϵ}_{\mathit{ab}}^2+{\delta }^2}\hfill \end{array},\end{array}$$ [12]

where $g_a^{\alpha \beta }$ and $G_a^{\alpha \beta }$ differ in the power in ${ϵ}_{\mathit{ab}}$ in the denominator. In particular, both qQM and QM take huge values when the conduction and valence bands are nearly degenerated. In later sections, we find that the giant eMChA conductivity in kagome metal is caused by the η-odd part of the QM ($G^{zz,-}$) at the Fermi level, which gives a resonantly enhanced contribution for $\eta \lesssim 0.02$.

Finally, we briefly discuss the reciprocal conductivities due to ${ϵ}_{\mathit{k}}^{-}$ based on Eqs. 9 and 10. The NLH conductivity is given as ${\sigma }_{\mathit{yzz}}\propto ⟨v^{\mathit{yzz}}⟩{\gamma }^{-2}$, where $v^{\mathit{yzz}}\equiv {\partial }_y{v}^{\mathit{zz}}$, by using the integration by parts. Apparently, $⟨v^{\mathit{yzz}}⟩$ is finite thanks to ${ϵ}_{\mathit{k}}^{-}$. (Note that the Bloch’s theory (85) does not prohibit a finite $⟨v^{\mathit{yzz}}⟩$.) Also, the eMChA conductivity is given as ${\sigma }_{zzz,x}\propto ⟨v^{\mathit{yzz}}{v}^{\mathit{zz}}-v^{\mathit{zzz}}{v}^{\mathit{yz}}⟩{\gamma }^{-3}$, whose η-odd term is finite according to Table 2. Thus, the TRS breaking dipole potential ${ϵ}_{\mathit{k}}^{-}$ gives finite NLH and eMChA conductivities.

Here, we demonstrate the important role of the resonantly enhanced QM on the eMChA conductivity. Here, we denote ${\sigma }_{zzz,x}\equiv \frac{1}{N}\sum _{\mathit{k}}{\sigma }_{zzz,x}(\mathit{k})$, where ${\sigma }_{zzz,x}(\mathit{k})\propto -\sum _a{[f_0^{\prime }({(v^{\mathit{zz}})}^2{v}^y-v^{\mathit{zz}}{v}^{\mathit{yz}}{v}^z)]}_a\frac{1}{{\gamma }^3}$ according to Eq. 10. η-even part ${\sigma }_{zzz,x}^{+}(\mathit{k})$ vanishes after the $\mathit{k}$-summation, so only η-odd part ${\sigma }_{zzz,x}^{-}(\mathit{k})$ contributes to ${\sigma }_{zzz,x}$. Fig. 3E shows ${\sigma }_{zzz,x}^{-}(\mathit{k})$ in the $k_z=\pi /2$ plane and the $k_x=0.08\pi$ plane, for $\eta =0.024$ and $T=0.01$. Here, we set $I_{\mathrm{st}}=0.01$, which is a few times larger than the onset temperature ($T^{\ast }\sim 35$ K), estimating from mean-field (BCS-type) theories. The obtained ${\sigma }_{zzz,x}$ is almost linear in $I_{\mathrm{st}}$.

It shows large, resonant-like peaks near the LC-induced band reconstruction between the pure-type and mix-type Fermi surfaces, denoted as red and blue broken lines in Fig. 3D. (The original Fermi surfaces at $\eta =0$ are expressed in SI Appendix, section D (84) in the folded BZ.) Therefore, giant eMChA conductivity in kagome metals originates from the “pure-mix band reconstruction.”

The resonant-like peaks in Fig. 3D mainly originate from ${[{(v^{\mathit{zz}})}^2]}^{-}$ in ${\sigma }_{zzz,x}^{-}(\mathit{k})$. Based on the ($a,b$)-band model, an approximate relation ${[{(v^{\mathit{zz}})}^2]}_a^{-}\approx {[{(g^{\mathit{zz}})}^2]}_a^{-}\approx 2G_a^{zz,-}·{(v_{\mathit{ab}}^z{v}_{\mathit{ba}}^z)}^{+}$ is derived. To verify this relation, we shows the η-odd part of the QM on the Fermi surface: $G_{\mathit{k}}^{zz,-}\equiv \sum _a{G}_{a,\mathit{k}}^{zz,-}(-f_0^{\prime }({ϵ}_{a,\mathit{k}}^0))$ in the ${\mathit{k}}_z=\pi /2$ plane in Fig. 3E. Its peak positions are almost the same as those of ${\sigma }_{zzz,x}^{-}(\mathit{k})$ shown in Fig. 3D. In fact, ${\sigma }_{zz,x}^{-}(\mathit{k})\sim \sum _a{f}_0^{\prime }{G}_a^{zz,-}{v}_a^{y,+}\frac{1}{{\gamma }^3}$ and $v_a^{y,+}$ is an odd-function of $k_y$. Therefore, the giant eMChA in kagome metal originate from the η-odd QM $G_{\mathit{k}}^{zz,-}$. Three-dimensional mix-type Fermi surface with large $k_z$-directional velocity gives rise to large $G_{\mathit{k}}^{zz,-}$.

Numerical Study of Nonreciprocal Conductivities

In the following, we perform the numerical study of the nonlinear conductivities, under the stripe CDW along the AB direction (along the y-axis). To achieve enough numerical accuracy, we use $480\times 480\times 240$ $\mathit{k}$-meshes in the original BZ. We set $\gamma =1$, $e=1$, and $ħ=1$ for simplicity.

Fig. 3 F and G shows the eMChA conductivity ${\sigma }_{zzz,x}$ and the NLH conductivity ${\sigma }_{\mathit{yxx}}$ in the v-LC state, $\mathit{\eta }=\eta {\mathit{e}}_{\mathrm{I}}$ and $\mathit{\theta }=(0,0,0)$, as functions of the electron filling n at $T=0.01$. (The obtained results are essentially unchanged for $T=0.001\sim 0.01$.) $n_{\exp }$ corresponds to the electron filling of CsV₃Sb₅, and $n_{\mathrm{vHS}}$ is the vHS filling for the $d_{\mathit{xz}}$-orbital band. Both ${\sigma }_{zzz,x}$ and ${\sigma }_{\mathit{yxx}}$ are odd-functions of η, so they are reversed by changing the direction of $B_z$. Unexpectedly, ${\sigma }_{zzz,x}$ increasing with decreasing the LC order for $\eta \ge 0.008$, and ${\sigma }_{\mathit{yxx}}$ is nearly independent of η. The same behaviors are obtained for $\eta =0.008\sim 0.064$; see SI Appendix, section C (84). Both ${\sigma }_{zzz,x}$ and ${\sigma }_{\mathit{yxx}}$ significantly decrease if $v_{b,\mathit{k}}^{\mu \nu }$ in Eqs. 9 and 10 is replaced with ${\tilde{v}}_{b,\mathit{k}}^{\mu \nu }$, that is, if the qQM $g_{b,\mathit{k}}^{\mu \nu }$ is dropped. Therefore, giant nonreciprocal conductivities originate from the qQM term, which is drastically enlarged near the band crossing points.

Considering that the LC transition temperature $T_{\mathrm{LC}}$ reported in kagome metals is $35\sim 130$K, the magnitude of the order parameter at $T=0$ will be $|\mathit{\eta }|=0.005\sim 0.02$ [eV] in the case of $|\mathit{\eta }|/T_{\mathrm{LC}}\sim 2$. Therefore, giant eMChA observed in kagome metals is given by the resonantly magnified QM. In the SI Appendix, section C (84), we find that ${\sigma }_{zzz,x}$ and ${\sigma }_{\mathit{yxx}}$ in the v-LC state in Fig. 3 F and G are almost unchanged in the s-LC state. They are qualitatively unchanged if the $2\times 2$ LC and BO coexist. In the SI Appendix, section E (84), we explain that the experimentally observed giant eMChA coefficient ${\gamma }_{\mathrm{eM}}^{\exp }\sim {10}^{-11}$ [A⁻¹T⁻¹m²] under $B_{x}a\approx 18$ T (37) can be consistently understood by setting $\gamma \lesssim {10}^{-3}$ [eV] and $I_{\mathrm{st}}\sim 0.01$[eV].

Physically, the switchable eMChA in kagome metals is closely related to the uniform magnetization $M_{\mathrm{orb}}^0$ and anomalous Hall effect. Fig. 4A shows the $M_{\mathrm{orb}}^0$ derived from SI Appendix, Eq. S8 (84) per V-site in the triple-$\mathit{q}$ LC state with $\eta =0.008$, as functions of n for $I_{\mathrm{st}}=\varphi =0$. Note that $M_{\mathrm{orb}}^0\propto {\eta }^3$ for $\varphi =0$; see SI Appendix, section A (84). Thus, $|M_{\mathrm{orb}}^0|\ll {10}^{-3}[{\mu }_{\mathrm{B}}]$ is realized for $\eta \sim 0.01$. The anomalous Hall conductivity ${\sigma }_{\mathit{xy}}^{\mathrm{AHE}}$ is also shown in Fig. 4A. Both $M_{\mathrm{orb}}^0$ and ${\sigma }_{\mathit{xy}}^{\mathrm{AHE}}$ change their signs for $\eta \to -\eta$. Both quantities exhibit an n-dependence similar to that of the eMChA shown in Fig. 3F.

Fig. 4.

Relation between eMChA, anomalous Hall effect, and orbital magnetization. (A) $M_{\mathrm{orb}}^0$ per V site and ${\sigma }_{\mathit{xy}}^{\mathrm{AHE}}$ as functions of n in the v-LC state. (B) LC-induced local magnetic fields at the center of the hexagons, H and H₂ in Fig. 2B, derived from classical Biot–Savart law. (C) η-dependences of the eMChA, NLH, $M_{\mathrm{orb}}^0$ and ${\sigma }_{\mathit{xy}}^{\mathrm{AHE}}$ as functions of η at $n=4.8$, in the presence of small BO $\varphi =0.004$. Only eMChA exhibits a resonance-like enhancement for $\eta \sim 0.005$. (D) Jump in the eMChA, NLH, $M_{\mathrm{orb}}^0$ and AHE as function of $B_z$ below $min\{T_{\mathrm{LC}},T_{\mathrm{st}}\}$, where the sign of η is switched by the direction of $B_z$ while $|\eta |$ is fixed.

In Fig. 4B, we also present the LC-induced local magnetic field $b^{\mathrm{LC}}(\mathit{r})$ at the H and H₂ points indicated in Fig. 2B. Here, $b^{\mathrm{LC}}$ is obtained from the interbond ($i,j$) charge current $J_{i,j}$, which is calculated following ref. 86, by applying Biot–Savart law to giant unit-cell model ($N\sim 5,000$). We find that the relation $b^{\mathrm{LC}}(\mathrm{H})=-3b^{\mathrm{LC}}({\mathrm{H}}_2)$ holds approximately, but deviates slightly due to the breaking of global TRS. In fact, the absolute value of $J_{i,j}$ differs between the hexagon around H and that surrounding H₂ in Fig. 2B, even though $|\delta t_{i,j}^{\mathrm{c}}|$ is equal for all nearest bonds (see figure 3 of ref. 86). Thus, we obtain $|M_{\mathrm{orb}}^0|\lesssim {10}^{-4}$ [${\mu }_{\mathrm{B}}$] and $b^{\mathrm{LC}}$ is on the order of 1 [mT] for $\eta \sim 0.01$ at $n\sim n_{\exp }$. (Note that $b^{\mathrm{LC}}\propto \eta$ (86), while $M_{\mathrm{orb}}^0\propto {\eta }^3$ (33).) The smallness of the LC-induced magnetism is consistent with the recent measurements using polarized neutron scattering (87) and tuning fork resonator (14).

Now, we explain the drastic effect of η-odd QM on eMChA. Fig. 4C shows ${\sigma }_{zzz,x}$, ${\sigma }_{\mathit{yzz}}$, $M_{\mathrm{orb}}^0$, and ${\sigma }_{\mathit{xy}}^{\mathrm{AHE}}$ in an arbitrary unit as functions of η. We set $n=4.8$ to avoid the exceptional η-dependence of ${\sigma }_{\mathit{yxx}}$ at $n=n_{\exp }$ shown in Fig. 3F. Here, we introduce the small BO $\varphi =0.004$ to make the numerical calculation stable for $\eta \sim 0$. Importantly, only ${\sigma }_{zzz,x}$ exhibits the resonance-like singular behavior $\propto {\eta }^{-1}$ for $\eta \gtrsim 0.005$. This resonance-like behavior appears for a wide range of n. (Although the maximum of ${\sigma }_{zzz,x}$ at $\eta \equiv {\eta }_{\max }$ decreases with ϕ, the value of ${\eta }_{\max }\approx 0.005$ is essentially independent of ϕ; see SI Appendix, section D.)

As shown in Fig. 3 D and E, ${\sigma }_{zzz,x}^{-}(\mathit{k})\sim \sum _a{f}_0^{\prime }{G}_a^{zz,-}{v}_a^{y,+}\frac{1}{{\gamma }^3}$ is resonantly enhanced by η-odd QM term for small η. In contrast, the NLH does not exhibit the resonance-like behavior because the qQM $g_{a,\mathit{k}}^{\mu \nu }$ in ${\sigma }_{\mathit{yzz}}(\mathit{k})$ drastically changes its sign around the Fermi surfaces as functions of $(a,\mathit{k})$, and its contributions are canceled out upon the $\mathit{k}$ integration. In SI Appendix, section F (84), we demonstrate the relations $v_{\mathit{k}}^{zz,-}\approx g_{\mathit{k}}^{zz,-}$ and ${({(g^{\mathit{zz}})}^2)}_{\mathit{k}}^{-}\propto G_{\mathit{k}}^{zz,-}$. It is found that $v_{\mathit{k}}^{zz,-}$ exhibits sudden sign changes near the resonant position.

Many-Body Effects on the eMChA, Other eMChA Mechanisms

In this study, we neglected the many-body effects on the nonreciprocal conductivities. However, in general, transport phenomena due to the quasiparticles at the Fermi surface are modified by the Fermi liquid effects, such as the band-renormalization factor $z={(1-\partial \mathrm{\Sigma }/\partial ϵ{|}_{ϵ=0})}^{-1}(<1)$ and the current vertex corrections (60). Here, $\mathrm{\Sigma }(ϵ)$ is the self-energy. In the following, we discuss the effect of z on the eMChA. In Fermi liquid systems, the quasiparticle energy ${ϵ}_{\mathit{k}}$ is renormalized as ${ϵ}_{\mathit{k}}^{\ast }\equiv z{ϵ}_{\mathit{k}}$. Also, the quasiparticle damping rate γ and the velocity $v_{\mu }$ are renormalized as ${\gamma }^{\ast }\equiv z\gamma$ and $v_{\mu }^{\ast }\equiv {\partial }_{\mu }{ϵ}_{\mathit{k}}^{\ast }=z{v}_{\mu }$, respectively (60). By definition, ${\gamma =-\mathrm{Im}\mathrm{\Sigma }|}_{ϵ=+i0}$. It is noteworthy that the mean-free-path $v_{\mu }^{\ast }/2{\gamma }^{\ast }$ is independent of z. The band-renormalization modifies the recurrence formula of the nonequilibrium distribution function, which is explained in the SI Appendix, section C (84) as $f_{n+1}=-{\tau }^{\ast }{B}^{\ast }{f}_n$, where ${\tau }^{\ast }=1/2{\gamma }^{\ast }$ and $B^{\ast }=(-\mathit{E}-{\mathit{v}}_{\mathit{k}}^{\ast }\times \mathit{B})·{\mathbf{\nabla }}_{\mathit{k}}$. (Here, we drop both ${\mathit{m}}_{\mathit{k}}$ and ${\mathbf{\Omega }}_{\mathit{k}}$ because we are interested in the most divergent terms with respect to τ. Note that both ${\mathit{m}}_{\mathit{k}}$ and ${\mathbf{\Omega }}_{\mathit{k}}$ are independent of z because the band-energy splitting in their denominators is normalized by z.)

First, we discuss how the formulae of ${\sigma }_{\alpha \alpha }$, ${\sigma }_{\alpha \beta \beta }$, and ${\sigma }_{\alpha \beta \beta ,\gamma }$ given in Eqs. 8–10 are modified, by applying the renormalized quantities $v_{\mu }^{\ast }$, ${\gamma }^{\ast }$ and ${ϵ}_{\mathit{k}}^{\ast }$ in $f_0^{\prime }$. ($v_{\alpha \beta }$ is also renormalized as $v_{\alpha \beta }^{\ast }\equiv z{v}_{\alpha \beta }$ because the qQM $g_a^{\alpha \beta }$ in Eq. 11 is renormalized as ${g_a^{\alpha \beta }}^{\ast }\equiv z{g}_a^{\alpha \beta }$.) Since $f_0^{\prime }({ϵ}_{\mathit{k}}^{\ast })=z^{-1}{f}_0^{\prime }({ϵ}_{\mathit{k}})$ at low temperatures, we obtain the conductivities for general z as follows:

$$\begin{array}{cc}\hfill {\sigma }_{\alpha \alpha }(z)& \hfill =\mathrm{Eq}.\mathbf{8},\hfill \end{array}$$ [13]

$$\begin{array}{cc}\hfill {\sigma }_{\alpha \beta \beta }(z)& \hfill =z^{-1}\times \mathrm{Eq}.\mathbf{9},\hfill \end{array}$$ [14]

$$\begin{array}{cc}\hfill {\sigma }_{\alpha \beta \beta ,\gamma }(z)& \hfill =z^{-1}\times \mathrm{Eq}.\mathbf{1}0.\hfill \end{array}$$ [15]

Therefore, both NLH and eMChA conductivities are strongly magnified in strongly correlated electron systems with $z^{-1}\gg 1$. In kagome metals, $z^{-1}$ is just ∼2 according to ARPES measurements (7). In contrast, $z^{-1}$ exceeds 1,000 in some heavy fermion systems. Therefore, strongly correlated electron systems with $z^{-1}\gg 1$ provide a promising platform of the nonreciprocal transport phenomena.

In the present intrinsic eMChA mechanism given by Eq. 10, the giant ${\tau }^3$-order conductivity arises from Bloch electrons on the Fermi surface. Due to the factor $f_0^{\prime }$ in Eq. 10, the band structure away from the Fermi level is not essential. In kagome metals, the relationship $\tau v_{\mathrm{F}}\gg a_0$ is satisfied at low temperatures. In this case, the present ${\tau }^3$-order mechanism dominates over both the ${\tau }^2$-order Berry curvature dipole mechanism (see the discussion in SI Appendix, section B) and the τ-linear QM dipole mechanism.

Recently, the extrinsic eMChA mechanism due to impurity skew scattering was analyzed in ref. 88, where the Lorentz force leads to a ${\tau }^3$-order eMChA conductivity, similar to the present intrinsic case. Given the pronounced enhancement mechanism driven by the QM revealed in this work, and the fact that the skew scattering mechanism involves the factor $\tau /{\tau }_{\mathrm{sk}}$ (${\tau }_{\mathrm{sk}}$: the skew scattering time) which is usually small, the intrinsic eMChA mechanism is expected to provide the dominant contribution in kagome metals. (In contrast, for the AHE, the skew scattering term ($\propto {\tau }^1$) can dominate over the intrinsic term ($\propto {\tau }^0$) in ultraclean systems.) The extrinsic eMChA mechanism in kagome metals warrants future study.

Summary and Discussions

In this paper, we have identified the microscopic mechanism underlying the giant eMChA originating from chiral LC in kagome metals, based on the derived exact formulas for eMChA and NLH with respect to the dominant terms for $\tau v_{\mathrm{F}}\gg 1$. Under the chiral LC order η, significant eMChA conductivity ($\propto \eta {\tau }^3{I}_{\mathrm{st}}$) and NLH one ($\propto \eta {\tau }^2{I}_{\mathrm{st}}$) manifest in the presence of $2a_0$ or $4a_0$ stripe CDW that is universally observed in CsV₃Sb₅. The eMChA originates from the TRS breaking dipole potential ${ϵ}_{\mathit{k}}^{-}\propto k_y\eta I_{\mathrm{st}}$. Notably, eMChA experiences a dramatic amplification (by ∼100 times) driven by the QM mechanism, arising from band reconstruction in the LC phase for a broad parameter range. This mechanism leads to substantial eMChA even when the LC order η is small (approximately $T_{\mathrm{LC}}=30\sim 100\mathrm{K}$), underscoring the critical role of quantum geometry in kagome metals.

The achievements of this study are summarized as follows: i) Uncovering the mechanism behind the giant and field-tunable eMChA (37) driven by the chiral loop-currents in kagome metals. ii) Identification of a QM mechanism responsible for giant eMChA, highlighting the importance of quantum geometry in kagome metals. iii) Development of nonreciprocal transport theory utilizing the QM in strongly correlated metals, like various kagome metals, chiral magnets (52, 53), and loop-current phase of cuprate superconductors (33).

Fig. 4D depicts the schematic picture of the present eMChA mechanism under the chiral LC state. At a fixed $T(<T_{\mathrm{LC}})$, the LC order η is fixed to $+{\eta }_0$ or $-{\eta }_0$, and its sign is switched by $B_z$ via the orbital Zeeman energy $-N_{\mathrm{V}}{M}_{\mathrm{orb}}{B}_z$, where $N_{\mathrm{V}}$ is the number of V-sites in a domain. This fact leads to the sign reversal of the NLH and eMChA by tiny $B_z$ below $min\{T_{\mathrm{LC}},T_{\mathrm{st}}\}$, where $T_{\mathrm{st}}$ is the stripe CDW transition temperature. This result is consistent with the experimental report of the eMChA in ref. 37. Importantly, in ref. 89, no eMChA is observed in KV₃Sb₅, in which the triple-$\mathit{q}$ LC emerges while no stripe CDW is observed experimentally. This observation is naturally explained by the present theory.

Here, we briefly discuss experimental evidence of the TRS breaking, IS breaking, and nematicity. Recent transport study of CsV₃Sb₅ microsample (34) reveals that small magnetic field $h_z(<10\mathrm{T})$ or small strain strongly magnifies the LC that accompanies the nematicity. The enhancement of the LC under $h_z$ is also observed by μSR (30, 32) and field-tuned chiral transport (37) measurements. The nematicity in the LC phase has been also reported by the STM study (10). These experiments are naturally explained based on the GL free-energy theory (29). Notably, a recent magnetotropic susceptibility study reveals a small but heavily anisotropic $M_{\mathrm{orb}}^0$ with tiny hysteresis, consistent with its chiral LC origin (14). Furthermore, the superconducting diode effect under $h_z=0$ (90) and the second-order optical response (91) strongly indicate the IS breaking in both normal and superconducting states. The present study paves the way toward resolving these outstanding problems in kagome metals.

This study shows that the experimentally observed stripe CDW (81–83) acts as a trigger for IS breaking in the LC phase. However, the origin of the stripe CDW, as well as its appropriate order parameter and strength, remains to be clarified in future studies. In addition, it would be valuable to study other potential density-wave states or rotation lattice distortions (92) that produce similar effects to those of the stripe CDW. As an interesting direction for future research, we briefly comment on a different kagome metal with double-layer structure ScV₆Sn₆, which exhibits the $\sqrt{3}\times \sqrt{3}$ CDW (93). It is expected that the CDW originates from the flat phonon modes with Sn vibrations (94, 95). Interestingly, ScV₆Sn₆ exhibits the TRS breaking in the CDW phase (96). The LC-induced QM effects in different kagome metals would be an interesting future problem.

Materials and Methods

Two-Dimensional d_xz-Orbital Kagome Lattice Model with LC+BO

First, we consider the original kagome lattice model with three sublattices A, B, and C. Fig. 1C shows the Fermi surface in the original BZ. The three nesting vectors are given as ${\mathit{q}}_1={\mathit{a}}_{\mathrm{AB}}\times \widehat{z}/S$, ${\mathit{q}}_2={\mathit{a}}_{\mathrm{BC}}\times \widehat{z}/S$, and ${\mathit{q}}_3={\mathit{a}}_{\mathrm{CA}}\times \widehat{z}/S$. Here, $S=\sqrt{3}{a}_0^2/2\pi$ is the area of the original unit cell divided by π, where $a_0=2|{\mathit{a}}_{\mathrm{AB}}|$. For example, ${\mathit{q}}_1$ is given by the vector $\overrightarrow{M}$ in Fig. 1C. These three vectors ${\mathit{q}}_n$ ($n=1,2,3$) are moved to the same Γ point in the folded BZ for the $2\times 2$ density wave.

In the main text, we introduced the form factor for the $2\times 2$ LC order, defined as the normalized $\delta t_{\mathit{ij}}^{\mathrm{c}}$. We introduce the form factor at the wavevector $\mathit{q}={\mathit{q}}_n$ by $f_{\mathit{ij}}^{(n)}$, only for the nearest-neighbor bonds ($i,j$) for simplicity. Then, $f_{\mathit{ij}}^{(1)}$ is finite only for the nearest-neighbor AB bonds: $f_{\mathit{ij}}^{(1)}=+i$ when the sites $(i,j)$ belongs to the sublattices $(l,m)=(1,2),(2,4),(4,5),(5,1)$, and $=-i$ for $(7,8),(8,10),(10,11),(11,7)$. Importantly, the odd parity relation $f_{\mathit{ij}}^{(1)}=-f_{\mathit{ji}}^{(1)}$ holds for the LC. Other form factors with ${\mathit{q}}_2$ and ${\mathit{q}}_3$, $f_{\mathit{ij}}^{(2)}$ and $f_{\mathit{ij}}^{(3)}$, are also derived from Fig. 1A. Using the short notation ${\mathit{f}}_{\mathit{ij}}=(f_{\mathit{ij}}^{(1)},f_{\mathit{ij}}^{(2)},f_{\mathit{ij}}^{(3)})$, the LC form factor is given as $\delta t_{\mathit{ij}}^{\mathrm{c}}=\mathit{\eta }·{\mathit{f}}_{\mathit{ij}}$. Here, $\mathit{\eta }\equiv ({\eta }_1,{\eta }_2,{\eta }_3)$ is the set of the current order parameters with the wavevector ${\mathit{q}}_n$ ($n=1,2,3$). Note that the LC form factors are Hermitian $\delta t_{\mathit{ij}}={(\delta t_{\mathit{ji}})}^{\ast }$, which leads to the relation $f_{\mathit{q}}^{\mathit{lm}}(\mathit{k})={(f_{-\mathit{q}}^{\mathit{ml}}(\mathit{k}+\mathit{q}))}^{\ast }$ (5).

Also, the BO form factor shown in Fig. 1B is given as $\delta t_{\mathit{ij}}^{\mathrm{b}}=\mathit{\varphi }·{\mathit{g}}_{\mathit{ij}}$, where $\mathit{\varphi }\equiv ({\varphi }_1,{\varphi }_2,{\varphi }_3)$ is the set of the BO parameters with the wavevector ${\mathit{q}}_n$ ($n=1,2,3$), and $g_{\mathit{ij}}^{(n)}=g_{\mathit{ji}}^{(n)}=+1$ or $-1$ as shown in Fig. 1B. For example, $g_{\mathit{ij}}^{(1)}=+1[-1]$ for the sites $(i,j)$ that belong to the sublattices $(1,2),(4,5),(8,10),(11,7)$ [$(2,4),(5,1),(7,8),(10,11)$]. Note that $g_{\mathit{ij}}^{(1)}$ is finite only for the nearest-neighbor AB bonds.

Inversion Symmetry Breaking in 3D Kagome Lattice Model

In the original 3D crystal structure shown in Fig. 5A, the IS centers are located at A, B, C, H (filled square symbols), and A^′, B^′, C^′, H^′ (open square symbols). Here, X^′ are the middle points between two layers. Fig. 5B shows a v-LC state with $\mathit{\theta }=(0,0,0)$. Here, only the IS centers H and H^′ survive as shown in Fig. 5B. In Fig. 5C, we introduce the $2a_0$ stripe CDW. Then, none of the IS centers remains as in Fig. 5C. Then, we find that the eMChA becomes finite. Fig. 5D shows the s-LC state with $\mathit{\theta }=(0,\pi ,\pi )$. Here, the triple-$\mathit{q}$ LC on the even-number layers and that on the odd-number layers are shifted by the vector ${\mathit{a}}_{\mathrm{AB}}$. In this $2\times 2\times 2$ s-LC state, only the IS center C^′ survives as shown in Fig. 5D. By introducing the $2a_0$ stripe CDW along AB direction, none of the IS centers remain as in Fig. 5E. Then, we find that the eMChA becomes finite. Note that the IS center C^′ remains intact if the $2a_0$ stripe CDW is along BC or CA directions, and therefore the eMChA disappears.

To summarize, by introducing the $2a_0$ stripe CDW that is generally observed in Cs-based compounds, IS is broken in both the v-LC state in Fig. 5C ($\mathit{\theta }=(0,0,0)$) and the s-LC state in Fig. 5E ($\mathit{\theta }=(0,\pi ,\pi )$). In both cases, finite eMChA is expected to be realized. (In contrast, the alternative v-LC state with $\mathit{\theta }=(\pi ,\pi ,\pi )$ preserves IS even after introducing the stripe, so the eMChA vanishes.) It is noteworthy that IS in the single-$\mathit{q}$ LC state is broken by introducing the stripe CDW that is parallel to the single-$\mathit{q}$ LC order. Therefore, the eMChA can appear even in the single-$\mathit{q}$ LC state.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Numerical data have been deposited in eMChA_numerical-data20250216 at https://doi.org/10.6084/m9.figshare.29320406 (97).