Charge-loop current order and Z3 nematicity mediated by bond order fluctuations in kagome metals

Recent experiments on geometrically frustrated kagome metal AV3Sb5 (A = K, Rb, Cs) have revealed the emergence of the charge loop current (cLC) order near the bond order (BO) phase. However, the origin of the cLC and its interplay with other phases have been uncovered. Here, we propose a novel mechanism of the cLC state, by focusing on the BO phase common in kagome metals. The BO fluctuations in kagome metals, which emerges due to the Coulomb interaction and the electron-phonon coupling, mediate the odd-parity particle-hole condensation that gives rise to the topological current order. Furthermore, the predicted cLC+BO phase gives rise to the Z3-nematic state in addition to the giant anomalous Hall effect. The present theory predicts the close relationship between the cLC, the BO, and the nematicity, which is significant to understand the cascade of quantum electron states in kagome metals. The present scenario provides a natural understanding.

More recently, the non-trivial time reversal symmetry breaking (TRSB) order at T TRSB attracts considerable attention.The magneto-chiral anisotropy observed by STM [6,14] strongly indicates the emergence of the chiral charge-loop-current (cLC) inside the BO phase.The Kerr rotation measurements suggest the appearance of the TRSB state at T ∼ T BO .On the other hand, the TRSB amplitude grows prominent at T ≈ 35 K < T BO according to the muon spin relaxation (µSR) studies [15][16][17] and the field-tuned chiral transport study [14].The chiral cLC is driven by the additional odd-parity hopping integral δt c ij (=imaginary), and the accompanied topological charge-current [18] give the giant anomalous Hall effect (AHE) below T ≈ 35 K [19,20].The correlation driven topological phase in kagome metals is very unique, and its mechanism should be revealed.
The cascade of rich quantum phases in kagome metals gives rise to various exotic electronic states.A notable example is the emergent nematic (C 2 ) order inside the BO and the cLC phases, which is clearly observed by the elastoresistance, [21] the scanning birefringence [17], and the STM [7] studies.In addition, nematic SC states have been reported [22,23].Thus, kagome metals provide a promising platform for exploring the interplay between the electron correlations and the topology.
Microscopically, the BO and the cLC are the non-local particle-hole (ph) condensation [8][9][10][11][12][13][24][25][26][27][28][29][30][31][32][33][34][35][36].Thus, the emergence of the BO/cLC indicates the existence of the large non-local effective interaction in kagome metals.It is given by the off-site Coulomb interaction V within the mean-field approximation (MFA) [10,11,37,38], while the necessary model parameters are severe.Recently, sizable off-site interaction is found to be induced by the beyond-MFA effects [32][33][34][35][36], and the realized BO fluctuations mediate non-BCS SC states in Kagome metals [13] and other strongly correlated metals [27][28][29][30][31][32]35].Now, it is interesting to investigate the BO-fluctuation-mediated cLC state because this idea would lead to a unified understanding of the cascade of rich quantum phases.In this paper, we propose a cLC mechanism driven by the developed BO fluctuations in kagome metals.It is FIG.2: (a) Expression of χ 0,lmm ′ l ′ g (q).(b) q-dependence of χ 0,ABAB g (q, 0).revealed that the quantum BO fluctuations in metals mediate the cLC order parameter (=imaginary δt c ij ).This cLC mechanism is universal because it is irrelevant to the origin of the BO.Furthermore, we discover that the coexistence of the BO and the cLC order give rise to the novel Z 3 nematicity along the three lattice directions reported in Refs.[7,17,21].The present theory reveals the close relationship between the cLC, BO, nematicity and SC state, which is significant to understand the unsolved quantum phase transitions in kagome metals.
Here, we introduce the kagome-lattice tight-binding model with b 3g -orbitals of the vanadium sites (A,B,C) shown in Fig. 1 (a).The kinetic term is given by Ĥ0 , where l, m denote the sublattices A,B,C, and h 0 lm (k) (= h 0 ml (k) * ) is the Fourier transform of the nearest-neighbor hopping integral t (= 0.5eV).Here, we set the unit of energy eV.The temperature is set to T = 0.05 in the numerical calculation.The Fermi surface (FS) at n = 0.875 per site is shown in Fig. 1 (c).The FS is close to the three van-Hove singular (vHS) points (k A , k B , k C ), each of which is composed of a single b 3g -orbital.This simple three-orbital model well captures the main pure-type FS in kagome metals [6,[39][40][41][42][43].The wavevectors of the BO correspond to the inter-sublattice nesting vectors q n (n = 1, 2, 3) in Fig. 1 (c).(The equivalent square lattice kagome model is convenient for the numerical study; see SM A [44].) Based on the kagome-lattice Hubbard model with onsite U , the experimental BO at q = q n is derived from the paramagnon-interference mechanism [13].Figure 1 (d) shows the corresponding hopping modulation along the A-C direction due to the BO, The obtained BO is mainly given by the nearest-neighbor components, and it exhibits the staggered pattern δt b AC (R) = −δt b AC (−R).The Fourier transform of the BO modulation, δt b ij , gives the even-parity BO form factor g lm q (k): [36,45]: where q is the wavevector of the BO.Here, we use the BO form factor derived in Ref. [13].
Below, we study the cLC mechanism due to the BO fluctuations.For this purpose, we introduce the following effective BO interaction to derive the BO susceptibility: where Ôg q ≡ k,l,m,σ g lm q (k)c † k+q,l,σ c k,m,σ is the BO operator [36,45,46].Here, v is the effective interaction, and the form factor g lm q (k) is normalized as max k,l,m |g lm q (k)| = 1 at each q.The interaction (2) would originate from the combination of (i) the paramagnon-interference due to on-site U [13], (ii) the Fock term of off-site V [11], and (iii) the bond-stretching phonon [47].As for (iii), g lm q (k) is given by the hopping modulation due to the stretching mode and v = 2η 2 /ω D , where η is the electron-phonon coupling constant and ω D is the phonon energy at q ≈ q n .As for (ii), v = 2V as we explain in the SM B [44].Thus, the effective interaction (2) is general.Possible driving force of the BO has been discussed experimentally [48,49].
Next, we calculate the BO susceptibility per spin given as χ g (q, ω l ) ≡ 1 2 , where ω l is boson Matsubara frequency.By applying the randomphase approximation (RPA), χ g (q, ω l ) is obtained as where the notation q ≡ (q, ω l ) is used.The BO irreducible susceptibility χ 0 g (q) is given by [36,45] where ǫ n is a fermion Matsubara frequency.Its diagrammatic expression and the numerical result are shown in Figs. 2 (a) and (b), respectively.Note that the spin susceptibility is not changed by v. Now, we explain the BO fluctuation-mediated cLC order.Although the cLC instability is smaller than others in the MFA as shown in SM B [44], its instability is strongly magnified by the beyond-MFA effects.In twodimensional metals, in general, spin-or charge-channel fluctuations induce prominent beyond-MFA effects that is called the vertex corrections [27][28][29][30][31][32][33][34][35][36].As revealed in Ref. [50], the Maki-Thompson (MT) vertex correction due to the spin-fluctuations induces the cLC state in a frustrated Hubbard model based on the functionalrenormalization-group theory.In kagome metals, instead of spin fluctuations, BO fluctuations are expected to mediate the cLC state.Therefore, we solve the following linearized DW equation [32,50,51] by considering the BO fluctuations in Eq. ( 3), and obtain the odd-parity cLC solution δt c ij = −δt c ji : where I L,M q (k, p) is the electron-hole pairing interaction due to the MT term.L ≡ (l, l ′ ) and M i represent the pair of sublattice indices.λ q is the eigenvalue that represents the instability of the DW at wavevector q, and max q {λ q } is unity at the transition temperature.f L q (k) is the Hermitian form factor that is proportional to the ph condensation or, equivalently, the symmetry breaking component in the self-energy.[26,27,31,32,52].The kernel function due to the single exchange of the BO fluctuations, I BO ∝ χ g , is given as which is diagrammatically expressed in Fig. 3 (a).In addition, we include the Hartree term due to U , . This term leads to the suppression of the net charge order, while the BO and cLC are not suppressed at all.Here, the kernel function is given by I = I BO + I U , and we set v = 1.53 and U = 1.27.Note that the coefficient y in Eq. ( 7) depends on the origin of the BO: y = 1/2 in the phonon mechanism, while y = 2 in the Fock term of off-site V because both charge-and three spin-channel BO fluctuations develop, as explained in SM B [44]. y ≫ 1 can be realized in the paramagnoninterference mechanism [13].Because we are interested in a general argument, we simply set y = 1 below.
Figure 3 (b) shows the q-dependence of the largest (second-largest) eigenvalue λ q by solving the DW equation in red solid (blue dashed) line.The largest eigenvalue exhibits the maximum value at q = q n (n = 1, 2, 3), and its form factor is odd-parity: f lm q (k − q/2) = −f ml q (−k − q/2).Then, the corresponding real-space hopping modulation is odd-parity δt c ij = −δt c ji and pure imaginary when δt c ij is Hermitian.The obtained δt c AC (R) ≡ δt c iAjC for the cLC at q = q 3 along the A-C direction is shown in Fig. 3 (c), where R ≡ a CA •(r C i −r A j ) is odd-integer.In addition, the odd-parity relation The obtained charge loop current pattern for the 3Q state is depicted in Fig. 3 (d).
Note that the second-largest eigenvalue at q = q n in Fig. 3 (b) corresponds to the charge order (n A = n B ), originating from the flatness of the FS between two vHS points in this simplified model in Fig. 1 (c).This instability is suppressed by U due to its Hartree term.Also, we discuss that the Aslamazov-Larkin (AL) term is unimportant in the SM C [44].
Here, we discuss why the cLC order is mediated by the BO fluctuations.Figures 4 (a) and (b) show the firstand the second-order scattering processes derived from the DW equation ( 6) at q = q 3 , respectively.The former umklapp term gives the repulsive interaction that leads to f AC q3 = −f CA q3 , and the latter backward term gives the attraction among the same f lm q3 .Thus, both interactions cooperatively induce the odd-parity current order form factor in Fig. 3 (c).The infinite series of such ladder diagrams are calculated by solving Eq. ( 6), and the obtained v-dependence of the cLC eigenvalue is shown in Fig. 4 (c).A schematic BO+cLC phase diagram derived from the present theory is depicted in Fig. 4 (d).This phase diagram is reminiscent of that of the d-wave SC state around the antiferromagnetic endpoint.
In the next stage, we discuss the electronic nematic states due to the coexistence of the BO (δt b ij ) in Fig. 1  (b) and the cLC order (δt c ij ) in Fig. 3 (d).Here, we set 05.In Fig. 5 (a), the hexagonal loop current pattern in the cLC coincides with the hexagonal bond pattern in the BO.Its folded FS has C 6 symmetry as shown in Fig. 5 (b).In Fig. 5 (c), we shift the cLC pattern by a BA .Then, the center of C 6 symmetry in the BO and that of the cLC do not coincide, and the realized FS has C 2 symmetry as depicted in Fig. 5 (d).This nematicity is rotated by 120 (240) degrees by displacing the cLC pattern by a CA (a BC ).Thus, the coexistence of the BO and cLC leads to the Z 3 nematic order.This is consistent with the nematic transition at T nem ≈ 35 K observed by the nematic susceptibility measurement [21].Here, we briefly discuss the realized phase diagram based on the Ginzburg-Landau (GL) free energy.(i) When the BO solely occurs, the 3Q BO state is more stable than the 1Q one due to the third-order GL term [12,13,53].(ii) When the cLC order occurs inside the 3Q BO state (T < T cLC < T BO ), the nematic BO+cLC state in Fig. 5 (c) is more stable than the hexagonal BO+cLC state (= Fig. 5 (a)) and the (1Q cLC)+(3Q BO) state due to the third-order GL term, as we discuss in the SM D [44].(iii) When the cLC solely occurs, the third-order GL term vanishes, and the 3Q (1Q) cLC order emerges when r ≡ d 2,b /2d 2,a < 1 (r > 1), where d 2,a , d 2,b are the forth order GL parameters [44].Note that the 1Q cLC order corresponds to a nematic state.
We note that the nematic state is also caused by the 3Q state composed of the three-dimensional (3D) BO at q 3D n with q 3D 1,z = q 3D 2,z = π and q 3D 3,z = 0, as discussed in Ref. [12]: This nematic 3D-BO state is different from the present TRSB nematic BO+cLC state.These two different nematic states would be realized at different temperatures.Finally, we discuss the transport phenomena that originate from the cLC [18,54].Using the general expression of the intrinsic conductivity [55][56][57][58][59], we calculate the Hall conductivity (σ xy and σ yx ) due to the Fermi-surface contribution in the BO+cLC state.The expression is When γ ≫ ∆, in contrast, σ H decreases with γ in proportion to γ −2 .This crossover behavior is universal in the intrinsic Hall effect, which was first revealed in heavy fermion systems [55], and found to be universal in later studies [56][57][58][59][60].Note that 1 2 (σ xy + σ yx ) is nonzero in the nematic state.To understand the origin of the intrinsic Hall effect, we plot A H (k) ≡ (A xy (k) − A yx (k))/2 at γ = 0.05 in Fig. 6 (b): It shows a large positive value mainly around the vHS points, due to the band-hybridization induced by the cLC order.The obtained σ H ∼ 1 corresponds to 4 × 10 3 Ω −1 cm −1 because the interlayer spacing is ∼ 0.6nm.Thus, giant AHE σ H ∼ 10 2 Ω −1 cm −1 reported in Refs.[19,20] is understood in this theory.
In summary, we proposed a cLC mechanism mediated by the BO fluctuations in kagome metals.This mechanism is universal because it is independent of the origin of the BO.Furthermore, we revealed that novel Z 3 nematicity emerges under the coexistence of the cLC and the BO reported in Refs.[7,17,21] in addition to the giant AHE [19,20].This theory presents a promising scenario for understanding the BO, the cLC and the nematicity in kagome metals in a unified way.The analysis of the kagome-lattice U -V Hubbard model based on the mean-field theory is presented in Section SF of Ref. [13].The charge (spin) channel eigenvalue λ c(s) in the mean-field theory is given by solving the linearized DW equation with the Hartree-Fock kernel function made of U and V .Its diagrammatic expression for V is shown in Fig. S2 (a).
Figure S2 (b) shows the obtained several largest eigenvalues, λ s SDW and λ c X (X=CDW, BO, cLC), as functions of V /U at U = 0.79 [13].These eigenvalues linearly increase with respect to U and V at a fixed V /U .When V /U ≪ 1, a simple SDW order (f = 1) at q ≈ 0 is realized for U ∼ 1.6.It originates from the Hartree term of U .When V /U ≫ 1, on the other hand, simple chargedensity-wave (CDW) order (f = 1) at q ≈ q n is realized.In this model, q 1 = (π, 0), q 2 = (π, π), q 3 = (0, π).It is caused by the Hartree term of V .For V /U = 0.4 ∼ 0.65, the BO is realized by the Fock term of V .Note that the non-local BO is not suppressed by U , while the simple CDW order due to the Hartree term of V is strongly suppressed by U .However, the cLC instability is smaller than other instabilities within the Hartree-Fock approximation.
The form factors of the BO between the nearest sites in the square kagome-lattice model in Fig. S1 (a which are normalized as max q {b lm (q)} = 1.Here, b lm (q) = b ml (q) * and b ll (q) = 0.
B-2: Effective interaction due to BO susceptibility In Fig. S2 (b), we found the development of the BO instability within the mean-field approximation.Next, we derive the effective interaction mediated by the BO susceptibility.The final result is given in Eq. (S15), which is essentially equivalent to Eq. ( 7) derived in the main text. in the BO susceptibility R ≡ χ b AB,BA (q 1 )/χ b AB,AB (q 1 ) is of order unity, when the BO Stoner factor α BO = 2V (χ 0b AB,AB (q 1 ) + χ 0b AB,BA (q 1 )) is close to unity.In fact, for q ≈ q 1 , the relation between the BO susceptibility and its irreducible susceptibility is χb = χb0 + 2V χb0 χb , where χb0 is the 2 × 2 matrix: χb0 Then, the BO susceptibility for q ≈ q 1 is obtained as where a ≡ χ 0b AB,AB (q), b ≡ χ 0b AB,BA (q), and According to Eqs. (S13) and (S14), R becomes b/a(≪ 1) when V = 0.In contrast, we obtain R ≈ 1 at α BO ≈ 1 (α BO = (a + b)2V ). Figure S3 shows the BO susceptibilities χ b AB,AB and χ b AB,BA as functions of V .We set a = χ b0 AB,AB (q 1 ) = 0.
where χ b (q) ≡ χ b AB,AB (q)+χ b AB,BA (q) ≈ χ b0 AB,AB (q)/(1− α BO (q)) for q ≈ q 1 , and g lm ≈ b lm is the normalized BO form factor derived from the DW equation [13].Equation (S15) is equal to Eq. ( 7) in the main text with y = 2.Note that both charge-and three spin-channel BO fluctuations develop in the present off-site V mechanism.
To summarize, the relation χ b AB,AB ≈ χ b AB,BA (i.e., R ≈ 1), which is assumed in the cLC mechanism in the main text, is well satisfied in the off-site V mechanism.Because this relation is also satisfied in the paramagnoninterference mechanism [13] and the phonon mechanism, these three different BO mechanisms will cooperate.We consider that the main mechanism of the BO in kagome metals is the paramagnon interference mechanism [13], and both the bond-stretching phonon mode and the offsite Coulomb interaction will assist the BO formation.

C: Effects of AL-type VCs
Here, we examine the role of the Aslamazov-Larkin (AL) vertex corrections (VCs) due to the interference between two bosonic susceptibilities (χ boson ) shown in Fig. S4.The AL terms are significant for the even-parity order parameter in the paramagnon-interference mechanism.This mechanism is responsible for the BO and the orbital order in Fe-based superconductors [27,31,32], high-T c cuprates [29,30,51,51], and kagome metals [13].In contrast, the AL term is unimportant for the oddparity order parameter, and instead, the Maki-Thompson (MT) term is significant for the current order in the frustrated Hubbard models [50] and non-Fermi liquid transport phenomena [59].
Here, we explain that the AL terms due to the bond-susceptibilities, which were neglected in the main text, are unimportant in the present cLC mechanism in kagome metals.Figure S4 exhibits the VC for f AB q1 (k) at k ≈ k A .These terms are almost canceled for the odd-parity cLC order because of the relation f AB q1 (k) = −f BA q1 (−k − q).In fact, the order parameter in the real space satisfies the relation δt ij = Pδt ji , where P = +1 (−1) for the even (odd) parity order and l, m = A, B, C.Then, its Fourier transform gives the form factor: In the main text, we explained that the coexistence of the 3Q BO and the 3Q cLC leads to one C 6 state and

FIG. 3 :
FIG. 3: (a) DW equation due to the single exchange term of χg.(b) q-dependent eigenvalue of the DW equation.Redsolid (blue-dashed) line denotes the largest (second-largest) eigenvalue, and both show peaks at q = qn.(c) Imaginary hopping modulation Imδt c AC (R).Its triple-q order gives the cLC pattern in (d).One can check that the clock-wise (anticlock-wise) loop currents on hexagons (triangles) in (d) are inverted and moved by aAC under the sign change of f AC q 3 .
FIG. 4: (a) The first-and (b) the second-order scattering processes with respect to χg.The repulsion (attraction) due to the odd-(even-) order terms cause the cLC ph pairs.(c) v-dependence of the eigenvalue of cLC λq 3 .(d) Schematic phase diagram predicted in the present study.The cLC is mediated by the BO fluctuations near the BO-endpoint.

FIG. 5 :
FIG. 5: (a) C6-symmetric BO+cLC state in real space and (b) its folded FS.The folded Brillouin zone (BZ) is shown by dotted lines.(c) Nematic BO+cLC state with the director parallel to aBA and (d) its nematic FS.Here, we changed the sign of f AB q 1 in (a).The nematic state parallel to a lm appears by changing the sign of f lm qn in (a).Thus, the Z3 nematic state is realized.Here, we use large |δt b,c ij | (= 0.05) to exaggerate the nematicity.

1 [
Supplementary Materials] Charge-loop current order and Z 3 -nematicity mediated by bond-order fluctuations in kagome metal AV 3 Sb 5 (A=Cs,Rb,K) Rina Tazai, Youichi Yamakawa, and Hiroshi Kontani Department of Physics, Nagoya University, Nagoya 464-8602, Japan A: Square-lattice kagome model In Ref. [13], the present authors found that the paramagnon-interference theory naturally explains the bond-order (BO) on the basis of the kagome-lattice Hubbard model.The used lattice structure with square unit cell and its Fermi surface (FS) are shown in Figs.S1 (a) and (b), respectively.The van-Hove singular point at k = k X (X=A,B,C) is composed of the X-sublattice orbitals.The form factor of the 3Q BO and that of the 3Q cLC are shown in Figs.S1 (c) and (d), respectively.
FIG. S1: (a) Square-lattice kagome metal model that is convenient for the numerical study.(b) Obtained FS in the square Brillouin zone.(c) Form factor of the 3Q BO in real space.The red bonds represent the Tri-Hexagonal pattern.(d) Form factor of the 3Q cLC.The coexistence of (c) and (d) leads to the C6 symmetry state.
35 and b/a = 0.2.Both χ b AB,AB and χ b AB,BA increase with V , while ∆χ b ≡ χ b AB,AB − χ b AB,BA is almost constant.This result means that the relation R ≈ 1 holds near the BO-endpoint, around which the relation α BO 1 holds.According to Eqs. (S10) and (S12), in the case of R ≈ 1, the MT kernel function for the DW equation is simply given as

j(
δt ij e −k•(ri−rj ) e −q•rj Pδt ji )e −k•(ri−rj ) e −q•rj = Pf ml q (−k − q).(S16)In addition, it is verified that each AL term in Fig.S4is small because the momentum summation is restricted by four g's.(Note that |g lm q (k)| ≤ 1.) D: Stability of the nematic BO+cLC state