^{†}

^{†}thanks: These authors contributed equally to this work.

^{†}

^{†}thanks: These authors contributed equally to this work.

Jyotirmoy SauDepartment of Condensed Matter and Materials Physics,S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India Sourav ChakrabortyDepartment of Condensed Matter and Materials Physics,S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India Sourabh SahaDepartment of Condensed Matter and Materials Physics,S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India Kalpataru Pradhankalpataru.pradhan@saha.ac.inTheory Division, Saha Institute of Nuclear Physics,A CI of Homi Bhabha National Institute, Kolkata-700064, India Anamitra Mukherjeeanamitra@niser.ac.inSchool of Physical Sciences, National Institute of Science Education and Research,a CI of Homi Bhabha National Institute, Jatni 752050, India Manoranjan Kumarmanoranjan.kumar@bose.res.inDepartment of Condensed Matter and Materials Physics,S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata 700106, India

###### Abstract

Understanding the interplay of strong correlation and temperature in nodal-line semimetals can offer novel ways to control spin currents. Here we consider the 3d-5d double-perovskite Ba_{2}CoWO_{6}, which features mirror-symmetry-protected nodal-lines, strong Co-site interactions, and spin-orbit coupling (SOC) at W sites. Our first principles and exact diagonalization results reveal a half-metallic ground state with high-spin Co and topologically non-trivial bands. We demonstrate that SOC gaps out nodal points, causes band-inversion and generates anomalous Hall response. A semi-classical Monte Carlo finite-temperature simulation of five-orbital Hubbard model uncovers an emergent Co-spin scalar chirality and colossal positive transverse-magnetoresponse. We predict the temperature and magnetic field scales for the tunability of scalar-chirality and magnetoresponse.

Introduction:— Crystalline-symmetry-protected topological insulators [1] has ushered a new paradigm of research on the coexistence of broken time-reversal and inversion symmetry along with band topology [2, 3]. Dirac [4, 5, 6], nodal-line[7] and Weyl [8, 9] semi-metal have been theoretically proposed and experimentally realized [10, 11, 12]. In particular, half-metals with fully spin-polarized conduction bands bring together correlation effects and crystalline symmetry-induced band-topology, two vital ingredients for realizing magnetic-topological metals. While the magnetic order can reduce the crystalline symmetry, it allows for the survival of some nodal lines, classified by generalized Chern numbers [13, 14]. In the presence of spin-orbit coupling (SOC), these remnant nodal lines can be gapped out along with band-inversion at nodal points [15]. When the nodal points are close to the Fermi energy, the band-inversion leads to anomalous Hall conductivity (AHC) in spin-polarized bands. The transverse spin-current generated as a consequence has potential applications in low-energy electronics [16], spintronics [17], and quantum computation [18]. Thus, understanding the interplay of correlation effects induced magnetism, crystalline symmetry protection, and spin-orbit effects is vital.

Double perovskites (DP) half-metals offer a unique platform where crystalline symmetry, strong Coulomb correlation, and SOC can compete. Unlike simple perovskites, in DP, the interaction and SOC effects occur at distinct atomic sites. In many DP of the form, A_{2}BB^{′}O_{6} half-metallicity typically originates from correlation-induced local moments at B(=3d) atom and carriers delocalizing in the B-B^{′}(=4d/5d) structure [19, 20]. The latter B^{′}(=4d/5d) is the source of SOC. DP’s are also known to host mirror planes and are natural candidates for crystalline-protected bands. From an experimental standpoint, the considerable charge transfer energy between 3d and 5d atoms allows anti-site defect-free growth [21, 22]. However, theoretical investigations of these important systems have been limited to first-principles [23, 7] and zero-temperature mean-field studies assuming classical spin-moments at B-site [24, 25]. While these studies have added valuable insights, both first-principles and mean-field approach studies miss out on strong correlation effects. Further, classical spin treatment cannot capture the formation and evolution of B-site local magnetic moment with temperature.Thus, the question of magnetization at the B site evolves when electrons traverse in a topological band, the impact of topology on their magnetic order and transport, and its temperature and magnetic field response has largely remained open. The dearth of results is due to the enormous complexity of modeling multi-orbital models and the associated computational demands. Nevertheless, addressing these questions is a fundamental theoretical challenge in materials theory and is paramount for future technological applications. In this letter, we combine first-principles study, exact diagonalization, and a semi-classical Monte-Carlo approach to address these questions.

We consider the double perovskite Ba_{2}CoWO_{6} (BCWO) containing 3d and 5d transition metal atoms, where the former introduces a strong local correlation effect, and the latter provides a spin-orbit coupling. Co^{2+} is nominally in a $d^{7}$ state with two $t_{2g}$ orbitals doubly occupied, the third $t_{2g}$ level, and two $e_{g}$ orbitals singly occupied, while W is in a $d^{0}$ state.Through first-principles calculations, we demonstrate a half-metallic ground state in BCWO with a Co high-spin state providing localized moments (majority spins) and delocalized minority carriers. Our analysis shows that, without spin-orbit coupling (SOC), the band structure supports mirror-symmetry-protected nodal lines, two of which are gapped out due to magnetic order. Introducing SOC on W gaps out the remaining nodal line induces band inversion at the nodal points and generates AHC due to Berry curvature effects.Next, we construct a down-folded five-orbital Hubbard model with full Kanamori parmeterization[26], confirm the ground state using exact diagonalization, and track the temperature evolution of magnetic order and charge transport with a semi-classical Monte Carlo approach. We uncover a remarkable manifestation of topology in the Co-site magnetic order as an emergent non-coplanar spin texture. We show that the non-coplanarity is generated without adding Dzyaloshinskii-Moriya interaction (DMI) and is consistent with non-zero Berry curvature and transverse conductivity. To demonstrate the topological origin of the non-coplanarity, we show that an external magnetic that suppresses the non-coplanarity also suppresses the AHC while minimally affecting the longitudinal response. The combined temperature and magnetic field investigation of transport leads to a colossal transverse magnet response.We predict the temperature scale for the survival of AHC and the magneto-response systematics, offering an experimental knob for controlling AHC.

Electronic Structure:— We compute the band structure of BCWO, which is a face-centered cubic crystal with the Co and W atoms surrounded by an octahedral cage of oxygen as shown in Fig. 1(a). Details are provided in Sec. IA of Supplemental Materials[27]. The space group of the crystal is Fm$\bar{3}$m (space group no. 225), which possesses O${}_{h}^{5}$ octahedral symmetry and a lattice parameter of a = 8.210 Å[28]. The structure exhibits three mirror planes (M_{x}, M_{y}, M_{z}), illustrated in Fig.1(b). The spin and orbital projected density of states (DOS) in Fig.1(c) without SOC shows that the metallic nature originates from the Co and W $t_{2g}$ $\downarrow$-spin (minority) channel electrons with dominant (small) spectral weight from Co (W) at $E_{F}$. The charge gap in the $\uparrow$ spin (majority) spin channel is 2.5 eV. The Bader charge analysis indicates that Co2^{+} (3d^{7}) is in the high spin state S=3/2 and W6^{+} (5d^{0}) has S $=$ 0, giving a net magnetic moment of 3.0 $\mu_{B}$/f.u [28]. The corresponding electronic level diagram in Fig.1(d). From the first-principles analysis, we extract the Co crystal field splitting $\Delta_{CF}^{Co}$ to be 1.2 eV, the W crystal field splitting $\Delta_{CF}^{W}$ to be 8eV to be and a Co-W charge transfer energy $\Delta_{CT}$ to be 2.5 eV. The high spin state of Co arises due to large Hund’s coupling ($J_{H}$). The hybridization between Co $t_{2g}\downarrow$ and W $t_{2g}$ stabilizes the half-metallic ground state as also seen from Wannierization calculations (see Supplemental Material [27] Sec IA).

Crystalline symmetries & band topology:— We show the spin polarized ($\uparrow$-blue, $\downarrow$-magenta) band structure along the the high-symmetry direction in Fig.2(a). A linear band crossing point (circle) between Co d_{xz} and d_{yz} orbitals (of the $\downarrow$-spin bands) is observed along the $K$ to $\Gamma$ direction, just below the Fermi energy ($E_{F}$). We provide an expanded view of the crossing in Sec. IB of the Supplemental Material[27]. The crossing of the singly degenerate minority band strongly indicates nodal points originating from crystalline symmetries[15]. BCWO crystal structure possesses M_{x}(k_{x}=0), M_{y}(k_{y}=0), M_{z}(k_{z}=0) and three C_{4} rotation axes, k_{x}, k_{y} and k_{z}. Similar symmetry has been reported for Heusler alloy nodal semi-metals[29, 15]. Magnetization along (001) direction preserves only M_{z}(k_{z}=0) mirror symmetry and C_{4z} rotational symmetry, and consequently, only a nodal line on the k_{z} = 0 plane survives. We show the surviving nodal line in the k_{x}-k_{y} in Fig.2(c) by a thin dashed line. We show only the projection in Fig.2(c) in the k_{x}-k_{y} plane; the nodal line forms a closed loop in the E-k_{x}-k_{y} space.

The mirror Chern number of the system is derived by adding the contributions from each nodal line. In the presence of SOC, the mirror Chern numbers are determined using the winding numbers of Wannier centers on the mirror invariant planes. The winding number on the (K_{z}=0) mirror plane is 1, whereas on the (k_{x}=0, k_{y}=0) mirror plane it is 0. For finite SOC, the spin-orbital mixing leads to gapping of the remaining nodal line and consequently opens up a gap at these degenerate points as shown in Fig.2(b) along the high-symmetry point $\Gamma$-K. In the Supplemental Material[27], Sec. IB, we show that the gapped nodal line causes a band inversion at the nodal points close to the Fermi energy and indicates the non-trivial topological nature of the electronic band structure. The non-trivial topology of the electronic bands gives rise to non-zero Berry curvature distribution around the $E_{F}$. From the z-component of Berry curvature ($\Omega_{z}$) along the same high-symmetry path shown in Fig.2(d), we find sharp peaks along the K-$\Gamma$ and L-$\Gamma$ directions and negligible contributions in the other directions. The origin of these large contributions is the small gaps in the neighborhood of the nodal line. We present the full $\Omega_{z}$ distribution in the k_{x}=0 plane in Supplemental Material[27] Sec IB. The Berry curvature induces AHC as a function of filling, as shown in the inset in Fig.2(d). At $E_{F}$ for BCWO, we predict $\sigma_{xy}$, to be 100 S/cm.

Multi-orbital Hubbard model for BCWO:— We now consider a material-realistic model with the full $d$-manifold on Co and W retaining multi-orbital interactions with Kanamori parameterization. The onsite intra-orbital repulsion between up and down electrons $U$, inter-orbital repulsion (spin-independent) $U^{\prime}$, Hund’s coupling preferring spin alignment ($J_{H}$) and pair-hopping term ($J^{\prime})$ are related by $U^{\prime}=U-2J_{H}$ and $J^{\prime}=J_{H}$ [26].To reduce computational complexity, we employ a down-folded hopping matrix $t_{\alpha,\beta}$ between Co and W orbitals, integrating out the oxygen. We further introduce SOC with coupling strength $\lambda$ on W. The full Hamiltonian, the down-folded hopping matrix, and the onsite SOC Hamiltonian are provided in Supplemental Material[27] Sec II and Sec III respectively. We find from the down-folding that the non-zero hopping elements are only between the Co and W $t_{2g}$ orbitals. We model the crystal field splitting between $t_{2g}$ and $e_{g}$ on Co and W to be $\Delta_{CF}^{Co}$=0.8eV and $\Delta_{CF}^{W}=4.0$eV respectively and set the onsite energy difference between Co and W $t_{2g}$ orbitals $\Delta_{CT}=2.5$eV. We choose $U$ in the range of 1.25eV to 2.5eV, $J_{H}=U/4$, and $U^{\prime}=U-2J_{H}$ for Co and vary the spin-orbit coupling (SOC) parameter $\lambda$ between 50meV to 80meV. The results below are robust for small non-zero $U$ on W and small SOC on Co. We have used the largest hopping matrix element ($t=0.2eV$) of the down-folded hopping matrix to convert the Hamiltonian parameters to energy dimensions. Our numerical conclusions discussed below also hold for $\Delta_{CF}^{Co}\in[0.5,1.0]$eV, $\Delta_{CF}^{W}\geq 3.0$eV and $\Delta_{CT}\in[2.0,3.5]$eV. The full parameter-dependent phase diagram will be reported elsewhere.

i. Exact diagonalization:— We first perform exact diagonalization (ED) on a linear chain containing two Co-W clusters. We consider five orbitals on Co and three $t_{2g}$ orbitals on W. Given the high energy location, the $e_{g}$ on W is neglected, as seen from the first-principles study. For hopping, we employ the down-folded Co-W hopping elements mentioned above. We provide the details of the ED calculations in Supplemental Material[27] Sec. IV. In Fig.3 we indicate the charge density $<n_{t_{2g}}>$ and $<S^{z}_{t_{2g}}>$ of individual $t_{2g}$ and $e_{g}$, orbitals of Co and W. Thus, $<S^{z}_{Co}>$ is 0.191$\times 3$+ 0.5$\times 2$ amounting to about 1.573 and $<S^{z}_{W}>$=$-0.022\times 3=-0.066$. The average magnetization per unit cell is close to 3/2, as found in the first-principles calculations. In addition, we notice that the Co-W hybridization induces a finite occupation on W, unlike the nominal W ($d^{0}$) count. The ED results, albeit with severe size limitations, thus indicate a Hund’s coupling induced high-spin Co and $t_{2g}$ minority ($\downarrow$-spin) Co-W hybridization, which leads to the half-metallic ground state. We construct a schematic representation of the ground state spin configuration in Fig.3. The ED calculation, albeit with strong finite size effects, confirms the first-principles ground state.

ii. Finite temperature properties:— We now employ a semi-classical Monte-Carlo (s-MC) approach that provides finite-temperature properties of the multi-orbital Hubbard model in three dimensions on 6$\times$6$\times$6 clusters with periodic boundary conditions. The s-MC results have been demonstrated to capture physics beyond finite-temperature mean-field theory [30, 31].We provide the methodological details in Supplemental Material[27] Sec.V.The low-temperature s-MC orbital and spin-resolved DOS in Fig.4(a) reproduces the half-metallic ground state and agrees qualitatively with the first-principles DOS in Fig.1(c). Fig.4(b) shows the ($\lambda=0$) temperature evolution of the orbital resolved magnetization $\langle S^{z}\rangle$(T) on the left axis, and the $\mathbf{q}=0$ static magnetic structure factor ($S_{0}(T)$) that quantifies the ferrimagnetic order, on the right axis. At low temperature, the $\langle S^{z}\rangle(T)$, for the Co $t_{2g}$ and $e_{g}$ orbitals are 0.54 and 0.7 respectively, while the $\langle S^{z}\rangle(T)$ for W is $\sim-0.012$. The ferrimagnetic $T_{c}$ coincides with the concomitant vanishing of $\langle S^{z}\rangle$ for all orbitals. The inset in (b) shows that increasing thermal energy overcomes the energy barrier between Co and W, allowing greater occupation of the W site at the expense of the Co occupation. This thermal fluctuation-driven evolution of Co and W filling reduces the Co magnetization and suppresses the half-metallic order.The ($\lambda=0$) longitudinal ($\sigma_{xx}$) and the band-topology-induced transverse ($\sigma_{xy}$) conductivity with temperature in Fig.4 (c) and (d), respectively, has a clear correlation with the magnetic order shown in (b). The conductivity calculations within the Kubo-Greenwood formalism are provided in Supplemental Material [27] Sec. VI. In (c), the minority (majority) spin components of $\sigma_{xx}$ show metallic, or $d\sigma_{xx}/dT<0$ (insulating or $d\sigma_{xx}/dT>0$) behavior. The spin-dependent conductivities coincide at $T_{C}$. The half-metallic behavior survives for finite $\lambda$. The direction of the ferrimagnetic order allows the definition of a global z-axis of defining $\uparrow/\downarrow$, even in the presence of spin-orbital mixing SOC.We see from Fig.4 (c) that the temperature where spin-polarized conductivities coincide, signaling loss of half-metallicity, is suppressed systematically with increasing $\lambda$.The $\sigma_{xy}$ or AHC in Fig.4 (d) is zero for $\lambda=0$ within numerical resolution. For $\lambda\neq 0$, we find that $\sigma_{xy}$ is finite and acquires the largest value at low temperatures. It monotonically decreases with temperature increase, with the overall magnitude being larger for greater $\lambda$. The temperature value of the inflection point in $\sigma_{xy}$ increases with decreasing $\lambda$. Compared with the $\sigma_{xx}$, there is a possibility for a temperature regime close to $T_{c}$ where the half-metallicity is lost, but AHC survives. Nonetheless, the $\sigma_{xy}$ is finite. However, due to the size limitations of the lattice sizes simulated, we leave the exploration of this interesting possibility for the future.

Emergent chiral spin-texture:— We now demonstrate that the SOC on W induces a non-coplanarity of the Co moments. We compute the scalar spin-chirality $\chi\equiv 1/N\sum_{i\langle j,k\rangle}\langle S_{i}.(S_{j}\times S_{k})\rangle$ in the x-y plane using standard definition[32, 33]. The spins in the above definition belong to the Co sites, and the angular brackets denote quantum and thermal averaging. $\chi$ is calculated in individual x-y planes and summed over the planes stacked along z.Fig5 (a) shows the temperature dependence of $\chi(T)$ and its dependence on SOC. The scalar chirality is zero, as is $\sigma_{xy}$ for $\lambda=0$, although the small contribution from intrinsic magnetization cannot be ruled out. Similarly the SOC and temperature dependence of $\chi(T)$ follow that of $\sigma_{xy}(T)$ seen in Fig.4 (d). Since scalar chirality is a measure of non-coplanarity, an external magnetic field could polarize the spins and suppress the non-coplanarity. The loss of non-coplanarity destabilizes the spin texture, which should be detrimental to $\sigma_{xy}$ if $\chi$ is indeed topological in origin. In Fig.5 (b), we show a remarkable suppression of AHC with a magnetic field. The sign of $d\sigma_{xy}/dT$ at low temperatures is reversed by the magnetic field value beyond 0.075$t$. Thus the $\chi(T)$ and the $\sigma_{xy}(T)$ show a strong interdependence in accordance with literature [34, 35].In contrast, the magnetic field slightly enhances the longitudinal conductivity in Supplemental Material [27] Sec. VII.

In terms of $t\sim 0.2$eV, $T_{c}\sim 0.02t$ corresponds to 40K, within the ballpark of experimental observation ($T_{C}\sim 20K)$ [36] and $h_{c}\sim 0.1t$ for switching of $\sigma_{xy}$ for $\lambda=50$meV is about 95T.We note that the s-MC calculations are performed on small clusters due to computational complexity. As a result, they suffer from finite-size effects. So, our finite critical temperature and magnetic field scales overestimate the actual scales of the problem. Nonetheless, our results demonstrate the feasibility of such phenomena qualitatively.

Conclusion:— In summary, we have demonstrated the fate of a nodal line semi-metal in a strongly correlated system at zero and finite temperature for the first time. Our study of BCWO reveals a half-metallic ground state with Co high-spin ferromagnetic background facilitating minority carrier delocalization in topologically non-trivial bands. SOC at W sites gaps nodal points, generating an anomalous Hall response. Using a five-orbital Hubbard model and semi-classical Monte Carlo simulations for the first time, we uncover an emergent non-coplanar spin texture at Co sites, resulting in non-zero Berry curvature and colossal transverse magneto-response controllable by an external magnetic field. This highlights the tunability of BCWO’s electronic properties, which are vital for spintronic applications. Our theoretical prediction holds for other 3d-5d semi-metallic metallic DP.Usually, spin-fermion-based half-metals have a colinear ferromagnetic or anti-ferromagnetic spin background. Here, we have modeled DP with a five-orbital Hubbard model and SOC at finite temperature for the first time. We have revealed emergent crystalline symmetry-induced scalar chirality, taking our results beyond any previous theoretical study of DP limited to spin-fermion models with classical spins coupled to fermions where transverse conductivity requires including Dzyaloshinskii-Moriya interaction.

## ACKNOWLEDGMENT

M.K and S.C. thanks DST for funding through grant no. CRG/2020/000754. J.S thanks University Grant Commission (UGC) for Ph.D. fellowship. S.S thanks DST-INSPIRE for financial support.

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