Vector-chirality driven topological phase transitions in noncollinear antiferromagnets and its impact on anomalous Hall effect

Magnetic materials showing topologically nontrivial quantum states with high tunability is an undoubtedly important topic in condensed matter physics and material science. Based on the first-principles electronic structure calculations and subsequent symmetry adapted effective low-energy k.p theory, we show in a noncollinear antiferromagnet (AFM), Mn₃Sn, that the switching of the vector-chirality, $\kappa$, is an unconventional route to topological phase transition from a nodal-ring to a Weyl point semimetal. Specifically, we find that the switching of $\kappa$ via $staggered$ rotation leads to gapping out an elliptic nodal-ring everywhere at the Fermi-level except for a pair of points on the ring. As a consequence, the topological phase transition switches the anomalous Hall conductivity (AHC) from zero to a giant value. Furthermore, we theoretically demonstrate how the controlled manipulation of the chiral AFM order keeping $\kappa$ unaltered favors unusual rotation of Weyl-points on the ring. In fact, without $staggered$ rotation, this enables us to tune and switch the sign of in-plane components of the AHC by a collective uniform rotations of spins in the AFM unit cell.

Over the past decade, a variety of topological semimetals (TSMs) RevModPhys.90.015001 ; doi:10.1146/annurev-matsci-070218-010049 ; Hasan_2021 ; Yang_2016 , namely, nodal point semimetals PhysRevLett.108.140405 ; PhysRevB.83.205101 ; PhysRevB.99.075116 , nodal line/ring semimetals PhysRevB.92.045108 ; Fang_2016 ; PhysRevLett.122.077203 ; Belopolski_2019 ; PhysRevB.104.235136 , nodal surface semimetals PhysRevB.97.075120 ; D0TC01978J ; PhysRevB.97.235150 have taken widespread attention due to the unusual transport phenomena related to the protected nontrivial boundary states. Most nodal point semimetals fall into two categories: Weyl semimetals (WSMs) huang2015weyl ; Yang_2017 ; Kuroda_2017 ; Yan_2017 and Dirac semimetals (DSMs) Liu_2014 ; doi:10.1021/jacs.8b09900 ; C3CP53257G ; hosen2018distinct , depending on the degeneracies and distribution of the band crossing nodes in the bulk band structures. A four-fold degenerate Dirac node can be thought of a point with a degenerate pair of Weyl points (WPs) carrying opposite chiral charges, where WPs can be derived by separating Dirac points in momentum space by breaking either time reversal symmetry or inversion symmetry. The nodal-ring semimetals (NRSMs) with two fold degenerate one-dimensional band crossings on the other hand can be gapped out except a pair of discrete touching points, an alternative approach to create WPs. The advantage here is the controlled generation of the discrete touching pairs on a loop. Recently, coexistence of magnetism, in particular ferromagnetism and topological quantum states in band structures has emerged as an important platform both for fundamental and technological interests 2019NatRP…1..126T ; naturematerial . In conventional topological ferromagnets, the ability to externally control the magnetism by magnetic, electrical and optical fields offers a remarkable way of transitions between topological phases PhysRevLett.115.036805 ; zhang2014electrically ; doi:10.1073/pnas.1713458114 ; article . Subsequently, attention has been drawn to nontrivial topology in magnetic materials without a net magnetization i.e. antiferromagnets (AFMs) PhysRevX.9.041040 ; Otrokov_2019 ; wangC ; liuH . Note, the recent developments of electrical manipulation and detection of AFM orders Wadley_2016 ; Zelezny , particularly in view of topological AFM spintronics Nature_anti ; prl118 ; diang-Fu2019 open up possibilities to find topological phase transitions in AFMs too. This is due to the fact that the current induced $staggered$ torque salemi2019orbitally allows unprecedented control on AFM orders which, in turn, may change symmetries of AFMs significantly to host various topological quantum phases. Nonetheless, no noncollinear AFM has been realized to show topological phase transitions under the change in the noncollinear order and possibly, we need an unconventional route to manipulate the AFM order.

The noncollinear AFM orders in Mn based kagome-like planes have been subsequently discovered in a series of Heusler materials, cubic Mn₃Y (Y = Rh, Ir, Pt) PhysRevLett.112.017205 ; PhysRevB.92.144426 and hexagonal Mn₃X (X = Ge, Sn, Ga) K_bler_2014 ; nakatsuji2015large ; doi:10.1126/sciadv.1501870 ; doi:10.1063/1.5143959 ; PhysRevB.95.075128 ; Ikhlas_2017 ; csingh ; PhysRevLett.76.4963 materials. Since then, such AFM orders become an important topic of research, as they are found to show various TSM phases Tsai_2020 ; PhysRevB.99.104428 , however, the noncollinear magnetization driven transitions between various TSM phases are missing. Here, the noncollinear AFM order can be classified based on defining a vector spin chiralityKawamura_2001 as,

$$\kappa=\frac{2}{3\sqrt{3}}\sum_{<ij>}[{\hat{n}_{i}}\times{\hat{n}_{j}}]_{z},$$

(1)

where $<\!{ij}\!>$ runs over the nearest neighbor spins, $\textbf{S}_{i}$ = S ​​$\hat{n}_{i}$ with $\hat{n}_{i}$ as the unit vector. The magnitude of $\kappa$ is unity while the signs ‘$+$’ and ‘$-$’ represent the chiral states with magnetic orders, “direct” and “inverse” triangular AFM, respectively. Notably, the magnetic ground state of Mn₃X and Mn₃Y show opposite $\kappa$ value in the kagome sublattices which remains unaltered with simultaneous uniform spin rotations within the plane. However, materials within Mn₃Y family are crystallized into a face-centered cubic structure (space group Pm$\overline{3}$m Zhangprb and $O_{h}^{1}$ magnetic point group PhysRevB.95.094406 ) where magnetic kagome planes are stacked in “$-ABC-ABC-$” order along the crystallographic (111) axis. This is different in compared to the hexagonal Mn₃X family ($D^{4}_{6h}$ magnetic point group PhysRevB.95.094406 ; prb101 ) where one finds “$-AB-AB-$” stacking order along the hexagonal (0001) axis. Therefore, for a fixed $\kappa$, one would expect that the band structures in both family of materials will be significantly different from the symmetry constraints. In this class of chiral AFMs, $\kappa$ becomes an important quantity which by switching between $+1$ and $-1$ can lead to control various symmetries and hence, offer a possible route to tailor nontrivial topological phases.

In this work, within a noncollinear chiral AFM, Mn₃Sn, we have shown a topological phase transition from NRSM to WSM through an unusual vector chirality, $\kappa$ switching from ‘+1’ to ‘-1’ via introducing a $staggered$ rotation. We note that earlier studies established the Weyl-points (WPs) physics in the “inverse” triangular AFM state Yang_2017 . In contrast, the switching to its competing chiral state carrying $\kappa=+1$ (“direct”) and exhibiting topological NRSM state is one of the key findings of the present study. The switching of vector chirality $\kappa$ from ‘+1’ to ‘-1’ tunes the anomalous Hall conductivity (AHC) value from zero to a giant value. Moreover, even in the absence of $staggered$ rotation, a simultaneous uniform rotation of Mn moments keeping $\kappa=-1$ serves to rotate the Weyl points and hence, tailor the anomalous Hall effect (AHE). This therefore switches the sign of the in-plane components of the AHC. Finally, we theoretically demonstrate by tuning the chiral AFM order how the TSM phases switch and tailor the nontrivial transport properties.

To uncover the topological phase transition within the electronic structure theory of noncollinear AFMs, we adopt the density functional theory (DFT) formalism as implemented in the full-potential linearized augmented plane wave (FLAPW) all-electron code, FLEUR fleur and the plane-wave projected augmented wave (PAW) pseudopotential code, Vienna ab initio Simulation Package (VASP) vasp ; paw2 . Details of the computational and numerical approaches are provided in the Method subsection. In the following section, we discuss the interplay between the vector chirality $\kappa$ and $\kappa$ induced nontrivial topological phases in band structures of a noncollinear AFM, Mn₃Sn. As a consequence, we show how the onset of various topological phases via controlled manipulation of the chiral AFM order drives the system into AHC switching and further tuning of its in-plane components.

In Mn₃Sn, the essential topological footprints near the $E_{f}$ are captured by constructing a low-energy k.p effective model Hamiltonian of two bands around the high-symmetric $K$ points in the BZ. Here, in the construction, we take into account all relevant symmetries in the noncollinear AFMs and the detailed construction of the theory is given in the Supplementary Note 3. The Hamiltonian for the existence of a NR solution takes the form

$$\mathcal{H}_{\textrm{NR}}=a_{1}k_{z}^{2}k_{x}\sigma_{x}+(m_{0}-m_{1}k_{x}^{2}-m_{2}k_{y}^{2}-m_{3}k_{z}^{2})\sigma_{z},$$

(2)

where {$\sigma_{i}$} are the Pauli matrices and the momentum, $\mathbf{k}$ = ($\textbf{k}_{||},k_{z}$), is measured relative to the $K$ point. $a$ and $m$ are the free parameters in the model. The eigenvalue solutions, $\mathcal{E}_{\pm\mathbf{k}}$, are degenerate in the $k_{z}=0$ plane, forming an elliptical NR state satisfying $m_{1}k_{x}^{2}+m_{2}k_{y}^{2}=m_{0}$. The constructed Hamiltonian is invariant under important symmetries, $\mathcal{C}_{3z}$, $\mathcal{M}_{x}$, $\mathcal{M}_{y}\mathcal{T}$ and $\mathcal{M}_{z}\mathcal{T}$, observed in the particular $\kappa=+1$ AFM state, $a_{1}^{+}$ in Fig. 4(a) (also see Fig. 1(b)). As stated earlier, a simultaneous local rotation of {$\textbf{S}_{i}$} in the unit cell keeps $\kappa$ value unchanged along with the $\mathcal{C}_{3z}$ rotation symmetry intact, however, can change the other symmetries. If we replace the $a_{1}k_{z}^{2}k_{x}\sigma_{x}$ in Eq. 2 with $a_{1}\mathbf{k}^{2}_{||}k_{y}\sigma_{x}$, the new k.p Hamiltonian on the hand preserves symmetries, $\mathcal{M}_{x}T$, $\mathcal{M}_{y}$, $\mathcal{M}_{z}T$ and $\mathcal{C}_{3z}$ which are protected in another $\kappa=+1$ AFM state, 3-in-3-out configuration. In that case, the NR solution disappears as the bands are completely gapped out for all $\textbf{k}_{||}$ points. The ab initio band structure is in good agreement with the model solution and the fully gapped state is clearly visible in the 3D energy band dispersion, see Supplementary Fig. 3(h). Therefore, the existence of NR solution on the $k_{z}=0$ plane is found for the coexistence of $\mathcal{C}_{3z}$, $\mathcal{M}_{x}$, $\mathcal{M}_{y}\mathcal{T}$ and $\mathcal{M}_{z}\mathcal{T}$ symmetries, particularly preserved in $a_{1}^{+}$ magnetic texture. Above combination of symmetries is very unique as moving to $\kappa=-1$ AFM state, $a_{1}^{-}$ in Fig. 4(b) (constructed via $120^{o}~{}staggered$ rotation), $\mathcal{M}_{x}$, $\mathcal{M}_{y}\mathcal{T}$ and $\mathcal{M}_{z}\mathcal{T}$ symmetries remain conserved while the $\mathcal{C}_{3z}$ symmetry is broken. The effective low energy Hamiltonian in that case takes the form

$$\mathcal{H}_{\textrm{WP}}=v_{x}k_{x}\sigma_{x}+(m_{0}-m_{1}k_{x}^{2}-m_{2}k_{y}^{2}-m_{3}k_{z}^{2})\sigma_{z},$$

(3)

where the $v_{x}k_{x}\sigma_{x}$ is the perturbing term. The energy dispersion solution for $k_{z}=0$ plane shows a pair of gapless points at $\mathbf{k}_{||}=(0,\pm\sqrt{m_{0}/m_{2}})$ along the $k_{y}$ axis which is the minor axis of the elliptic NR solution earlier. Therefore, we infer that the perturbation leads to generate two crossing points on the minor axis of the ellipse, consistent with the ab initio band structure. On the other hand, by replacing $v_{x}k_{x}\sigma_{x}$ to $v_{y}k_{y}\sigma_{x}$ in Eq. 3, we find a pair of gapless points at $\textbf{k}_{||}=(\pm\sqrt{m_{0}/m_{1}},0)$ along the $k_{x}$ axis. This Hamiltonian is invariant under $\mathcal{M}_{x}T$, $\mathcal{M}_{y}$ symmetries which are conserved for $a_{4}^{-}$ AFM state in Fig. 4(e). This is hence consistent with the location of the ab initio calculated WPs, lying along the major axis of the ellipse. The rotation of WPs related to the collective spin rotations, therefore, has led to choose a perturbation of the form ($\mathbf{v\cdot k}_{||})\sigma_{x}$ in Eq. 3. Most importantly, a pair of gapless points is always found as the solution of the $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian in the $k_{x}$-$k_{y}$ plane. Depending on the vector $\mathbf{v}$ in the Hamiltonian, a pair of band touching points in the solution change their positions on an elliptic ring. This is consistent with the rotation of the WPs around $K$ point in the band structure of various magnetic textures with $\kappa=-1$. The vector $\mathbf{v}$ here can be closely associated with the magnetic octupole moments in Mn₃Sn which is believed to break the time reversal symmetry Tsai_2020 ; PhysRevB.95.094406 .

In our results, we have established the intimate connection between vector chirality driven topological phase transition and AHE in antiferromagnets. Importantly, the noncollinear AFM spin texture has a significant impact on its band structure, Berry curvature and hence, on the intrinsic AHE. The NR state in case of $a_{1}^{+}$ is found to exhibit net zero Berry curvature in its vicinity and thus does not generate any AHE. We also find that the fully gapped NR state found in 3-in-3-out AFM texture does not induce Berry curvature, which is responsible for the absence of AHE. This therefore, is in contrast with the ferromagnetic topological semimetal Co₂MnAl article that generate large AHE only by gapping out NRs in the presence of net magnetization in the system. On the other hand, the elliptic NR has evolved into a pair of WPs when the $a_{1}^{+}$ texture turns into $a_{1}^{-}$ texture by $120^{o}$ $staggered$ rotation. The WPs contribute in generating large Berry curvature, which is responsible for the observed giant AHC. The absence of intrinsic AHE may arise either from nontrivial topological NR phase or from the gapped phase, depending on the underlying noncollinear AFM texture. Nonetheless, the switching of AHC from zero to a giant value may indirectly justify our claim of topological phase transition from the NRSM to the WSM phase. Moreover, the smooth variation of $\sigma_{x}$ or $\sigma_{y}$ component (including switching between positive value to negative value and vice-versa) is consistent with the rotation of the WPs on an elliptic ring, compare the results in Fig. 4 and Table 1. The important point to be noticed is that the giant $\sigma_{x}$ ($\sigma_{y}$) value, particularly in case of $a_{1}^{-}$ ($a_{4}^{-}$) AFM state, arises due to the odd nature of the Berry curvature $\Omega_{y}$ ($\Omega_{x}$) under the symmetry operation $\mathcal{M}_{y}T$ ($\mathcal{M}_{x}T$). Therefore, the odd Berry curvature does not contribute in the AHE. In case of intermediate $120^{o}$ AFM configurations ($e.g.$ $a_{5}^{-}$ and $a_{6}^{-}$ connected by LR), the relevant symmetries are broken which then lead to nonzero values of $\sigma_{x}$ and $\sigma_{y}$ both.

In summary, based on the detailed electronic structure calculations and low energy $\mathbf{k}\cdot\mathbf{p}$ effective theory, we systematically reveal that multiple nontrivial TSM phases can be realized in $120^{o}$ noncollinear AFM, Mn₃Sn, by controlled alternation of the spin configurations. The analyses demonstrate that the NR and the gapless nodes in the form of WPs are strongly rely on the chiral orders characterized by the vector chirality, $\kappa=+1$ and $-1$, respectively. This work suggests that a $staggered$ rotation involving two spins in the kagome triangle is an unconventional route to topological phase transition from an elliptical NR to a pair of WPs through the switching of $\kappa$ from ‘$+1$’ to ‘$-1$’. It is worth to mention that the $staggered$ torque with sufficient strength required for switching $\kappa$ is difficult to generate at low temperature. However, at elevated temperatures when multiple magnetic phases may coexist apl2018 , the required torque can be reduced significantly. Notably, such types of $staggered$ rotations, albeit small in magnitude, have been recently realized in this class of materials via an external magnetic field miwa and uniaxial strain pizo . Later, depending on the nature of rotation (CR or LR) of spins in the $\kappa=-1$ AFM texture, the WPs are found to rotate as if the topological charges can be created on preferred locations on the ring. The symmetry adapted $\mathbf{k}\cdot\mathbf{p}$ theory moreover captures our findings in good agreement and, hence, brings an unconventional way of generating a pair of WPs from a ring in noncollinear AFMs. We further find a remarkable switching of AHC from zero to a giant value associated with the TSM phase transition in the system carrying nonzero vector chirality. Finally, it shows that the components of AHC, $\sigma_{x}$ and $\sigma_{y}$, can be tuned smoothly depending on the location of the WPs on an elliptic ring. These topological features in the band structures are also observed in the sister compounds in the hexagonal Mn₃X family (see the discussion in Supplementary Note 4 and Supplementary Figs. 5 and 6 for Mn₃Ge and Mn₃Ga, respectively). We thus believe that the TSM phase dynamics in hexagonal $120^{o}$ noncollinear AFM will offer a new avenue to develop concepts on $staggered$ torques for the manipulation of vector chiral order in AFMs and it possibly adds an alternative component to antiferromagnetic spintronics.

The data of this study are available from the corresponding author upon reasonable request via email: [email protected].

The DFT calculations are performed with public codes. Their input files are available upon reasonable request.

A.K.N, S.P. and K. Saha acknowledge the support from the Department of Atomic Energy, Government of India. A.K.N. and S.P. acknowledge the computational resources, Kalinga cluster, at National Institute of Science Education and Research, Bhubaneswar, India. A.K.N. thanks Prof. P. M. Oppeneer for the Swedish National Infrastructure for Computing (SNIC) facility. A.K.N. thanks Prof. P. Mahadevan for critical reading of the manuscript and stimulating discussions. A.K.N. and S.P. acknowledge Dr. Ajaya K. Nayak, Charanpreet Singh, Dr. Hirak K. Chandra, Sandip Bera, Sayan Banik, Arghya Mukherjee for fruitful discussion. K. Samanta acknowledge the computing resources granted by JARA-HPC from RWTH Aachen University and Forschungszentrum Jülich, Germany.

A.K.N. has conceptualized and supervised the work. The ab initio simulations are performed by S.P., K.Samanta and A.K.N. The k.p theory is constructed by S.P., K.Saha and A.K.N. The paper was written by A.K.N with inputs from S. P., K.Samanta and K.Saha. All authors have contributed to the discussions and analyses of the data and approved the final version.

The authors declare no competing financial interests.