Phase transitions associated with magnetic-field induced topological orbital momenta in a non-collinear antiferromagnet

Resistivity measurements are widely exploited to uncover electronic excitations and phase transitions in metallic solids. While single crystals are preferably studied to explore crystalline anisotropies, these usually cancel out in polycrystalline materials. Here we show that in polycrystalline Mn3Zn0.5Ge0.5N with non-collinear antiferromagnetic order, changes in the diagonal and, rather unexpected, off-diagonal components of the resistivity tensor occur at low temperatures indicating subtle transitions between magnetic phases of different symmetry. This is supported by neutron scattering and explained within a phenomenological model which suggests that the phase transitions in magnetic field are associated with field induced topological orbital momenta. The fact that we observe transitions between spin phases in a polycrystal, where effects of crystalline anisotropy are cancelled suggests that they are only controlled by exchange interactions. The observation of an off-diagonal resistivity extends the possibilities for realising antiferromagnetic spintronics with polycrystalline materials.


Introduction
The combined approach of experimental and theoretical research has recently led to a deeper understanding of electronic transport phenomena in antiferromagnetic (AFM) materials, fostering the development of future antiferromagnetic spintronic devices for which the manipulation of antiferromagnetic order by external means is a key issue [1][2][3][4][5][6] .
In particular, antiferromagnets with non-collinear magnetic structure are quantum materials that exhibit unique spin-dependent properties 7 , and compounds with chiral magnetic order [8][9][10] have been intensely investigated due to intriguing topological electronic features.Weyl points close to the Fermi level, serving as sinks or sources of Berry curvature, give rise to an anomalous Hall effect (AHE) [11][12][13] that enables electrical read-out of the magnetic state.Electrical read-out is also possible via the spin-orbit (SO) coupling induced anisotropic magnetoresistance (AMR), but this is usually small on the order of 0.1%.Large or even 'colossal' AMR values have been observed at magneticfield induced transitions between different AFM phases 14 or by current-induced internal fields relying on the strong coupling between charge current and magnetic order 15 .
Theoretical calculations of the electronic transport properties are most often carried out for single-crystalline materials, where the various anisotropies have to be taken into account.From a materials science point of view, polycrystalline materials are just as interesting as single crystals, since they are used more frequently in applications and crystallographic dependencies are balanced out by the randomly arranged crystallites.Moreover, in the case of a polycrystalline non-collinear antiferromagnet, it is not a priori clear that all non-diagonal elements of the resistivity tensor cancel when subjected to an applied magnetic field, which can change the AFM structure in each crystallite.
Prime examples of compounds where the non-collinear AFM order is susceptible to small variations of composition, strain, and magnetic field are Mn-based Mn3AX (A = Ga, Ge, Zn, Ag, Ni; X = C, N) compounds of cubic antiperovskite structure.They are well known for their large magnetovolume, piezomagnetic, and barocaloric effects and negative thermal expansion due to a strong spin-lattice coupling [16][17][18][19][20][21][22] .The mutual relation between the crystal lattice and magnetic order is based on the sensitivity of the Mn magnetic moment to the local atomic environment 23 .This leads to a geometrically frustrated lattice with Mn atoms arranged into a kagome lattice in the (111) plane, forming triangles with magnetic moments on equivalent Mn sites 10,24 The long-range AFM order is often of the coplanar Γ 5g (Fig. 1a,b) or Γ 4g type 25 .In the latter case, an octupole driven AHE response occurs when the magnetic field is rotated in the kagome plane along related crystallographic directions, which is fundamentally different to the conventional dipole driven AHE 26 .These materials often show rich phase diagrams, and transitions between the phases can be controlled by the magnetic field and current 27 .
Here we report an investigation on A-site doped Mn3(Zn0.5Ge0.5)Npowder samples where spontaneous changes occur in the longitudinal and transverse resistivity at low temperatures which are susceptible to weak magnetic fields.This is attributed to a magnetic phase transition between different non-collinear phases.The subtle change in magnetic structure is supported by neutron diffraction demonstrating its high sensitivity to minute changes in magnetic structure.The fact that the magnetic phase transitions between different spin phases are observed in a powder sample, where effects from crystalline anisotropy are cancelled suggests that they are controlled only by exchange interactions, in contrast to the SO interactions controlling, e.g., the  5g - 4g transition.This can be adequately described in a phenomenological model revealing the presence of a magnetic-field induced topological orbital momentum.The unexpected and surprising result that a finite transverse resistivity remains in a polycrystal of noncollinear AFM order is not due to symmetry effects or the AHE and provides information about the nontrivial magnetic phase diagram when properly investigated.

Experimental Results
The Mn3Zn0.5Ge0.5Nsintered-powder samples with a crystallite size of ~ 400 nm exhibit the coplanar Γ 5g type antiferromagnetic order below the Néel temperature TN = 411 K derived from neutron diffraction and magnetization measurements 21,28 (Fig. 1a, Supplementary Fig. S1a and Tables S1-S3).Refinement of the neutron scattering data at T = 10 K confirms the same space group Pm-3m and magnetic space group R-3m (166,97) 20 as isostructural Mn3AN (A = Ni, Cu, Zn, Ga, Ge, Pd, In, Sn, Ir, Pt) 23 independent of the doping.Figures 2a and b show the temperature dependence of the longitudinal and transverse resistivities ρ║ and ρ┴, respectively, of two samples at various magnetic fields H.During cooling from room temperature, the sample shows a metallic behavior (inset of Figure 2a).In zero field, ρ║(T) and ρ┴(T) clearly change upon cooling across a temperature T* = 3.7 K.The non-zero transverse resistivity below T* even in the absence of a magnetic field is surprising because in a powder sample with an equiprobable distribution of grain orientations, this component should average out to zero.The step-like transition shifts to lower temperatures and broadens with increasing magnetic field of the order of tens of mT (#1: H0 = 30 mT, #2: H0=120 mT).This extraordinary behavior is unexpected considering the usual robustness of AFM order against moderate magnetic fields.Several samples cut from the same batch exhibit an overall similar behavior regarding the step-like behaviour of ρ║ or ρ┴ at T*.This transition is not observed for isostructural Mn3Ag0.93N(Supplementary Fig. S5), i.e., in undoped samples and seems to originate from the doping or from the chemical disorder of the A atoms.The corresponding transverse conductivity is σ┴(0) = -ρ┴ /( ρ║ 2 + ρ┴ 2 ) = 20 Ω -1 cm -1 at T = 1.8K in zero field.This is smaller than the transverse conductivities of 150 -500 Ω -1 cm -1 arising from an AHE observed for other noncollinear antiferromagnets with triangular magnetic order 8,29 .
The magnetic field dependences of ρ║ and ρ┴ below T* are even functions of the magnetic field (Fig. 2c,d) with a saturation above H0.ρ┴(H) is quite different from the magnetic field behavior expected for an AHE which would appear as an odd function of magnetic field due to the broken time-reversal symmetry.1][32] By investigating several samples cut from the same batch in different directions we observe either an increase or decrease of the resistivities while crossing T*.We could not check whether this behaviour could be changed by, e.g., magnetic-field cooling of the sample from above TN 30,31 because the sample degraded at higher temperatures close to TN and the effect disappeared.Apart from the different signs, both longitudinal and transverse resistivity components exhibit the same temperature and field dependence.
The size of ┴ is independent of the magnetic field orientation (Fig. S4) when the field is rotated from out of plane to in plane (angle ) or in the plane around the surface normal (angle ) of the sample.A sinusoidal dependence of the AMR amplitude on the angle between current and magnetic field is not observed possibly due to the polycrystallinity of the sample strongly reducing the orientation dependence of the AFM 1 .
The observed transition cannot be ascribed to superconductivity 33 or to a structural phase transition inferred from the nonzero ρ║ and neutron scattering.Effects arising from weak localization or weak antilocalization in a 3D metal or even in a 3D Weyl metal do not play a role 34,35 because the expected corrections to the conductivity in dependence of T and H are not observed.The step-like behavior of ρ║(T,H) and ρ┴(T,H) and its saturation at moderate magnetic fields indicate a transition of magnetic origin rather than weak localization effects.
Measurements of the magnetoresistance down to 0.1 K establish a H-T phase diagram with phase boundaries appearing between different spin configurations (Fig. 3 and Supplementary Fig. S3) discussed in detail below.Interestingly, the transverse component of resistivity is zero at high magnetic fields above a critical value, as well as in the fully compensated antiferromagnetic phase Γ 5g .This observation seems contradictory, but our analysis of the temperature-and field-dependencies of resistivity points to the existence of two phase-transition lines (Fig. 3, red and blue lines) instead of one.
The integral magnetization M of Mn3Zn0.5Ge0.5N is very small, corresponding to a tiny average magnetic moment of 3.6×10 -5 μB/Mn and does not show a transition at low temperatures at 2 K (Supplementary Fig. S1 b,c) apart from a shallow increase below 5 K.Alternatively, neutron diffraction was employed to confirm the subtle change of the magnetic phase with temperature.The positions of the reflections in the diffraction patterns of Mn3Zn0.5Ge0.5Nare independent of temperature between 2 K and 10 K (Supplementary Fig. S6), implying that variations of the crystal and/or magnetic structures with temperature are very small.However, an intensity difference I(T) -I( 10K) of a few percent can be attributed to a change of magnetic texture with decreasing temperature (Fig. 4a).In particular, the considerable increase of intensities at P3 and P4 below 4 K is correlated with the appearance of the additional contribution to the resistivities below T* = 3.7 K.We estimate the relative contributions of nuclear and magnetic scattering by Rietveld analysis, revealing no magnetic contributions for P1, P2 but approximately 4% and 12% for the intensities of P3, and P4, respectively, at 2 K (Fig. 4b, inset).In Fig 4b, the integrated intensities at P3 and P4 -including a magnetic contribution -first only gradually increase with cooling but then strongly increase below 5 K while intensities at P1 and P2 remain almost zero.This indicates that the local magnetic moments of Mn atoms slightly change as the temperature drops below 3.7 K without a change of the crystalline lattice.The neutron diffraction pattern at 10 K (Table S2) can be fitted well with a structural model of cubic symmetry (Fig. 1a) and a superlattice magnetic model of Γ 5g type displayed in Fig. 1b, resulting in a refined moment M2x = 3.26(2) μB/Mn.However, if we use the same model to fit the patterns below 10 K, the refinement becomes worse.Assuming a finite magnetic moment component M2y along the y direction (Fig. 1d) clearly improves the profile difference.
M2y is determined from the minimalization of the refinement parameter χ 2 (Supplementary Fig. S8).The evolution of spin texture with cooling below 3.7 K is shown in Fig. 4c.M2y increases towards low temperatures while M2x remains almost constant.The moment reorientation from the perfect triangle configuration (Γ 5g , =0) towards the new configuration ( >0) exemplified by the angle , gradually grows from zero at 4 K to 6.5˚ at 2 K (Fig. 4 c) and the net magnetic moment increases by 1%.
Hence, the coplanar triangle Γ 5g phase (Fig. 1b) changes to a coplanar magnetic phase FO (Fig. 1d) of lower symmetry.For isostructural Mn3AgN, we do not observe a variation of the intensities Pi (i = 1-4) upon cooling to 2 K within the accuracy of the measurement in agreement with the lack of a transition in the resistivity (Supplementary Figs.S5, S10, S11).In this case, the triangle configuration remains stable.Application of a magnetic field at T = 3 K < T*, gives rise to a broad dip around 20 mT in the intensity difference I(μ0H)-I(20 mT) of the integrated intensity P4 while P2 remains constant (Fig. 4d, Supplementary Fig. S11).The decrease of the integrated intensity P4 in from zero to 20 mT is attributed to a field induced change of the magnetic phase in accordance with the phase diagram in Fig. 3.These results confirm the change of the coplanar triangular spin arrangement with decreasing temperature and/or increasing field and demonstrate the high sensitivity of the neutron powder diffraction to small changes in the magnetic structure.

Theoretical Model
The fact that we experimentally observe temperature and/or magnetic field induced changes in the diagonal and off-diagonal elements of the resistivity tensor and in the magnetic neutron scattering lets us to preliminary conclude that these changes occur due to transitions between three magnetic phases with different spin symmetries, i.e., with different relative angles between the neighboring spins 36 .One of these phases has lower symmetry compared to the others, which is reflected in additional components of the resistivity tensor.The appearance and disappearance of these components indicates a transition between low-and high-symmetry phases.We associate the observed phase transitions with a topological orbital momentum that can be induced by the external magnetic field 37 .Our results suggest that the energy associated with the topological orbital momentum gives rise to an effective ponderomotive force.This force is proportional to the square of the magnetic field and serves to stabilize the highly symmetric phases, even in the presence of the magnetic field as discussed below.
In the following, we develop a phenomenological model to describe the observed field and temperature dependencies of the resistivity tensor.The model is based on analyzing the spin symmetries of the existing phases, as illustrated in Fig. 1b, d, and f.We distinguish between two high-symmetry phases: coplanar ('triangle' Γ 5g , Fig. 1b) and noncoplanar ('flower' FL, Fig. 1f), as well as one coplanar low-symmetry phase ('fork' FO, Fig. 1d).Formally, these phases can be described in terms of three spin (or magnetization) vectors  1,2,3 localised at different magnetic atoms within unit cell.
From the symmetry point of view it is convenient to introduce linear combinations   = (  +   − 2  )/√6 and   = (−  +   )/√2 , that form a multidimensional order parameter of the antiferromagnetic phases, and the combination  = (  +   +   )/√3, which is the total magnetization 38,39 .The properties of the phases are summarized in Table 1.
Our model is based on the analysis of the magnetic energy F which, in the spirit of the Landau's theory of phase transitions, we introduce in a following way (for details see Supplementary Materials Note 5) The first three terms of the equation represent the bilinear (Heisenberg) and the biquadratic exchange coupling.The strength of coupling is characterized by phenomenological constants J(T), D, and D'.We assume a linear temperature temperature dependencies of the AMR.We also neglect any variations in the component   and consider its value to be the same in all three phases.We compute the resistivity tensor for each phase by considering all possible orientations of the magnetic field with respect to crystallographic axes.After computing the average over the particle distribution, we obtain a phase diagram that can be divided into three distinct regions (Fig. 3).

Discussion
Within the first region, denoted as Γ 5g , either the triangle or FL high-symmetry phase remains stable.These two phases have identical resistivity tensors, which makes it impossible to distinguish between them and resolve the transition line (Fig. 3, dashed line) separating them.In the FO phase, all particles underwent a transition from the triangle phase to the FO phase, followed by a smooth growth of in-plane magnetization My, and the associated field and temperature dependencies of the AMR exhibit a smooth dependence.Conversely, in the intermediate phase I, some of the particles transform from the FL phase into the low symmetry phase with decreasing magnetic field.
Consequently, the AMR exhibits a kink (Fig. 2d) when the contribution of these particles becomes noticeable in the signal.
In the polycrystalline material, different grains within the sample are affected by different components of the magnetic field.The transition from the fully compensated Γ 5g phase to the low-symmetry FO phase is induced by the field component that is parallel to the {111} ordering planes.On the other hand, the transition to the high symmetry non-coplanar FL phase is induced mainly by the field component that is perpendicular to the ordering plane, and additionally stabilized by the in-plane component in the vicinity of the triple point   (2.8 ).Importantly, we find that the transition between the noncoplanar FL and low-symmetry phases occurs at a higher field strength compared to the transition from Γ 5g to the low-symmetry FO phase.
Consequently, the phase diagram of the polycrystalline sample comprises two distinct phase transition lines that delineate the transitions into the low-symmetry phase from Γ 5g and from the FL phase into the I phase, respectively.Finally, we discuss the origin of the nonzero transverse resistivity component.The spin symmetry, which describes the effects of exchange interactions, can only account for diagonal components of the resistivity tensor.However, the contribution of magnetic symmetry, which captures the effects of SO interactions, cannot be ignored as it can lead to magnetic order and subsequently affect the resistivity tensor.Therefore, the influence of both spin and magnetic symmetries on the resistivity tensor needs to be considered in order to fully understand the transverse resistivity component.We performed a qualitative analysis of the magnetic symmetry in all three phases.The magnetic symmetry group of the high-symmetry triangle and flower phases still includes third-order rotations, which results in the exclusion of transverse resistivity components in these cases.However, in the low-symmetry fork phase, the appearance of in-plane magnetization is related to the rotation of spins out of the {111} crystallographic planes 38,39 .This symmetry reduction allows for the presence of transverse components of resistivity.Moreover, the magnitude and direction of these components are dependent on the orientation of the magnetic field and do not average out.

Sample preparation
Sintered polycrystalline samples of the composition Mn3Zn0.5Ge0.5Nand Mn3Ag0.95Nwere prepared by solid-state reaction using fine powders of Mn2N, Zn, Ge, and Ag as starting materials 21 .The well-mixed powders in stoichiometric proportion were pressed into pellets and then sealed in a quartz tube under vacuum (10 -5 Pa) by wrapping in a Ta foil.The quartz tube was sintered in a box furnace at 1073 K for 80 hours, and then cooled down to room temperature.

Sample characterization
The crystalline and magnetic structure was characterized by the time-of-flight (T.O. F.) diffractometer GPPD (general purpose powder diffractometer) at CSNS (the China Spallation Neutron Source), Dongguan, China.The neutron beam of GPPD with the high intensity and good spatial resolution makes it well suited for the accurate analysis of magnetic materials with complex orderings 40 .Samples are loaded in a vanadium holder and installed in a liquid helium cryostat that achieves temperatures down to 2 K and magnetic fields up to 70 mT.The neutron diffraction patterns are collected in the T. O. F. mode with wavelength bands of 0.1 -4.9 Å between 2 K and 300 K.The General Structure Analysis System (GSAS) program was used for Rietveld refinement with scattering lengths of -0.373×10 -12 , 0.568×10 -12 , 0.818×10 -12 , 0.592×10 -12 , and 0.936×10 -12 cm for Mn, Zn, Ge, Ag, and N, respectively 41 .The compositions determined by the Rietveld analysis of the neutron diffraction data at 300 K is Mn3Zn0.54Ge0.46Nand Mn3Ag0.93Nrespectively.Electronic-transport properties and the specific heat were measured in a physical property measurement system (PPMS, Quantum Design) for temperatures 1.8 -400 K and in magnetic fields up to 9 T. Measurements down to 0.1 K have been taken in a 3 He/ 4 He dilution refrigerator using a Lakeshore 370 ac resistance bridge.The longitudinal resistivity ║ and the transverse resistivity ┴ were measured on samples of rectangular shape with 0.12-mm thick Cu wires glued by silver epoxy.A current I was passed through the sample along the long side of the sample and voltages V║ and Vy were measured along and perpendicular to the current direction, respectively, to yield ║ = V║wt/I l and ┴ = V┴ t/I, where w, t, and l are the width, thickness, and distance between the voltage contacts, respectively.The magnetic field H was applied perpendicularly to both ║ and ┴ directions except for angle-dependent measurements where it was rotated towards the ║ and ┴ directions.Temperature-dependent magnetization data between 2 and 30 K and isothermal magnetization curves were recorded in a superconducting quantum-interference device (SQUID) magnetometer for magnetic fields up to 5 T. In addition, the temperature dependence of magnetization between 300 and 580 K was also measured by a vibration sample magnetometer (VSM).S3).The boundaries H0 and H1 between the different spin configurations are determined from the change of slope in resistivity as indicated for T = 1.8K in Fig. 2d, specifying the regions of spin configurations "triangle" Γ 5g , "fork" FO, "intermediate" I, and "flower" FL.Solid lines represent the theoretical predictions of the phase diagram.The FL phase is above the highest transition line.The I phase is a mixture of two low-symmetry phases that are obtained due to different components of the magnetic field (in-plane and outof-plane).Dashed line represents the transition between Γ 5g and FL which is not observed in the resistivities.Table 1 | Properties of the spin phases.Spin symmetry group operations and components of the resistivity tensor are defined in the spin-related frame: the z axis is perpendicular to the ordering plane spanned by the vectors   and   (or is along 3 rd order axis), the x axis is parallel to the vector   (or is along 2 nd order axis).For the resistivity tensor only non-zero components are given.

Fig. 2 |
Fig. 2 | Longitudinal and transverse resistivity.a, b, Temperature dependence of the longitudinal resistivity ρ║(T) and the transverse resistivity ρ┴(T) for various magnetic fields H applied perpendicularly to the current direction x for two samples cut from the same batch.Inset shows the ρ║(T) up to room temperature.c, d, ρ║(H) and ρ┴(H) in perpendicular magnetic field H at various temperatures T. The transition field H0 was determined from the resistivity at 1.8 K to 30 mT for a, c and 120 mT for b, d, respectively.Solid lines in b and d represent calculations according to the model, see text.

Fig. 3 |
Fig. 3 | H-T phase diagram derived from the transition in ρ║(T,H) and ρ┴(T,H) measured down to 0.1 K (see Supplementary Fig.S3).The boundaries H0 and H1 between the different spin configurations are determined from the change of slope in resistivity as indicated for T = 1.8K in Fig.2d, specifying the regions of spin configurations "triangle" Γ 5g , "fork" FO, "intermediate" I, and "flower" FL.Solid lines represent the theoretical predictions of the phase diagram.The FL phase is above the highest transition line.The I phase is a mixture of two low-symmetry phases that are obtained due to different components of the magnetic field (in-plane and outof-plane).Dashed line represents the transition between Γ 5g and FL which is not observed in the resistivities.

Fig. 4 |
Fig. 4 | Neutron diffraction patterns of Mn3Zn0.5Ge0.5N.a, Intensity difference IT-I10 = I(T)-I(10 K) of neutron diffraction patterns.The data are shifted with respect to each other for clarity.b, Temperature dependence of the integrated intensity of reflections P1, P2, P3, and P4, occurring at 0.78 Å, 0.93Å, 1.60 Å, and 2.77 Å, respectively, in a.For clarity, data of P3 and P4 are shifted upward by 0.1 and 0.2, with respect to P1 and P2, respectively.The amount of magnetic phase estimated from the Rietveld refinement is displayed in the inset.c, Temperature dependence of the local magnetic moments M2x and M2y of Mn atoms and the angle  determined by Rietveld refinement where a nonzero M2y indicates the direction of a deviation from the triangle Γ 5g -type order.The temperature range where the spontaneous magnetoresistivity occurs is indicated by a cyan colored background.The lines in the figure are guides to the eye.d, Field dependence of the integrated intensity of reflections P2, P3, and P4 at 3 K < T*.Intensities of P2 and P3 have been shifted upwards by 3.95 and 1.07 with respect to P2 and P3 respectively, for clarity.