Transport evidence of current-induced nematic Dirac valleys in a parity-time-symmetric antiferromagnet

Domain: physics
Published: 2026-01-07
Editorial outcome: minor_revision

Abstract
Itinerant antiferromagnets with broken time-reversal symmetry have recently attracted attention, since their spin-split bands enable large magnetotransport responses comparable to ferromagnets despite the negligible spontaneous magnetisation. When the inversion symmetry is further broken by the antiferromagnetic order, the emerging odd-parity multipole order render the bands spin-degenerate but asymmetric in the momentum space. For such parity-time-symmetric antiferromagnets, it has been predicted that electronic nematicity is induced by current, allowing unconventional nonlinear transport phenomena. However, their experimental evidence has been lacking. Here, we report two-fold nonreciprocal angular magnetoresistance in the tetragonal layered Dirac material SrMnBi2 with parity-time-symmetric antiferromagnetic order in its Mn-Bi layers. By quantitatively modelling the angular and field dependencies using a phenomenological framework, we reveal that the observed nonreciprocal interlayer resistivity arises from the current-induced breaking of four-fold rotoinversion symmetry of the Dirac valleys in the Bi square net adjacent to the Mn-Bi layer. Furthermore, we demonstrate the alignment of parity-time-symmetric antiferromagnetic domains via current-field cooling, achieving electric-magnetic control of the f-wave polarity in momentum space. The observed switchable nonreciprocal transport associated with current-tuneable valley nematicity paves the way for novel antiferromagnetic spintronic and valleytronic applications.

Introduction
Antiferromagnets have many advantages for spintronic applications, such as resistance to perturbing magnetic fields and the absence of stray fields. For device operation, it is necessary that antiferromagnets exhibit clear magnetotransport responses like ferromagnets (e.g. magnetoresistance and Hall effects), although they do not in principle have spontaneous magnetisation. Recently, large anomalous Hall and Nernst effects comparable to ferromagnets were discovered in several antiferromagnets in spite of the negligibly small spontaneous magnetisation. Their electrically switchable properties have garnered significant attention, both for their potential applications and for their relevance to fundamental physics. Notable examples include non-collinear magnets hosting cluster multipoles, as well as the so-called altermagnets. A defining characteristic of these antiferromagnets is that their time-reversal (T) symmetry is intrinsically broken due to their unique magnetic and crystal structures. This causes spin splitting in the band structure, which is the source of their distinct magnetotransport phenomena.

In those antiferromagnets, the parity (P) symmetry associated with the spatial inversion is usually preserved. However, when the P-symmetry is also broken by the magnetic order, the band structure undergoes significant changes. In such cases, both the P- and T-symmetries are broken, while the product PT-symmetry is preserved. The impact of PT-symmetry on band structure was initially discussed for insulating magnetoelectric materials, such as Cr2O3, and has recently been extended to metallic magnets. Based on the electric/magnetic multipole theory, the PT-symmetric magnetic order is characterised by the odd-parity magnetic multipole order, resulting in a spin-degenerate band structure that is asymmetric in the momentum space. This unique band structure was theoretically predicted to give rise to various intriguing transport and optical phenomena, including nonlinear electrical conductivity and photocurrent generation. While a few related experimental observations, such as nonlinear Hall effects and bulk photogalvanic effects, have been reported, more direct manifestations of the asymmetric band structure have yet to be fully investigated. One such phenomenon is the current-induced shift of asymmetric bands, which breaks the original symmetry and results in nematic distortion of the Fermi surface. This electronic deformation can generate the lattice deformation via the electron-phonon coupling, which is a magnetic and metallic counterpart of piezoelectricity (called magnetopiezoelectric effect). In fact, it was experimentally observed that a very small change in sample length was caused by current in several PT-symmetric antiferromagnets. However, intrinsic electronic responses (e.g. transport or optical changes) directly linked to the nematic deformation of the Fermi surface remain experimentally unexplored.

In this study, we have investigated transport phenomena associated with current-induced nematic deformation in the layered PT-symmetric antiferromagnet SrMnBi2. In this material, the local Mn spins surrounded by the Bi tetrahedron exhibits a staggered antiferromagnetic order. Therefore, the spin-up Mn is located at the centre of the red-coloured Bi tetrahedron, while the spin-down Mn is located at the centre of the blue-coloured Bi tetrahedron (Fig. 1a). This leads to the simultaneous breaking of the P- and T-symmetries in a nonpolar manner, as the up-spin state in the red tetrahedron and the down-spin state in the blue tetrahedron are not related by either P- or T-operations (Fig. 1a, c). Although this Mn-Bi layer is a Mott insulator, SrMnBi2 also has a highly-conductive Bi square net layer sandwiched between the Mn-Bi layers, forming quasi-2D (gapped) Dirac bands. Since a small amount of hole-type carriers are naturally doped, these bands allow the metallic conduction not only in the in-plane direction, but also in the out-of-plane direction by forming coherent cylindrical Fermi surfaces at low temperatures (see the metallic interlayer resistivity below ~150 K in Fig. 1b). Therefore, SrMnBi2 provides an ideal arena to study the current effects on a PT-symmetric antiferromagnet. Furthermore, the Dirac bands in this material possess valley degrees of freedom (see below), enabling unconventional valley symmetry breaking in conjunction with PT-symmetric antiferromagnetic order.

We here concretely explain the asymmetric nature of the quasi-2D Dirac bands in SrMnBi2. On the kz=0 plane, these bands form exactly four-fold elliptic valleys (Fig. 1d). However, the kz dispersion, namely the warping of the cylindrical Fermi surface, is asymmetric with respect to kz=0 due to the antisymmetric spin-orbit coupling arising from the odd-parity multipole order in the Mn-Bi layer. First-principles calculations show that the cylindrical Fermi surface in the kz>0 (kz<0) regime contracts (expands) for the valley A elongated along [110] direction, while the opposite kz dispersion occurs for the valley B elongated along the [1-10] direction, as schematically shown in Fig. 1e (For the quantitative kz dispersion, see Fig. S1). As a result, the total Fermi surfaces composed of both valleys preserve four-fold rotoinversion (combination of 90° rotation and spatial inversion) symmetry.

Results
To achieve high current density without significant current heating effects, we fabricated micro-devices (Fig. 2a) capable of reaching a current density of jz~5×107 A/m2 with an applied current of Iz=200 μA. Fig. 2c shows the AMR profile (i.e. azimuth angle φ dependence) of the first-harmonic component of interlayer resistivity ρωzz at 1.5 K at 12 T. As expected for the four-fold symmetry of the Fermi surfaces at jz=0, the observed ρωzz(φ) shows narrow dips for B||[110] and [1-10] (φ=0°, 90°, ... ) while it shows broad peaks for B||[100] and [010] (φ=45°, 135°, ... ), as reported previously. The observed four-fold symmetry of ρωzz(φ) is consistent with the four-fold rotoinversion symmetry of the Fermi surfaces at jz=0, since ρωzz(φ) involves integration over the full kz range (see Methods for a theoretical expression of the first-harmonic component of conductivity). Intuitively, the direction of the current-induced nematic deformation switches by 90° depending on the sign of jz, and its overall effect on ρωzz(φ) averages out, yielding four-fold symmetry in the linear transport. In contrast, the second-harmonic component of interlayer resistivity ρ2ωzz(φ) exhibits distinct two-fold symmetry: sharp dips are located at φ=0°, 180° (B||[110], [1-10]), while sharp peaks are located at φ=90°, 270° (B||[1-10], [110]). This clearly indicates that the valleys A and B are non-equivalent when nonzero jz is applied, breaking the four-fold symmetry in the nonlinear transport. Reflecting that the deformation of each valley is driven by current, the peak height of ρ2ωzz (denoted by Δρ2ωzz in Fig. 2d) is almost proportional to jz (Fig. 2e). It should be noted here that essentially the same two-fold symmetric ρ2ωzz(φ) was observed in a micro-fabricated device with a different surface orientation (Fig. S3), indicating that the present results are independent of the crystal cutting geometry.

The profile of ρ2ωzz(φ) strongly depends on magnetic field (Fig. 3b). At 2 T, it exhibits a sine-like curve with two-fold symmetry. However, with increasing field, the dip (peak) structures located at φ=0°, 180° (φ=90°, 270°) progressively evolve, accompanied by the gradual change in the weak background component. As discussed below, the observed variation in ρ2ωzz(φ) is explained by the field dependence of ρωzz(φ); the dips of ρωzz(φ) (at φ=0°, 90°, 180°, ...) become deeper and narrower with increasing field (Fig. 3a).

To formulate the relation between ρ2ωzz(φ) and ρωzz(φ), we here employ a phenomenological model of interlayer magnetoconductivity σzz (=1/ρωzz) taking account of the in-plane anisotropy of quasi-2D Dirac valley:

σzz(φ)= 1+2rσ2Dcos2φ / 1+rcos2φ + 1+rσ2Dcos2(φ+π/2) / 1+rcos2(φ+π/2) + σ3D, (1)

where σ2D (σ3D) is the relative contribution of each quasi-2D Dirac valley (all 3D Fermi surfaces from the parabolic bands). r is the anisotropic factor of magnetoconductivity, resulting in the maximum (minimum) conductivity for the field along the shorter (longer) axis of the elliptic Dirac valley. Note here that the first (second) term corresponds to the contribution from the valley A (B), which gives σzz peaks, i.e. ρzz dips at φ=90°, 270° (φ=0°, 180°). The experimental profiles of ρωzz(φ) at various magnetic fields are nicely fitted by Eq. (1), as demonstrated in Fig. 3c. Reflecting the deeper and narrower dips in ρωzz(φ) at increasing field, the values of σ2D and r are monotonically increased with the field (the same is true for σ3D, see Fig. S2). These fitted results are consistent with those reported in the literature.

We now take account of the impact of the current-induced nematicity on Eq. (1). Considering that the valleys A and B are non-equivalent in the presence of jz, the nematicity can be incorporated as current-induced changes in r and σ2D as follows

r -> r(1±ϵr), (2)

σ2D -> σ2D(1±ϵ2D) (3)

where ϵr(∝jz) and ϵ2D(∝jz) are dimensionless variations in r and σ2D, respectively. Note here that the + sign corresponds to the first term (valley A) in Eq. (1), while the - sign corresponds to the second term (valley B) in Eq. (1). The resultant variation in σzz is given to the first order of ϵr and ϵ2D by (see “Methods”)

σzz(φ) -> σzz(φ)+δσzz(φ), (4)

δσzz(φ) = ( (cosφ)^2 / 1+rcos2φ - (cos(φ+π/2))^2 / 1+rcos2(φ+π/2) ) * 2σ2D * ϵr + ( 1 / 1+rcos2φ + 1 / 1+rcos2(φ+π/2) ) * 2σ2D * ϵ2D (5)

Since ϵr and ϵ2D are proportional to jz, δσzz is also proportional to jz, corresponding to the nonreciprocal component of the interlayer conductivity. By converting the nonreciprocal conductivity to resistivity using the relation ρ2ωzz = -(δσzz/σzz^2) * jz (see “Methods”), we fit the ρ2ωzz(φ) profiles at various fields using the same parameters, while σ2D and r were fixed to the values obtained from fitting ρωzz(φ) at each field (Fig. 3c). The experimental ρ2ωzz(φ) profiles are well reproduced by Eq. (5) regardless of field, as indicated by the solid curves in Fig. 3b. This demonstrates that the field dependence of ρ2ωzz(φ) arises solely from the field dependence of ρωzz(φ). The extracted values of ϵ2D and ϵr are nearly field-independent and are of the order of 10−5 (Fig. 3d). Consequently, the magnitude of current-induced nematicity is estimated to be of the order of 10−5 for jz~5×107 A/m2 (Iz=200 μA).

In the previous study, current-induced lattice displacement was measured in EuMnBi2, where the largest strain was ~1.4×10−8 at jz~4×104 A/m2 at the lowest temperature. This corresponds to the strain of ~2×10−5 at jz~5×107 A/m2 (the same jz used in this study). Thus, the nematicity estimated from the AMR profile is of the same order of magnitude as that revealed by the lattice displacement measurements.

Here, we discuss the temperature dependence of ρ2ω(ϕ). The amplitude of the ρ2ω(ϕ) peaks and dips decreases with increasing temperature and nearly vanishes above 30 K (Fig. S4d), which is well below the Néel temperature TN=295 K. This arises primarily from the suppression of the AMR ρωzz(ϕ) at elevated temperatures (Fig. S4c), since ρ2ωzz(ϕ) originates from the current-induced modulation of ρωzz(ϕ). Additionally, a decrease in the current-induced nematicity (i.e. ϵ2D and ϵr) with increasing temperature, as was previously reported in the lattice displacement measurements in EuMnBi2, may contribute to the suppression of ρ2ω(ϕ) compared to ρωzz(ϕ). These results suggest that the temperature dependence of ρ2ω(ϕ) involves multiple underlying mechanisms.

Finally, we demonstrate the detection and control of magnetic domains in the present PT-symmetric antiferromagnet. Fig. 4c, d show the profiles of ρ2ωzz(φ) for the two adjacent devices #1 and #2, respectively, after cooling from room temperature without applying any field or current. Notably, the positions of the sharp peaks and dips of ρ2ωzz(φ) are reversed between #1 and #2, apart from the relatively large background component in #2. This clearly indicates that the sign of ϵr and ϵ2D, i.e. the domain of antiferromagnetic order, differs between the two devices, even within the same micro-fabricated crystal (inset to Fig. 4c, d).

To align the domains, we employed a poling procedure conducted above TN. Symmetry analysis predicts that the domains of PT-symmetric antiferromagnetic order in this nonpolar system can be aligned by applying an electric current and a magnetic field orthogonal to each other, as illustrated in Fig. 4b (see discussions below). To achieve this experimentally, we applied a current of 1 mA (jz~3×108 A/m2) along [001] and a field of 18 T along [110] at 305 K (>TN), and then cooled the sample to 4.2 K. After poling, we repeated the AMR measurements at 1.5 K and 12 T. In device #1, the peak and dip of ρ2ωzz(φ) are inverted compared to the previous measurement without poling (Fig. 4e). On the other hand, in device #2, the magnitude of the peak and dip of ρ2ωzz(φ) is enhanced while keeping their positions (Fig. 4f). These results indicate that the antiferromagnetic domains in both devices were aligned to the original domain in device #2 through the poling procedure, thereby demonstrating electric-magnetic manipulation of the nonpolar PT-symmetric antiferromagnetic order. We note that the gradual background component in device #2 does not change before and after the poling, suggesting that it may arise from pinned domains and/or other extrinsic effects.

We next discuss the mechanisms underlying this control of antiferromagnetic domains in the present material. Recently, several current-driven domain control methods have been developed for metallic antiferromagnets, although their variety remains limited. For T-odd antiferromagnets with spin-split bands, conventional control mechanisms, similar to those in ferromagnets, can be employed. Examples includes spin-orbit torque switching at the interfaces of artificial heterostructures. In contrast, the control mechanisms for PT-symmetric antiferromagnets with spin-degenerate bands are significantly more intricate. In the polar systems, such as CuMnAs and Mn2Au, current-induced switching was achieved via the sublattice-dependent spin-momentum locking. However, in nonpolar systems like SrMnBi2, the absence of polarity (p-wave) in momentum space prevents the domain switching using current alone. Instead, the higher-order f-wave polarity in momentum space can be utilised. To this end, we have applied an in-plane field, effectively rendering the system polar and enabling current-induced switching. This combined electric and magnetic manipulation of PT-symmetric antiferromagnetic domains provides a novel protocol distinct from those employed for T-odd or polar antiferromagnets currently under active investigation.

For PT-symmetric antiferromagnets, a variety of nonlinear conduction phenomena, classified by the dependence on the relaxation time, were theoretically predicted. In this context, our study focuses on the Drude component to provide direct evidence of current-induced electronic nematicity, manifested as nonlinear interlayer conduction along the high-symmetry c-axis under a magnetic field. In addition, SrMnBi2 is also predicted to exhibit other types of nonlinear in-plane conduction along the ab plane, such as the (zero-field) nonlinear Hall effect arising from the quantum geometric component. Exploring in-plane nonlinear responses may thus represent an important direction for future research in nonpolar PT-symmetric antiferromagnets.

To conclude, we report the observation of current-induced electronic nematicity in the nonpolar PT-symmetric antiferromagnet SrMnBi2 by measuring the nonreciprocal angular magnetoresistance (AMR) effect. The breaking of the original four-fold rotoinversion symmetry is detectable via the valley degrees of freedom of Dirac fermions in the Bi square net, which is adjacent to the Mn-Bi tetrahedral layer exhibiting PT-symmetric antiferromagnetic order. By employing a phenomenological model of the AMR effect attributed to elliptic Dirac valleys, we quantitatively reproduce the two-fold nonreciprocal AMR signal, demonstrating the current-induced lifting of the valley degeneracy. The layered structure incorporating a Dirac fermion layer in this material offers a novel platform for the electrical control of valleys by the PT-symmetric antiferromagnetic order. Our findings may pave the way for developing unconventional spintronic and valleytronic devices, thereby broadening the scope of antiferromagnetic spintronics.

Methods
Sample preparation and device fabrication
Single crystals of SrMnBi2 were grown by a Bi self-flux method. The alumina crucible filled with Sr, Mn, and Bi metal ingots (Sr:Mn:Bi=1:1:9) was sealed in an evacuated quartz tube. The ampoule was heated at 1050 °C and then cooled to 400 °C at the rate of 2 °C/h, where excess Bi flux was decanted by centrifuge.

To measure the interlayer resistivity with a high current density, we fabricated micro-devices using a focused ion beam (FIB) technique. The typical dimensions of the fabricated device are 2 μm × 2 μm × 3 μm. The interlayer resistivity ρzz and ρ2ωzz were measured by the conventional AC-lockin measurement with a typical frequency of 23 Hz. ρ2ωzz was obtained as ρ2ωzz=(V2ωzz/Iz)=(l/S), where V2ωzz is the imaginary part of the second-harmonic voltage drop, Iz is the RMS value of AC current, l is the length of voltage terminals, and S is the cross section of current path. To obtain the B dependence, we symmetrise V2ωzz with respect to B. The measurements were performed using commercial superconducting magnets, where the direction of magnetic field was controlled using a probe equipped with a two-axis rotating stage in Institute for Materials Research, Tohoku University.

Theoretical expression of first-harmonic conductivity along the c-axis for in-plane magnetic fields
From the Boltzmann equation, the conductivity along the c||z-axis under the magnetic field B in the present material is given by:

σzz(B) ∝ τ Σ_valley ∫ (vz(k,t=0) * vz(k,t)) * (-∂f/∂E) dk,

where f is the Fermi distribution function, τ is the constant scattering time. vz(k) is the z component of Fermi velocity, which is nonzero due to kz dispersion E(k) of quasi 2D Fermi surfaces. In the semiclassical picture, electron motion in a magnetic field is described by the equation of motion:

ℏk_dot = -ev×B, v = 1/ℏ * ∂E(k)/∂k,

which determines the time evolution of k(t). The term 〈vz(k, t)〉 represents the time-averaged z component of the velocity within the relaxation time τ.

Formulation of the fitting function of ρ2ωzz
Considering the current-induced nematicity described by Eq. (2), the variation of the first term (associated with valley A) in Eq. (1) can be expressed, to the first order of ϵr and ϵ2D, as:

2σ2D(1+ϵ2D) / 1+r(1+ϵr)cos2φ ∝ 2σ2D / 1+rcos2φ * ϵ2D - 2σ2D * r * cos2φ / (1+rcos2φ)^2 * ϵr

The second and third terms in the above equation originate from the current-proportional variations in r and σ2D, respectively. Similarly, the variation of the second term in Eq. (1) can be derived to the first order of ϵr and ϵ2D, resulting in the total variation given by δσzz(φ) in Eq. (5).

The relation between the current density jz and the applied electric field Ez is expressed as:

jz = (σzz+δσzz)Ez

From this, we obtain

Ez = jz / (σzz+δσzz) ∝ jz/σzz * (1 - δσzz/σzz)

ρ2ωzz ∝ -δσzz/σzz^2 * jz

where σzz and δσzz in the final equation correspond to ρzz and ρ2ωzz, respectively.

First-principles calculation
We performed first-principles band structure calculations based on the density functional theory with the generalised gradient approximation with the Perdew-Burke-Ernzerhof parametrization and the projector augmented wave (PAW) method as implemented in the Vienna ab initio simulation package. We used the experimental crystal structure with the antiferromagnetic ordering shown in Fig. 1a. Core-electron states in PAW potentials were [Ar]3d10, [Ar], and [Xe]5d10 for Sr, Mn, and Bi, respectively. We used a plane-wave cutoff energy of 400 eV for Kohn-Sham orbitals with including the spin-orbit coupling. A 16×16×16 k-mesh was used. After first-principles band calculation, we extracted Mn-d + Bi-p Wannier orbitals using Wannier90 software. Fermi surfaces were depicted using the tight-binding model consisting of these Wannier orbitals.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Referee report on Sakai et al. Nature Communications\nThe manuscript reports on current induced symmetry breaking in a PT symmetric antiferromagnet. It also shows how, with a combination of current and magnetic field, domains can be controlled in this system.\nI evaluate this as a very strong paper that brings the field of antiferromagnetic spintronics a step forward. I strongly support publication in Nature Communications.\nThe paper is well written. Figures are clear, the data of high quality and the line of arguments convincing.\nIn principle, the paper can be published after addressing the minor suggestions/questions for clarification below.\n1. In figure 1e, the authors say that the 4-fold symmetry is preserved. However, in the sketches of the FS, it looks like it is broken. The authors should explain better, why the 4-fold symmetry is not broken here.\n2. When writing about the results of Fig. 2, the authors should already make the statement that the results are independent of the geometry of the sample as they show in the SI\n3. The authors should mention the Neel temperature of this material and comment on the temperature dependence with respect to that temperature. Why does the effect die off with increasing temperature so quickly?\nIn the paragraph starting with “In this study, …”\nIt should be “is located at…”\nThere is a bracket with (vide in fra). What does that mean?", "reviewer_2": "Itinerant antiferromagnets with broken time-reversal symmetry provide powerful platforms for creating and controlling novel quantum states of matter. In this manuscript, the author reports nonreciprocal angular magnetoresistance in the layered SrMnBi2 and attributes it to the broken four-fold symmetry of the Dirac valleys. I found this work interesting. However, I have some comments that need to be addressed by the author, before I recommend the publication of this manuscript in Nature Communications.\n1. Fermi surfaces for SrMnBi2 at Kz = 0 possesses four-fold symmetry, while at Kz <0 and > 0 shows two-fold symmetry. Although SrMnBi2 is of layered structure, the obvious Kz shows 3D electronic structure. Under Iz, how to understand the angular-dependent magnetoresistance?\n2. Experimental implementation of weak symmetry is difficult. A high-quality micro-fabricated device is critically important, the weak symmetry of the angular-dependent magnetoresistance becomes increasingly prominent as the current increases. Angle dependence of the first-harmonic interlayer resistivity show four-fold symmetry, and the second-harmonic resistivity show two-fold symmetry. Here, the weak enhancement of the two-fold symmetry is attributed to the broken four-fold symmetry. Why doesn’t this refer to an intrinsically weak two-fold symmetry in the electronic structure?\n3. The angular-dependent magnetoresistance for the current flowing in different directions within the ab plane may help to understand the present results.\n4. Further experimental observations is required to confirm the conclusions presented in the paper."}, "author_response": "--- Authors’ response to Reviewer #1---\nReviewer #1’s comment:\nThe manuscript reports on current induced symmetry breaking in a PT symmetric\nantiferromagnet. It also shows how, with a combination of current and magnetic field,\ndomains can be controlled in this system.\nI evaluate this as a very strong paper that brings the field of antiferromagnetic spintronics a\nstep forward. I strongly support publication in Nature Communications.\nThe paper is well written. Figures are clear, the data of high quality and the line of arguments\nconvincing.\nIn principle, the paper can be published after addressing the minor suggestions/questions\nfor clarification below.\nAuthors’ response:\nWe sincerely thank the Reviewer #1 for carefully reading our manuscript and\nrecommending the publication in Nature Communications. We also appreciate the\nconstructive suggestions, which have helped us to improve the quality and clarity of our\nwork. Below, we provide detailed responses to each of the reviewer’s comments. All\nrevisions made in response to the comments are summarized in “Summary of the\nRevisions,” and are clearly marked in red in both the revised manuscript and the\nSupplementary information.\nReviewer #1’s comment (1):\n1. In figure 1e, the authors say that the 4-fold symmetry is preserved. However, in the\nsketches of the FS, it looks like it is broken. The authors should explain better, why the 4-fold\nsymmetry is not broken here.\nAuthors’ response (1):\nWe thank the Reviewer #1 for this important comment. As the reviewer points out, the\nfour-fold symmetry is broken in the Fermi surface (FS) cross-sections on the 𝑘 >0 and\n!\nk <0 planes due to the antisymmetric spin–orbit interaction arising from the\n!\nPT-symmetric antiferromagnetic order. However, it should be noted that the manner in\nwhich the four-fold symmetry is broken is opposite between the 𝑘 >0 and 𝑘 <0 regions.\n! !\nSpecifically, valley A (B) contracts (expands) for 𝑘 >0, whereas valley A (B) expands\n!\n(contracts) for 𝑘 <0, as shown in the right panels of Fig. 1e. As a result, the four-fold\n!\nrotoinversion symmetry of the FSs is exactly preserved over the full 𝑘 range in the Brillouin\n!\nzone.\nFrom the conventional transport theory for the c-axis conductivity under the in-plane fields,\nthis four-fold rotoinversion symmetry of the full FS ensures the four-fold 𝜑 dependence of\nfirst-harmonic component 𝜌\"(𝜑) as follows. In the semiclassical picture, the electron\n!!\norbital motion under magnetic field 𝑩 is described by\n1𝜕𝐸(𝒌)\nℏ𝒌̇ =(−𝑒)𝒗×𝑩, 𝒗= .\nℏ 𝜕𝒌\nUsing the Boltzmann equation, the first-order conductivity along the c axis is calculated as\n𝜕𝑓\n𝜎 (𝑩)∝ : :𝜏𝑣 (𝒌,𝑡 =0)〈𝑣 (𝒌,𝑡)〉(− ),\n!! ! ! 𝜕𝐸\n$%&&'( 𝒌\nwhere 𝑓 is the Fermi distribution function, t the constant scattering time. 𝑣 (𝒌) is the z\n!\ncomponent of Fermi velocity, which is nonzero due to the 𝑘 dispersion 𝐸(𝒌) of quasi 2D\n!\nFermi surface. 〈𝑣 (𝒌,𝑡)〉 is its time average within the scattering time. The key point is that\n!\nthe summation is taken over all k-points and valleys, which means that the symmetry of the\nentire FSs over the Brillouin zone determines that of the first-order (i.e., first harmonic\ncomponent of) conductivity. Therefore, the four-fold symmetry of 𝜑 dependence of 𝜌\"(𝜑)\n!!\nis preserved in SrMnBi .\nFS as illustrated in Fig. 1e and its relation to the four-fold 𝜑 dependence of 𝜌\"(𝜑).\n!!\nPlease see the Summary of the Revisions (1), (3), (7), and (10).\nReviewer #1’s comment (2):\n2. When writing about the results of Fig. 2, the authors should already make the statement\nthat the results are independent of the geometry of the sample as they show in the SI\nAuthors’ response (2):\nWe thank the Reviewer #1 for this helpful suggestion. Following this, we have added a\nstatement addressing the reproducibility among the different sample geometries to the\nparagraph discussing the results presented in Fig. 2 [Summary of the Revisions (4)].\nReviewer #1’s comment (3):\n3. The authors should mention the Neel temperature of this material and comment on the\ntemperature dependence with respect to that temperature. Why does the effect die off with\nincreasing temperature so quickly?\nAuthors’ response (3):\nWe appreciate this insightful comment. The Neel temperature of SrMnBi is 295 K, below\n2\nwhich the current-induced nematicity should emerge in principle. However, experimentally\nthe peak/dip features of second-harmonic component 𝜌)\"(𝜑) disappear above ~30 K.\n!!\nThis\narises primarily from the suppression of the AMR of first-harmonic component 𝜌\"(𝜑) [≡\n!!\n𝜌\"(𝜑 =45°)/ 𝜌\"(90°)] with increasing temperature. At 50 K, for instance, the magnitude of\n!! !!\nthe AMR drops to less than one-tenth of its value at 1.5 K. Under such conditions, the\nsecond-harmonic component 𝜌)\"(𝜑) becomes extremely difficult to detect, since it\n!!\noriginates from the current-induced modulation of 𝜌\"(𝜑) as described in Eqs. (4) and (5).\n!!\nTo reveal the relationship between the temperature dependence of 𝜌)\"(𝜑) and 𝜌\"(𝜑),\n!! !!\nwe have added data on the temperature dependence of AMR of 𝜌\"(𝜑) to the revised\n!!\nsupplementary Fig. S4 [Summary of the Revisions (8)]. Interestingly, this data (Fig. S4c and\nFig. 4d) reveals that the 𝜌)\"(𝜑) signal seems to diminish more rapidly with increasing\n!!\ntemperature than the 𝜌\"(𝜑) signal, particularly in the low-temperature range. This\n!!\nsuggests that the current-induced nematicity itself (quantified by 𝜖 and 𝜖 ) also weakens\n* )+\nwith increasing temperature. In support of this, the previous study (Ref. [36]) reported that\ncurrent-induced lattice displacement in EuMnBi₂ rapidly decreases with temperature,\nespecially below ~30 K. At present, understanding the origin of this temperature\ndependence remains an important subject for future theoretical and experimental\ninvestigation.\nWe sincerely thank the reviewer for highlighting this point. In the revised manuscript, we\nhave added a paragraph to discuss the temperature dependence of 𝜌)\"(𝜑) signals\n!!\npresented above [Summary of the Revisions (5)].\nReviewer #1’s comment (4):\nIn the paragraph starting with “In this study, …”\nIt should be “is located at…”\nThere is a bracket with (vide in fra). What does that mean?\nAuthors’ response (4):\nWe thank the Reviewer #1 for kindly pointing out these typos. They have been corrected\nin the revised manuscript. [Summary of the Revisions (11)]\n--- Authors’ response to Reviewer #2---\nReviewer #2’s comment:\nItinerant antiferromagnets with broken time-reversal symmetry provide powerful platforms\nfor creating and controlling novel quantum states of matter. In this manuscript, the author\nreports nonreciprocal angular magnetoresistance in the layered SrMnBi2 and attributes it to\nthe broken four-fold symmetry of the Dirac valleys. I found this work interesting. However, I\nhave some comments that need to be addressed by the author, before I recommend the\npublication of this manuscript in Nature Communications.\nAuthors’ response:\nWe thank the Reviewer #2 for carefully reading our manuscript and for giving fruitful\ncomments to improve our paper. Below we respond all the comments and accordingly\nrevised the manuscript. We believe that, with these revisions, the manuscript is now suitable\nfor publication in Nature Communications. All revisions made in response to the comments\nare summarized in “Summary of the Revisions,” and are clearly marked in red in both the\nrevised manuscript and the Supplementary information.\nReviewer #2’s comment (1):\n1. Fermi surfaces for SrMnBi2 at Kz = 0 possesses four-fold symmetry, while at Kz <0 and >\n0 shows two-fold symmetry. Although SrMnBi2 is of layered structure, the obvious Kz shows\n3D electronic structure. Under Iz, how to understand the angular-dependent\nmagnetoresistance?\nAuthors’ response (1):\nWe thank the Reviewer #2 for this comment. In this study, we applied the current along\nthe z direction and measured resistance along the z direction, while rotating magnetic field\nwithin the xy plane. Such interlayer conductivity under the in-plane fields can be understood\nin the following semiclassical picture. The electron orbital motion under magnetic field 𝑩 is\ndescribed by\n1𝜕𝐸(𝒌)\nℏ𝒌̇ =(−𝑒)𝒗×𝑩, 𝒗= .\nℏ 𝜕𝒌\nUsing the Boltzmann equation, the first-order conductivity along the 𝑧||𝑐 axis is calculated\nas\n𝜕𝑓\n𝜎 (𝑩)∝ : :𝜏𝑣 (𝒌,𝑡 =0)〈𝑣 (𝒌,𝑡)〉(− ),\n!! ! ! 𝜕𝐸\n$%&&'( 𝒌\nwhere 𝑓 is the Fermi distribution function, t the constant scattering time. 𝑣 (𝒌) is the z\n!\ncomponent of Fermi velocity, which is nonzero due to the 𝑘 dispersion 𝐸(𝒌) of quasi 2D\n!\nFermi surface (i.e., 3D electronic structure). 〈𝑣 (𝒌,𝑡)〉 is its time average within the\n!\nscattering time.\nIn the above expression of the first-order conductivity, the summation is taken over\nall k-points and valleys. This means that the symmetry of the full Fermi surfaces directly\ndetermines the symmetry of 𝜑 dependence of the first-order (i.e., first-harmonic component\nof) resistivity 𝜌\"(𝜑). In SrMnBi , as the reviewer points out, the four-fold symmetry is\n!! 2\nbroken in the Fermi surface cross-sections on the 𝑘 >0 and 𝑘 <0 planes due to the\n! !\nantisymmetric spin–orbit interaction arising from the PT-symmetric antiferromagnetic order.\nHowever, it should be noted that the manner in which the four-fold symmetry is broken is\nopposite between the 𝑘 >0 and 𝑘 <0 regions. Specifically, valley A (B) contracts\n! !\n(expands) for 𝑘 >0, whereas valley A (B) expands (contracts) for 𝑘 <0, as shown in the\n! !\nright panels of Fig. 1e. As a result, the four-fold rotoinversion symmetry of the full Fermi\nsurfaces is exactly preserved over the full 𝑘 range in the Brillouin zone, leading to the\n!\nfour-fold symmetry of the 𝜑 dependence of 𝜌\"(𝜑).\n!!\nIn the revised manuscript, we have added theoretical expression of c-axis conductivity in\nan in-plane magnetic field. Furthermore, we have provided a detailed explanation of the\npreserved four-fold symmetry of the 𝜑 dependence of 𝜌\"(𝜑) and the full Fermi surfaces.\n!!\nPlease see the summary of revisions (1), (2), (3), (7), and (10).\nReviewer #2’s comment (2):\n2. Experimental implementation of weak symmetry is difficult. A high-quality micro-fabricated\ndevice is critically important, the weak symmetry of the angular-dependent\nmagnetoresistance becomes increasingly prominent as the current increases. Angle\ndependence of the first-harmonic interlayer resistivity show four-fold symmetry, and the\nsecond-harmonic resistivity show two-fold symmetry. Here, the weak enhancement of the\ntwo-fold symmetry is attributed to the broken four-fold symmetry. Why doesn’t this refer to\nan intrinsically weak two-fold symmetry in the electronic structure?\nAuthors’ response (2):\nWe thank Reviewer #2 for raising this important question. We agree that if the electronic\nstructure inherently possesses a weak two-fold symmetry, it may become more pronounced\nwith increasing current, as the signal-to-noise ratio improves.\nFirst, we would like to point out that such intrinsic two-fold symmetry would manifest itself\nin the AMR of first-harmonic component 𝜌\"(𝜑). This corresponds to the case where the\n!!\ncase where 𝜖 and 𝜖 are constant (independent of current 𝑗 ) in Eq. (5). Under this\n* )+ !\ncondition, 𝛿𝜎 (𝜑) in Eq. (4) gives rise to weak two-fold symmetry in the first-harmonic\n!!\ncomponent of interlayer conductivity (or resistivity). To test this, we examined the current\ndependence of 𝜌\"(𝜑) and included the corresponding experimental data as new\n!!\nSupplemental figure Fig. S5 [Summary of the Revisions (9)]. As shown, all the 𝜌\"(𝜑)\n!!\ncurves almost perfectly overlap with each other, with only a slight upward shift that is likely\ndue to weak heating effects. No signature of four-fold symmetry breaking or enhancement in\ntwo-fold symmetry is discernible. Therefore, the enhancement of the two-fold symmetry with\nincreasing current is evident only in the second-harmonic component 𝜌)\"(𝜑). This\n!!\nindicates that 𝜖 and 𝜖 are proportional to 𝑗 and hence the two-fold symmetry is\n* )+ !\ninduced by the applied current.\nWe would also like to re-emphasise that the electronic structure, i.e., the Fermi surface, of\nSrMnBi retains exactly tetragonal symmetry, consistent with its crystallographic and\n2\nmagnetic structures (space group I4/mmm and magnetic space group I 4'/m' 2/m' 2'/m). As\ndescribed in our previous response, although the Fermi surface cross-section on the 𝑘 ≠0\n!\nplane has a two-fold symmetry due to the antisymmetric spin-orbit interaction associated\nwith the PT-symmetric antiferromagnetic order, the four-fold (rotoinversion) symmetry is\npreserved over the entire Brillouin zone. As a possible extrinsic source of symmetry\nbreaking of the electronic structure, we also considered the geometry of the micro-fabricated\ndevice. To rule this out, we checked that the results are reproduced among the\nmicro-fabricated devices with different cutting surfaces, as shown in Fig. S3. We have\nadded the relevant statements in the revised manuscript to make clear this point [Summary\nof the Revisions (4)].\nWith these revisions together with new supporting data, we believe it is now clarified that\nthe observed two-fold second-harmonic component 𝜌)\"(𝜑) represents direct evidence of\n!!\ncurrent-induced nematicity.\nReviewer #2’s comment (3):\n3. The angular-dependent magnetoresistance for the current flowing in different directions\nwithin the ab plane may help to understand the present results.\nAuthors’ response (3):\nWe thank the Reviewer #2 for this insightful suggestion. We totally agree that the\nelectrical current flowing along the ab plane may also give rise to novel nonlinear responses\ncharacteristic of the PT-symmetric magnetic order in SrMnBi\n2\n. As discussed in Ref. [23],\nthese includes nonlinear Hall effect associated with Berry curvature dipole (induced by\nmagnetic field), which has been actively studied in other materials.\nHowever, for the specific purpose of detecting the current-induced nematicity, probing\ntransport phenomena within the ab-plane presents significant experimental challenges. First,\nmeasuring the in-plane resistivity while applying a current along the c-axis would be\ntechnically difficult. Moreover, the angular-dependent magnetoresistance (AMR) becomes\nmore complicated for in-plane resistivity, since the in-plane current itself breaks the four-fold\nsymmetry (i.e., the longitudinal and transverse configurations are inequivalent), which\nmakes it further difficult to extract the four-fold symmetry breaking purely induced by the\nc-axis current. For these reasons, our study focuses on applying current along the\nhigh-symmetry c-axis while rotating the magnetic field within the ab plane, enabling us to\ndetect the four-fold symmetry breaking without the influence from in-plane current.\nNevertheless, we completely agree with the reviewer on the importance of investigating\nin-plane nonlinear transport phenomena in this material. In the revised manuscript, we have\nadded a new paragraph to discuss such transport phenomena, along with the theoretical\nclassification introduced in Ref. [23]. Please see the Summary of the Revisions (6). We\nsincerely appreciate the Reviewer #2 for suggesting this valuable direction for future\nresearch.\nReviewer #2’s comment (4):\n4. Further experimental observations is required to confirm the conclusions presented in the\npaper.\nAuthors’ response (4):\nWith the above revisions and responses, we believe that we have provided compelling\nexperimental evidence of current-induced nematicity in SrMnBi , through a detailed analysis\n2\nof angular magnetoresistance for interlayer resistivity. While other nonlinear transport\nphenomena involving current along the different directions contains rich physics related to\nthe quantum geometry, as explored in other PT-symmetric antiferromagnets, they do not\ndirectly reflect the current-induced nematicity. We therefore consider them to be beyond the\nscope of the present study. The central purpose of this work is to demonstrate, for the first\ntime, transport-based detection of current-induced nematicity in a PT-symmetric\nantiferromagnet.\nWe hope that the revised manuscript is now suitable for publication in Nature\nCommunications.\n--- Summary of the Revisions ---\n(1) 2nd paragraph on page 5 and caption of Fig. 1e: Adding an explanation about four-fold\n(rotoinversion) symmetry of the entire Fermi surface of SrMnBi .\n2\n“Importantly, this opposite asymmetry of the valleys between the kz > 0 and at kz <\n0 regions (right panels of Fig. 1e) cancels out over the entire kz range, thereby\npreserving the exact four-fold (rotoinversion) symmetry of the total Fermi surfaces.\n…”\n“…, preserving the four-fold (rotoinversion) symmetry of the total Fermi surfaces.”\n(2) 3rd paragraph on page 5: Clarifying the current direction for angular-dependent\nmagnetoresistance.\n“…(AMR) measured along the c axis.…”\n(3) 1st paragraph on page 7: Adding an explanation about the four-fold symmetry of\nfirst-harmonic component based on a theoretical expression.\n“From theoretical expressions of the first-harmonic component of conductivity (see Methods),\nthe observed four-fold symmetry of ρω(φ) reflects the preservation of tetragonal symmetry of\nthe total Fermi surfaces for jz = 0. Intuitively, …”\n(4) 1st paragraph on page 7: Adding a statement regarding the reproducibility among the\ndevices with different crystal geometry.\n“It should be noted here that essentially the same two-fold symmetric ρ2ω(φ) was\nobserved in a microfabricated device with a different surface orientation (Fig. S3),\nindicating that the present results are independent of the crystal cutting geometry. ”\n(5) 2nd paragraph on page 10: Adding a new paragraph to discuss the temperature\ndependence of.\n“Here, we discuss the temperature dependence of ρ2ω(φ). The amplitude of the\nρ2ω(φ) peaks and dips decreases with increasing temperature and nearly vanishes above 30 K\n(Fig. S4d), which is well below the Neel temperature T = 295 K [40]. … ”\nN\n(6) 2nd paragraph on page 12: Adding a new paragraph to discuss the nonlinear transport\nphenomena for the current flowing different directions.\n“For PT -symmetric antiferromagnets, a variety of nonlinear conduction phenomena,\nclassified by the dependence on the relaxation time, were theoretically predicted\n[23]. In this context, … ”\n(7) Methods part on page 13: Adding a new paragraph “Theoretical expression of\nfirst-harmonic conductivity along the c-axis for in-plane magnetic fields”.\n“From the Boltzmann equation, the conductivity along the c||z-axis under the\nmagnetic field B in the present material is given by [56–59] … ”\n(8) New Supplementary Fig. S4c (on page 5 in the revised Supplementary information):\nShowing the temperature dependence of AMR of first-harmonic component 𝜌\"(𝜑).\n!!\n(9) New Supplementary section “Current dependence of first-harmonic component” and\nSupplementary Fig. S5 (on pages 4-5 in the revised Supplementary information):\nShowing the four-fold AMR of first-harmonic component 𝜌\"(𝜑) under elevated applied\n!!\ncurrent.\n(10) References: Adding the following references for the theoretical expression of interlayer\nconductivity.\n[56] T. Osada, S. Kagoshima, and N. Miura, Phys. Rev. Lett. 77, 5261-5264 (1996).\n[57] I. J. Lee and M. J. Naughton, Phys. Rev. B 57, 7423-7426 (1998).\n[58] N. Hanasaki, S. Kagoshima, T. Hasegawa, T. Osada, and N. Miura, Phys. Rev. B 57,\n1336-1339 (1998).\n[59] H. Takatsu, J. J. Ishikawa, S. Yonezawa, H. Yoshino, T. Shishidou, T. Oguchi, K. Murata,\nand Y. Maeno, Phys. Rev. Lett. 111, 056601 (2013).\n(11) Corrections of typos (1st paragraph on page 5):\n“ … while the spin-down Mn is located at the centre of … ”\n“ … in this material possess valley degrees of freedom (see below), … ”\n(12) Adding the sections of data availability and code availability"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have replied to all requests/questions in a satisfying manner and adjusted their manuscript accordingly.\nSo the reply (also to the second referee) confirmed my judgement of strongly supporting publication in Nature Communications.", "reviewer_2": "The present manuscript is not suitable for publication in Nature Communications.\n1. Fermi surfaces for SrMnBi2 at Kz = 0 possesses four-fold symmetry, while at Kz <0 and > 0 shows two-fold symmetry. The two-fold symmetry of the second-harmonic interlayer resistivity is attributed to the broken four-fold symmetry and enhanced by the current. The thermal effects should be carefully taken into consideration.\n2. Again, further experimental observations are required to confirm the conclusions presented in the paper.", "reviewer_3": "I think authors have shown convincingly that going from linear transport to second harmonic transport they see a reduction of symmetry -> going from four fold (in the transport) to two fold (e.g., Fig. 2 but also many other figures). They’ve also shown that this reduction (induced by current) has a “magnetic” nature. Not only can they control it by B field (Fig 3), pooling to different magnetic structure yields different second order resistivity. So from this point of view reduction of demonstrated 4- fold rotational symmetry (in transport) in going from first to second order transport: this claim I feel is well evidenced in their paper and in essence is the main claim of the paper. As a such, personally, i am more or less OK with this paper as it is.\nNow to the tricky bit that i've been asked to comment on: is there any basis to Reviewer 2’s apprehension. Yes and no. I don’t believe any additional data is really needed. My feeling of about how referee 2 feels (that the two-fold symmetric second order transport is just an enhancement of a weak two-fold symmetry breaking) is the following: the system actually doesn’t have four fold rotational symmetry strictly. Instead it is roto-inversion symmetric, see e.g, Figure 1e composite symmetry (a type of r-> -r and rotate by 90 degrees). This C4 vs rotoinversion distinction doesn’t show up in linear transport along k_z because you sample all the k_zs equally so it manifests as a C4 symmetric transport (note that linear transport reflects the ground state symmetry). But at second order it is C4 inversion that gets broken because current is running at linear order. In fact, the four data points of Figure 2e indicate that it is this current that is responsible for the (roto-inversion) symmetry breaking. So I think there is quite a bit of evidence for this. So both authors and Referee are right: the referee is right to suspect there is a weak 2 fold symmetry (it is there in the bandstructure). But the authors are also right that this is just not accessible in k_z linear transport and has to be activated by current flowing. Authors do use the word roto-inversion here are there (and even in the reply).\nNow whether to call it current induced nematicity or not? This as with many things comes down to how one chooses to use terminology. On face value, the linear to second order transport shows current inducing a four-fold to two-fold pattern (well evidenced). On the other hand, this is sort of expected from the bandstructure (this does not take away from the novelty of current paper) since it is roto-inversion symmetric. I tend to be OK with the use of nematicity provided the authors make it super clear that it is rotoinversion (not just writing rotational symmetry breaking and then using brackets for \"rotoinversion\") that is being broken by current."}, "author_response": ""}]

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