Introduction

4f-based materials often exhibit complex magnetic ground states1,2,3, which include spin helix4, conical spiral5, and more exotic spin textures with nontrivial topology, such as skyrmions6,7,8. Magnetic textures play a central role in understanding material’s properties because of the coupling between spin, charge, and lattice degrees of freedom. For instance, spin textures may influence the electronic transport by partially gapping out states near the Fermi level, and incommensurate textures can cause localization of the electronic wave functions in the strong coupling regime9,10,11,12. Notably, spin textures can also produce a nonzero Berry curvature and Hall conductivity by breaking certain symmetries13,14,15,16.

Magnetic cerium-based compounds are a particularly relevant platform to understand the role of spin textures because their magnetic ground states can be tuned by using modest nonthermal parameters, such as pressure and magnetic field. One example is tetragonal CeAuSb2, in which a conventional single-Q (1Q) magnetic order, as well as a more exotic double-Q (2Q) magnetic order are stabilized as a function of magnetic fields (H) applied along the c-axis. In both phases, magnetic moments point along the c-axis due to the strong Ising spin anisotropy. At zero field, CeAuSb2 orders antiferromagnetically below TN = 6.3 K in a 1Q stripe structure with wave vector Q1 = (η, η, 1/2)[η = 0.136(2)]17,18,19,20,21. A 2Q phase emerges in the region between Hc1 = 2.8 T and Hc2 = 5.6 T before magnetic moments in CeAuSb2 becomes fully polarized. A tricritical point of Hc2 has been previously identified and signals the change in the field-induced transition from second to first order, as temperature is decreased19.

Notably, the stripe pattern with twofold rotational symmetry in the ab-plane has been recently shown to be associated with a nematic state—an electronic state that breaks the rotational symmetry of the underlying lattice, but not its translational symmetry22. This nematic state, which sets in just above TN, is accompanied by a structural transition that is strongly coupled to the 1Q stripe phase below Hc1 (ref. 22). Whether this nematic state survives in the 2Q phase remains an open question, and whether the 2Q structure is checkerboard (C4-symmetric) or woven (C2-symmetric) remains a matter of contention20,23.

Electrical transport data also pose intriguing questions. The electrical resistivity of CeAuSb2 increases when the material transits from the 1Q state to the 2Q state18,19. In the fully polarized state, the resistivity drops to a value smaller than that in zero field. A recent explanation for the enhanced resistivity in the 2Q state relies on quasi-nesting of the itinerant Fermi surface20. Nonetheless, the field dependence of the resistivity in CeAuSb2 remains poorly understood, and whether it is better described by a localized or itinerant 4f-electron picture remains controversial20,24.

To address these questions, we first map out the H–T phase diagram of CeAuSb2 by means of high-resolution electrical resistivity and thermal expansion measurements (Fig. 1a). The presence of two nearby transitions as a function of temperature for fields between Hc1 and Hc2 suggests that the coupled structural–magnetic transition survives in the 2Q phase. Supported by calculations of a Kondo lattice model, we attribute the puzzling field dependence of the electrical resistivity (Fig. 1b) in different magnetic states to the localization of electronic wave functions due to the spin textures. Importantly, our analysis reveals that it is not possible to simultaneously nest two pairs of hot spots by the 2Q state. We therefore propose that the field-dependent electronic transport of CeAuSb2 acts as a fingerprint of the different spin textures.

Fig. 1: Phase diagram of CeAuSb2 its unusual magnetoresistance.
figure 1

a Temperature–magnetic field phase diagram of CeAuSb2. The legend identifies how transition temperatures and fields are obtained. The solid squares, circles, and diamonds indicate the higher transition temperature Ts in three regions. The open squares, circles, and diamonds indicate the lower transition temperature TN in three regions. The solid triangles indicate the first critical field Hc1 separating single-q and multi-q phases. The open and solid stars indicate the lower and higher second critical fields, respectively. The open up triangles, down triangles, and left triangles indicate TN in three regions. The sign of the peaks in the derivative are indicated in brackets: N negative, P positive. Colors represent the local exponent, \(n=\partial {\rm{ln}}{{\Delta }}\rho /\partial {\rm{ln}}T\) and Δρ = ρab − ρ0 = ATn. The contour map shows the presence of three distinct regions in the phase diagram; the green, light-blue, and deep-blue regions denote the single-Q stripe phase (1Q), multi-Q phase (2Q) below \(\tilde{H}\), and multi-Q phase (2Q) above \(\tilde{H}\). Dashed vertical line inside 2Q phase denotes a crossover boundary at \(\tilde{H}\). b Electrical resistivity as a function of applied field along the c-axis. The inset shows an anomaly at the crossover field \(\tilde{H}\). Error bars are smaller than the size of the data points.

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Results

Thermal expansion and electrical resistivity

First, we determine the H–T phase diagram of CeAuSb2 by thermal expansion and electrical resistivity measurements. The results are compiled in Fig. 1a. The overall structure of the phase diagram agrees with previous results; however, our thermal expansion measurements reveal an additional phase transition that has been overlooked19,21,23. Figure 2a shows the temperature dependence of the thermal expansion, ΔL/L, of CeAuSb2 along [001] at various magnetic fields applied parallel to the c-axis. The anomalies in ΔL/L in the vicinity of TN are qualitatively different from the typical anomaly expected for a single magnetic phase transition. In fact, two peaks are observed in the linear thermal expansion coefficient, α = (1/L)(dL/dT), along [001] (Fig. 2b) and [100] (Supplementary Fig. 1b). Recent X-ray diffraction data point to the presence of a structural transition nearly coincident with the zero-field TN, which suggests that the presence of two phase transitions in thermal expansion is a signature of the coupled structural–magnetic phase transition22. Therefore, our thermal expansion measurements reveal that structural and magnetic transitions remain strongly coupled in 2Q phase, as two phase transitions are also detectable above Hc1.

Fig. 2: Field evolution of phase transitions and ordering wave vector in CeAuSb2.
figure 2

a Temperature dependence of the thermal expansion of CeAuSb2 along [001] at various magnetic fields applied along the c-axis. Labels in bold indicate Hc1, \(\tilde{H}\), and Hc2. b Temperature dependence of the linear thermal expansion coefficient along [001] with 1 and 5 T applied along the c-axis. The estimated pressure is 0.07 and 0.02 GPa for high-(dashed line) and low-(solid line) strain measurements, respectively. Solid arrows indicate peak positions: blue and magenta arrows for the Ts and TN, respectively. Dashed arrows indicate points of minor peaks under low and high strains. c Temperature dependence of the in-plane electrical resistivity, ρab, of CeAuSb2 for applied magnetic fields along c-axis. Data at different fields are shifted for clarity. Arrows indicate antiferromagnetic transition temperatures: black, red, and blue arrows for the TN1, TN2, and TN3, respectively. d–f Temperature derivative of ρab under applied fields as a function of temperature at 5.5 T (d), 3.5 T (e), and 0 T (f). g Magnetic ordering wave vector η[r, l, u. ] for Q1 = (η, η, 1/2) in CeAuSb2 as a function of magnetic field applied along the c-axis at 100 mK. Data from ref. 20 were obtained by neutron diffraction measurement. Dashed lines at Hc1, \(\tilde{H}\), and Hc2 separate three distinct regions. Error bars are smaller than the size of the data points.

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Figure 2b shows the temperature dependence of the longitudinal linear thermal expansion coefficient when fields of 1 and 5 T are applied along the c-axis. A small amount of uniaxial c-axis pressure was applied to the sample when mounting it in a capacitance dilatometer. The estimated stress is 0.07 and 0.02 GPa for high- and low-strain conditions, respectively. Under high strain, the amplitude of the largest peak in αc is reduced by 16% at 1 T and 7% at 5 T compared to the low-strain data, whereas the amplitude of smallest peaks are remarkably suppressed by 66% at 1 T and 89% at 5 T. Though the field-dependent magnetic phases are different and the sign of the two peaks are reversed between 1 and 5 T, the smaller peaks at higher temperature are always more sensitive to [001] strain than the largest peaks at lower temperature. Therefore, these results indicate that in CeAuSb2 (i) the higher transition temperature at Ts is consistent with a structural transition, as structural transitions are naively expected to be more sensitive to lattice distortions caused by uniaxial strain than magnetic transitions, and (ii) the coupled phase transitions are preserved at high field, namely the zero-field coupled structural–magnetic transition survives in the 2Q phase. Lastly, the thermal expansion signatures of the phase transitions in CeAuSb2 dramatically change not only at Hc1 and Hc2, but also at 4.75 T, suggesting a new field boundary at \(\tilde{H}=4.75\) T (see Fig. 2a and Supplementary Fig. 1a).

Next, we turn to the field dependence of the electrical transport in CeAuSb2. Figure 2c shows the temperature dependence of the in-plane resistivity, ρab, under various magnetic fields applied along the c-axis. At low fields, a sharp drop at TN1 (black arrows) marks the magnetic transition temperature to a 1Q stripe phase. TN1 decreases with increasing field, and above Hc1 a shallower kink-like anomaly (red arrows) occurs at TN2, which is the magnetic transition temperature to a 2Q phase. As the field is further increased, TN2 decreases, and the temperature dependence of ρab below TN2 remains virtually unchanged to 4.5 T.

Above 4.5 T, however, the signature of the magnetic transition, indicated by blue arrows at TN3, changes qualitatively. Previous reports have missed this crossover field likely because its signatures occur in a very narrow field region. Above \(\tilde{H}\) ≈ 4.75 T, ρab increases on cooling through TN3, which indicates the opening of a gap. As TN3 is suppressed toward zero temperature with field, the magnitude of the jump increases. Finally, there is no evidence for a phase transition above Hc2 = 5.6 T.

Figure 2d–f shows the derivative of the electrical resistivity, dρ/dT, as a function of temperature at 0, 3.5, and 5.5 T. A large positive peak in dρ/dT is observed at TN1, whereas a small positive peak is observed at TN2 and 3.5 T. At 5.5 T, however, a sharp negative peak is observed at TN3, indicating that the behavior of dρ/dT crosses over at \(\tilde{H}\). Even though a detectable upturn is observed at \(\tilde{H}\) in the magnetoresistance of CeAuSb2 at low temperature (inset of Fig. 1b), the absence of an anomaly at \(\tilde{H}\) in the Hall resistivity25 and magnetostriction, which is a thermodynamic probe, suggests the presence of a crossover at \(\tilde{H}\) rather than a phase transition (see Supplementary Note 1). The contour map of the local exponent n of ρab shows a change in magnetic scattering below TN in the vicinity of \(\tilde{H}\) (Fig. 1a), in agreement with a crossover field boundary at \(\tilde{H}\) in the 2Q phase. Previous reports have been overlooked the crossover behavior in 2Q phase due to a narrow field range between \(\tilde{H}\) and Hc2 (~0.7 T). In fact, the color map of the exponent n in ref. 22 does not display the signature of crossover in ρ(T) above \(\tilde{H}\) because of the lack of the data in the vicinity of Hc2. Nonetheless, the tricritical point of Hc2 has been previously located near \(\tilde{H}\) and 4 K, which strongly suggests that \(\tilde{H}\) coincides with the tricritical point of Hc2. As a result, \(\tilde{H}\) not only represents a crossover as a function of magnetic fields at fixed temperature, but it also marks a change in the nature of the magnetic phase transition as a function of temperature at fixed fields. For fixed fields below \(\tilde{H}\), the antiferromagnetic phase transition at TN is of the second order, whereas the transition becomes first order at TN (i.e., hysteretic) when \(H\,> \,\tilde{H}\) (ref. 19). We note that the tricritical point of Hc2 does not change under pressure22. Importantly, the wave vector component η decreases as a function of field when \(H\,> \,\tilde{H}\) as shown in Fig. 2g (ref. 20). As discussed below, the upturn of resistivity for \(H\,> \,\tilde{H}\) stems from the intertwined effects of a varying ordering wave vector and the electronic localization of wave functions.

Spin Hamiltonian

After determining the experimental H–T phase diagram of CeAuSb2, we now provide a theoretical model to understand it. In CeAuSb2, Ce3+ ions carry a magnetic moment and interact with the conduction electrons, which in turn mediate interaction between cerium moments. Therefore, a sensible starting point to describe the magnetic properties of CeAuSb2 is the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. In the paramagnetic phase, the system has C4 rotation symmetry, and the appearance of a 1Q magnetic state implies that the Fermi surface is quasi-nested with nesting wave vectors Q1 and Q2, which are related by C4 rotation. To second order in the local exchange coupling between the conduction electrons and localized moments, the Hamiltonian of the system reads \(-{\sum }_{Q}\tilde{J}(Q){{\bf{S}}}_{Q}\cdot {{\bf{S}}}_{Q}\), in which we have assumed an isotropic exchange coupling (\(\tilde{J}\)) among spins. Here SQ is the Fourier component of the localized moment. The functional form of \(\tilde{J}(Q)\) depends on the Fermi surface and in principle can be obtained from density functional calculations. The ground state magnetic texture implies that \(\tilde{J}(Q)\) is maximized at Q1 and Q2. From experiments, a strong easy-axis anisotropy is also known to exist, which forces the spins to point along the c axis. In the presence of an external magnetic field, the system Hamiltonian can thus be written as:

$${\mathcal{H}}=-\mathop{\sum}\limits _{Q}\tilde{J}(Q){{\bf{S}}}_{Q}\cdot {{\bf{S}}}_{Q}-A\mathop{\sum}\limits _{i}{S}_{i,z}^{2}-\mathop{\sum}\limits _{i}{\bf{H}}\cdot {{\bf{S}}}_{i}.$$
(1)

In triangular lattices, \({\mathcal{H}}\) is known to support triple-Q spin textures (e.g., skyrmion lattice)26,27,28,29; however, in tetragonal crystals such as CeAuSb2, 2Q order is not favored by \({\mathcal{H}}\) because the harmonic Q3 = Q1 + Q2 is not an optimal wave vector and costs energy. As an example, our Monte Carlo simulation of a particular form of \(\tilde{J}(Q)\) within the frustrated J1–J2 or J1–J3 model on a square lattice shows a 1Q spiral phase followed by a fully polarized state in field29.

In cerium-based compounds, the exchange interaction between 4f and conduction electrons is often substantial. In the strong coupling regime, the system becomes nonmagnetic due to Kondo singlet formation. In the weak coupling regime, the system becomes magnetic and the dominant exchange coupling is the conventional RKKY interaction. Importantly, even in this weak coupling regime, higher-order spin exchange interactions beyond the conventional RKKY interaction can be important30. By expanding the spin–charge coupling to quartic order, a four-spin interaction term emerges, which was demonstrated to favor multiple-Q magnetic ordering31.

The existence of the 2Q order in CeAuSb2 thus requires higher-order coupling, and the spin Hamiltonian is written as:

$${\mathcal{H}}= \; 2\mathop{\sum}\limits _{\nu }\left[-\tilde{J}{{\boldsymbol{S}}}_{{{\boldsymbol{Q}}}_{{\boldsymbol{\nu }}}}\cdot {{\boldsymbol{S}}}_{-{{\boldsymbol{Q}}}_{{\boldsymbol{\nu }}}}+\tilde{K}{({{\boldsymbol{S}}}_{{{\boldsymbol{Q}}}_{{\boldsymbol{\nu }}}}\cdot {{\boldsymbol{S}}}_{-{{\boldsymbol{Q}}}_{{\boldsymbol{\nu }}}})}^{2}\right]\\ -A\mathop{\sum}\limits _{i}{S}_{i,z}^{2}-\mathop{\sum}\limits _{i}{\bf{H}}\cdot {{\bf{S}}}_{i}.$$
(2)

In this model, \(\tilde{J}(Q)\) is assumed to peak sharply at Qv, and therefore only the exchange coupling at Qv is taken into account. Because the biquadratic interaction \(K=N\tilde{K}\) (N is the system size) is always positive, the four-spin term favors multiple-Q ordering by distributing the static spin structure factor weight equally on the symmetry related Qv. In CeAuSb2, spins along the c-axis are simply antiferromagnetically coupled. Therefore, we will restrict to the two dimensional limit in the following discussions.

Figure 3 shows the numerical results of (a) the uniform magnetization \(M=(1/N){\sum }_{i}{S}_{i}^{z}\) and (b) the Qη-component magnetization \({M}_{{{\boldsymbol{Q}}}_{\eta }}=(1/N)\sqrt{\scriptstyle| {\sum }_{i}{S}_{i}^{z}{e}^{i{{\boldsymbol{Q}}}_{\eta }\cdot {{\boldsymbol{r}}}_{i}}{| }^{2}}\) as a function of an external field. The field-dependent real-space spin configurations are shown in Fig. 3c–e, and their corresponding spin structure factors are shown in Fig. 3f–h. The optimal spin configuration at zero field is a 1Q collinear state whose spin configuration and spin structure factor are shown in Fig. 3c, f. Three magnetization jumps are identified by increasing field in Fig. 3a. The first jump at H ~ 0.46 represents a crossover from the up-up-up-down-down-down configuration in Fig. 3c to the up-up-up-up-down-down configuration in Fig. 3d, g, which results from the approximation of a sharply peaked \(\tilde{J}(Q)\). This jump disappears for a smooth function of \(\tilde{J}(Q)\), in agreement with experiments. The other two jumps in magnetization represent phase transitions from the 1Q collinear to the 2Q bubble state and from the 2Q bubble to the fully field polarized (FP) state, which are caused by the interplay between the multiple-spin interaction (K) and the easy-axis single-ion anisotropy (A). The spin configuration of the 2Q state is characterized by the collinear bubble structure without an xy spin component, as shown in Figs. 3e, h. The equivalent four peaks in the spin structure in Fig. 3h also indicate the formation a square bubble crystal. Note that the 2Q bubble state vanishes by taking K = 0 or A = 0.

Fig. 3: Magnetization and spin configurations of CeAuSb2.
figure 3

H dependences of the magnetization for a uniform component, and b Q1 (squares) and Q2 (circles) components at \(\tilde{J}=1\), K = 0.4, and A = 0.7. The vertical dashed lines show the phase boundaries. c–e The real-space spin configurations in c the 1Q collinear state at H = 0, d the 1Q collinear state at H = 0.6, and e the 2Q bubble state at H = 0.8. The contour shows the z-component of the spin moment. f–h The square root of the spin structure factor. The solid squares represent the Brillouin zone.

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By including temperature, we obtain a magnetic phase diagram consistent with experiments. It is important to note that the antiferromagnetic transition at zero magnetic field is second order, whereas the transition at Hc2 at T = 0 is strongly first order19. At finite temperatures, we therefore generically expect a tricritical point of Hc2 at which the nature of the phase transition at TN changes from second to first order, as reported previously19.

Theory: transport

In this section, we show that the resistivity enhancement in the 2Q phase can be fully modeled theoretically. Our model is based on two elements: (1) relatively strong coupling between localized moments and conduction electrons, and (2) incommensurate spin texture with respect to the atomic lattice.

The presence of an incommensurate magnetic texture is known to result in band folding. More specifically, an ordering wave vector Q = (p/q, 0, 0), wherein q and p are coprime integers, causes band folding q times, resulting in a smaller Brillouin zone. To illustrate the band folding mechanism, the inset of Fig. 4 displays the simplest example of a commensurate wave vector that folds the bands once (i.e., q = 2). The presence of exchange coupling may open a gap at the Brillouin zone boundary causing an increase in resistivity known as the superzone mechanism32,33,34.

Fig. 4: Inverse participation ratio of CeAuSb2.
figure 4

Inverse participation ratio (IPR) averaged over all the eigenvalues at 1Q (circles) and 2Q (squares) states. Error bars are smaller than the size of the data points. Inset illustrates the band folding and gap opening at the Brillouin zone boundary in the presence of a commensurate spin texture.

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Incommensurate magnetic textures may have more profound effects on the electronic transport. Rigorously speaking, the band structure picture is no longer a good description of the electronic state because of the lack of translational invariance. The incommensurate potential induced by the magnetic texture to a certain extent works as random disorder, but with a weaker effect. Importantly, random disorder causes Anderson localization when the disorder potential is strong enough9,11,12. In the Anderson localized phase, in which the electronic wave functions are strongly localized in space, the system behaves as an insulator even though there exists a finite electronic density of states (DOS) at the Fermi energy. An intertwined insight into this problem comes from the realization that the incommensurate potential can also cause electronic localization. This type of electronic localization was demonstrated in the Kondo lattice model hosting an incommensurate magnetic texture12, and can be understood in terms of the band folding picture. Any incommensurate wave vector can be approximated by a rational number Q ≈ p/q with p, q → ∞. The folded bands can be extremely flat in the folded Brillouin zone when levels repel each other due to the local exchange coupling. The flat band limit therefore corresponds to the electronic localization.

More specifically, we consider the following Kondo lattice Hamiltonian to model the electronic transport in CeAuSb2:

$${{\mathcal{H}}}_{0}= -{t}_{1}\mathop{\sum}\limits _{{\rm{NN}}}{c}_{i}^{\dagger }{c}_{j}-{t}_{3}\mathop{\sum}\limits _{{\rm{NNN}}}{c}_{i}^{\dagger }{c}_{j}-\mu \mathop{\sum}\limits _{i}{c}_{i}^{\dagger }{c}_{i}\\ -J\mathop{\sum}\limits _{i}{c}_{i,\alpha }^{\dagger }{\sigma }_{\alpha ,\beta }\cdot {{\bf{S}}}_{i}{c}_{i,\beta },$$
(3)

where NN and NNN denote nearest-neighbor and next-nearest-neighbor hopping, and J is the coupling between the conduction electron and spin texture. We choose the hopping strength t3 = −0.5t1 and chemical potential μ = 0.98t1, such that the Fermi surface is quasi-nested. The corresponding electronic filling per spin is 0.638, which is fixed in the calculations. Equation (2) can be obtained from Eq. (3) by integrating out conduction electron degrees of freedom and expanding the exchange interaction to quartic order in J (ref. 31). We choose experimentally measured incommensurate Q = 0.136(2) and neglect the variation of Q under magnetic field. We fix the spin configurations in the calculations, and we take \({{\bf{S}}}_{i}=[0,0,\cos ({{\bf{Q}}}_{1}\cdot {{\bf{r}}}_{i})]\) for the 1Q phase, \({{\bf{S}}}_{i}=[0,0,\cos ({{\bf{Q}}}_{1}\cdot {{\bf{r}}}_{i})+\cos ({{\bf{Q}}}_{2}\cdot {{\bf{r}}}_{i})]/2\) for the 2Q phase, and Si = [0, 0, 1] for the FP state.

The degree of localization of electronic wave functions can be characterized by the inverse participation ratio (IPR) defined as \({I}_{n}={\sum }_{{{\bf{r}}}_{i}}| {\psi }_{n}({{\bf{r}}}_{i}){| }^{4}{({\sum }_{{{\bf{r}}}_{i}}| {\psi }_{n}({{\bf{r}}}_{i}){| }^{2})}^{-2}\), where ψn(ri) is the nth eigenfunction of \({{\mathcal{H}}}_{0}\). Because the IPR magnitude is a measure of the spread of the electronic wave function in space, higher IPR means a more localized state and hence smaller electrical conductivity. In is finite for a localized state but vanishes as 1/Ld for an extended state. Here L is the linear system size and d is the spatial dimension. The results of In and the profiles of the wave functions are displayed in Fig. 4. In the 1Q phase, the wave function along the Q direction is localized for large J, but remains extended in the direction transverse to Q because of translational invariance. For the 2Q phase, the wave function is very localized at large J and becomes more localized than the 1Q counterpart for J > 3.7t1.

Next, we calculate the electronic conductance under the influence of the magnetic texture. We focus only on the effect of the wave function localization by neglecting the scattering of electrons by impurities, magnetic fluctuations, and the off-diagonal conductivity caused by magnetic field. In fact, there is no sudden change in the off-diagonal resistivity when the spin texture changes from 1Q to 2Q in CeAuSb2 (ref. 25). As a result, the conductance depends on the DOS at the Fermi energy and the degree of the localization of the electronic wave function. The DOS for both 1Q and 2Q states, displayed in the inset of Fig. 5, indicates that the 2Q DOS is actually larger than the 1Q DOS for most Js, particularly for large J.

Fig. 5: Conductance vs J in different magnetic states and density of states of CeAuSb2.
figure 5

J/t1 dependence of the conductance for the 1Q (circles), 2Q (squares), and field polarized (FP) (triangles) states. Here, the conductance is averaged over domains with different Q orientations. Inset shows the density of state at the Fermi energy for the 1Q (circles) and 2Q (squares) states. Error bars are smaller than the size of the data points.

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To compare our simulations to experiments, in which multiple domains of spin texture with different Q orientations coexist, we take the average of the longitudinal conductance over a random distribution of different Q domains (see Supplementary Note 3). Note that the dependence of the conductance remains qualitatively the same, if we assume that Q orientations are locked to the four equivalent crystal directions.

The conductance as a function J at T = 0 for three different spin textures is shown in Fig. 5. Overall, the conductance decreases with J due to electronic localization in the 1Q and 2Q phases. The decrease in the FP state is caused by the shift of the electronic spectrum due to the coupling to the FP spin arrangement, which results in the reduction of DOS at Fermi energy. Interestingly, in the region J > 3.2t1, the conductance of the FP state is highest, followed by 1Q and 2Q states. This result is fully consistent with experimental observations in CeAuSb2, namely, the resistivity in 2Q state is the largest and the resistivity in FP state is smallest (see Fig. 1b). Therefore, our theory model supports the notion that the incommensurate magnetic state is responsible for the increased resistivity in 2Q state. Though it is challenging to estimate J/t1 without the experimental data or a microscopic model that can account for the strength of J in CeAuSb2, our experimental and theoretical results suggest that CeAuSb2 exhibits a large exchange coupling parameter between f and conduction electrons within the magnetically ordered regime.

Discussion

Enlightened by our model calculations, we are now positioned to understand the behavior of the electrical resistivity in CeAuSb2. There are two competing effects at play: (1) the suppression of magnetic fluctuations when magnetic moments order below TN, which reduces the resistivity, and (2) the enhancement of the degree of localization of 4f electronic wave function, which increases resistivity. The temperature- and field-dependent experiments in CeAuSb2 indicate that the former factor is dominant in the 2Q phase above Hc1.

As recognized in ref. 20, spin textures might modify the resistivity by gapping out states near the Fermi level when the ordering wave vector corresponds to a nesting vector of the Fermi surface, i.e., the 1Q state would nest a pair of hot spots in the Fermi surface, whereas the 2Q state would nest two pairs. We note that this mechanism does not require the ordering wave vector to be incommensurate, and it also does not depend on whether the magnetic texture is of itinerant or localized origin.

The key result revealed by our analysis is that it is not possible to simultaneously nest two pairs of hot spots by the 2Q state. To exemplify this issue, let us consider four Fermi surface sheets ϵ(k1), ϵ(k2), ϵ(k1 + Q1), and ϵ(k2 + Q2), where k1 and k2 are related by C4 rotation, and ϵ(k1) and ϵ(k1 + Q1) are quasi-nested by the vector Q1. For a 1Q collinear magnetic state with ordering wave vector Q1, the magnetic texture gaps out states in the Fermi sheets at ϵ(k1) and ϵ(k1 + Q1), and deforms the Fermi sheet at ϵ(k2) and ϵ(k2 + Q2). For the 2Q state with Q1 and Q2 ordering wave vectors, the electronic states at ϵ(k1), ϵ(k1 + Q1), and ϵ(k1 + Q2) are mutually connected. However, this does not allow fully gapped states at ϵ(k1) and ϵ(k1 + Q1) as in the case of 1Q magnetic texture. Instead, the presence of 2Q only deforms the Fermi surface. We therefore reach the important conclusion that the electronic DOS within the 2Q phase is not necessarily smaller than that in the 1Q phase. This is demonstrated explicitly using the model in Eq. (3), in which the DOS for the 2Q phase is indeed comparable to that in the 1Q phase, as shown in Fig. 5. As a consequence, the enhanced resistivity in the 2Q phase is naturally explained by the enhanced localization of wave functions due to the incommensurate structure.

Further, an increase in the temperature-dependent resistivity at TN is observed in experiments above \(H\,> \,\tilde{H}\). We note that the magnetic transition at TN becomes first order, magnetic fluctuations are not critical, and the ordering wave vector Q decreases continuously with applied magnetic fields. Importantly, a change in Q simultaneously affects the degree of localization of wave functions, the spin-dependent disorder potential, and band folding (e.g., the superzone mechanism), particularly when Q < 2kF, wherein 2kF is the Fermi momentum35. These intertwined factors therefore result in the observed upturn in ρ(T) on cooling through TN.

Finally, we turn to the discussion of the symmetry of the 2Q phase in CeAuSb2. A neutron diffraction report showed that the 2Q magnetic order at high fields has two possible magnetic structures, i.e., checkered pattern with fourfold symmetry or woven pattern with twofold symmetry20. From their analysis, the authors conclude that the woven order might be favored because the field dependence of the maximum Ce3+ moment in this structure is consistent with the nature of the magnetization plateau of \({M}_{{{\bf{Q}}}_{1}+{{\bf{Q}}}_{2}}\) in the 2Q phase. From uniaxial strain measurements23, however, it was proposed that the 2Q magnetic order most likely preserves the symmetry between the (100) and (010) directions, suggesting that the 2Q order is close to a checkerboard structure.

At zero field, a structural transition connected with 1Q magnetic order was recently discovered22. The H–T phase diagram and strain dependence of these coupled transitions suggest that the structural transition occurs above TN and is coupled to the stripe magnetic order below Hc1. Here we observe that this coupling survives within the 2Q phase above Hc1, which indicates that a structural transition above TN is still present and that the C4 symmetry may be broken at the structural transition. Our results are therefore consistent with a woven structure or a deformed checkerboard. The effect of the structural transition can be modeled by using spatially anisotropic exchange coupling in the spin Hamiltonian (Eq. (2)), and it is expected to yield a distorted 2Q spin texture, e.g., by elongating the spin texture in Fig. 3e in the vertical direction, akin to the woven structure. Nevertheless, high-resolution spectroscopic measurements are required to unambiguously determine the magnetic structure of the 2Q phase. Here, we focus on c-axis strain because the structural and magnetic transitions exhibit opposite signs in thermal expansion along this direction. This enables the deconvolution of the phase transitions, in contrast to in-plane thermal expansion. In addition, to obtain meaningful information about in-plane nematicity, the applied in-plane strain must be sufficient to detwin the crystals, as shown for the case of K-doped BaFe2As2 (ref. 36). Importantly, in the case of iron-based superconductors, crystals under [100] strain remain well twinned, which makes [110] strain necessary for detwinning them. We expect that [110] strain may be also required in the case of CeAuSb2 because its magnetic order spontaneously lifts the (110)/(1\(\bar{1}\)0) degeneracy37. Our results will stimulate further measurements to investigate the nature of nematicity in both 1Q and 2Q phase, e.g., nematic susceptibility with in-plane [110] strain, similar to previous reports on iron-based superconductors38.

Going beyond the electrical transport signatures investigated in the present work for CeAuSb2, incommensurate multiple-Q spin textures are generally expected to significantly affect emergent quantum states. An incommensurate multiple-Q state breaks translation invariance, and it realizes a scenario similar to that of quasicrystals, in which a standard band structure theory based on crystal momentum is no longer applicable. Novel phenomena that do not have a counterpart in translationally invariant systems can thus appear39,40. Furthermore, the localization of electronic wave functions caused by incommensurate spin textures in the strong coupling regime can significantly enhance the Coulomb interaction or attractive interaction between electrons, which could lead to pronounced effects on many-body quantum states.

Here, we investigate CeAuSb2, an f-electron incommensurate antiferromagnet, via electrical transport and thermal expansion measurements under applied fields along the c-axis. Our field–temperature phase diagram shows that the coupled structural–magnetic transition in the low-field 1Q phase survives in the high-field 2Q. We also identify a crossover magnetic field, \(\tilde{H}=4.75\) T, above which the electrical resistivity increases upon cooling through the antiferromagnetic transition temperature. Our theoretical model demonstrates the electrical resistivity behavior of CeAuSb2 in 1Q, 2Q, and fully polarized states. Supported by calculations of a Kondo lattice model, we attribute the resistivity enhancement in the 2Q phase to localization of the electronic wave functions caused by the incommensurate magnetic textures.

Methods

Crystal synthesis and experiments in magnetic fields

Single crystals of CeAuSb2 were synthesized by a standard self-flux technique described in ref. 19 with Au excess to eliminate deficiency in the Au site. The orientation of the polished sample was verified by X-ray and Laue diffraction at room temperature. The thermal expansion and magnetostriction were measured using a capacitance cell dilatometer, as described by Schmiedeshoff et al.41. This design uses a CuBe spring to hold secure the sample. The spring constant was estimated using finite element analysis, which was used to calculate the uniaxial pressure applied to sample during measurement. A standard four-probe technique was employed to measure the in-plane electrical resistivity of CeAuSb2, using a Model 372 AC Resistance Bridge. Two different cryostats were used to control temperature and magnetic field: a 4He cryostat for measurement of temperature dependence of resistivity from 10 to 1.8 K and for applied magnetic fields along c-axis, and a 3He cryostat for measurement of magnetoresistance at 0.3 K as a function of magnetic field applied along the c-axis to 7 T.

Monte Carlo simulation and quantum transport calculation

We perform Monte Carlo simulations for systems with N = 96 × 96 spins by using standard approaches based on the Metropolis algorithm at target low temperatures. We numerically anneal the system to reach the ground state. The details of the simulation are discussed in Supplementary Note 2. To simulate quantum transport of the system, we consider a two-terminal setup. The details of the calculation are discussed in Supplementary Note 3.