Nematic State in CeAuSb 2

At ambient pressure and zero field, tetragonal CeAuSb 2 hosts stripe antiferromagnetic order at T N ¼ 6 . 3 K. Here, we first show via bulk thermodynamic probes and x-ray diffraction measurements that this magnetic order is connected with a structural phase transition to a superstructure that likely breaks C 4 symmetry, thus signaling nematic order. The temperature-field-pressure phase diagram of CeAuSb 2 subsequently reveals the emergence of additional ordered states under applied pressure at a multicritical point. Our phenomenological model supports the presence of a vestigial nematic phase in CeAuSb 2 akin to iron-based high-temperature superconductors; however, superconductivity, if present, remains to be discovered.


I. INTRODUCTION
Heavy-fermion systems, which consist of a lattice of f electrons that hybridize with the conduction electron sea, host prototypical examples of strongly correlated electronic states [1][2][3][4]. In particular, tetragonal Ce-based compounds often reveal novel quantum states of matter in the vicinity of a quantum critical point (QCP) at which a magnetic transition is suppressed to zero temperature by nonthermal parameters, e.g., pressure, magnetic field, or chemical doping [5,6]. Non-Fermi liquid (NFL) behavior, complex magnetic order, charge order, and unconventional superconductivity (SC) are examples of these states, which have also been observed in copper-oxide and iron pnictide high-T c superconductors, although often accompanied by electronic nematicity-an electronic state that breaks the rotational symmetry of the underlying lattice but not its translational symmetry [7][8][9]. The origin of the nematic state, as well as its role on the superconducting state, remains controversial [10]. More recently, evidence for nematicity at high magnetic fields has been found in the heavy-fermion CeRhIn 5 [11], indicating that superconductivity and nematicity may also be intertwined in this class of strongly correlated materials [12]. Recent magnetostriction and nuclear magnetic resonance experiments point to the importance of crystalline electric field (CEF) effects in the putative nematic state of CeRhIn 5 [13,14]. CeTSb 2 (T ¼ transition metal) is also a dense Kondo lattice system with pronounced CEF effects [15,16]. The magnetic and CEF ground states of CeTSb 2 depend on the transition metal T [16,17], and CeAuSb 2 orders antiferromagnetically below T N ¼ 5-6.8 K depending on the occupancy of the Au site [18][19][20]. Previous x-ray and neutron diffraction measurements showed that CeAuSb 2 crystallizes in a tetragonal crystal structure (P4=nmm) [21,22]. Pressurizing CeAuSb 2 does not suppress T N to zero temperature but rather induces new phases hindering the appearance of a QCP or SC [20]. The nature of the pressure-induced phases in CeAuSb 2 remains an open question but possibly stems from competing magnetic interactions known to exist in this series of compounds [23,24]. In view of field-dependent properties of CeAuSb 2 at atmospheric pressure [19], application of magnetic fields could shed light on the evolution of the magnetic interactions in CeAuSb 2 under pressure. Unlike pressure, however, magnetic fields are symmetry breaking and tend to localize 4f electrons. In CeAuSb 2 , T N is gradually suppressed by magnetic fields applied along the c axis, and two metamagnetic (MM) transitions are observed at H c1 ∼ 2.8 T and H c2 ∼ 5.6 T at low temperatures [18]. The H-T phase diagram constructed by magnetoresistance measurements reveals that both MM transitions are first order at low temperature [19]. In addition, H c2 has a tricritical point at 3.7 K, which may be suppressed under applied pressure [19]. A recent neutron diffraction study revealed that at H c1 the magnetic structure changes from a single-q striped phase with wave vector Q 1 ¼ ðη; η; 1=2Þ [η ¼ 0.136ð2Þ to a multi-q (woven or checkered) phase with Q 1 ¼ ðη; η; 1=2Þ and Q 2 ¼ ðη; −η; 1=2Þ [22]. This ground-state competition again stems from competing magnetic interactions and resembles the single-q to double-q magnetic transition observed in pnictides [25]. Furthermore, striped phases appear to be ubiquitous in strongly correlated systems including manganites, cuprates, nickelates, and cobaltites [26][27][28][29].
Remarkably, both striped and woven magnetic phases in CeAuSb 2 have twofold rotational symmetry in the ab plane even though the underlying lattice has fourfold rotational symmetry at high temperatures. As a result, one expects that the lattice will also break tetragonal C 4 symmetry in the presence of magnetoelastic coupling. In fact, in-plane uniaxial pressure experiments show that the magnetic transitions of CeAuSb 2 are sensitive to strain [30,31]. The situation thus seems analogous to iron-based high-T c superconductors such as NaFeAs ("111") and LaFeAsO ("1111"), which share the same P4=nmm space group as CeAuSb 2 [32,33]. There, the stripe magnetic order is generally preceded by a nematic phase with broken C 4 symmetry of the underlying lattice, in agreement with general theoretical expectations [34][35][36][37].
Here, we investigate whether such a nematic state exists in heavy-fermion CeAuSb 2 . Bulk thermodynamic probes show the presence of two transitions at about 6.5 K and 6.3 K in CeAuSb 2 at ambient pressure. X-ray diffraction measurements reveal that the development of zero-field striped magnetic order is connected to a structural transition. Our phenomenological model proposes that this structural transition must be a nematic transition at T nem ≥ T N . Our results support the scenario in which CeAuSb 2 hosts a nematic phase that is not intertwined with superconductivity.

II. METHODS
Single crystals of CeAuSb 2 were synthesized by a selfflux method described in Ref. [19]. The highest T N of 6.8 K is achieved in the electrical resistivity when the crystals are close to being stoichiometric, i.e., an average site occupancy of 100% of Au and residual resistivity ratio (RRR) of about 20. Samples with Au deficiency tend to have lower T N [18,38]. Single crystals were pressurized to 2.66 GPa using a hybrid Be-Cu/NiCrAl clamp-type pressure cell. Daphne oil 7373 was used as a pressure-transmitting medium, and lead was used as a manometer [39]. The in-plane electrical resistivity of CeAuSb 2 was measured using a conventional four-probe technique with an LR700 Resistance Bridge in 4 He and 3 He cryostats from 300 K to 0.3 K. Magnetization measurements were performed using a commercial superconducting quantum interference device (SQUID), and the specific heat was measured using a commercial small-mass calorimeter that employs a quasiadiabatic thermal relaxation technique. Thermal expansion along the c axis was measured at atmospheric pressure using a capacitance cell dilatometer with a resolution in ΔL=L of 10 −8 . X-ray powder diffraction (XPD) measurements were performed at the x-ray diffraction and spectroscopy (XDS) beamline of the Brazilian Synchrotron Light Laboratory in Campinas, which uses a 4-T superconducting multipolar wiggler source [40]. The XPD patterns were collected at 20 keV in a transmission geometry using an area detector (MAR345). The sample was mounted at the cold finger of an open-cycle He cryostat (base temperature 2.3 K).

III. RESULTS
Specific heat, magnetic susceptibility, and thermal expansion measurements in CeAuSb 2 reveal two closely lying transitions. As shown in Fig. 1(a), and highlighted in the inset of Fig. 1(b), two peaks occur in the specific heat at T nem ¼ 6.48 K and T N ¼ 6.33 K, indicated by arrows. Figure 1(c) shows the temperature dependence of the magnetic susceptibility for Hjj½001, and the presence of two distinct phase transitions is evident in the derivative dχ=dT at T nem ¼ 6.53 K and T N ¼ 6.33 K shown in the inset. We note that the signatures of the two phase transitions in CeAuSb 2 in both specific heat and dχ=dT are similar to those found in BaFe 2 As 2 due to the nematic and magnetic transitions [41,42]. Figure 1(d) shows the temperature dependence of the c-axis length change, ΔL c =L c , with an anomaly in the vicinity of 6.5 K. This anomaly is not typical of purely magnetic phase transitions [43,44], and the linear thermal-expansion coefficient, α c ¼ ð1=L c ÞðdL c =dTÞ, further reveals a negative peak at T nem ¼ 6.55 K as well as a positive peak at T N ¼ 6.33 K, respectively [see inset of Fig. 1(d)]. Here, T nem and T N decrease together when fields are applied along the c axis, suggesting a strong coupling of the order parameters associated with these transitions. As shown below, T nem is associated with a structural transition that likely breaks C 4 symmetry and T N with a magnetic transition to the single-q stripe phase.
X-ray powder diffraction patterns of CeAuSb 2 measured at T ¼ 7 K and T ¼ 3 K are shown in Fig. 2. Rietveld refinement of the 7-K data in the published structure [46] provides a satisfactory fit in the P4=nmm cell with lattice parameters a ¼ 4.3954ð2Þ Å and c ¼ 10.318ð1Þ Å. As indicated by the black difference line [Ið3KÞ − Ið7KÞ] in Fig. 2, the intensities of most Bragg peaks change (i.e., increase or decrease) below T N =T nem , whereas a number of intensities also appear unaffected. Within the limited resolution of this experiment (Δd=d > 10 −4 ), we observe no evidence for a lattice distortion. As illustrated by the insets in Fig. 2, however, four peaks appear at low temperatures, which are not indexed in the parent phase and thus confirm that the true symmetry of the low-temperature phase of CeAuSb 2 is lower than that of the P4=nmm space group. Because of the powder average and the limited number of observed intensities in the present data set, the full solution of the low-temperature modulated structure will have to await a single crystal diffraction study. Nevertheless, the absence of nuclear satellite peaks in neutron diffraction measurements [22] suggests that the x-ray diffraction superstructure peaks are due to a modulation of charge density, which is intimately coupled to the magnetic order parameter.
The existence of a magnetic ordered state that breaks a point-group symmetry (i.e., tetragonal symmetry), besides time-reversal and translational symmetry, suggests that the phase transition can happen in two stages [37]: At T nem , tetragonal symmetry is broken, whereas at T N , both timereversal and translational symmetries are broken. While it is challenging to develop a microscopic model for CeAuSb 2 that can account for the magnetic and nematic phase transitions in this compound, the thermodynamic properties such as specific heat can be obtained from an appropriate phenomenological model. We consider such a model by using a uniform stripe magnetic order with a twocomponent order parameter m ¼ ðm 1 ; m 2 Þ, with ordering wave vectors Q 1 ¼ ðη; η; 1=2Þ and Q 2 ¼ ðη; −η; 1=2Þ for m 1 and m 2 , respectively. Note that these two ordering vectors are not equivalent in the P4=nmm group, but they are related by a 90°rotation. Up to quartic terms, the Ginzburg-Landau free energy compatible with the symmetries present in CeAuSb 2 is given by This result is identical to the free energy employed to understand the nematic phase of the pnictides [36]. Here, for simplicity, we consider both m 1 and m 2 to be real and that contributions from higher harmonics are neglected. Within a mean-field approximation, at T < T 0 N , the free energy is minimized for 0 < g < u by developing a stripe magnetic order, with either hm 1 i or hm 2 i becoming nonzero. Such a magnetic order also breaks C 4 symmetry down to C 2 . Therefore, T 0 N marks a simultaneous second-order nematic and magnetic phase transition.
So far, this analysis has neglected the gradient terms in Eq. (1), which describe contributions of magnetic fluctuations to the free energy. Going beyond mean-field analysis and including the effects of magnetic fluctuations, this model generally predicts a nematic transition at T nem ≥ T N , where T N is the renormalized magnetic transition temperature to the stripe phase [36]. Such a paramagnetic-nematic state is characterized by anisotropic magnetic fluctuations, hm 1 2 − m 2 2 i ≠ 0, which spontaneously break the C 4 symmetry of the system even though The fate of the coupled magnetic-nematic transitions depends on the parameters of the phenomenological model, which in our case are the degree of anisotropy of the magnetic fluctuations and the nematic coupling strength. The former is given by the ratio ζ ≡ J z =J between the outof-plane and in-plane effective exchange interactions, whereas the latter is given by the ratioᾱ ≡ u=g of the coefficients of Eq. (1). As shown previously in Ref. [36], for a fixed ζ, largeᾱ gives two second-order transitions at T nem > T N , whereas smallᾱ gives a simultaneous firstorder transition. In the intermediateᾱ range, two transitions remain, but one of them becomes first order [36] (see Supplemental Material [45] for details). We computed the entropy and heat capacity for a fixed ζ and decreasingᾱ towards the critical valueᾱ c , below which the magnetic transition becomes first order but remains distinct from the nematic transition (see Supplemental Material [45] for details). Figure 1(b) shows the calculated heat capacity C=T as a function of reduced temperaturẽ t ∝ ðT − T N Þ=T N . Upon approachingᾱ c , the specific heat peak at T N increases at a faster rate than the jump at T nem , and the separation between the two transition temperatures decreases, signaling the approach of a first-order magnetic transition. In order to compare our model with the experimental results in Fig. 1(a), which do not show sharp jumps, we include weak disorder by considering a distribution of T 0 N values, yielding a smooth temperature dependence of the derivative of C=T [see inset of Fig. 1(b)]. The comparison shows that dðC=TÞ=dt ¼ 0 at T N and dðC=TÞ=dt has a minimum at T nem and suggests that CeAuSb 2 at ambient conditions is close toᾱ ¼ 2.7. Our model predicts that belowᾱ c , both nematic and magnetic phase transitions become first order and simultaneous at T nem ¼ T N . Without a full microscopic theory, it is difficult to determine how the Ginzburg-Landau coefficients depend on the experimentally relevant parameters such as pressure and magnetic field. However, as we show below, our experimental data under applied pressure suggest that hydrostatic pressure might reduceᾱ, thus making the transition first order and simultaneous. Now, we turn our attention to the field and pressure dependence of the coupled transitions in CeAuSb 2 . Figure 3(a) shows the temperature dependence of the inplane resistivity (ρ ab ) of CeAuSb 2 under various pressures up to 2.66 GPa. At ambient pressure, a sharp drop in ρ ab occurs at about 6.8 K. As shown in Fig. 3(b), at low pressure, a single peak marked by orange arrows occurs in dρ=dT. Within the resolution of these data (ΔT ∼ 0.08 K), this single peak likely encloses both T N and T nem found in C=T, dχ=dT, and ΔL=L (ΔT ∼ 0.01, 0.02, and 0.005 K, respectively). With increasing pressure, T N þ T nem is suppressed to 5 K at 2.24 GPa. Above 2.24 GPa, however, the single peak splits into two peaks in dρ=dT, which are indicated by black and red arrows, respectively. Note that T N2 marks a shoulderlike broad peak that increases with increasing pressure and T nem ¼ T N marks a sharp symmetric peak that decreases up to 2.52 GPa but increases above 2.52 GPa. We discuss the origin of these transitions below. Above 2.16 GPa, the results from increasingand decreasing-temperature ramps are shown together in Fig. 3(b), which are indicated by solid and open symbols, respectively. Interestingly, a hysteresis appears at T nem ¼ T N in dρ=dT, pointing to a first-order phase transition. Hysteresis is not typical of a naked antiferromagnetic (AFM) transition but rather suggests the development of a multicomponent state. At T N2 and T N þ T nem , however, hysteresis is not observed in the resistivity. To shed light on the temperature-pressure phase diagram shown in Fig. 3(c), we turn to the pressure dependence of the initially coupled transitions estimated via the Ehrenfest relation where V m is the molar volume, and Δβ and ΔC=T are changes of the volume thermal expansion coefficient and the specific heat divided by temperature at the transition. By using the experimental values of β and C=T at ambient pressure, we obtain dT N =dP ¼ 0.37 K=GPa and dT nem =dP ¼ −0.77 K=GPa (see Fig. S3   of the two transitions would be opposite [47,48]. These two transitions, however, also decrease together with pressure due to their coupling [49,50]. In our results, the fact that T N2 increases with pressure above the critical pressure suggests that it is associated with a different ordered state not coupled to a tetragonal-symmetry breaking, whose origin will be discussed below. The temperature-pressure false-color map of the local exponent n of the temperature dependence of ρ ab is shown in Fig. 3(c), where n ¼ ∂ ln Δρ=∂ ln T and Δρ ¼ ρ ab − ρ 0 ¼ AT n . The contour map in Fig. 3(c) provides a proxy for the magnetic scattering in different regions of the phase diagram. Above P c , the green region (n ≈ 2) below T nem ¼ T N displays the same behavior as the zero-pressure phase, indicating that the magnetic phase is striped AFM, whereas below T N2 the light-blue region indicates a distinct behavior (n ≈ 1.2). In Fig. 5, we show that the power-law behavior in the light-blue region at high pressure is similar to the behavior of the multi-q AFM phase at ambient pressure and high fields. Therefore, our results suggest that at P c , the coupled transition at T N þ T nem becomes (1) a second-order AFM transition to the multi-q phase at T N2 and (2) a simultaneous first-order structural and magnetic phase transition to the stripe phase at T nem ¼ T N due to a finite magnetoelastic coupling. This case suggests the existence of a multicritical point at P c indicated by a yellow circle in Fig. 3(c) at which four phases meet: a disordered phase PM, a paramagnetic-nematic phase PMnem, an antiferromagnetically ordered stripe-nematic phase AFM(S)nem, and an antiferromagnetically ordered multi-q phase AFM(M).
Evidence for a hysteretic first-order transition is further confirmed in Fig. 4(a), which shows the field dependence of the in-plane resistivity under various pressures, for increasing-and decreasing-field ramps applied along the c axis. At 0.3 K, the first MM transition field H c1 and the second MM transition field H c2 do not change with increasing pressure as shown in Fig. 4(a). At low pressure, the magnetoresistance between H c1 and H c2 monotonically increases with increasing field, in agreement with Ref. [19]. Above 2.16 GPa, however, a hysteretic steplike anomaly at H Ã is observed between H c1 and H c2 . Note that H Ã decreases with increasing pressure and temperature (see Fig. S4 [45]). Figures 4(b) and 4(c) show the temperature dependence of ρ ab and dρ=dT at representative pressures for CeAuSb 2 in an applied magnetic field. At 1.85 GPa, the coupled transition at T N þ T nem turns into two transitions at T Ã and T N2 above H c1 . The new field-induced phase transition at T Ã (H > H c1 ) matches the transition at H Ã (see Fig. S7 [45]). Figure 5 displays the temperature-magnetic-field falsecolor map of the local exponent n of ρ ab for representative pressures, where n ¼ ∂ ln Δρ=∂ ln T and Δρ ¼ ρ ab − ρ 0 ¼ AT n . The contour maps show the presence of two distinct regions in the AFM phase diagrams. The single-q region (S) displays a local exponent n ≈ 2, whereas the multi-q region (M) reveals n ≈ 1 at ambient pressure. The similar color map for the exponent n in Ref. [18] does not display the difference of magnetic scattering between distinct magnetic phases because the color scale of that map is not segmented enough to show the difference of n between 1 and 2. At 2.52 GPa, the contour map indicates that the pressure-induced new phase in zero field below T N2 is related to the field-driven multi-q phase at ambient pressure. In order to understand the origin of T Ã and the other phases denoted by (M', M", and M*) under pressure, the underlying magnetic structures will have to be determined by neutron or nuclear magnetic resonance measurements under applied pressure and field.
Interestingly, experiments on CeAuSb 2 under uniaxial strain along the ½100 direction also suggest additional magnetic phases [30,31]. In particular, the H − T phase diagram under applied compression along the ½100 direction resembles our phase diagram in Fig. 5(d), but it is not precisely the same [31]. Different effects caused by uniaxial and hydrostatic applied pressure have also been observed in the Fe-based superconductors [51]. Finally, we note that the tricritical point of H c2 indicated by yellow circles in Fig. 5 is not suppressed to zero temperature with pressure (see Ref. [45]).

IV. DISCUSSION
Striped magnetic phases are ubiquitous in strongly correlated materials [26,29,52,53], and unconventional superconductivity is arguably intertwined with spin and charge stripe correlations in the copper oxides and ironbased materials [27]. However, superconductivity is absent in, e.g., nickel oxides, which also host a stripe pattern within the NiO 2 planes analogous to the CuO 2 planes in copper oxides [28]. Here, we show that the f-electron system CeAuSb 2 also supports a nematic state in the absence of superconductivity. In CeAuSb 2 , the signatures of two phase transitions in various thermodynamic measurements are very similar to signatures for nematic and magnetic transitions in the Fe-based superconductors. Despite the opposite sign of the calculated dT=dP for the different transitions, the coupled transitions decrease together with pressure initially, as in the iron-pnictide compounds. Unlike the Fe-based materials, however, in CeAuSb 2 , an additional transition appears at P c . Above P c , this new transition temperature increases with increasing pressure, in contrast to the lower-temperature transition.    The key difference between CeAuSb 2 and the Fe-based materials is the absence of superconductivity under pressure. Though nematicity may boost Cooper pairing in the iron-based superconductors, nematic fluctuations may be weak in CeAuSb 2 due to the first-order nature of the nematic transition and the change in the ground-state wave function of the system at P c . Previous pressure work revealed that two energy scales, one associated with Kondo coherence and the other with crystalline electric field (CEF) splitting, become similar at P c . As a result, wave functions of the excited crystal-field levels become admixed into the ground state, hindering formation of a fully Kondo coherent ground state, yet allowing new magnetic orders that lead to a magnetic state that persists to over 4 GPa [20]. Because of the competing magnetic interactions known to be present in the class of localized materials, various magnetic orders compete under pressure and magnetic field. The family member CeAgSb 2 also shows a new magnetic phase above P c ∼ 2.7 GPa, and T max exhibits unexpected negative pressure dependence above P c , which may be due to the influence of a low-lying CEF level Δ 1 ∼ 50 K [44,54]. Neutron and x-ray scattering studies would be valuable to shed light not only on the crystal-field wave functions at ambient conditions but also on their evolution under applied pressure and magnetic fields.

V. CONCLUSIONS
In summary, we have constructed the temperaturemagnetic-field phase diagrams of CeAuSb 2 under applied pressure. Bulk thermodynamic probes reveal two closely lying phase transitions at T N ¼ 6.3 K and T nem ¼ 6.5 K, and x-ray diffraction measurements verify that striped magnetic order is connected to a structural phase transition at ambient pressure. Our theoretical model suggests that stripe magnetic order is preceded by a nematic phase in CeAuSb 2 . The discovery of a putative nematic phase in an f-electron material at zero magnetic field provides an unexplored framework for nematicity, though superconductivity, if present, remains to be discovered.