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Element-specific field-induced spin reorientation and an unusual tetracritical point in MnCr2S4

The ferrimagnetic spinel MnCr2S4 shows a variety of magnetic-field-induced phase transitions owing to bond frustration and strong spin-lattice coupling. However, the site-resolved magnetic properties at the respective field-induced phases in high magnetic fields remain elusive. Our soft x-ray magnetic circular dichroism studies up to 40 T directly evidence element-selective magnetic-moment reorientations in the field-induced phases. The complex magnetic structures are further supported by entropy changes extracted from magnetocaloric-effect measurements. Moreover, thermodynamic experiments reveal an unusual tetracritical point in the H-T phase diagram of MnCr2S4 due to strong spin-lattice coupling.

Frustrated magnets with competing magnetic interactions offer an exceptional playground for studying a variety of exotic magnetic states and emergent magnetic excitations [1]. The competing interactions between the constituent magnetic ions can be fine-tuned by applying an external magnetic field, which stabilizes various field-induced states with complex magnetic structures [2,3]. This leads to a rich magnetic-field-temperature (H-T ) phase diagram that also can be used as a test bed for validating various theoretical models [4,5]. Cubic spinels AB 2 X 4 , where A and B represent magnetic cations, belong to paradigmatic frustrated systems. Geometrical frustration results from the B magnetic cation being located at the vertices of a pyrochlore lattice with antiferromagnetic (AFM) exchange interactions, whereas bond frustration is caused by the competing exchange interactions of J A-A , J A-B , and J B-B [6]. In addition to the magnetic frustration, a strong spinlattice coupling gives rise to a robust magnetization plateau, to unconventional metastable magnetostructural [7], and to spin-driven multiferroic states [8].
MnCr 2 S 4 crystallizes in a normal cubic-type spinel structure (space group F d3m) where the Mn 2+ (S = 5/2, 3d 5 ) and Cr 3+ (S = 3/2, 3d 3 ) ions reside on the tetrahedral A and octahedral B sites, respectively. This results in a bond frustration with competing AFM interactions between J Mn-Cr and J Mn-Mn . The title compound shows two consecutive transitions at T C ≈ 65 K to a ferrimagnetic state and at T YK ≈ 5 K to a canted spin configuration [9]. It is suggested that at T YK , the Yafet-Kittel (YK) phase is stabilized. This phase is composed of two nonequivalent manganese sites, Mn1 and Mn2, with the magnetic moments rotated by +120 • and -120 • with respect to the chromium moment that is parallel to the external magnetic field caused by the strong J Cr-Cr FM interaction [10]. A variety of magneto-structural phases have been identified in the H-T phase diagram spanning up to 100 T and below 20 K, including a robust plateau between 25-50 T [11] (see Sec. I of the Supplemental Material [12]). In addition, from a quantum lattice-gas model, the YK state and the intermediate phase, which is located between the YK and plateau phases, were suggested to be spin-superfluid and spin-supersolid states, respectively [13][14][15]. Very recently, multiferroicity has been reported in the YK and intermediate phases of MnCr 2 S 4 [16,17]. However, a detailed understanding of the magnetic structure in the field-induced phases in MnCr 2 S 4 is still missing. Magnetic spinels with a single magnetic ion could be studied in details by use of bulk magnetization and neutron scattering measurements [18,19]. MnCr 2 S 4 , however, possesses two magnetic ions featured by the site-specific magnetic properties at the A and B sites. This makes bulk-magnetization studies quite challenging to reveal the microscopic nature of the field-induced phases.
In this work, we investigate the microscopic nature by performing soft x-ray magnetic circular dichroism (XMCD) measurements of a MnCr 2 S 4 single crystal in pulsed magnetic fields along with macroscopic thermodynamic experiments. The element-selective XMCD study, complemented by the extracted entropy changes, reveals a complex sublattice-moment rotation at the manganese and chromium sites in fields up to 40 T. Furthermore, the observed magnetocaloric effect (MCE) together with ultrasound and specific-heat measurements reveal a tetracritical point (TP) owing to spin-lattice coupling in the H-T phase diagram, which is a rather surprising finding in an isotropic Heisenberg magnet. We performed various experiments including soft XMCD at the Mn and Cr L 3 edges as well as magnetocaloric effect complemented by specific-heat and ultrasound experiments. Details of the experimental methods are described in Sec. II of the Supplemental Material [12]. We note the very weak magnetic anisotropy due to half-filled e g and t 2g (Mn 2+ ) and t 2g (Cr 3+ ) states with zero orbital moment. Thus ensures that the present results in high magnetic fields are independent on the field direction. This was verified in previous studies that showed no magnetic anisotropy above 0.5 T [9,20]. Figure 1 shows zero-field X-ray absorption spectroscopy (XAS) and XMCD results at the Mn and Cr L 3 edge measured at 15 K. In order to prepare a single-magnetic domain, a pulsed field of up to 20 T along the [110] axis (hereafter called initial field) was applied before taking the zero-field spectra. The XAS spectra with multiplet structures agree with those previously reported for Mn 2+ in tetrahedral [21] and Cr 3+ in octahedral symmetry [22,23]. Note that the Mn XAS and XMCD give an average information on the two nonequivalent Mn sublattices. The largest XMCD features near the L 3 edge (Mn at ∼ 639.8 eV and Cr at ∼ 576 eV) show an opposite sign of the XMCD amplitude, documenting the antiparallel coupling between the Mn and Cr 3d longitudinal moments (m z ). In static-field bulk magnetization measurements, no remanence has been observed at tem- show bulk magnetization data and element-selective L 3 -edge XMCD up to 40 T, respectively, recorded at the initial temperature T ini of 8 K in pulsed magnetic fields. The XMCD amplitude is proportional to the total moment (m z ) of the respective atom [24,25]. The sign of the Mn XMCD amplitude at 639.8 eV is reversed to provide correspondence with the direction of the moment of the Mn sublattice with respect to the field direction as shown in the illustration of Fig. 1 and which is also valid for Fig. 3. The bulk magnetization and its field derivative signal a sequence of successive phase tran- The sign of the XMCD amplitudes is inverted to provide correspondence with the relative Mn sublattice magnetization. The data for Tini = 8 K are the same as that shown in Fig. 2 The results for the field scans between 10 and 25 K are vertically shifted for clarity. The horizontal dashed lines show the zero level, which is close to the maximum field at each temperature. The red, blue, and black arrows denote the anomalies, which signal the phase transitions of II-III, III-IV , and I-IV , respectively.
sitions ranging from I to IV as shown at the top of Fig. 2. These phases are suggested to be I (ferrimagnetic order with disordered transverse components of the moments of the Mn ions), II (YK state), III (lowfield asymmetric state), and IV (plateau state) [15]. In Ref. [15], the low-field asymmetric state was described by a spin configuration with a tilted triangular structure, when compared to the YK configuration, however, with the chromium moments still aligned strictly parallel to the external magnetic field. In earlier work, this asymmetric spin state was thought to result from the manganese spins located at the two nonequivalent tetrahedral sites bearing magnetic moments of different length and sign [26] or this spin state was described by an "oblique" spin configuration with deviations of the chromium moments from the external field direction, a configuration which was thought to be stabilized by biquadratic exchange [27]. Similar spin structures, with the chromium moments not parallel to the external field were proposed and were shown to result from strong spin-lattice coupling effects [11]. Figure 2(b) shows that the Cr and Mn XMCD amplitudes remain almost constant in phase I and II. The Mn XMCD amplitude markedly starts to On the other hand, the Cr XMCD amplitude decreases slightly in phase III (Fig. 2). This could indeed reflect the deviation of the Cr magnetic moments from the original direction parallel to the field in the phase II, which was predicted in a recent study and was explained by strong spin-lattice coupling [11]. In phase IV , the Cr XMCD amplitude further increases and reaches its saturation significantly larger than the amplitude in phase II. The magnetic structures deduced from the XMCD data are shown in Fig. 2(b), which provides element-selective information on the proposed YK (II), low-field asymmetric (III), and plateau (IV ) states. As discussed in the following, this microscopic picture is further supported by our pulsed-field MCE measurements (Fig. 4).
In the temperature range below 20 K, the moment of the Mn sublattice is anticipated to be responsible for the main contributions to the field-induced changes in the bulk magnetization, rather than the Cr counterpart [9]. Figure 3 shows the field dependence of the Mn L 3 -edge XMCD amplitude at selected T ini . At T ini = 8 and 10 K, the system evolves through the phases I, II, III, and IV as shown in Fig.  4. The field-dependent XMCD amplitude signals the transitions of II-III and III-IV . At T ini = 15 and 17.5 K, anomalies are identified that correspond to the phase transition from I to IV . At T ini = 25 K, no anomaly is detected up to 40 T, which well agrees with the previous magnetization data [15]. We note that Monte Carlo simulation suggests subtle displacements of the sulfur atoms that bridge the Mn and Cr cations [11]. In future, it will be important to investigate the lineshapes of XAS/XMCD spectra at high magnetic fields in phase III and IV , which could provide further insights into such subtle changes of the local environments. Now, we present the thermodynamic results. Figure 4 shows the contour plot of the magnetic entropy S M extracted from the isentropic T (H) data (black solid lines) obtained in pulsed fields up to 60 T. The phase boundaries are drawn by purple lines deduced from ultrasound and specific-heat anomalies (see Sec. III of the Supplemental Material [12]), which separate the phases I-V (see Sec. I of the Supplemental Material [12]). The absolute values of the entropy were calibrated with data at 10 T from literature [9].
Each MCE trace shows multiple kinks that signal successive magnetic phase transitions. Obviously, the MCE exhibits the anomalies at the same positions as those deduced from ultrasound and specific-heat measurements. Starting from the phase I, the temperature decreases in the adiabatic processes due to the increase of S M toward the phases II and III. After entering the phase IV , the adiabatic temperature remarkably increases resulting from the decrease of S M up to 40 T. Then S M increases toward the phase V and levels off. The overall phase boundaries are qualitatively in good agreement with the previous works based on other macroscopic experiments [11,15]. The smaller entropy values S M in the phase IV compared to the phases II and III, probably signal the larger noncollinearity of the magnetic structures in the latter states.
Our element-selective XMCD results complemented by this MCE data provide compelling evidence for the proposed noncollinear (collinear) magnetic structures in the phases II and III (IV ). This is in line with a recent report suggesting a spin current or inverse Dzyaloshinskii-Moriya mechanism to be responsible for the multiferroic properties observed in the phases II and III [16]. Furthermore, our MCE results indicate the importance of the adiabatic temperature changes for mapping the H-T phase diagram from pulsed-field experiments (see Sec. IV of the Supplemental Material [12]).
Remarkably, our observations signal the presence of a tetracritical point in the phase diagram, which was already suggested from the phase diagram determined from pyrocurrent measurements [16]. The transition lines that cross at the TP presumably belong to second-order transitions, because of the absence of any divergent and/or hysteretic behavior in the experimental results. Although thermodynamically controversial, the TP, where four second-order transition lines meet, has been theoretically predicted [13,14,28,29] for AFM systems with competing magnetic anisotropies, and experimentally observed as well [30,31]. On the other hand, MnCr 2 S 4 has three spin degrees of freedom (n = 3), dominated by rather isotropic Mn-Mn, Mn-Cr, and Cr-Cr Heisenberg exchange interactions. According to renormalizationgroup theory, the TP does not emerge in n = 3 spin systems under uniform or staggered magnetic fields with either longitudinal or transverse component [14,29]. In MnCr 2 S 4 , the intermediate phase (III) is stabilized via spin-lattice coupling (see Sec. V of the Supplemental Material [12]). Monte-Carlo calculations show that strong spin-lattice interaction in MnCr 2 S 4 spontaneously breaks the Z 2 symmetry of the Mn-Cr bonds in the phases III, IV , and V [11] and effectively generates staggered magnetic fields at the site of the manganese spins. The effective staggered field in MnCr 2 S 4 has both longitudinal and transverse components with respect to the uniform external magnetic field, originating from a slight canting of the Cr moment observed in our experiments [ Fig. 2(b)]. This stabilizes the asymmetric intermediate phase and results in the TP, which is rather unusual for conventional Heisenberg systems [29].
Finally, we discuss the symmetry of the phases I-IV based on the magnetic structures deduced from the current study. At zero field, with decreasing temperature, the O(3) and O(2) symmetries are successively broken at T C and T YK , respectively. In finite magnetic fields, the O(3) symmetry is always broken. In the low-field intermediate phase III, both the O(2) and Z 2 symmetries are broken. On the other hand, in the plateau phase IV , the Z 2 symmetry is kept broken, while the O(2) symmetry is restored. A series of symmetry breakings at the respective phase transitions is consistent with the earlier theoretical studies which predict the TP [14,29].
In summary, element-specific XMCD together with the observed entropy variations provide direct evidence of the proposed noncollinear magnetic structures in the YK (II), low-field asymmetric (III), and plateau (IV ) phases. Because the H-T phase diagram of MnCr 2 S 4 is symmetric with respect to a field of approximately 40 T [11,15], the present results provide a plausible description of the magnetic structures beyond 40 T, in phase V , and the higher-field phase (called inverse YK phase in Ref. [11]) close to saturation. In addition, our thermodynamic results hint toward an alternative route to the tetracritical point (TP) due to significant spin-lattice couplings in frustrated systems. We believe the understanding of the relation between the TP and spin-strain interactions might give further insight into the spin-superfluid and supersolid states suggested in analogy to bosonic systems [15,32].
We acknowledge the support of the HLD at HZDR, member of the European Magnetic Field Laboratory (EMFL), the Deutsche Forschungsgemeinschaft  Fig. S1 shows predicted magnetic structures for MnCr 2 S 4 composed of the chromium moment at the A site and two nonequivalent manganese moments at the B site in the fieldinduced phases reported in earlier studies [1,2]. The phases of I -V are located up to 60 T below 20 K (Fig. 4 in the main text). The phases V I and V II are reached at a magnetic field of 75 and 85 T, respectively [2].
Ultrasound measurements were conducted using a pulse-echo phase-sensitive detection technique [6] in static fields up to 17 T, and in pulsed fields up to 60 T with the sound propagation vector, k, parallel to [111]. Both, longitudinal (k || u) and transverse (k ⊥ u) modes with u||[110] were studied, where u is the displacement vector. These modes are related to the elastic constants c L = (c 11 + 2c 12 + 4c 44 )/3 and c T = (c 11 + c 44 − c 12 )/3, respectively. In the static-field measurements, the ultrasound frequency below (above) 12 T was 60.7 (55.6) MHz.
Specific-heat (C) measurements were performed using a 25 T cryogen-free superconducting magnet. C vs T was obtained using both, a standard relaxation technique in fields above 17 T and a dual-slope method [7] in fields below 14.5 T. MCE in pulsed magnetic fields was measured by using a RuO 2 thermometer (900 Ω, 0.6 × 0.3 × 0.1 mm 3 ) under adiabatic conditions [8,9]. A standard ac four-probe method with a numerical lock-in technique at a frequency of 50 kHz was applied to measure the resistance of the thermometer. High-field magnetization was investigated using a coaxial pickup coil system [10]. Specific heat, MCE, ultrasound, and bulk magnetization measurements were performed in magnetic fields applied along the [111] direction.

III. ULTRASOUND AND SPECIFIC-HEAT RESULTS
In Figs. S2(a) and S2(b), the relative sound velocity ∆v/v and sound attenuation ∆α are plotted on a logarithmic temperature scale for the longitudinal acoustic mode at different static magnetic fields. The strong exchange-striction coupling makes the ultrasonic properties sensitive to the maganese spin structures [1,2]. At zero field, a well-defined deep minimum in ∆v/v and a λ-type anomaly in ∆α appear at T ≈ 5 K. These anomalies shift to higher temperatures with increasing field, which matches well with the field dependence of T YK previously reported [1,11]. Above 12 T, where phase III is the zero-temperature ground state, the anomalies are seen near T ≈ 9.5 K, representing the high-field branch of the phase boundary recently suggested [12]. Above 15 T, ∆v/v (∆α) shows an additional kink (broad maximum) at higher temperatures between 12 and 16 K.
In Fig. S2(c), the temperature dependence of the molar specific heat C is shown in fields up to 20 T, together with the lattice contribution, C l , by assuming a Debye temperature of 410 K [11]. Earlier conductivity studies reported negligible electronic contributions to C in this temperature range [11]. Therefore, magnetic degrees of freedom play the dominant role

IV. MCE AND THE H-T PHASE DIAGRAM
The H-T phase diagram constructed from the pulsed-field results obtained under (quasi)adiabatic conditions can lead to shifted phase boundaries if the adiabatic temperature changes caused by magnetocaloric effect (MCE) are not properly taken into account [13].
Earlier pulsed-field measurements indicated that the phase boundaries of I-II and III-IV extend up to ∼ 13 -14 K [1,2,12]. However, the present MCE results provide corrected phase boundaries at lower temperatures, which forms the dome-like structure from the phases II and III in the H-T phase diagram [ Fig. 4 in the main text]. These results imply that the recently suggested ferroelectric phase (called FE3 in Ref. [12]) resides in the plateau state, IV . Figure S4 shows the influence of the MCE effect on the phase diagram of MnCr 2 S 4 . The closed symbols are obtaind from static-field (DC) ultrasound (US) and specific-heat (C) measurements, while the open symbols are taken from pulsed-field US and magnetization (M) measurements. In Fig. S4(a), initial temperatures T ini are used for plotting the anomalies from the pulsed-field data (open symbols). On the other hand, in Fig. S4(b), the temperature position of the anomalies from pulsed-field data are corrected by an interpolation using the T (H) curves, which are shown by black solid lines, obtained from the adiabatic MCE measurements. When the MCE is not considered [ Fig. S4(a)], additional phases seem to exist at higher temperatures than the phases II and III, which was assumed in Ref. [12]. Figure S4(b) indicates that the anomalies obtained from the pulsed-field data map well onto the dome-like boundary that separates the phases I-II and III-IV as measured under static-field condition.
Furthermore, the anomalies at higher fields above 50 T are also located at positions closer to the kinks observed in the T (H) curves in Fig. S4(b) than those in Fig. S4(a).