Magnetization plateaux cascade in the frustrated quantum antiferromagnet Cs$_2$CoBr$_4$

We have found an unusual competition of two frustration mechanisms in the 2D quantum antiferromagnet Cs$_2$CoBr$_4$. The key actors are the alternation of single ion planar anisotropy direction of individual magnetic Co$^{2+}$ ions, and their arrangement in a distorted triangular lattice structure. In particular, uniquely oriented Ising-type anisotropy emerges from competition of easy plane ones, and for a magnetic field applied along this axis one finds a cascade of five ordered phases at low temperatures. Two of these phases feature magnetization plateaux. The low field one is supposed to be a consequence of collinear ground state stabilized by the anisotropy, while the other plateau bears characteristics of an ``up-up-down'' state endemic for lattices with triangular exchange patterns.

We have found an unusual competition of two frustration mechanisms in the 2D quantum antiferromagnet Cs2CoBr4. The key actors are the alternation of single ion planar anisotropy direction of individual magnetic Co 2+ ions, and their arrangement in a distorted triangular lattice structure. In particular, uniquely oriented Ising-type anisotropy emerges from competition of easy plane ones, and for a magnetic field applied along this axis one finds a cascade of five ordered phases at low temperatures. Two of these phases feature magnetization plateaux. The low field one is supposed to be a consequence of collinear ground state stabilized by the anisotropy, while the other plateau bears characteristics of an "up-up-down" state endemic for lattices with triangular exchange patterns.
A conventional picture of frustrated quantum magnet implies a competition between the Heisenberg terms in S = 1/2 Hamiltonian. A Heisenberg magnet on a generic triangular lattice is an archetype example [1]. Anisotropy, if present, is usually just a weak perturbation stemming from the spin-orbit interactions. Alternatively, like in triangular lattice XXZ model, it acts in the same way on every bond and this situation is not drastically different from the Heisenberg case [2]. However, recently emerging topics of quantum spin ice [3,4] or Kitaev magnets [5,6] teach us a very different approach. In those anisotropy is the key player and the main ingredient creating frustration. Interestingly, this physics stemming from strong spin-orbit coupling is not endemic to the 4d and 5f magnets, but is also possible in 3d magnets, for instance cobalt-based ones [7]. In fact, in low symmetry Co 2+ magnets (S = 3/2 and quenched orbital momentum) the single ion anisotropy that splits the |±1/2 and |±3/2 spin states may not be uniform between the sites. If no unique anisotropy axis is present, the interactions between the spins become frustrated automatically. If the spins are at the same time residing on a non-bipartite lattice such as a triangular one, geometrical frustration is also there. Two frustration mechanisms are present simultaneously and this results in a complicated interplay. This possibility is relatively well explored for a perfect triangular lattice [8], but much less so for less symmetric cases.
The subject material of the present Letter, Cs 2 CoBr 4 , possesses an interesting combination of geometric frustration and anisotropy very much in line with the above discussion. It is the last unexplored member of otherwise well known family of quantum magnets with the distorted triangular lattice Cs 2 M X 4 , where M is copper or cobalt and X is chlorine or bromine. The other three materials, essentially chain-like magnets Cs 2 CuCl 4 [9,10], Cs 2 CoCl 4 [11][12][13] and more twodimensional Cs 2 CuBr 4 [14,15] demonstrate very rich phase diagrams in applied magnetic fields. Although the existence of the last material in this quartet, Cs 2 CoBr 4 , was documented a long time ago [16], it was never investigated in a context of quantum magnetism. In this Letter we report the highly unusual magnetic phase diagram of Cs 2 CoBr 4 , that is very anisotropic and features a cascade of magnetization plateaux for one particular direction of the magnetic field. One of these plateaux is found at zero magnetization, while the other corresponds to a field induced "up-up-down" phase that is characteristic for the triangular lattice systems. The plateaux are well compatible with the effective Hamiltonian, which at the same time creates a lot of uncertainty for the nature of the remaining phases due to the unusual interplay of different frustration mechanisms.
Transparent, cerulean-coloured single crystals of Cs 2 CoBr 4 were grown using the Bridgman method [16]. Its structure is isomorphic to that of the other Cs 2 M X 4 materials, orthorhombic P nma (space group 62) with a = 10.181, b = 7.723, c = 13.492Å. The unit cell shown in Fig 1(a) contains four Co 2+ S = 3/2 ions within four CoBr 4 distorted tetrahedra, related to each other by mirror reflections in ab and bc planes. Mirror ac plane is the only symmetry of an individual distorted tetrahedron. As local symmetry at Co 2+ site is lower than cubic, the single ion anisotropy D(n·Ŝ) 2 should be present. The anisotropy axis is n = (± cos β, 0, sin β) on different tetrahedra, as the symmetry dictates. The angle β and sign of anisotropy constant D are not known a priori (in a sister material Cs 2 CoCl 4 they are estimated as β 51 • and D 7 K [11]). Further idea about interactions between the cobalt spins can be derived from comparison with Cs 2 CoCl 4 , Cs 2 CuCl 4 and Cs 2 CuBr 4 . All of them have dominant interaction J within the chains running along b direction, while the weaker zig-zag exchange J connects the chains into distorted triangular lattice in the bc plane, see Fig. 1(c). The exchange along the a direction is negligibly small. Value of J /J may vary from almost zero (Cs 2 CoCl 4 case) to 0.3 − 0.5 in copper based members of the family.
The key parameters of the Hamiltonian, such as D, β and mean field exchange coupling J 0 = 2J + 4J can be straightforwardly extracted from the magnetic susceptibility data for fields, applied along the three principal directions of the crystal. was measured with an MPMS SQUID magnetometer in a field of 0.1 T. This data is shown in Fig. 1(b). The H b (perpendicular to n) susceptibility is quite different from H a, c directions that look rather similar (angles β and π/2 − β between the field and n). All of them show typical "Curie tail" behavior at high temperatures, that becomes suppressed at low temperatures as antiferromagnetic correlations take over. Susceptibility along b shows a rounded maximum close to 4 K -a picture, typical for low-dimensional magnets with suppressed magnetic order. No signs of ordering are found down to 1.8 K. The simultaneous fitting of the data for all three directions, based on a single ion model with mean field interactions (see the Supplemental Material for details [17]) yields D = 14(1) K, β = 44(1) • and J 0 = 5.5(2) K, with gfactors being 2.42(1), 2.47(2) and 2.37(1) along a, b and c directions. This means that i) single ion anisotropy is of easy plane type, so at low temperature only pseudospin-1/2 degrees of freedom are active, and ii) easy planes, while being uniform within the chains, have alternating orientation between the chains and the neighboring ones are nearly orthogonal to each other. We can consider β = π/4 for practical purposes. Then, utilizing the "ro- tated" xyz coordinate system [ Fig. 1(d)], the approximate Hamiltonian for the S = 3/2 cobalt spins can be written as: To construct the effective low-energy Hamiltonian, one needs to project out the high-spin states that are inaccessible at low temperatures due to large D. This is achieved by SchriefferWolff transformation [12,18], where to the zeroth order we can simply replace spin-3/2 operators with spin-1/2 ones asŜ x,y → 2Ŝ x,y andŜ z →Ŝ z in even chains;Ŝ z,y → 2Ŝ z,y andŜ x →Ŝ x in odd chains. The resulting Hamiltonian is:   3. (a,b) Magnetization of Cs2CoBr4 at T = 0.1 K measured with Faraday balance technique. Thick dashed line is the reference SQUID data at 1.8 K [same experiment as in Fig. 1(b)]. Horizontal dashed line shows the expected saturation moment gµBS. Inset in (b) shows the relative magnetization of pseudospin-1/2 degrees of freedom derived from the magnetization curve. Three plateaux are identified as corresponding to collinear antiferromagnetic, collinear up-up-down and fully polarized arrangements of pseudospins.
A graphical representation of this Hamiltonian is also given in Fig. 1(d). Here the intrachain exchange is of strongly XY nature, with the easy plane direction alternating between the chains. In contrast, the frustrated interchain interaction is now Ising-like, with the easy axis given by the only common direction of two adjacent easy planes.
This difference between the emergent Ising axis b and the other directions is clearly manifest in the specific heat data. Measurements on m = 0.81(6) mg Cs 2 CoBr 4 sample were carried out on a 9 T PPMS system with a 3 He-4 He dilution refrigerator insert. A standard relaxation calorimetry method was used, also in combination with so-called "longpulse" technique [19]. The resulting cumulative specific heat dataset for H a, b directions is shown in Fig. 2(a,b). For the transverse field direction a (as well as for the others, see Supplemental Material [17]) the phase diagram essentially contains a single ordered "A" phase and a small "F" satellite. In contrast, field along the emergent Ising axis results in a sequence of five different phases, from "A" to "E". In either case the phase diagram is terminated around 5.5−6 T and above this field the system is simply a semipolarized anisotropic paramagnet. This is in line with the effective exchange coupling J 0 5.5 K determined from the mean field analysis. Anomalies corresponding to the phase transitions are well visible in the C p (H) scans, with examples given in Fig. 2(c,d). Additional insight is brought by capacitive torque magnetometry. Similarly to [20], sample is placed on a flexible cantilever and the force that it experiences is measured by the cantilever deflection that translates into the setup's capacity change. The torque data [ Fig. 2(e,f), the original curves are given in the Supplemental Material [17]] ensures that all the observed transitions are of magnetic origin: each transition results in a strong anomaly in total force that acts on the sample in the magnetic field.
The low temperature specific heat in A-and C-phases is vanishingly small. This suggests their gapped nature (see the Supplemental Material for some details [17]). While for small magnetic fields the gap seems a natural consequence of Ising-like anisotropy, its presence in the magnetized C-state is not so trivial. A candidate gapped state in a system featuring triangular bond pattern is the famous "up-up-down" (abbreviated as uud) collinear spin arrangement. This is further confirmed by a direct measurement of Cs 2 CoBr 4 magnetization curve at 100 mK. This measurement is performed on the same sample as the specific heat with the help of miniature home-built Faraday balance magnetometer with a twistresistant cantilever [21]. The resulting curves are demonstrated in Fig. 3 together with the reference data from SQUID magnetometer at 1.8 K that was also used for calibration. While for the transverse H a direction the measured magnetization curve is relatively smooth and shows only weak kinks at the two phase transitions, the situation is very different for longitudinal H b magnetization. Most of the transitions are marked with discontinuities. Moreover, the slope of the magnetization curve is clearly reduced in A-and C-phases. But are these the real magnetization plateaux? We argue that they are. The extra slope dM/dH is originating from admixing of single ion high energy |±3/2 states to the ground state by a non-commuting magnetic field, and it is also pronounced at high fields when the pseudospin degrees of freedom are fully polarized. The slope of magnetization curve slightly above 6 T [22] should provide a reasonable estimate of the effect. The corrected data representing the relative magnetization of the pure pseudospin-1/2 is shown in the inset of Fig. 3(b). The plateau character of the A-and C-phases is much more pronounced in this representation.
For the low field A-phase a suitable candidate structure might be a collinear antiferromagnetic "stripe" state (as it was observed in a sister material Cs 2 CoCl 4 [11], see Fig. 4). This state automatically satisfies the anisotropic exchange interactions within both even and odd chains. On the mean field level the chains remain decoupled for any strength of J , and the overall collinear structure must be fixed by some kind of "order from disorder" mechanism [23]. This state is significantly more robust in Cs 2 CoBr 4 than in Cs 2 CoCl 4 : it is destabilized by a field strength that is nearly 0.3 of the saturation field for the 1/2-pseudospins, while in the latter material the corresponding number is only 0.1. This may reflect the increased J /J ratio in Cs 2 CoBr 4 . To summarize, the A-plateau is naturally explained as the non-magnetized state of a collinear magnet in a small field applied along the effective easy axis.
The magnetization C-plateau is close to 1/3 of the full saturated value for the pseudospin, validating it as a collinear uud structure. This is again pointing to the importance of J bonds. The uud collinear structures are specific to the systems with triangular exchange patterns. Being stabilized by the quantum fluctuations at low temperatures, they represent another example of "order from disorder" [1,24]. Again, effectively easy axis character of the system will be in favor of such structure too. Anisotropy and quantum fluctuations are playing together in this scenario, and the resulting 1/3 magnetization plateau is rather wide: it occupies almost 0.25 of the full phase diagram width in the magnetic field. For comparison, in Cs 2 CuBr 4 the relative width of the uud phase is just 0.05 [14,15], and in the ideal triangular Heisenberg case the expected number is 0.2 [24]. Remarkably, a group or recently reported delafossite-like anisotropic triangular lattice antiferromagnets, such as NaYbO 2 [25], NaYbSe 2 [26] and NaYbS 2 [27], were found to exhibit unusually wide uud phase as well. A property they share with Cs 2 CoBr 4 is the significant anisotropy that varies between the bonds.
The nature of the remaining phases, B,D,E and F, is unknown at the moment. While in XXZ models or in the presence of weak spin-orbit interactions various (nearly) coplanar phases are known to occur in a magnetized triangular lattice [2,24,28,29], heavy frustration created by the competing single ion anisotropy directions would probably be prohibitive for their formation in Cs 2 CoBr 4 . Spins in neighbouring chains strongly prefer to be confined in two orthogonal planes, and, while allowing collinear states (as shown in Fig. 4), this circumstance impedes coplanar ones.
The phase diagram of the sister material Cs 2 CoCl 4 , demonstrating some incommensurate and multi-Q states [11,30], is of limited guidance too. It misses the aspect of significant frustration by J interactions, as it can be concluded from absence of uud state. Thus, neither conventional triangular lattice, nor chain-based approach seems to be fully appropriate for discussion of Cs 2 CoBr 4 phase diagram. The situation that we encounter here according to Eq. (2) is more akin, although not fully identical, to a triangular Kitaev-Heisenberg model that can host much more exotic phases like vortex crystals and spin nematics [31]. Although the proposal for extremely exotic physics is too preliminary at the moment, Hamiltonian (2)  FIG. 4. Sketches of the plausible collinear magnetic phases (stripe, uud and saturated) in Cs2CoBr4 for H b field direction. Intervening phases "B", "D" and "E" remain to be clarified.
phase diagram in Fig. 2(b) is suggestive of some nontrivial spin textures that may be present among the many magnetic phases. We would also like to stress that the proposed Hamiltonian (2) is the most basic one, and does not include further symmetry-allowed terms such as second single-ion anisotropy constant E and multiple Dzyaloshinskii-Moriya interactions (that are very important in Cs 2 CuCl 4 , for instance [10,32,33]).
To summarize, S = 3/2 quantum antiferromagnet Cs 2 CoBr 4 is found to feature an unusual type of frustration that stems from both the geometry of exchange bonds and geometry of the strong single-ion anisotropies. The "spin space" component of frustration creates an effective S = 1/2 Hamiltonian with an emergent special direction. Application of the field along this direction results in a cascade of phase transitions, with two phases being M 0 and M 1/3 magnetization plateaux. While the plateau states can be preliminarily identified as collinear antiferromagnetic and uud structures naturally compatible with the effective Hamiltonian, the situation is much less certain for the magnetizable phases. Both J − J character of exchange interactions and frustrated anisotropies are equally important here. This makes the scenarios derived from both XXZ-like triangular lattice or XY-like chain equally problematic for the description of the possible states. To the best of our knowledge, frustrated Hamiltonians of this type were not considered before in the literature. At the same time, the prototype material is already there and the corresponding parameters can easily be tuned by chemical composition or pressure. We believe that further experimental and theoretical effort aimed at exploring this specific frustration mechanism may yield some novel exotic magnetic states. This work was supported by Swiss National Science Foundation, Division II. We would like to thank Dr. George Jackeli (University of Stuttgart) for illuminating discussions. * povarovk@phys.ethz.ch † zhelud@ethz.ch; http://www.neutron.ethz.ch/ [1] O. A. Starykh, "Unusual ordered phases of highly frus-