Spin liquids in geometrically perfect triangular antiferromagnets

Yuesheng Li,1, 2 Philipp Gegenwart,1 and Alexander A. Tsirlin1, ∗ 1Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany 2Wuhan National High Magnetic Field Center and School of Physics, Huazhong University of Science and Technology, 430074 Wuhan, China The cradle of quantum spin liquids, triangular antiferromagnets show strong proclivity to magnetic order and require deliberate tuning to stabilize a spin-liquid state. In this brief review, we juxtapose recent theoretical developments that trace the parameter regime of the spin-liquid phase, with experimental results for Co-based and Yb-based triangular antiferromagnets. Unconventional spin dynamics arising from both ordered and disordered ground states is discussed, and the notion of a geometrically perfect triangular system is scrutinized to demonstrate non-trivial imperfections that may assist magnetic frustration in stabilizing dynamic spin states with peculiar excitations.

I. INTRODUCTION is expected to freeze, whereas its quantum counterpart remains dynamic by virtue of quantum fluctuations. Ex- Frustrated magnets entail competing exchange inter- perimentally, this presence or absence of spin freezing actions and behave differently from their non-frustrated serves as a useful diagnostic tool along with magnetic counterparts, where all couplings act to stabilize a mag- excitations that bear signatures of many-body entangle- netic order. It is then natural that in the presence of ment in quantum spin liquids although may be exotic in frustration any ordering effects are impeded, and a non- classical spin liquids too [1,2]. ordered, paramagnetic-like state survives down to much The gap between these two definitions – phenomeno- lower temperatures than in conventional magnets. Even- logical and microscopic – has led, and still leads to a tually, frustrated spin systems may not show any long- common confusion of whether a given material should be range magnetic order at all, and enter instead a pecu- interpreted as a spin liquid. If it is, how to juxtapose liar low-temperature state known as spin liquid. On the the experimental magnetic response with theory, and if most basic level, this state can be matched with ordi- it is not, is there still a room for the spin-liquid physics? nary liquids in the sense that spins develop short-range It is exactly these controversial but pertinent issues that correlations but lack any long-range magnetic order, and we seek to address in the present brief review using tri- the system does not undergo symmetry breaking upon angular antiferromagnets as an example. For a more de- cooling. tailed introduction into the physics of spin liquids beyond An exact definition of a spin liquid is, however, more the triangular systems we refer readers to excellent sum- complex than that and involves a fair amount of ambigu- maries [3,4], as well as more technical and elaborate ity or even contention. Key ingredients of the spin-liquid overviews of the field [5,6] that were recently published. state are the absence of long-range magnetic order and Historically, triangular antiferromagnets were first sys- the presence of persistent spin dynamics down to zero tems where magnetic frustration was encountered, and temperature. These two aspects essentially distinguish early ideas of quantum spin liquid state were estab- the spin liquid from magnetically ordered states with lished [7]. On the experimental side, the field has seen symmetry breaking, and from spin glasses, where spin several revivals related to active studies of organic charge- fluctuations slow down upon cooling, so that spins even- transfer salts and, more recently, Co2+ and Yb3+ oxide tually become static. On the other hand, without pos- compounds that are the main topic of our present review. tulating any microscopic aspects of this disordered and For practical purposes, we restrict this review to geomet- dynamic (liquid-like) state, such a definition allows fun- 1 rically perfect spin- 2 triangular antiferromagnets, where damentally different systems with similar phenomenology magnetic ions form regular triangular framework, and ex- to be classified as spin liquids. clude systems with higher spin as well as systems with Microscopically, one distinguishes between quantum three-fold frustrated loops comprising non-equivalent ex- arXiv:1911.11157v2 [cond-mat.str-el] 12 Mar 2020 spin liquids where spins are quantum-mechanically en- change pathways (an excellent overview of all those can tangled, and classical spin liquids that can be naively be found in Ref.8). In particular, organic charge-transfer seen as a “soup” of different magnetic orders, all hav- salts that are often discussed in the context of spin-liquid ing the same energy. This classical scenario leaves spins physics [9] are beyond the scope of our present review, to fluctuate as long as the temperature is high enough because their spin lattices entail distorted triangles, and to overcome transition barriers between different ordered low-temperature structural instabilities abound. states. At very low temperatures, classical spin liquid As simple and natural as it seems, the definition of the geometrically perfect material also appears to be contro- versial – perhaps even more controversial than the defi- ∗ [email protected] nition of the spin liquid itself. The natural development 2

FIG. 1. Spin states in triangular antiferromagnets. (a) Valence-bond solid with valence bonds between nearest neighbors; a superposition of all such states makes the nearest-neighbor RVB state. (b) 120◦ and stripe orders. (c) Magnetic phase diagram [10] with lines corresponding to different ∆ values in Eq. (2); SL stands for the spin liquid. Panel (c) is reprinted with permission from Ref. 10, c American Physical Society, 2018. of the field requires that every new spin-liquid candidate pairs and not to individual spins. In real materials, is claimed to be more perfect and ideal than its predeces- this order becomes even more tangible, because VBS sors, which raises a question of whether any material is states are intertwined with lattice distortions that sta- truly “geometrically perfect”. We shall discuss this issue bilize singlet pairs on bonds with stronger exchange in- at some length to show non-trivial departures from the teractions [9]. “perfection”, and argue that they may strongly influence Another, and more crucial difference is that only the the physics, sometimes in a very interesting way. 1 RVB states show fractional, fermionic (spin- 2 ) statis- tics of magnetic excitations that can be contrasted with the bosonic (spin-1) statistics of magnons in long-range- II. THEORY ordered (anti)ferromagnets. Excitation brings spin dimer into a triplet state, which can be thought as a superpo- A. Resonating valence bonds sition of classical states with two parallel spins. In the RVB states, these spins can separate from each other Early ideas of the quantum spin liquid appeared in the and move through the crystal independently, because context of Heisenberg spins on the triangular lattice. An- the ground-state wavefunction embraces all possible con- derson and Fazekas [7, 11] conjectured that long-range- figurations of valence bonds. In this case, the spin-1 ordered 120◦ N´eelstate (Fig.1b) obtained by classical excitations break down (fractionalize) into two spinons minimization is not the lowest-energy state of a quan- (Fig.2) that become elementary excitations of an RVB tum system, where spins gain additional energy by form- state [16, 17]. Emergent spinon excitations reflect the ing pairs, quantum-mechanical singlets. These singlet highly non-trivial, entangled nature of RVB quantum pairs were termed valence bonds and treated using for- spin liquids and bear direct relation to charge fraction- malism initially developed by Rumer for molecules [12– alization of fractional quantum Hall state in electronic 14]. The intuitive chemical analogy was further backed systems with quantum entanglement [18–20]. Topologi- by the fact that valence bonds not only occur as spin- cal aspects behind this relation are elaborated in a recent pairs but can also form resonant configurations akin to review article [21]. Pauling’s idea of resonating bonds in molecules [15]. The possibility of spinon and other fractionalized ex- The presence or absence of the resonance effect distin- citations triggered major interest in theories of quan- guishes valence-bond solid (VBS), the combination of tum spin liquids. Experimental work inspired by their static and non-resonating valence bonds on a lattice, from predictions initially focused on the search for materials the resonating-valence-bond (RVB) states obtained by a that evade magnetic order as best candidates for real- superposition of different VBS states. It is this RVB world hosts of spinon excitations, although some of the state with singlet pairs restricted to nearest neighbors recent results suggest that even long-range-ordered mag- (Fig.1a), that was proposed as the lowest-energy, liquid- nets may show non-trivial spectral features [22, 23] po- like ground state of triangular antiferromagnets [7, 11]. tentially related to fractionalization. The question of Both VBS and RVB states would fall under the most whether these complex excitations are truly fractional- general definition of a spin liquid in the sense that they ized is, however, much more subtle [24], and we shall see lack long-range magnetic order and show persistent spin this below. dynamics. However, the VBS state involves in fact some Different types of spin liquids exist under the umbrella symmetry breaking, although it now relates to singlet of RVB physics. In triangular systems, short-ranged va- 3

FIG. 2. Spinon excitation in the nearest-neighbor RVB state of triangular antiferromagnets. The excitation is created by 1 breaking one of the valence bonds. Two unpaired spins can propagate independently and constitute spin- 2 (spinon) excitations. lence bonds like those shown in Fig.1a give rise to a continuum may be a strong, if not compelling signature gapped Z2 quantum spin liquid [25, 26] characterized by of the spin-liquid physics. a topological order [27] with two types of excitations. Those at higher energies are spinons due to the breaking of valence bonds, whereas low-energy excitations are vi- B. Spin Hamiltonians sons related to the mutual arrangement of the valence bonds and topological order therein [27, 28]. Among Further evidence for the spin-liquid behavior may come many peculiar properties of these excitations, we note from the fact that material in question lies close to the their potential usage in topological quantum comput- interaction regime where theory predicts breakdown of ing [29]. magnetic order with the formation of a spin liquid. For When valence bonds occur between all atoms in the the purpose of the current review, we restrict ourselves crystal, including distant neighbors, gapless spin liquid to spin models with pair-wise interactions typical for in- characterized by purely spinon excitations is formed [17]. sulating systems. Materials like organic charge-transfer Spinons develop a Fermi surface and interact with the salts approach or even undergo metal-insulator transi- U(1) gauge field [30]. The main lure of this U(1) or, tions and require more complex spin models that also in- colloquially, “spinon-metal” state is associated with high- clude multi-spin terms, most notably ring exchange, on temperature superconductivity, because fractionalization top of the pair-wise interactions. These additional terms separates spin and charge and may allow charge propa- play crucial role in stabilizing quantum spin liquids [9]. gate unhindered by magnetic effects. A comparatively Anisotropic spin Hamiltonian of triangular antiferro- recent review of this topic can be found in Ref. 31. magnets comprises several terms, Experimentally, both gapless and gapped quantum X  XXZ ±± z± spin liquids are characterized by a continuum of spinon H = Hm + Hm + Hm , (1) excitations conveniently probed by inelastic neutron scat- m tering. Thermodynamic and transport measurements of- fer additional diagnostic tools. The “short-ranged” RVB where m = 1 describes nearest-neighbor interactions, states are gapped and should give rise to an exponen- m = 2 describes second-neighbor interactions, etc. The tial behavior of the magnetic susceptibility and specific first term of Eq. (1) stands for the XXZ Hamiltonian [38], heat. In contrast, the gapless U(1) quantum spin liq- XXZ X x x y y z z uid in a triangular system is identified by the sub-linear Hm = Jm (Si Sj + Si Sj + ∆Si Sj ), (2) 2 T 3 power-law behavior of the specific heat [32], although hiji linear or even exponential (gapped) behavior may occur too when spinons interact, eventually reducing the gauge and ∆ is the extent of the XXZ anisotropy in the notation of Refs. 10 and 39. symmetry to Z2 [33, 34]. Predictions for the magnetic susceptibility [35] as well The second term describes diagonal components of the as optical [36] and thermal [37] conductivities of spin liq- exchange beyond the XXZ model, uids with spinon Fermi surfaces are available too. How- ±± X ±± x x y y ever, one has to be aware that by no means these pre- Hm = 2Jm [(Si Sj − Si Sj ) cos ϕα− dictions exhaust all possible experimental responses, nor hiji x y y x − (Si Sj + Si Sj ) sin ϕα], (3) the family of RVB states embraces all possible instances of quantum spin liquids. Many other types of spin-liquid whereas the third term stands for the off-diagonal phases have been envisaged by theory [5,6], and their ex- anisotropy, perimental identification should be taken pragmatically. Any unconventional behavior, such as linear or sublinear z± X z± y z z y Hm = Jm [(Si Sj + Si Sj )] cos ϕα− evolution of the magnetic specific heat, coupled with the hiji x z z x absence of magnetic order and presence of an excitation − (Si Sj + Si Sj ) sin ϕα]. (4) 4

Here, ϕα = 0, ±2π/3 is the bond-dependent pre-factor tion to the aforementioned physics, while others entail that reflects the rotation of the local coordinate frame different types of structural deformations. We first dis- under three-fold symmetry of the lattice. cuss Co-compounds that fulfill our criterion of geometri- ±± z± The ∆ = 1 and Jm = Jm = 0 regime implies cal perfection, do not yet enter the spin-liquid state, but isotropic (Heisenberg) interactions. In systems of our can be described by model Hamiltonians in the vein of interest, departures from this regime can be caused by Eq. (1) and reveal highly non-trivial excitations. the easy-plane anisotropy (∆ < 1), anisotropy in the ±± xy plane (Jm 6= 0), or the off-diagonal anisotropy z± (Jm 6= 0). A. Single-ion physics

2+ 1 Co proved convenient for studying spin- 2 antifer- C. Survey of magnetic ground states romagnets and especially triangular systems. In the 2+ 7 3 octahedral environment, Co (3d ) features S = 2 , Contrary to Anderson’s conjecture, Heisenberg anti- but its orbital moment remains unquenched. Spin- ferromagnets with purely nearest-neighbor interactions orbit coupling then splits this manifold and separates feature the long-range-ordered 120◦ ground state [40, 41] the lower-lying Kramers doublet that becomes predom- that also survives (sometimes with weak modifications) in inantly occupied at low temperatures, acting as an ef- 1 the presence of easy-axis (∆ > 1) or easy-plane (∆ < 1) fective spin- 2 [62]. Orbital degeneracy is thus lifted by anisotropies [42–44]. Only in the pure Ising limit will a the spin-orbit coupling without lowering the symmetry, partially disordered state occur [45]. This state is, how- and a regular atomic arrangement with the strong geo- ever, not a quantum spin liquid, because Ising spins are metrical frustration can be preserved, in stark contrast 1 2+ 3+ 4+ classical. Large residual entropy indicative of a classical to other spin- 2 ions. For example, Cu , Ti , and V spin liquid is found indeed [45]. are all subject to strong Jahn-Teller effects that distort Quantum spin liquid can be stabilized by interactions triangular frameworks or even lead to their defragmenta- beyond nearest neighbors. The J1 − J2 model of Heisen- tion [63, 64]. berg spins on the triangular lattice received ample at- The typical splitting between the ground-state 1 tention and was shown to host a spin-liquid phase at (jeff = 2 ) Kramers doublet and lowest excited state is of J2/J1 ' 0.07 − 0.15 [46–50], although the nature of this the order of 10−20 meV [65], suggesting that at least be- 1 phase remains vividly debated, with both gapless [51–54] low 50 K the magnetism is purely spin- 2 . In the trigonal and gapped [46, 55] scenarios being likely proposals. At and hexagonal symmetries typical for triangular antifer- 2+ higher J2/J1, a collinear stripe order (Fig.1b) becomes romagnets, Co acts as an anisotropic magnetic ion with stable [56–58]. different g-values and different exchange interactions for Exchange anisotropy offers another route to the spin- the in-plane and out-of-plane directions [62]. The ex- ±± z± liquid state(s). The effect of the J1 and J1 terms of tent of this XXZ exchange anisotropy is relatively weak, Eqs. (3) and (4) largely resembles that of J2, because though (TableI). they stabilize stripe order too [59, 60]. A region of the Many of the Co2+-based triangular materials are non- quantum spin liquid phase separates these stripe states frustrated, because closely spaced Co2+ ions feature fer- from the 120◦ order [10, 61] and may thus appear even romagnetic interactions arising from nearly 90◦ superex- in the absence of interactions beyond nearest neighbors change pathways, as in CoCl2 and CoBr2 [66] or β- (Fig.1c). Interestingly, this quantum spin liquid of the Co(OH)2 [67]. Robust antiferromagnetism is only pos- anisotropic J1-only model is connected to the correspond- sible in materials with large Co–Co separations that ing regime of the isotropic J1 − J2 model [10, 61], indi- naturally eliminate any couplings beyond J1, because cating the same (conceivably, Dirac [54, 61]) type of a second-neighbor Co–Co distances of at least 10 A˚ are pro- quantum spin liquid anticipated in triangular antiferro- hibitively large for the superexchange. With negligible ±± z± ±± z± 2+ magnets if either of the J2, J1 , or J1 are properly J and J terms, Co triangular antiferromagnets tuned – a positive message for the experiment. Another are doomed to remain in the 120◦-ordered state, but can (dual) spin-liquid region has been identified in the limit of be used to probe its interesting dynamics. z± large J1 [39] but may be harder to reach in real materi- ±± ± als, where J1 and Jz are usually smaller than J1. It is also worth noting that none of these putative spin-liquid B. Hexagonal perovskites and magnetic excitations phases is directly related to the RVB states discussed previously. Best material prototypes of Co-based triangular an- tiferromagnets are found among hexagonal perovskites of the 6H-Ba3CoX2O9 family with X = Sb [68–70], III. CO-BASED MATERIALS Nb [71, 72], and Ta [73, 74] (Fig.3a). They share many similarities, but only the Sb compound was so far studied Many triangular antiferromagnets were studied over in detail thanks to the availability of large single crys- the years, but only a handful of them show close rela- tals [75]. It behaves as a typical easy-plane triangular 5

TABLE I. Microscopic parameters of Co-based triangular antiferromagnets. All N´eeltemperatures TN and exchange constants J are given in Kelvin. Two parameter sets for Ba3CoSb2O9 are obtained from fits to the magnetization data [77] and to the inelastic neutron scattering data in the 1 3 -plateau phase [103], respectively. z J1 ∆ TN TN /J gk g⊥ Ref.

Ba3CoSb2O9 19.5 0.95 3.8 0.21 3.87 3.84 [77] 20.3 0.86 3.8 0.22 3.95 [103]

Ba2La2CoTe2O12 22 3.8 0.17 3.5 4.5 [104] FIG. 3. Excitations of the 120◦ ordered state. (a) Crys- Ba8CoNb6O24 1.7 1.0 0.1? – 3.84 [105] tal structure of 6H-Ba3CoSb2O9 with the triangular layers of Co2+ ions in the ab plane. (b) Excitation spectrum probed by inelastic neutron scattering, with the white lines showing one- magnon dispersion from linear spin-wave theory [84]. Note given the nearly 60 % reduction of the ordered moment the excitation continuum that appears above 1 meV at the with respect to its classical value [98, 99]. Moreover, K-point and above 1.8 meV at the M-point, while at lower non-collinear spin arrangement facilitates magnon de- energies quasiparticle bands repelled by the continuum [85] cays [100, 101] that eventually account for the continuum are observed [76, 84]. Panel (a) was prepared using the VESTA at ~ω > J1 [102]. software [86], and Ba atoms were omitted for clarity. Panel Two aforementioned scenarios – exotic fractionalized (b) is reprinted from Ref. 84, c CC-BY-4.0. excitations vs. interacting magnons – are in fact encoun- tered in several materials of current interest, such as the Kitaev candidate α-RuCl3 [23, 24]. On the experimental antiferromagnet with the 120◦ magnetic order in zero side this leads to a large deal of ambiguity, because even field [68, 70, 76] and anisotropic magnetization process the observation of an excitation continuum, the main fin- 1 revealing the 3 -plateau at 10−15 T for in-plane fields, but gerprint and ultimate signature of the spin-liquid physics, no plateau when the field is applied perpendicular to the appears to be inconclusive when it comes to the question easy plane [77–79]. High-field behavior of Ba3CoSb2O9 of whether spinons occur or magnons decay. was of significant interest, because the coplanar V -phase Experiments on Ba3CoSb2O9 confirmed that only at stabilized by quantum fluctuations [80, 81] could be de- low energies, ω = J1 ≤ 1.6 meV, do the excitations re- 1 ~ tected experimentally for the first time in a spin- 2 mag- semble magnons (Fig.3b). At higher energies, a broad net [82, 83]. However, the main draw of this material continuum is observed [70, 76, 84]. Magnons are strongly relates to its non-trivial magnetic excitations that were renormalized and damped [76], in agreement with non- recently juxtaposed with theoretical predictions for tri- linear spin-wave calculations. As for the continuum part angular antiferromagnets. of the spectrum, the bulk of the spectral weight is ob- Already the first theoretical studies of spin dynamics served below 4 meV [84] corresponding to ~ω < 2.5J1 in exposed salient deviations from non-interacting magnon fair agreement with both spinon [92] and magnon [102] scenario of linear spin-wave theory. Not only the spin scenarios. The aspect missing in both is the double-band waves are renormalized, sometimes changing their shape structure around the M-point at 1.3− 1.6 meV, (Fig.3b) to produce roton minima [87] around the M-points [88– the energy range where excitation continuum already ap- 90], but also the spectrum is washed out into a broad pears in other parts of the Brillouin zone [84]. Recent continuum at energies ~ω > J1 [90–93]. These findings theory work ascribed this feature, avoided quasiparticle were initially interpreted as signatures of spinon excita- decay, to an interaction between the one-magnon band tions – an idea inspired by the work on square-lattice and continuum [85]. Instead of smearing out the for- Heisenberg antiferromagnets. mer, strong interaction separates the two. The contin- 1 Spin- 2 antiferromagnets, especially in 2D, show re- uum shifts to higher energies, while the “one-magnon” duced ordered moments indicative of spin fluctuations (quasiparticle) band is pushed down in agreement with in the ordered state. This fluctuating component can the experimental observations. be represented by an RVB state and held responsible for This strong-interaction scenario relies on the presence various spectral features [94], including the broadening of an excitation continuum, while making no assump- of spectral lines at high energies interpreted as a spinon tions regarding its magnon or spinon origin. Indeed, continuum [22, 95]. While a similar physics could be whereas magnons are a good starting point for under- envisaged in the triangular case [89], and spinons may standing all spectral features of large-spin triangular an- 1 indeed account for a large part of the calculated spec- tiferromagnets [106], the spin- 2 case of Ba3CoSb2O9 is tral weight [92], non-linear spin-wave theory offered an more involved. First, vestiges of the continuum are seen alternative explanation in terms of interacting magnons. up to much higher energies (~ω ' 6J1) than any theory The roton minima can be reproduced by including the would predict. Second, exact parametrization obtained 1 1/S corrections [96, 97] that are crucial in this case, by inelastic neutron scattering in the 3 -plateau phase of 6

XXZ antiferromagnet. An important lesson from these materials is that not only the interlayer couplings but also J1 can become very small (TableI), thus shifting the onset of spin-spin correlations and TN to very low temperatures. Recently reported materials like glaserite- type Na2BaCo(PO4)2 lacking magnetic order down to 50 mK [110] should also be scrutinized from this per- spective and do not necessarily feature spin-liquid be- havior of any kind. Co2+ vanadates of the same glaserite family entail ferromagnetic J1 [111–113], suggesting the presence of a ferromagnetic exchange component that in phosphates may nearly cancel the antiferromagnetic one leading to a material with a very weak J1 and only a weak frustration.

FIG. 4. Crystal structures of triangular Co-based antiferro- magnets. The c axis is along the vertical direction, whereas IV. YBMGGAO triangular layers are in the ab plane. VESTA software [86] was 4 used for crystal structure visualization. The Co2+ compounds give access to only a limited part of the general parameter space of triangular antiferro- ◦ Ba3CoSb2O9 does not allow an adequate quantitative de- magnets and remain in the region of the 120 magnetic scription of the zero-field spectra even on the level of non- order. Other phases should be probed by systems with linear spin-wave theory [103]. This leaves the problem of larger second-neighbor couplings and/or with a more pro- calculating spectral properties of 120◦-ordered quantum nounced exchange anisotropy. Anisotropic exchange in- antiferromagnet largely open, and the magnon-spinon di- teractions with sizable off-diagonal terms occur between chotomy unresolved. 4f ions [114], for example in spin-ice compounds with the three-dimensional pyrochlore lattice [115]. The work on 4f magnets in 2D remained scarce until 2015 when syn- C. New materials thesis and investigation of YbMgGaO4 [116] suggested the possibility of a spin-liquid state and spurred interest Zero-field 120◦ magnetic order generic to the 6H- in 4f-based triangular antiferromagnets. In contrast to the Co2+ compounds, where XXZ interaction regime is Ba3CoX2O9 materials appears to be one obstacle in the realization of a spin-liquid phase. This motivated several known with all certainty, the 4f ions with their different attempts to suppress magnetic order by increasing the electronic configurations and crystal-field ground states distance between the Co2+ layers. offer (at least potentially) a much broader diversity of mi- From the structural standpoint, triangular layers of croscopic scenarios. An inevitable drawback is that each Co2+ are separated by perovskite-type slabs with non- material features many exchange parameters leaving the magnetic ions. The thickness of these slabs can be in- microscopic parametrization ambiguous and the physical creased (in theory, arbitrarily) leading to compounds scenario controversial. like Ba La CoX O (X = Te, W) with the inter- Most of the 4f-based triangular antiferromagnets re- 2 2 2 12 3+ layer Co–Co distance of about 9.8 A˚ and, eventually, to ported to date utilize Yb as the magnetic ion. We shall 2+ discuss its behavior in greater detail using the best stud- Ba8CoNb6O24, where the triangular planes of Co are as far as 19 A˚ apart (Fig.4). With the exception of the ied material YbMgGaO4 as an example. A more detailed latter, these compounds still develop long-range magnetic and technical overview of this material can be found in a 5 recent progress report [117]. order [104, 107]. In fact, even the Mn-based (spin- 2 ) ana- ◦ log of Ba8CoNb6O24 reveals the 120 order [108], whereas Ba CoNb O itself also shows a characteristic peak in 8 6 24 3+ the spin-lattice relaxation rate at 0.1 K reminiscent of A. Nature of the Yb magnetism the magnetic ordering transition, although the NQR line 3+ does not broaden below this temperature [109]. These The central part of YbMgGaO4 are Yb ions with observations suggest that residual interlayer couplings their valence electrons residing in the 4f shell, which is (likely of dipolar nature) always remain in place and tend subject to a strong spin-orbit coupling. While formally a 7 3+ 1 to induce magnetic order with a sizable TN /J = 0.1−0.2. J = 2 ion, Yb reveals an effective spin- 2 physics at low Despite the increased interlayer spacing and reduced temperatures, because crystal electric fields (CEFs) split 7 TN , even the most 2D materials show the same dynamics the J = 2 multiplet into four Kramers doublets, similar as their less 2D analogs [105]. They can be most natu- to the effect of spin-orbit coupling on Co2+ (Sec.IIIA). rally understood as systems approaching eventual long- Only the lowest CEF level is relevant to cooperative mag- range order, which is indeed anticipated in any J1-only netism observed at low temperatures. This level is a 7

FIG. 5. Determination of the exchange parameters in YbMgGaO4 [116]. The Curie-Weiss fitting of the inverse susceptibility, −1 χ (a), is performed in the temperature range of the R ln 2 plateau in the magnetic entropy, Smag (b), whereas the temperature- independent van Vleck contribution χ0 can be cross-checked by the slope of the magnetization isotherm, M(H), above the 3+ 1 saturation field (c). The saturated magnetization is Ms =gµ ¯ B S ' 1.6 µB /Yb compatible with S = 2 andg ¯ ' 3.29, the powder-averaged g-value determined from ESR [118].

Kramers doublet and can be directly mapped onto a 1 spin- 2 problem. More precisely, one refers to the mag- netic moment of Yb3+ associated with this doublet as a 1 pseudospin- 2 , because it combines strongly intertwined spin and orbital moments. Its principal feature is mag- netic anisotropy caused by the complex nature of the ground-state wavefunction and by the influence of the higher-lying CEF levels on the exchange. Another crucial feature of Yb3+ is the strong localiza- tion of its 4f electrons. Despite this ultimate localiza- FIG. 6. Low-temperature properties of YbMgGaO4. (a) tion, magnetic interactions between the pseudospins are Power-law scaling of the magnetic specific heat Cmag mea- not of purely dipolar nature and involve orbital overlap. sured in zero field [116], and (b) thermal conductivity κ sup- pressed with respect to the non-magnetic reference compound For example, in YbMgGaO4 with the Yb–Yb distance of 3.85 A,˚ the dipolar interaction of 0.25 K makes only 14% LuMgGaO4 [120]. Dotted lines are extrapolations that high- of the total interaction J ' 1.8 K determined experi- light the absence of the linear contribution, which would be 1 expected in a gapless quantum spin liquid. mentally from the Curie-Weiss temperature [118]. This puts forward superexchange as the main mechanism of magnetic couplings in Yb-based triangular antiferromag- nets and even allows a microscopic evaluation of magnetic C/(T −θ)+χ0, must be done in a prudently chosen tem- interactions based on superexchange theory [119]. perature range. At high temperatures (above 30 − 50 K 3+ The superexchange is weak, though, so Yb3+ oxides do in Yb oxides), 1/χ deviates from the linear behavior not reveal their interesting cooperative magnetism un- due to CEF excitations (Fig.5a), while at low temper- less cooled down to temperatures of the order of 1 K. atures, typically below 10 K, magnetic interactions come Many of these compounds were known since decades but into play. Therefore, both upper and lower limits of the traditionally described as simple paramagnets until mea- fit require a careful consideration. sured at low enough temperatures. This naturally sets a Magnetic entropy guides the choice. Between 10 and question of the relevant temperature scale. Which tem- about 30 K, it shows a plateau at R ln 2 corresponding to perature is low enough, and how to decide whether the the ground-state doublet (Fig.5b). The plateau implies absence of magnetic order down to a certain tempera- that, on one hand, magnetic interactions have been over- ture is a signature of the spin-liquid behavior, or simply ridden by thermal fluctuations and, on the other hand, a feature of very weak magnetic interactions that are eas- the temperature is not high enough to trigger CEF ex- ily overridden by thermal fluctuations within the selected citations, although the very presence of the higher-lying temperature range? CEF levels causes the non-zero van Vleck term χ0 in the susceptibility. Fitting three parameters (χ0, θ, and the Curie constant C) in such a narrow temperature win- dow certainly becomes ambiguous, but the estimate of B. Low-temperature behavior χ0 can be cross-checked by measuring field dependence of the magnetization, because above saturation M(H) YbMgGaO4 serves as a good illustration of how this still shows a small linear slope caused by the χ0H contri- temperature scale can be determined experimentally. bution due to the CEF excitations (Fig.5c). As for the The strength of magnetic interactions is gauged by the Curie constant C, it is proportional to the square of the Curie-Weiss temperature θ, but the Curie-Weiss fit, χ = g-factor that, in turn, can be independently determined 8

cusp and without any associated peak in the specific heat, although in spin glasses entropy change at the freezing point is usually detectable [124]. Moreover, no splitting between field-cooled and zero-field-cooled curves mea- sured in dc-field has been observed [125]. Magnetic re- sponse of YbMgGaO4 is thus very different from that of a canonical spin glass – perhaps not surprising, given persistent spin dynamics probed by muons even below the alleged freezing point [121]. All these observations suggest that YbMgGaO4 does show dynamic spins and hosts a spin liquid of some kind, whereas the peak in the ac-susceptibility may reflect only a minor frozen compo- nent.

FIG. 7. Excitation continuum in YbMgGaO4 at energies C. Spin dynamics of 0.3 meV (~ω = 2J1) and 1.5 meV (~ω = 10J1)[122]. Right panel shows the calculation based on the spinon-metal scenario. Reprinted from Ref. 122 with the permission, c Magnetic excitations of YbMgGaO4 do not contribute Macmillan Publishers Limited, 2016. to heat transport. Nevertheless, they do form a contin- uum that has been extensively studied by neutron scat- tering [122, 125–128]. It extends to about 2 meV corre- from electron spin resonance (ESR). This leaves θ as the sponding to energies as high as ~ω ' 13J1. With a very only independent fitting parameter and eventually leads similar momentum dependence of the spectral weight at to the estimates of J1 ' 1.8 K and ∆ ' 0.55 from the low (0.3 meV) and high (1.5 meV) energies [122], this inverse susceptibility measured for different field direc- spectrum (Fig.7) is drastically different from the typ- tions [118]. ical response of 120◦-ordered Co-based antiferromagnets The interaction strength of about 2 K also manifests (Sec.IIIB), where the spectral weight is restricted to itself in other physical quantities. Magnetic specific heat much lower energies and shows different momentum de- (Fig.6a) shows a maximum at 2.5 K due to short-range pendence at different energies, even inside the continuum. order [116], whereas muon relaxation rate gradually in- In YbMgGaO4, the spectral weight accumulates at creases around the same temperature, indicating the on- the zone boundary and appears to be nearly absent at set of spin-spin correlations [121]. At temperatures well the zone center, similar to the kagome spin liquid pin- below 2 K, one expects that spin-spin correlations dom- pointed experimentally in herbertsmithite [129]. Gap- inate over thermal fluctuations, and experimental re- less spinon excitations of the U(1) quantum spin liq- sponse of the magnetic ground state can be probed. Only uid can explain this distribution of the spectral weight via measurements in this low-temperature range can one down to at least 0.3 meV [122]. This scenario was fur- assure that the given material hosts a spin-liquid state or ther supported by a V-shaped band splitting in the ap- at least bears relation to the spin-liquid physics. plied magnetic field [126] predicted theoretically for non- In YbMgGaO4, several experimental techniques interacting spinons [130], although interactions between probed the nature of the ground state. First, mag- spinons change the scenario qualitatively [131]. netic specific heat shows no anomalies down to at least An alternative scenario was offered in Refs. 125 and 128 50 mK [116], whereas muons reveal persistent spin dy- that interpret the same broad continuum as excitations namics [121]: two necessary conditions of the spin-liquid out of a generic valence-bond state. Neutron scatter- behavior. Below 0.4 K, magnetic specific heat follows the ing experiments down to 0.07 meV [125], about 4 times γ Cm(T ) ∼ T power law with γ ' 0.7 [116], indicating lower energy than in Ref. 122, suggest a change in the gapless spin excitations (Fig.6). Moreover, the γ value width of the continuum with the threshold value around 2 is compatible with 3 expected for the U(1) quantum spin 0.2 meV, which is comparable to the interaction strength liquid in triangular antiferromagnets (Sec.IIA). J1 (Fig.8). Above this energy, the excitations can be as- Fractionalized (spinon) excitations of the U(1) state signed to the breaking of nearest-neighbor valence bonds, should also manifest themselves in the magnetic response while below 0.2 meV the spectral weight is described by and thermal transport. These experiments did not ar- processes that involve a re-arrangement of valence bonds rive at a consistent picture, though. YbMgGaO4 shows and orphan spins, with valence bonds up to third neigh- no thermal transport due to spinons (Fig.6b). More- bors included in the model [125]. 1 over, its thermal conductivity is suppressed with respect Interestingly, both pictures entail spin- 2 excitations to the non-magnetic Lu-based analog suggesting the ab- albeit with a very different physics behind them. The sence of mobile spinons [120]. ac-susceptibility measure- spinon-metal scenario of Ref. 122 postulates fractional- ments even revealed signatures of spin freezing around ized nature of the excitations without explicating their 100 mK [123], but with only a small magnitude of the microscopic origin. It can be better traced in the valence- 9

FIG. 8. Valence-bond model of the YbMgGaO4 excitations [125, 128]. (a) Cartoon representation of the excitation continuum comprising two types of processes: (b) breaking of nearest-neighbor valence bonds, and (c) re-arrangement of valence bonds and orphan spins, where only nearest-neighbor valence bonds are shown for the sake of clarity, but valence bonds up to third neighbors are included in the actual model. (d) Momentum dependence of the spectral weight at two energies in different parts of the continuum [125].

1 bond scenario, although here not all spin- 2 excitations within these planes, the YbMgGaO4 case appears to be are fractionalized. The continuum of high-energy excita- different. tions indicates that, upon breaking a valence bond, two Unequal charges of Mg2+ and Ga3+ prove to be cru- unpaired spins can separate, similar to spinon excitations cial. Depending on the local arrangement of these non- of an RVB state. These spinon-like excitations show a magnetic species, the Yb3+ ions experience different gap of about 0.2 meV, which is on the order of J1 and CEFs. Inelastic neutron scattering reveals three CEF- in agreement with theory (Sec.IIA). On the other hand, related peaks, each of them being much broader than the lower-energy excitations caused by the re-arrangement instrumental resolution (Fig.9a). Moreover, a shoulder of valence bonds fill this gap, extend to lowest energies, around 87 meV would indicate a “fourth” CEF excitation and stem from the presence of orphan spins in the ground forbidden for Yb3+, since only three Kramers doublets state. These excitations are fractional but not fraction- are available for excitations. Ref. 133 offered an atomistic alized. Their presence – or, more precisely, the inelastic interpretation of this strange CEF spectrum. The local nature of the re-arrangement process – further indicates a arrangement of Mg2+ and Ga3+ creates an uneven charge departure from the simple RVB state, where interchang- distribution and causes local displacements of both Yb3+ ing of unpaired spins and valence bonds should cost no and surrounding oxygens (Fig.9c). The magnitude of energy. The finite energy cost of this process may stem these displacements being as large as 0.1 A˚ implies that from a local non-equivalence of different lattice bonds the CEF energies may change compared to the undis- that, in turn, has to be caused by structural inhomo- torted scenario. A superposition of several local config- geneities. urations obtained within this approach leads to a decent description of the experimental spectrum (Fig.9b). Whereas CEF excitation energies do not determine D. Structural randomness magnetic interactions per se, they influence the g-values that are spread over finite ranges ∆g⊥/g⊥ ' 0.1 and Whether or not YbMgGaO4 is prone to structural in- ∆gk/gk ' 0.3, respectively. Moreover, local displace- homogeneities has been a matter of significant discussion. ments change the Yb–O–Yb angles that may not affect On one hand, this material was put forward as a geomet- relative values of the exchange parameters, but do change rically perfect triangular antiferromagnet based on its ro- their absolute values. For example, from the superex- bust trigonal symmetry that renders all nearest-neighbor change theory of Ref. 119 one expects that the absolute Yb–Yb distances as well as relevant superexchange path- value of J1 varies by about 50% throughout the crystal. ways equal [116]. The absence of any detectable magnetic Other experiments support not only the presence of impurities [118] and the lack of spin freezing (beyond the this structural randomness, but also its tangible effect effects discussed in Sec.IVB) would also imply that this on the magnetism. First, absent magnetic contribution material is less likely to suffer from inhomogeneities and to the thermal conductivity [120] is naturally explained disorder than other spin-liquid candidates, such as her- by the random exchange couplings that will cause local- bertsmithite [132]. On the other hand, YbMgGaO4 is ization of magnetic excitations regardless of their exact still imperfect in the sense that Mg and Ga are randomly origin. Second, inelastic scattering in the fully polarized distributed in the non-magnetic slabs that separate the state above 7.5 T shows an abnormally broad distribution triangular planes of Yb3+ (Fig. 11). Although such a of the spectral weight and hardly resembles spin wave of mixture of non-magnetic atoms between the magnetic a ferromagnet [127, 133]. Third, the spin-liquid state of places would normally have little effect on the magnetism YbMgGaO4 is remarkably insensitive to pressure [134], 10

FIG. 9. CEF excitations of YbMgGaO4 [133]. (a) Inelastic neutron scattering reveals an additional feature around 87 meV 3+ incompatible with four Kramers doublets of Yb (momentum dependence and the data for LuMgGaO4, the non-magnetic reference compound, exclude possible phonon origin of this excitation). (b) Simulation of the experimental spectrum combined from several local configurations obtained with different distributions of Mg2+ and Ga3+ around Yb3+; note that the 87 meV feature appears as the side peak of the highest CEF excitation. (c) Sample local configuration with opposite displacements of Yb3+ and O2− caused by the uneven charge distribution. suggesting that structural inhomogeneities may be in- strumental in destabilizing magnetic order and facilitat- TABLE II. Exchange parameters for YbMgGaO4 estimated by fitting different sets of the experimental data: i) Curie- ing spin dynamics. All these observations serve as the Weiss temperatures and ESR linewidths [118]; ii) inelastic most direct evidence that structural randomness is cen- neutron scattering in the fully polarized state and diffuse scat- tral to the magnetism of YbMgGaO4. tering in zero field modeled with [127] and without [135] J2; iii) inelastic neutron scattering and THz spectra [136]. The exchange parameters are introduced in Sec.IIB and given in E. Possible scenarios Kelvin with error bars where available. I[118]II[127]III[136] IV [135] Several microscopic parameterizations reported for J1 1.8(2) 2.54(5) 1.98(7) 2.5 YbMgGaO4 are summarized in TableII. Whereas all ∆ 0.54(5) 0.58(2) 0.88(3) 0.76 ±± studies agree on the presence of easy-plane anisotropy J1 /J1 0.09(1) 0.06 0.4(3) 0.26 (∆ < 1), more subtle (but crucial) details of the second- z± J1 /J1 0.02(4) 0 0.6(6) 0.45 neighbor interactions and off-diagonal anisotropy remain J2/J1 0 0.22 0.18(7) 0 controversial and illustrate challenges in the experimen- tal determination of the exchange parameters for 4f mag- netic ions. As many as four independent parameters have to be used for nearest-neighbor interactions, another four On the other hand, it was argued that these terms induce parameters can be envisaged for second-neighbor inter- a large magnon gap [138] that is incompatible with the actions, etc. apparent gapless behavior [116]. By disregarding J2, the authors of Ref. 118 used The accuracy of the neutron-based parametrization Curie-Weiss temperatures and electron-spin-resonance was improved by adding THz data that probe excita- (ESR) linewidths to determine all four components of tions at the zone center, as opposed to neutron scatter- the nearest-neighbor exchange tensor. This leads to a ing that is more sensitive to the zone boundary. This sizable XXZ anisotropy with the relatively weak but non- combined approach [136] hardly improves estimates of ±± z± the J ±± and J z± terms, but lends additional confidence negligible additional terms J1 and J1 (TableII, set 1 1 I). Fits to the inelastic neutron data [127] suggest a in the finite-J2 scenario (set III). The estimated value of roughly similar interaction regime [137], but with a siz- J2 = 0.3 − 0.4 K largely exceeds the dipolar coupling of able second-neighbor coupling J2/J1 ' 0.2 that is as- 0.07 K and serves as the first experimental evidence of 3+ sumed to be isotropic (set II). The primary reason for the long-range superexchange in Yb magnets. including J2 is the peak of the neutron spectral weight With J2/J1 ' 0.2, YbMgGaO4 may not be far from at the M-points (Fig. 10b), as typical for the stripe phase the quantum spin liquid region of J1 − J2 triangular an- at J2/J1 > 0.15. Alternatively, this stripe phase can be tiferromagnets (Sec.IIC). An optimistic scenario would ±± z± stabilized by the J1 and J1 terms (set IV) that de- deem YbMgGaO4 the first material prototype of this scribe the neutron data even in the absence of J2 [135]. quantum spin liquid, but several experimental observa- 11

bond glass appears [144]. This phase can be represented as a valence-bond state with short-range spin singlets and a small fraction of unpaired (orphan) spins [145]. Alter- natively, by analyzing a valence-bond state with random bond strengths, one can show that it is intrinsically un- 1 stable toward nucleation of spin- 2 topological defects, orphan or unpaired spins [146]. The above scenario is remarkably similar to the phe- nomenological model developed for the interpretation of magnetic excitations in Ref. 125. Indeed, a suit- able parameterization of the valence-bond state with or- phan spins allows quantitative description of the low- 2 temperature thermodynamics, including the peculiar T 3 power law of the specific heat [146] that was initially ascribed to the U(1) quantum spin liquid. Ref. 145 de- FIG. 10. Neutron scattering from Yb-based triangular an- scribes this new, randomness-induced phase as a “many- tiferromagnets. (a) Static structure factor obtained theoret- body localized RVB state”, but it clearly deviates from ically (DMRG) for the spin-liquid phase of the anisotropic the conventional RVB scenario, because gapped vison ex- spin Hamiltonian, Eq. (1)[10]. (b) Momentum dependence citations are preempted by the gapless low-energy dy- of the spectral weight for YbMgGaO4 (single crystal) with namics of orphan spins. An interesting question is peaks at the M-points [127]. (c) Momentum dependence of whether the mixed state of orphan spins and valence the spectral weight for NaYbO2 (powder sample) with the bonds, while not being an RVB state in the original sense, highest intensity at the K-point [140]. Here, K stands for the still shows quantum entanglement. First numerical re- zone corner and M for the midpoint of the zone edge. Panel sults suggest that this may be the case [144], and give cer- (a) is reprinted with permission from Ref. 10, c American Physical Society, 2018. tain hope that the randomness-induced spin-liquid-like phase is not yet another case of the spin-liquid mimicry, but on a longer run may disengage itself from the pre- tions speak against such an interpretation. First, low- cautionary “-like” ending. energy spectral weight peaks at the M-point [127], while theory expects the peak at K [10] (Fig. 10). Second, ab- sent magnetic contribution to the thermal transport [120] V. OTHER 4f MATERIALS leaves little room for a genuine, macroscopically entan- gled quantum state. Third, the randomness effect ex- The complexity of YbMgGaO4 arises from the intri- plicated in Sec.IVD raises serious doubts about inter- cate combination of magnetic frustration and structural preting YbMgGaO4 within the framework of any model randomness. The former is essential, and the latter un- having uniform exchange parameters. avoidable, but a crucial problem on the materials side The peak of the spectral weight at the M-point led is whether this structural randomness can be reduced to to an idea that the ground state of YbMgGaO4 may the level that it does not change the physics qualitatively, be a melted stripe order, where stripes are stabilized by leaving room for the “genuine” behavior of regular trian- the sizable J2, but their directions are random under a gular antiferromagnets described in Sec.IIC. ±± 3+ change in the sign of J1 [138] or a variation of all ex- Close-packed layers of Yb , the main building block change parameters throughout the crystal [139], the lat- of YbMgGaO4, are common for many structure types, ter scenario being consistent with the predictions of the including the abundant family of delafossites, where the superexchange theory [119]. Both types of disorder lead presence of only one sort of non-magnetic species should to a liquid-like classical phase, although it remains un- eliminate the randomness effect (Fig. 11). The absence clear whether excitations of this random stripe state [138] of randomness was indeed confirmed in NaYbO2 [140, may be responsible for the peculiar spin dynamics ob- 147, 148] by the observation of three resolution-limited served experimentally (Sec.IVC). peaks of the CEF excitations and the absence of extra In fact, all ordered phases of triangular antiferro- features in high-energy inelastic neutron spectra [140]. magnets are rather unstable toward randomness effects. No significant randomness is expected in the isostruc- Models with random distribution of nearest-neighbor ex- tural materials NaYbS2 [149], NaYbSe2 [150, 151], and change couplings were considered in the literature [141, CsYbSe2 [152] as well. Interestingly, all these compounds 142] even before YbMgGaO4 made a compelling exper- reveal main signatures of the spin-liquid behavior, absent imental case for their relevance. Already weak random- magnetic order [140, 150–152] and persistent spin dynam- ness transforms the 120◦ order into a glassy state [143], ics [140, 149, 153] down to low temperatures. Moreover, but the most interesting behavior is found in the limit of in NaYbO2 the spectral weight of the excitation contin- strong randomness and/or finite J2, where a gapless spin- uum peaks at the K-point of the Brillouin zone [140, 148] liquid-like phase distinct from either spin glass or valence- (Fig. 10c) in agreement with theoretical results for the 12

trast, out-of-plane fields cause magnetic order only above 8 T with no plateau feature in the magnetization. This drastic difference in the transition fields reflects the siz- able easy-plane anisotropy (TableIII), whereas the pres- ence (absence) of the magnetization plateau for the in- plane (out-of-plane) fields resembles the response of Co- based XXZ triangular antiferromagnets [77, 78], where interactions are restricted to nearest neighbors. Indeed, in the fully isotropic (Heisenberg) case, the up-up-down 1 phase and associated 3 -plateau occur only at J2/J1 < 0.125 [155, 156], suggesting that in the Yb3+ delafossites the J2/J1 ratio should be lower than in YbMgGaO4, in agreement with the fact that the absolute value of J1 FIG. 11. Crystal structures of Yb-based triangular anti- increases. ferromagnets: YbMgGaO4, NaYbO2 as representative of the 3+ Other Yb-based triangular materials include the family Yb delafossites, and KBaYb(BO3)2 as representative of the Yb3+ borates. All structures are trigonal, with c chosen as of borate compounds ABaYb(BO3)2 (A = Na, K) [157– the vertical direction, whereas triangular layers are in the ab 159], where YbO6 octahedra are not directly linked to plane. VESTA software [86] was used for crystal structure vi- each other, but connected via BO3 triangles, with the sualization. nearest-neighbor Yb–Yb distance increasing to 5.3−5.4 A˚ (Fig. 11). These materials may serve as interesting ref- erence systems, where J2 is effectively suppressed, giv- TABLE III. g-values and exchange parameters of the Yb3+ ing way to the purely nearest-neighbor triangular model. delafossites. The g-values are estimated by electron spin reso- The main question at this juncture is whether at least J1 nance [154], whereas J and ∆ are calculated from Curie-Weiss can be strong enough to produce any tangible magnetism. temperatures obtained for two different field directions and The Curie-Weiss temperatures well below 0.3 K [158, 159] represented as θ = 3 J, where J is the cumulative (nearest- 2 suggest that an extremely cold environment would be neighbor and next-nearest-neighbor) coupling between spins needed to access frustrated behavior of these triangu- pointing along this direction. Only the J value is given for lar antiferromagnets. With the shortest interlayer Yb– NaYbO2 due to the lack of single crystals for this compound. Yb distance (5.3 − 5.4 A)˚ approaching the intralayer one gk g⊥ J ∆ Ref. (6.6−6.7 A),˚ 2D nature of the magnetism also comes into NaYbO2 1.75 3.28 6.0 – 147 question, whereas mixing of A+ and Ba2+ in the same NaYbS2 0.57 3.19 9.0 0.13 149 crystallographic site creates random electric fields acting 3+ NaYbSe2 1.01 3.13 4.7 0.49 151 on Yb , similar to the YbMgGaO4 case (Sec.IVD). Both delafossites [160], YbMgGaO4-type [161], and mixed-cation borate [158, 162] compounds exist for many quantum-spin-liquid phase of triangular antiferromag- if not all 4f ions. The properties of these materials re- nets [10]. These promising observations suggest that the main to be explored, but a few literature cases suggest Yb3+ delafossites may give experimental access to the that at least Ce-based compounds will likely reveal an ul- spin-liquid phase in the absence of structural disorder. timate XY anisotropy reported for Ce3+ in the trigonal So far little is known about the microscopic regime CEF [163]. Only a moderate easy-plane anisotropy has 3+ 3+ of the Yb delafossites. Average exchange couplings been observed in Er selenides, AErSe2 (A = Na, K, gauged by their Curie-Weiss temperatures and saturation Cs) [164, 165], but in this case the first crystal-field exci- fields are at least 2-3 times stronger than in YbMgGaO4 tation lies below 1 meV in the selenides [165] or around (TableIII), which shifts the relevant temperature scale 2 meV in the isostructural sulphide [166] and is likely to toward higher temperatures, but simultaneously pushes impact the low-energy physics. the fully polarized phase above 14 − 16 T [151], the fea- Triangular magnetic layers are also featured by some sibility limit of neutron-scattering experiments. The g- of the non-chalcogenide 4f compounds. Most notably, tensors determined by electron spin resonance are indica- CeCd3P3 shows signatures of quasi-2D magnetism and tive of an easy-plane anisotropy (TableIII) that may be a long-range-ordering transition around 0.4 K, although even more pronounced than in the case of YbMgGaO4. it undergoes a structural phase transition already at 3+ Another generic property of the Yb delafossites is 127 K [167]. YbAl3C3 belongs to the same structural their field-induced phase transition between the puta- family and is known to develop singlet ground state [168, tive spin-liquid phase in zero field and one or several 169] as a consequence of a similar structural phase transi- states with long-range magnetic order. In-plane mag- tion [170, 171]. Metallicity of these compounds [167] may netic fields trigger the transition already around 2 T ac- be another important ingredient. Itinerant electrons will 1 companied by the 3 magnetization plateau [151, 152], generally mediate long-range interactions, both within which is compatible with the up-up-down magnetic or- and between the triangular planes, thus rendering the der confirmed experimentally for NaYbO2 [148]. In con- mapping onto simple short-ranged spin models impossi- 13 ble or at least ambiguous. With the phase diagram of anisotropic J1 −J2 triangu- Non-Kramers ions can be accommodated in the trian- lar antiferromagnets fully charted and disorder-free ma- gular geometry too, although they are known to produce terials like Yb3+ delafossites already available, experi- Ising spins [172] that are quite interesting on their own mental access to the quantum spin liquid phase outlined right but leave little room for quantum fluctuations and in Sec.IIC becomes imminent. It makes, however, only spin-liquid behavior, at least in zero field. TmMgGaO4 is one out of many interesting questions in the field. We an example of a triangular Ising antiferromagnet, where outline a few others below. structural randomness plays a role as crucial as in the Yb3+ analog, and the nature of magnetism remains con- First, with the exception of Ba3CoSb2O9 field-induced troversial [173, 174]. behavior is largely unknown, and even theory studies of the anisotropic J1 − J2 Hamiltonians in the applied field remain scarce [155, 156]. NaYbO2 shows a quite VI. SUMMARY AND OUTLOOK unusual transition between the spin-liquid phase in zero field and the collinear up-up-down phase in the applied field [147, 148] (a similar transformation was claimed Triangular antiferromagnets are no longer a land in randomness-influenced YbMgGaO4 too [175]). Both of static non-collinear magnetic structures of ever- phases may be quantum in nature, and their evolution increasing complexity [8]. Recent theory work charted is opposite to the typical scenario of Kitaev materials, stability regions of the quantum spin liquid (Sec.IIC) where magnetic order is suppressed by the applied field and identified unusual quantum features in the excitation ◦ giving way to a disordered state, possibly a spin liq- spectra of the 120 -ordered state (Sec.IIIB). Structural uid [176]. A quantum critical point separating the spin- randomness brought yet another dimension into this al- liquid and up-up-down phases can be envisaged, with the ready rich phase diagram by making unexpected connec- Yb3+ delafossites offering direct experimental access to tions to valence-bond states that were historically pro- it. posed for triangular antiferromagnets and led to impor- tant conceptual developments but have been discarded as Second, structural randomness could be tuned. possible ground states of any realistic spin Hamiltonian. YbMgGaO4 lies in the limit of strong randomness, where On the experimental side, the research reviewed in valence bonds coexist with a significant fraction of or- this article leads us to reconsider the notion of the ge- phan spins. Ramifications of this coexistence remain to ometrically perfect spin-liquid material and the role that be fully understood, but the most clear one is that abun- randomness or structural disorder play therein. The dant disorder renders magnetic excitations localized. An YbMgGaO4 case shows that neither lattice symmetry nor opposite limit of weak randomness, with valence bonds complete site occupations are sufficient conditions of a and only a minute fraction of orphan spins, may be, in “perfect material”, because even subtle structural effects contrast, the closest experimental realization of an RVB far away from the magnetic planes can strongly affect state. It can be conceived in Yb-based delafossites or magnetic interactions and spin dynamics. This imposes other disorder-free triangular compounds that are suit- very hard requirements for the material characterization, ably doped to induce weak randomness. and renders probes like CEF excitations central to decid- ing whether a given material is “perfect” or not. On the Third, all Yb-based triangular antiferromagnets stud- more positive side, it also offers a convenient experimen- ied so far are not magnetically ordered, which is even tal knob for tuning the material toward a dynamically surprising because in other classes of materials finding disordered state, potentially a spin liquid. an ordered state is by far easier than achieving a spin The quest for “structurally perfect” spin-liquid mate- liquid. Tuning one of these materials toward an ordered rials was largely motivated by the common notion of im- state (or finding another Yb-based triangular compound perfections being detrimental for the genuine spin-liquid with long-range magnetic order) may be an interesting state. A material with imperfections and without mag- endeavor. The primary goal here will be to probe mag- netic order was supposed to be a classical spin liquid netic excitations at different energy scales, keeping in that will freeze at low enough temperatures and show mind the experience with Co-based compounds, where mundane spin dynamics. YbMgGaO4 reveals this may exact nature of the excitation continuum remains con- not be the case, and even signatures of spin freezing troversial. The proclivity of Yb3+ and other 4f ions (Sec.IVB) do not preclude dynamic behavior for the to the anisotropic exchange implies strong tendency to majority of spins that, in fact, show highly unusual exci- magnon breakdown with unusual spectral features that tations. Taken together, these ideas suggest that struc- 1 will benchmark theoretical studies of quantum spin- 2 tri- tural imperfections should not always be avoided, but angular antiferromagnets. can be congenially used to facilitate spin dynamics. The general questions of which structural imperfections can Using theoretical map chart and appreciating the two- be used in this way, and how to identify their influence faced role of structural randomness will undoubtedly lead on magnetic interactions, remain interesting avenues for to new discoveries in the field of spin liquids and trian- future research in spin-liquid materials. gular antiferromagnets. 14

ACKNOWLEDGMENTS and Qingming Zhang for common work on triangular antiferromagnets, to Martin Mourigal for his insightful Intriguing and inspiring, triangular antiferromagnets and critical remarks, to Shiyan Li for providing thermal- conductivity data of YbMgGaO for this review, and to and especially YbMgGaO4 have triggered many fruit- 4 ful collaborations and insightful discussions. Our spe- Michael Baenitz for sharing unpublished results on the 3+ cial thanks goes to Sasha Chernyshev for his critical Yb delafossites. Last but not least, we thank Ger- vision and comprehensive theoretical development of man Science Foundation (DFG) for supporting our re- anisotropic triangular models, and to Sebastian Bachus search on frustrated magnets via Project No. 107745057 and Yoshi Tokiwa for their impressive work in the milli-K (TRR80). AT further thanks Alexander von Humboldt lab. We are grateful to Devashibhai Adroja, Lei Ding, Foundation for the financial support via the Sofja Ko- Franziska Grußler, Pascal Manuel, Astrid Schneidewind, valevskaya Award.

[1] L. Balents, Spin liquids in frustrated magnets, Nature [17] P. Anderson, G. Baskaran, Z. Zou, and T. Hsu, 464, 199 (2010). Resonating-valence-bond theory of phase transitions [2] M. Gingras and P. McClarty, Quantum spin ice: a and superconductivity in La2CuO4-based compounds, search for gapless quantum spin liquids in pyrochlore Phys. Rev. Lett. 58, 2790 (1987). magnets, Rep. Prog. Phys. 77, 056501 (2014). [18] V. Kalmeyer and R. Laughlin, Equivalence of the [3] J. Knolle and R. Moessner, A field guide to spin liquids, resonating-valence-bond and fractional quantum Hall Ann. Rev. Condens. Matter Phys. 10, 451 (2019). states, Phys. Rev. Lett. 59, 2095 (1987). [4] C. Broholm, R. Cava, S. Kivelson, D. Nocera, M. Nor- [19] R. Laughlin, The relationship between high- man, and T. Senthil, Quantum spin liquids, Science 367, temperature superconductivity and the fractional eaay0668 (2020). quantum Hall effect, Science 242, 525 (1988). [5] L. Savary and L. Balents, Quantum spin liquids: a re- [20] V. Kalmeyer and R. Laughlin, Theory of the spin liquid view, Rep. Prog. Phys. 80, 016502 (2017). state of the Heisenberg antiferromagnet, Phys. Rev. B [6] Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liq- 39, 11879 (1989). uid states, Rev. Mod. Phys. 89, 025003 (2017). [21] X.-G. Wen, Choreographed entanglement dances: Topo- [7] P. Anderson, Resonating valence bonds: A new kind of logical states of quantum matter, Science 363, eaal3099 insulator?, Mater. Res. Bull. 8, 153 (1973). (2019). [8] O. Starykh, Unusual ordered phases of highly frustrated [22] B. Dalla Piazza, M. Mourigal, N. Christensen, magnets: a review, Rep. Prog. Phys. 78, 052502 (2015). G. Nilsen, P. Tregenna-Piggott, T. Perring, M. Enderle, [9] B. Powell and R. McKenzie, Quantum frustration in D. McMorrow, D. Ivanov, and H. Rønnow, Fractional organic Mott insulators: from spin liquids to uncon- excitations in the square-lattice quantum antiferromag- ventional superconductors, Rep. Prog. Phys. 74, 056501 net, Nature Phys. 11, 62 (2015). (2011). [23] A. Banerjee, J. Yan, J. Knolle, C. Bridges, M. Stone, [10] Z. Zhu, P. Maksimov, S. White, and A. Chernyshev, M. Lumsden, D. Mandrus, D. Tennant, R. Moessner, Topography of spin liquids on a triangular lattice, Phys. and S. Nagler, Neutron scattering in the proximate Rev. Lett. 120, 207203 (2018). quantum spin liquid α-RuCl3, Science 356, 1055 (2017). [11] P. Fazekas and P. Anderson, On the ground state [24] S. Winter, K. Riedl, P. Maksimov, A. Chernyshev, properties of the anisotropic triangular antiferromagnet, A. Honecker, and R. Valent´ı,Breakdown of magnons in Phil. Mag. 30, 423 (1974). a strongly spin-orbital coupled magnet, Nature Comm. [12] G. Rumer, Zur Theorie der Spinvalenz, Nachrichten 8, 1152 (2017). von der Gesellschaft der Wissenschaften zu G¨ottingen, [25] R. Moessner and S. Sondhi, Resonating valence bond Mathematisch-Physikalische Klasse , 337 (1932). phase in the triangular lattice quantum dimer model, [13] G. Rumer, E. Teller, and H. Weyl, Eine f¨urdie Valen- Phys. Rev. Lett. 86, 1881 (2001). ztheorie geeignete Basis der bin¨arenVektorinvarianten, [26] R. Moessner and S. Sondhi, Resonating valence bond Nachrichten von der Gesellschaft der Wissenschaften liquid physics on the triangular lattice, Prog. Theor. zu G¨ottingen,Mathematisch-Physikalische Klasse , 499 Phys. Supplement 145, 37 (2002). (1932). [27] A. Ioselevich, D. Ivanov, and M. Feigelman, Ground- [14] L. Pauling, The calculation of matrix elements for Lewis state properties of the Rokhsar-Kivelson dimer model on electronic structures of molecules, J. Chem. Phys. 1, 280 the triangular lattice, Phys. Rev. B 66, 174405 (2002). (1933). [28] D. Ivanov, Vortexlike elementary excitations in the [15] L. Pauling and G. Wheland, The nature of the chemical Rokhsar-Kivelson dimer model on the triangular lattice, bond. V. The quantum-mechanical calculation of the Phys. Rev. B 70, 094430 (2004). resonance energy of benzene and naphthalene and the [29] L. Ioffe, M. Feigel’man, A. Ioselevich, D. Ivanov, hydrocarbon free radicals, J. Chem. Phys. 1, 362 (1933). M. Troyer, and G. Blatter, Topologically protected [16] S. Kivelson, D. Rokhsar, and J. Sethna, Topology of quantum bits using Josephson junction arrays, Nature the resonating valence-bond state: Solitons and high-Tc 415, 503 (2002). superconductivity, Phys. Rev. B 35, 8865(R) (1987). [30] G. Baskaran and P. Anderson, Gauge theory of high- temperature superconductors and strongly correlated 15

Fermi systems, Phys. Rev. B 37, 580(R) (1988). (2015). [31] P. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott in- [50] S. Saadatmand and I. McCulloch, Symmetry fractional- 1 sulator: Physics of high-temperature superconductivity, ization in the topological phase of the spin- 2 J1 −J2 tri- Rev. Mod. Phys. 78, 17 (2006). angular Heisenberg model, Phys. Rev. B 94, 121111(R) [32] O. Motrunich, Variational study of triangular lattice (2016). spin-1/2 model with ring exchanges and spin liquid state [51] R. Kaneko, S. Morita, and M. Imada, Gapless spin- in κ-(ET)2Cu2(CN)3, Phys. Rev. B 72, 045105 (2005). liquid phase in an extended spin-1/2 triangular Heisen- [33] S.-S. Lee, P. Lee, and T. Senthil, Amperean pairing in- berg model, J. Phys. Soc. Jpn. 83, 093707 (2014). stability in the U(1) spin liquid state with Fermi surface [52] Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and and application to κ-(BEDT-TTF)2Cu2(CN)3, Phys. F. Becca, Spin liquid nature in the Heisenberg J1 − J2 Rev. Lett. 98, 067006 (2007). triangular antiferromagnet, Phys. Rev. B 93, 144411 [34] V. Galitski and Y. Kim, Spin-triplet pairing instability (2016). of the spinon Fermi surface in a U(1) spin liquid, Phys. [53] S.-S. Gong, W. Zheng, M. Lee, Y.-M. Lu, and D. Sheng, Rev. Lett. 99, 266403 (2007). Chiral spin liquid with spinon fermi surfaces in the 1 [35] C. Nave, S.-S. Lee, and P. Lee, Susceptibility of a spinon spin- 2 triangular Heisenberg model, Phys. Rev. B 100, Fermi surface coupled to a U(1) gauge field, Phys. Rev. 241111(R) (2019). B 76, 165104 (2007). [54] S. Hu, W. Zhu, S. Eggert, and Y.-C. He, Dirac spin [36] T.-K. Ng and P. Lee, Power-law conductivity inside the liquid on the spin-1/2 triangular Heisenberg antiferro- Mott gap: Application to κ-(BEDT-TTF)2Cu2(CN)3, magnet, Phys. Rev. Lett. 123, 207203 (2019). Phys. Rev. Lett. 99, 156402 (2007). [55] D.-V. Bauer and J. Fjærestad, Schwinger-boson mean- [37] C. Nave and P. Lee, Transport properties of a spinon field study of the J1 − J2 Heisenberg quantum antifer- Fermi surface coupled to a U(1) gauge field, Phys. Rev. romagnet on the triangular lattice, Phys. Rev. B 96, B 76, 235124 (2007). 165141 (2017). z z z [38] Same interaction terms are often written as JmSi Sj [56] T. Jolicoeur, E. Dagotto, E. Gagliano, and S. Bacci, ± + − − + and Jm(Si Sj + Si Sj )[2], where in the isotropic case Ground-state properties of the S = 1/2 Heisenberg an- ± z tiferromagnet on a triangular lattice, Phys. Rev. B 42, (∆ = 1) Jm is twice smaller than Jm. This difference should be taken into account when comparing exchange 4800 (1990). parameters reported in different publications. [57] A. Chubukov and T. Jolicoeur, Order-from-disorder [39] P. Maksimov, Z. Zhu, S. White, and A. Chernyshev, phenomena in Heisenberg antiferromagnets on a trian- Anisotropic-exchange magnets on a triangular lattice: gular lattice, Phys. Rev. B 46, 11137 (1992). Spin waves, accidental degeneracies, and dual spin liq- [58] P. Lecheminant, B. Bernu, C. Lhuillier, and L. Pierre, uids, Phys. Rev. X 9, 021017 (2019). J1 −J2 quantum Heisenberg antiferromagnet on the tri- [40] D. Huse and V. Elser, Simple variational wave func- angular lattice: A group-symmetry analysis of order by 1 disorder, Phys. Rev. B 52, 6647 (1995). tions for two-dimensional Heisenberg spin- 2 antiferro- magnets, Phys. Rev. Lett. 60, 2531 (1988). [59] Y.-D. Li, X. Wang, and G. Chen, Anisotropic spin [41] T. Jolicoeur and J. Le Guillou, Spin-wave results for model of strong spin-orbit-coupled triangular antiferro- the triangular Heisenberg antiferromagnet, Phys. Rev. magnets, Phys. Rev. B 94, 035107 (2016). B 40, 2727(R) (1989). [60] Q. Luo, S. Hu, B. Xi, J. Zhao, and X. Wang, Ground- 1 [42] B. Kleine, E. M¨uller-Hartmann, K. Frahm, and state phase diagram of an anisotropic spin- 2 model on P. Fazekas, Spin-wave analysis of easy-axis quantum an- the triangular lattice, Phys. Rev. B 95, 165110 (2017). tiferromagnets on the triangular lattice, Z. Phys. B 87, [61] J. Iaconis, C. Liu, G. Hal´asz,and L. Balents, Spin liq- 103 (1992). uid versus spin orbit coupling on the triangular lattice, [43] D. Yamamoto, G. Marmorini, and I. Danshita, Quan- SciPost Phys. 4, 003 (2018). tum phase diagram of the triangular-lattice XXZ model [62] A. Abragam and B. Bleaney, Electron paramagnetic res- in a magnetic field, Phys. Rev. Lett. 112, 127203 (2014). onance of transition ions (Clarendon Press, Oxford, [44] D. Sellmann, X.-F. Zhang, and S. Eggert, Phase dia- 1970). gram of the antiferromagnetic XXZ model on the trian- [63] H. Pen, J. van den Brink, D. Khomskii, and gular lattice, Phys. Rev. B 91, 081104(R) (2015). G. Sawatzky, Orbital ordering in a two-dimensional tri- [45] G. H. Wannier, Antiferromagnetism. the triangular angular lattice, Phys. Rev. Lett 78, 1323 (1997). Ising net, Phys. Rev. 79, 357 (1950). [64] T. McQueen, P. Stephens, Q. Huang, T. Klimczuk, [46] Z. Zhu and S. White, Spin liquid phase of the S = 1/2 F. Ronning, and R. Cava, Successive orbital ordering transitions in NaVO , Phys. Rev. Lett. 101, 166402 J1 −J2 Heisenberg model on the triangular lattice, Phys. 2 Rev. B 92, 041105(R) (2015). (2008). [47] P. Li, R. Bishop, and C. Campbell, Quasiclassical mag- [65] R. Bo˘ca,Zero-field splitting in metal complexes, Coord. 1 Chem. Rev. 248, 757 (2004). netic order and its loss in a spin- 2 Heisenberg antifer- romagnet on a triangular lattice with competing bonds, [66] M. Wilkinson, J. Cable, E. Wollan, and W. Koehler, Phys. Rev. B 91, 014426 (2015). Neutron diffraction investigations of the magnetic or- [48] S. Saadatmand, B. Powell, and I. McCulloch, Phase di- dering in FeBr2, CoBr2, FeCl2, and CoCl2, Phys. Rev. 1 113, 497 (1959). agram of the spin- 2 triangular J1 −J2 Heisenberg model on a three-leg cylinder, Phys. Rev. B 91, 245119 (2015). [67] D. Hunt, G. Garbarino, J. Rodr´ıguez-Velamaz´an, [49] W.-J. Hu, S.-S. Gong, W. Zhu, and D. Sheng, Compet- V. Ferrari, M. Jobbagy, and D. Scherlis, The magnetic 1 structure of β-cobalt hydroxide and the effect of spin- ing spin-liquid states in the spin- 2 Heisenberg model on the triangular lattice, Phys. Rev. B 92, 140403(R) orientation, Phys. Chem. Chem. Phys. 18, 30407 (2016). 16

[68] Y. Doi, Y. Hinatsu, and K. Ohoyama, Structural and Ba3CoSb2O9, Phys. Rev. B 91, 024410 (2015). magnetic properties of pseudo-two-dimensional triangu- [83] X. Liu, O. Prokhnenko, D. Yamamoto, M. Bartkowiak, lar antiferromagnets Ba3MSb2O9 (M = Mn, Co, and N. Kurita, and H. Tanaka, Microscopic evidence of a 1 Ni), J. Phys.: Condens. Matter 16, 8923 (2004). quantum magnetization process in the S = 2 triangular- [69] Y. Shirata, H. Tanaka, A. Matsuo, and K. Kindo, Ex- lattice Heisenberg-like antiferromagnet Ba3CoSb2O9, perimental realization of a spin-1/2 triangular-lattice Phys. Rev. B 100, 094436 (2019). Heisenberg antiferromagnet, Phys. Rev. Lett. 108, [84] S. Ito, N. Kurita, H. Tanaka, S. Ohira-Kawamura, 057205 (2012). K. Nakajima, S. Itoh, K. Kuwahara, and K. Kaku- [70] H. Zhou, C. Xu, A. Hallas, H. Silverstein, C. Wiebe, rai, Structure of the magnetic excitations in the I. Umegaki, J. Yan, T. Murphy, J.-H. Park, Y. Qiu, spin-1/2 triangular-lattice Heisenberg antiferromagnet J. Copley, J. Gardner, and Y. Takano, Successive Ba3CoSb2O9, Nature Comm. 8, 235 (2017). phase transitions and extended spin-excitation contin- [85] R. Verresen, R. Moessner, and F. Pollmann, Avoided 1 uum in the S = 2 triangular-lattice antiferromagnet quasiparticle decay from strong quantum interactions, Ba3CoSb2O9, Phys. Rev. Lett. 109, 267206 (2012). Nature Phys. 15, 750 (2019). [71] M. Lee, J. Hwang, E. Choi, J. Ma, C. Dela Cruz, [86] K. Momma and F. Izumi, VESTA 3 for three- M. Zhu, X. Ke, Z. Dun, and H. Zhou, Series of dimensional visualization of crystal, volumetric and phase transitions and multiferroicity in the quasi-two- morphology data, J. Appl. Crystallogr. 44, 1272 (2011). 1 dimensional spin- 2 triangular-lattice antiferromagnet [87] These roton minima strongly influence thermodynamic Ba3CoNb2O9, Phys. Rev. B 89, 104420 (2014). properties of triangular antiferromagnets. Their relation [72] K. Yokota, N. Kurita, and H. Tanaka, Magnetic phase to rotons in superfluid 4He is discussed in Ref. 89. 1 diagram of the S = 2 triangular-lattice Heisenberg an- [88] W. Zheng, J. Fjærestad, R. Singh, R. McKenzie, and tiferromagnet Ba3CoNb2O9, Phys. Rev. B 90, 014403 R. Coldea, Anomalous excitation spectra of frustrated (2014). quantum antiferromagnets, Phys. Rev. Lett. 96, 057201 [73] K. Ranjith, K. Brinda, U. Arjun, N. Hegde, and (2006). R. Nath, Double phase transition in the triangular an- [89] W. Zheng, J. Fjærestad, R. Singh, R. McKenzie, and 1 tiferromagnet Ba3CoTa2O9, J. Phys.: Condens. Matter R. Coldea, Excitation spectra of the spin- 2 triangular- 29, 115804 (2017). lattice Heisenberg antiferromagnet, Phys. Rev. B 74, [74] M. Lee, E. Choi, J. Ma, R. Sinclair, C. Dela Cruz, and 224420 (2006). H. Zhou, Magnetic and electric properties of triangular [90] F. Ferrari and F. Becca, Dynamical structure factor of lattice antiferromagnets Ba3ATa2O9 (A = Ni and Co), the J1 − J2 Heisenberg model on the triangular lattice: Mater. Res. Bull. 88, 308 (2017). Magnons, spinons, and gauge fields, Phys. Rev. X 9, [75] D. Prabhakaran and A. Boothroyd, Crystal growth of 031026 (2019). the triangular-lattice antiferromagnet Ba3CoSb2O9,J. [91] A. Mezio, C. Sposetti, L. Manuel, and A. Trumper, Crystal Growth 468, 345 (2017). A test of the bosonic spinon theory for the triangular [76] J. Ma, Y. Kamiya, T. Hong, H. Cao, G. Ehlers, W. Tian, antiferromagnet spectrum, Europhys. Lett. 94, 47001 C. Batista, Z. Dun, H. Zhou, and M. Matsuda, Static (2011). and dynamical properties of the spin-1/2 equilateral [92] E. Ghioldi, A. Mezio, L. Manuel, R. Singh, J. Oitmaa, triangular-lattice antiferromagnet Ba3CoSb2O9, Phys. and A. Trumper, Magnons and excitation continuum in Rev. Lett. 116, 087201 (2016). XXZ triangular antiferromagnetic model: Application [77] T. Susuki, N. Kurita, T. Tanaka, H. Nojiri, A. Mat- to Ba3CoSb2O9, Phys. Rev. B 91, 134423 (2015). suo, K. Kindo, and H. Tanaka, Magnetization process [93] E. Ghioldi, M. Gonzalez, S.-S. Zhang, Y. Kamiya, and collective excitations in the S = 1/2 triangular- L. Manuel, A. Trumper, and C. Batista, Dynami- lattice Heisenberg antiferromagnet Ba3CoSb2O9, Phys. cal structure factor of the triangular antiferromagnet: Rev. Lett. 110, 267201 (2013). Schwinger boson theory beyond mean field, Phys. Rev. [78] G. Quirion, M. Lapointe-Major, M. Poirier, J. Quil- B 98, 184403 (2018). liam, Z. Dun, and H. Zhou, Magnetic phase diagram [94] C.-M. Ho, V. Muthukumar, M. Ogata, and P. Anderson, of Ba3CoSb2O9 as determined by ultrasound velocity Nature of spin excitations in two-dimensional Mott in- measurements, Phys. Rev. B 92, 014414 (2015). sulators: Undoped cuprates and other materials, Phys. [79] A. Sera, Y. Kousaka, J. Akimitsu, M. Sera, T. Kawa- Rev. Lett. 86, 1626 (2001). 1 mata, Y. Koike, and K. Inoue, s = 2 triangular-lattice [95] N. Christensen, H. Rønnow, D. McMorrow, A. Harri- antiferromagnets Ba3CoSb2O9 and CsCuCl3: Role of son, T. Perring, M. Enderle, R. Coldea, L. Regnault, spin-orbit coupling, crystalline electric field effect, and and G. Aeppli, Quantum dynamics and entanglement Dzyaloshinskii-Moriya interaction, Phys. Rev. B 94, of spins on a square lattice, Proc. Nat. Acad. Sci. 104, 214408 (2016). 15264 (2007). [80] A. Chubukov and D. Golosov, Quantum theory of an [96] O. Starykh, A. Chubukov, and A. Abanov, Flat spin- antiferromagnet on a triangular lattice in a magnetic wave dispersion in a triangular antiferromagnet, Phys. field, J. Phys.: Condens. Matter 3, 69 (1991). Rev. B 74, 180403(R) (2006). [81] C. Griset, S. Head, J. Alicea, and O. Starykh, Deformed [97] A. Chernyshev and M. Zhitomirsky, Spin waves in a triangular lattice antiferromagnets in a magnetic field: triangular lattice antiferromagnet: Decays, spectrum Role of spatial anisotropy and Dzyaloshinskii-Moriya in- renormalization, and singularities, Phys. Rev. B 79, teractions, Phys. Rev. B 84, 245108 (2011). 144416 (2009). [82] G. Koutroulakis, T. Zhou, Y. Kamiya, J. Thompson, [98] L. Capriotti, A. Trumper, and S. Sorella, Long-range H. Zhou, C. Batista, and S. Brown, Quantum phase N´eelorder in the triangular Heisenberg model, Phys. 1 diagram of the S = 2 triangular-lattice antiferromagnet Rev. Lett. 82, 3899 (1999). 17

[99] S. White and A. Chernyshev, Neel order in square and of glaserite systems with triangular magnetic lattices, triangular lattice Heisenberg models, Phys. Rev. Lett. Inorg. Chem. 58, 2813 (2019). 99, 127004 (2007). [114] W. Wolf, Anisotropic interactions between magnetic [100] A. Chernyshev and M. Zhitomirsky, Magnon decay ions, J. Phys. Colloques 32, C1 (1971). in noncollinear quantum antiferromagnets, Phys. Rev. [115] J. Rau and M. Gingras, Frustrated quantum rare-earth Lett. 97, 207202 (2006). pyrochlores, Ann. Rev. Condens. Matter Phys. 10, 357 [101] M. Zhitomirsky and A. Chernyshev, Colloquium: Spon- (2019). taneous magnon decays, Rev. Mod. Phys. 85, 219 [116] Y. Li, H. Liao, Z. Zhang, S. Li, F. Jin, L. Ling, (2013). L. Zhang, Y. Zou, L. Pi, Z. Yang, J. Wang, Z. Wu, and [102] M. Mourigal, W. Fuhrman, A. Chernyshev, and Q. Zhang, Gapless quantum spin liquid ground state in M. Zhitomirsky, Dynamical structure factor of the the two-dimensional spin-1/2 triangular antiferromag- triangular-lattice antiferromagnet, Phys. Rev. B 88, net YbMgGaO4, Sci. Reports 5, 16419 (2015). 094407 (2013). [117] Y. Li, YbMgGaO4: A triangular-lattice quantum [103] Y. Kamiya, L. Ge, T. Hong, Y. Qiu, D. Quintero-Castro, spin liquid candidate, Adv. Quantum Technologies 2, Z. Lu, H. B. Cao, M. Matsuda, E. Choi, C. Batista, 1900089 (2019). M. Mourigal, H. Zhou, and J. Ma, The nature of spin [118] Y. Li, G. Chen, W. Tong, L. Pi, J. Liu, Z. Yang, excitations in the one-third magnetization plateau phase X. Wang, and Q. Zhang, Rare-earth triangular lattice of Ba3CoSb2O9, Nature Comm. 9, 2666 (2018). spin liquid: A single-crystal study of YbMgGaO4, Phys. [104] Y. Kojima, M. Watanabe, N. Kurita, H. Tanaka, Rev. Lett. 115, 167203 (2015). A. Matsuo, K. Kindo, and M. Avdeev, Quantum mag- [119] J. Rau and M. Gingras, Frustration and anisotropic ex- 1 netic properties of the spin- 2 triangular-lattice antifer- change in ytterbium magnets with edge-shared octahe- romagnet Ba2La2CoTe2O12, Phys. Rev. B 98, 174406 dra, Phys. Rev. B 98, 054408 (2018). (2018). [120] Y. Xu, J. Zhang, Y. Li, Y. Yu, X. Hong, Q. Zhang, and [105] R. Rawl, L. Ge, H. Agrawal, Y. Kamiya, C. Dela S. Li, Absence of magnetic thermal conductivity in the Cruz, N. Butch, X. Sun, M. Lee, E. Choi, J. Oit- quantum spin-liquid candidate YbMgGaO4, Phys. Rev. maa, C. Batista, M. Mourigal, H. Zhou, and J. Ma, Lett. 117, 267202 (2016). 1 Ba8CoNb6O24: A spin- 2 triangular-lattice Heisenberg [121] Y. Li, D. Adroja, P. K. Biswas, P. J. Baker, Q. Zhang, antiferromagnet in the two-dimensional limit, Phys. J. Liu, A. Tsirlin, P. Gegenwart, and Q. Zhang, Muon Rev. B 95, 060412(R) (2017). spin relaxation evidence for the U(1) quantum spin- [106] T. Kim, K. Park, J. Leiner, and J.-G. Park, Hy- liquid ground state in the triangular antiferromagnet bridization and decay of magnetic excitations in YbMgGaO4, Phys. Rev. Lett. 117, 097201 (2016). two-dimensional triangular lattice antiferromagnets,J. [122] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan, Phys. Soc. Jpn. 88, 081003 (2019). Q. Wang, H. Walker, P. Steffens, M. Boehm, Y. Hao, [107] R. Rawl, M. Lee, E. Choi, G. Li, K. Chen, R. Baumbach, D. L. Quintero-Castro, L. Harriger, M. Frontzek, C. dela Cruz, J. Ma, and H. Zhou, Magnetic properties L. Hao, S. Meng, Q. Zhang, G. Chen, and J. Zhao, Ev- of the triangular lattice magnets A4B’B2O12 (A = Ba, idence for a spinon Fermi surface in a triangular-lattice Sr, La; B’ = Co, Ni, Mn; B = W, Re), Phys. Rev. B quantum-spin-liquid candidate, Nature 540, 559 (2016). 95, 174438 (2017). [123] Z. Ma, J. Wang, Z.-Y. Dong, J. Zhang, S. Li, S.-H. [108] R. Rawl, L. Ge, Z. Lu, Z. Evenson, C. Dela Cruz, Zheng, Y. Yu, W. Wang, L. Che, K. Ran, S. Bao, Q. Huang, M. Lee, E. Choi, M. Mourigal, H. Zhou, and Z. Cai, P. Cerm´ak,A.˘ Schneidewind, S. Yano, J. S. 5 J. Ma, Ba8MnNb6O24: A model two-dimensional spin- 2 Gardner, X. Lu, S.-L. Yu, J.-M. Liu, S. Li, J.-X. Li, and triangular lattice antiferromagnet, Phys. Rev. Mater. 3, J. Wen, Spin-glass ground state in a triangular-lattice 054412 (2019). compound YbZnGaO4, Phys. Rev. Lett. 120, 087201 [109] Y. Cui, J. Dai, P. Zhou, P. Wang, T. Li, W. Song, (2018). J. Wang, L. Ma, Z. Zhang, S. Li, G. Luke, B. Nor- [124] B. Schmidt, H. Aoki, T. Cichorek, J. Custers, P. Gegen- mand, T. Xiang, and W. Yu, Mermin-Wagner physics, wart, M. Kohgi, M. Lang, C. Langhammer, A. Ochiai, (H,T ) phase diagram, and candidate quantum spin- S. Paschen, F. Steglich, T. Suzuki, P. Thalmeier, 1 liquid phase in the spin- 2 triangular-lattice antiferro- B. Wand, and A. Yaresko, Low-energy excitations of the magnet Ba8CoNb6O24, Phys. Rev. Materials 2, 044403 semimetallic one-dimensional S = 1/2 antiferromagnet (2018). Yb4As3, Physica B 300, 121 (2001). [110] R. Zhong, S. Guo, G. Xu, Z. Xu, and R. Cava, Strong [125] Y. Li, S. Bachus, B. Liu, I. Radelytskyi, A. Bertin, quantum fluctuations in a quantum spin liquid can- A. Schneidewind, Y. Tokiwa, A. Tsirlin, and P. Gegen- didate with a Co-based triangular lattice, Proc. Nat. wart, Rearrangement of uncorrelated valence bonds ev- Acad. Sci. 116, 14505 (2019). idenced by low-energy spin excitations in YbMgGaO4, [111] A. M¨oller,N. Amuneke, P. Daniel, B. Lorenz, C. de la Phys. Rev. Lett. 122, 137201 (2019). Cruz, M. Gooch, and P. Chu, AAg2M[VO4]2 (A = Ba, [126] Y. Shen, Y.-D. Li, H. Walker, P. Steffens, M. Boehm, Sr; M = Co, Ni): a series of ferromagnetic insulators, X. Zhang, S. Shen, H. Wo, G. Chen, and J. Zhao, Frac- Phys. Rev. B 85, 214422 (2012). tionalized excitations in the partially magnetized spin [112] G. Nakayama, S. Hara, H. Sato, Y. Narumi, and H. No- liquid candidate YbMgGaO4, Nature Comm. 9, 4138 jiri, Synthesis and magnetic properties of a new series of (2018). triangular-lattice magnets, Na2BaMV2O8 (M = Ni, Co, [127] J. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, and Mn), J. Phys.: Condens. Matter 25, 116003 (2013). M. Stone, H. Zhou, and M. Mourigal, Continuous ex- [113] L. Sanjeewa, V. Garlea, M. McGuire, C. McMillen, and citations of the triangular-lattice quantum spin liquid J. Kolis, Magnetic ground state crossover in a series YbMgGaO4, Nature Phys. 113, 117 (2017). 18

[128] Y. Li, D. Adroja, D. Voneshen, R. I. Bewley, Q. Zhang, (2019). A. Tsirlin, and P. Gegenwart, Nearest-neighbour res- [145] H. Kawamura and K. Uematsu, Nature of the onating valence bonds in YbMgGaO4, Nature Comm. randomness-induced quantum spin liquids in two di- 8, 15814 (2017). mensions, J. Phys.: Condens. Matter 31, 504003 (2019). [129] T.-H. Han, J. Helton, S. Chu, D. Nocera, J. Rodriguez- [146] I. Kimchi, A. Nahum, and T. Senthil, Valence bonds in Rivera, C. Broholm, and Y. Lee, Fractionalized excita- random quantum magnets: Theory and application to tions in the spin-liquid state of a kagome-lattice antifer- YbMgGaO4, Phys. Rev. X 8, 031028 (2018). romagnet, Nature 492, 406 (2012). [147] K. Ranjith, D. Dmytriieva, S. Khim, J. Sichelschmidt, [130] Y.-D. Li and G. Chen, Detecting spin fractionalization S. Luther, D. Ehlers, H. Yasuoka, J. Wosnitza, A. Tsir- in a spinon Fermi surface spin liquid, Phys. Rev. B 96, lin, H. K¨uhne,and M. Baenitz, Field-induced instability 1 075105 (2017). of the quantum spin liquid ground state in the Jeff = 2 [131] L. Balents and O. Starykh, Collective spinon spin wave triangular-lattice compound NaYbO2, Phys. Rev. B 99, in a magnetized U(1) spin liquid, Phys. Rev. B 101, 180401(R) (2019). 020401(R) (2020). [148] M. Bordelon, E. Kenney, C. Liu, T. Hogan, [132] M. Norman, Colloquium: Herbertsmithite and the L. Posthuma, M. Kavand, Y. Lyu, M. Sherwin, search for the quantum spin liquid, Rev. Mod. Phys. N. Butch, C. Brown, M. Graf, L. Balents, and S. Wil- 88, 041002 (2016). son, Field-tunable quantum disordered ground state in [133] Y. Li, D. Adroja, R. Bewley, D. Voneshen, A. Tsir- the triangular-lattice antiferromagnet NaYbO2, Nature lin, P. Gegenwart, and Q. Zhang, Crystalline electric- Phys. 15, 1058 (2019). field randomness in the triangular lattice spin-liquid [149] M. Baenitz, P. Schlender, J. Sichelschmidt, Y. Onyki- YbMgGaO4, Phys. Rev. Lett. 118, 107202 (2017). ienko, Z. Zangeneh, K. Ranjith, R. Sarkar, L. Hozoi, [134] M. Majumder, G. Simutis, I. Collings, J.-C. Orain, H. C. Walker, J.-C. Orain, H. Yasuoka, J. van den Brink, T. Dey, Y. Li, P. Gegenwart, and A. Tsirlin, Persistent H. Klauss, D. Inosov, and T. Doert, NaYbS2: A planar 1 spin dynamics in the pressurized spin-liquid candidate spin- 2 triangular-lattice magnet and putative spin liq- YbMgGaO4, arXiv:1902.07749. uid, Phys. Rev. B 98, 220409(R) (2018). [135] Y.-D. Li, Y. Shen, Y. Li, J. Zhao, and G. Chen, Effect [150] W. Liu, Z. Zhang, J. Ji, Y. Liu, J. Li, X. Wang, H. Lei, of spin-orbit coupling on the effective-spin correlation G. Chen, and Q. Zhang, Rare-earth chalcogenides: A in YbMgGaO4, Phys. Rev. B 97, 125105 (2018). large family of triangular lattice spin liquid candidates, [136] X. Zhang, F. Mahmood, M. Daum, Z. Dun, J. Paddi- Chin. Phys. Lett. 35, 117501 (2018). son, N. Laurita, T. Hong, H. Zhou, N. Armitage, and [151] K. Ranjith, S. Luther, T. Reimann, B. Schmidt, M. Mourigal, Hierarchy of exchange interactions in the P. Schlender, J. Sichelschmidt, H. Yasuoka, A. Stry- triangular-lattice spin liquid YbMgGaO4, Phys. Rev. X dom, Y. Skourski, J. Wosnitza, H. K¨uhne,T. Doert, and 8, 031001 (2018). M. Baenitz, Anisotropic field-induced ordering in the [137] Note that the data of Ref. 127 were rather insensitive to triangular-lattice quantum spin liquid NaYbSe2, Phys. ±± z± the J1 and J1 terms, so the values of Ref. 118 were Rev. B 100, 224417 (2019). employed there without further refinement. [152] J. Xing, L. Sanjeewa, J. Kim, G. Stewart, [138] Z. Zhu, P. Maksimov, S. White, and A. Cherny- A. Podlesnyak, and A. Sefat, Field-induced mag- shev, Disorder-induced mimicry of a spin liquid in netic transition and spin fluctuations in the quantum YbMgGaO4, Phys. Rev. Lett. 119, 157201 (2017). spin-liquid candidate CsYbSe2, Phys. Rev. B 100, [139] E. Parker and L. Balents, Finite-temperature behavior 220407(R) (2019). of a classical spin-orbit-coupled model for YbMgGaO4 [153] R. Sarkar, P. Schlender, V. Grinenko, E. Haeussler, with and without bond disorder, Phys. Rev. B 97, P. Baker, T. Doert, and H.-H. Klauss, Quantum spin 184413 (2018). liquid ground state in the disorder free triangular lat- [140] L. Ding, P. Manuel, S. Bachus, F. Grußler, P. Gegen- tice NaYbS2, Phys. Rev. B 100, 241116(R) (2019). wart, J. Singleton, R. Johnson, H. Walker, D. Adroja, [154] J. Sichelschmidt, P. Schlender, B. Schmidt, M. Baenitz, A. Hillier, and A. Tsirlin, Gapless spin-liquid state in and T. Doert, Electron spin resonance on the spin-1/2 the structurally disorder-free triangular antiferromag- triangular magnet NaYbS2, J. Phys.: Condens. Matter net NaYbO2, Phys. Rev. B 100, 144432 (2019). 31, 205601 (2019). [141] K. Watanabe, H. Kawamura, H. Nakano, and T. Sakai, [155] M. Ye and A. Chubukov, Half-magnetization plateau Quantum spin-liquid behavior in the spin-1/2 random in a Heisenberg antiferromagnet on a triangular lattice, Heisenberg antiferromagnet on the triangular lattice,J. Phys. Rev. B 96, 140406(R) (2017). Phys. Soc. Jpn. 83, 034714 (2014). [156] M. Ye and A. Chubukov, Quantum phase transitions in [142] T. Shimokawa, K. Watanabe, and H. Kawamura, Static the Heisenberg J1 − J2 triangular antiferromagnet in a 1 and dynamical spin correlations of the S = 2 random- magnetic field, Phys. Rev. B 95, 014425 (2017). bond antiferromagnetic Heisenberg model on the tri- [157] M. Sanders, F. Cevallos, and R. Cava, Magnetism in angular and kagome lattices, Phys. Rev. B 92, 134407 the KBaRE(BO3)2 (RE = Sm, Eu, Gd, Tb, Dy, Ho, Er, (2015). Tm, Yb, Lu) series: materials with a triangular rare [143] S. Dey, E. Andrade, and M. Vojta, Destruction of long- earth lattice, Mater. Res. Express 4, 036102 (2017). range order in noncollinear two-dimensional antiferro- [158] S. Guo, T. Kong, F. Alex Cevallos, K. Stolze, magnets by random-bond disorder, Phys. Rev. B 101, and R. Cava, Crystal growth, crystal structure and 020411(R) (2020). anisotropic magnetic properties of KBaR(BO3)2 (R = [144] H.-Q. Wu, S.-S. Gong, and D. Sheng, Randomness- Y, Gd, Tb, Dy, Ho, Tm, Yb, and Lu) triangular lattice 1 induced spin-liquid-like phase in the spin- 2 J1 − J2 tri- materials, J. Magn. Magn. Mater. 472, 104 (2019). angular Heisenberg model, Phys. Rev. B 99, 085141 19

[159] S. Guo, A. Ghasemi, C. Broholm, and R. Cava, Mag- tiferromagnet YbAl3C3, J. Phys. Soc. Jpn. 76, 123703 netism on ideal triangular lattices in NaBaYb(BO3)2, (2007). Phys. Rev. Materials 3, 094404 (2019). [169] Y. Kato, M. Kosaka, H. Nowatari, Y. Saiga, A. Yamada, [160] J. Xing, L. Sanjeewa, J. Kim, G. Stewart, M.-H. Du, T. Kobiyama, S. Katano, K. Ohoyama, H. Suzuki, F. Reboredo, R. Custelcean, and A. Sefat, Crystal syn- N. Aso, and K. Iwasa, Spin-singlet ground state in the thesis and frustrated magnetism in triangular lattice two-dimensional frustrated triangular lattice: YbAl3C3, CsRESe2 (RE = La-Lu): Quantum spin liquid candi- J. Phys. Soc. Jpn. 77, 053701 (2008). dates CsCeSe2 and CsYbSe2, ACS Materials Lett. 2, 71 [170] T. Matsumura, T. Inami, M. Kosaka, Y. Kato, (2020). T. Inukai, A. Ochiai, H. Nakao, Y. Murakami, [161] F. Alex Cevallos, K. Stolze, and R. Cava, Structural dis- S. Katano, and H. Suzuki, Structural phase transition order and elementary magnetic properties of triangular in the spin gap system YbAl3C3, J. Phys. Soc. Jpn. 77, lattice ErMgGaO4 single crystals, Solid State Comm. 103601 (2008). 276, 5 (2018). [171] D. Khalyavin, D. Adroja, P. Manuel, A. Daoud-Aladine, [162] S. Guo, T. Kong, and R. Cava, NaBaR(BO3)2 (R = M. Kosaka, K. Kondo, K. McEwen, J. Pixley, and Q. Si, Dy, Ho, Er and Tm): structurally perfect triangular Field-induced long-range magnetic order in the spin- lattice materials with two rare earth layers, Mater. Res. singlet ground-state system YbAl3C3: Neutron diffrac- Express 6, 106110 (2019). tion study, Phys. Rev. B 87, 220406(R) (2013). [163] J. Banda, B. Rai, H. Rosner, E. Morosan, C. Geibel, and [172] V. Nekvasil and I. Veltrusk´y,Effective Hamiltonian and M. Brando, Crystalline electric field of Ce in trigonal ground state properties of rare-earth ions in iron gar- symmetry: CeIr3Ge7 as a model case, Phys. Rev. B 98, nets, J. Magn. Magn. Mater. 86, 315 (1990). 195120 (2018). [173] Y. Shen, C. Liu, Y. Qin, S. Shen, Y.-D. Li, R. Bewley, [164] J. Xing, L. Sanjeewa, J. Kim, W. Meier, A. May, A. Schneidewind, G. Chen, and J. Zhao, Intertwined Q. Zheng, R. Custelcean, G. Stewart, and A. Sefat, Syn- dipolar and multipolar order in the triangular-lattice thesis, magnetization, and heat capacity of triangular magnet TmMgGaO4, Nature Comm. 10, 4530 (2019). lattice materials NaErSe2 and KErSe2, Phys. Rev. Ma- [174] Y. Li, S. Bachus, H. Deng, W. Schmidt, H. Thoma, terials 3, 114413 (2019). V. Hutanu, Y. Tokiwa, A. Tsirlin, and P. Gegenwart, [165] A. Scheie, V. Garlea, L. Sanjeewa, J. Xing, and A. Sefat, Partial up-up-down order with the continuously dis- Crystal field hamiltonian and anisotropy in KErSe2 and tributed order parameter in the triangular antiferromag- CsErSe2, arXiv:2001.03175. net TmMgGaO4, Phys. Rev. X 10, 011007 (2020). [166] S. Gao, F. Xiao, K. Kamazawa, K. Ikeuchi, D. Biner, [175] W. Steinhardt, Z. Shi, A. Samarakoon, S. Dissanayake, K. Kr¨amer,C. R¨uegg,and T. Arima, Crystal-electric- D. Graf, Y. Liu, W. Zhu, C. Marjerrison, C. Batista, and field excitations in a quantum-spin-liquid candidate S. Haravifard, Field-induced transformation of the spin KErS2, arXiv:1911.10662. liquid state in triangular antiferromagnet YbMgGaO4, [167] J. Lee, A. Rabus, N. Lee-Hone, D. Broun, and E. Mun, arXiv:1902.07825. The two-dimensional metallic triangular lattice antifer- [176] S. Winter, A. Tsirlin, M. Daghofer, J. van den Brink, romagnet CeCd3P3, Phys. Rev. B 99, 245159 (2019). Y. Singh, P. Gegenwart, and R. Valent´ı, Models and [168] A. Ochiai, T. Inukai, T. Matsumura, A. Oyamada, and materials for generalized Kitaev magnetism, J. Phys.: K. Katoh, Spin gap state of S = 1/2 Heisenberg an- Condens. Matter 29, 493002 (2017).