Fingerprinting Triangular-Lattice Antiferromagnet by Excitation Gaps
K. E. Avers, P. A. Maksimov, P. F. S. Rosa, S. M. Thomas, J. D. Thompson, W. P. Halperin, R. Movshovich, A. L. Chernyshev
FFingerprinting Triangular-Lattice Antiferromagnet by Excitation Gaps
K. E. Avers,
1, 2, 3
P. A. Maksimov, P. F. S. Rosa, S. M. Thomas, J. D. Thompson, W. P. Halperin, R. Movshovich, and A. L. Chernyshev Los Alamos National Laboratory, Los Alamos, NM 87545, USA Department of Physics and Astronomy, Northwestern University, Evanston, IL, USA Center for Applied Physics & Superconducting Technologies, Northwestern University, Evanston, IL, USA Bogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia Department of Physics and Astronomy, University of California, Irvine, California 92697, USA (Dated: February 8, 2021)CeCd As is a rare-earth triangular-lattice antiferromagnet with large inter-layer separation. Ourfield-dependent heat capacity measurements at dilution fridge temperatures allow us to trace thefield-evolution of the spin-excitation gaps throughout the antiferromagnetic and paramagnetic re-gions. The distinct gap evolution places strong constraints on the microscopic pseudo-spin model,which, in return, yields a close quantitative description of the gap behavior. This analysis providescrucial insights into the nature of the magnetic state of CeCd As , with a certainty regarding itsstripe order and low-energy model parameters that sets a compelling paradigm for exploring andunderstanding the rapidly growing family of the rare-earth-based triangular-lattice systems. Rare-earth-based quantum magnets are of great cur-rent interest, as they naturally combine the effectsof strong spin-orbit coupling (SOC), which entanglesmagnetic degrees of freedom and orbital orientations[1, 2], with that of the geometric frustration of thelattices bearing a triangular motif, long anticipated asleading to exotic ground states and excitations [3, 4].A remarkable recent surge in the studies of quantumanisotropic-exchange magnets in general and rare-earthbased triangular-lattice (TL) materials in particular ispropelled by an avalanche of the newly synthesized com-pounds [5–22] and theoretical insights into their models[4, 23–28]. Rare-earth materials provide an ideal plat-form for the search of novel phases as they host short-range and highly anisotropic exchange interactions oftheir effective spin degrees of freedom due to strong SOCand highly-localized nature of the f -orbitals.In this Letter, we present low-temperature heat ca-pacity measurements of a Ce-based representative mem-ber of this family of materials, CeCd As . These mea-surements enable us to construct the magnetic field–temperature ( H – T ) phase diagram of CeCd As and de-termine the field-evolution of the spin-excitation gaps inits spectrum throughout the transition from a frustratedantiferromagnetic (AF) to a paramagnetic (PM) state.We augment these results by a theoretical analysis thatyields a close agreement with a distinct field-dependenceof the gaps and allows us to unequivocally identify theground state of CeCd As as being in a stripe phase.The phenomenological constraints on the parameters ofthe microscopic model result in robust certainty regard-ing the ground state and parameter region of the generalphase diagram of the anisotropic-exchange TL systemsto which CeCd As likely belongs. Our approach is ex-pected to enable a better understanding of the rapidly growing family of rare-earth-based TL materials. Material and Methods. —CeCd As crystallizes as thin( ∼ × × P /mmc space group in which Ce +3 ions form 2D tri-angular lattices that are widely spaced along the c-axis,as shown in Supplementary Materiel (SM) [29]. Theirlow-temperature magnetization is characteristic of aneasy-plane magnetic anisotropy with a ratio of Landeg-factors g ab /g c ∼ H – T phase diagram. —In zero field, CeCd As ordersat T N =412(20) mK in agreement with a recent report[31] of AF ordering at T N ∼
420 mK. At T N , only aboutone quarter of the entropy of the ground state doublet R ln 2 is recovered [29]. The substantial difference be-tween T N and the ab -plane Curie-Weiss (CW) tempera-ture θ CW = − . As [32], although with the caveat thatthe CW temperature can only serve as a crude estimateof the spins’ exchange strength in rare-earth materials.For magnetic fields applied in the ab -plane, T N in-creases, reaches its maximum of 496(30) mK near 2 T,and is suppressed at higher fields, see Fig. 1. Above 3.5 T,at temperatures above the sharp feature that is identifiedwith the magnetic ordering, specific heat also exhibits ashoulder-like anomaly denoted as T U in Fig. 2(b). InFig. 1, we highlight the crossover (XO) region betweenthese two features as a shaded area. Both anomalies aresuppressed to zero temperature by a magnetic field near4.7–4.8 T, suggesting a quantum critical point (QCP)within that field range. As is shown in Fig. 1, this regionof the phase diagram demonstrates a significant enhance-ment of the specific heat at a reference low-temperatureof 100 mK. a r X i v : . [ c ond - m a t . s t r- e l ] F e b .10.20.30.40.50.6 FIG. 1. Open symbols are T N and T U vs in-plane field H fromthe heat capacity data in Figs. 2(a) and (b); closed circles are C/T values at 100 mK, lines are guides to the eye. A crossover(XO) region between T N and T U is highlighted. Insert: sketchof the stripe- x AF phase in zero and finite field, see text.
Specific heat, T N and T U . —Specific heat data forCeCd As is presented in Fig. 2 for several field andtemperature regimes. The data up to 3 T is shown inFig. 2(a). It demonstrates the non-monotonic T N field-dependence, characterized by an initial increase followedby a suppression. Such an increase indicates an enhance-ment of the AF order with field and is known to occurin several quantum magnets and their models [33–35],wherein this effect is associated with the field-inducedsuppression of quantum or thermal spin fluctuations or areduction of frustration. Given our subsequent analysisof the nature of its magnetic ordered state, we ascribethe increase of T N in the case of CeCd As to a field-induced suppression of critical fluctuations related to thephase transition, see SM [29].As is shown in Fig. 2(b) for the fields 4 T and above,specific heat acquires an additional shoulder-like featureat T U > T N , giving rise to an XO region between the twotemperatures, see also Fig. 1. It appears that the lineof T N transitions continues through 3.5 T with no inflec-tion. Both T U and T N decrease towards zero temperatureat higher fields, in agreement with the expected suppres-sion of the AF order parameter to zero at a QCP. Forthe XO region of the phase diagram, we note that whilesome frustrated TL models consistently have a two-peakstructure in their specific heat [36], the second, higher-temperature anomaly at T U in Fig. 2(b) could also berelated to a field-induced high density of states in themagnon spectrum, see SM [29]. Specific heat, low- T . —Fig. 2(c) focuses on the low- T heat capacity. As is already indicated in Fig. 1, the be-havior of C/T in this temperature range near a presump-tive QCP is drastically different from that in other fieldsregions. It should also be noted that the QCP regionis where universal scalings are expected to dictate the
Linear
FIG. 2. Specific heat
C/T vs T shows: (a) the non-monotonicfield-dependence of T N for H < . T U and the XO region for H > . T dependence of C/T away and at the QCPtogether with the fits from Eqs. (1) and (2) (lines). T -dependence of all thermodynamic quantities [37].Fig. 2(c) also displays the specific heat as a functionof T for several representative fields away from the QCPregion (solid symbols). At T = 100 mK, C/T is small upto 4.25 T and is suppressed again in fields greater than5 T. Moreover, the temperature dependence of
C/T inFig. 2(c) in fields outside of the critical region clearly in-dicates activated behavior characteristic of gapped sys-tems, as we elaborate below.The enhancement in low- T entropy, manifested as abuild up of area under the C/T vs T curves, occurs forfields between 4.25 T and 5 T. This results in the large C/T values at 100 mK shown in Fig. 1. The value of
C/T at 100 mK is substantially enhanced, reaching amaximum of about 1 J/mol-K at 4.6 T. It is also ac-companied by a distinct change in the temperature de-pendence, shown in Fig. 2(c) for 4.6 T and 4.75 T (opensymbols), that is indicative of a power-law in T . The risein C/T below 100 mK for H =4.6 T is likely due to ourcalorimeter setup not accurately accounting for longerinternal thermal relaxation times for this H — T range. Low- T asymptotes. —The leading contribution to theheat capacity from a 2D gapped excitation can be ob-tained by approximating its energy as ε k ≈ ∆ + J k near2he minimum gap ∆ and J parametrizing the bandwidth C ( T ) /T = A (cid:0) x + 2 x + 2 (cid:1) e − x + γ + O (cid:0) e − x (cid:1) , (1)where x = ∆ /T , A ∼ /J , and γ is a background Som-merfeld term that we fix at 0.029 J/mol-K for all fits[29]. The 2D activated behavior of Eq. (1) fits very wellall C/T data up to 200 mK for fields away from the QCPregion, as is demonstrated in Fig. 2(c), see also Ref. [29].Since the fit is controlled by a single parameter ∆, thisanalysis allows us to trace the field-dependence of thelowest excitation gap in CeCd As .For systems with continuous spin symmetries, field-induced PM to AF transition is of the Bose-Einstein con-densation type [37, 38]. In our case, because SOC leavesno continuous symmetries intact, this transition is of adifferent universality class, characterized by the closingof the excitation spectrum gap in a relativistic manner, ε k ≈ √ ∆ + J k [37]. This implies an acoustic mode atthe QCP, ε k ∝ | k | (dynamical exponent z = 1), leading toa universal 2D scaling, C ( T ) ∝ T , at the transition field.The gap closure also explains the peak in C/T vs field inFigs. 1 and 2(c) near the QCP.At 4.75 T, CeCd As appears to be close to the QCP,as is indicated by the linear fit of C/T (dashed line) inFig. 2(c), which is indicative of gapless excitations. Forsmall gaps, we obtain a modified scaling C ( T ) /T ≈ A T (cid:0) − x /α (cid:1) + γ , (2)with α = 12 ζ (3), valid down to T ∼ ∆ / .
90 mK and of ≈ .
19 K at 4.6 T.
Gaps and other phenomenologies. —The spectrum gap∆, extracted from the specific heat data using Eqs. (1)and (2), is shown in Fig. 3 vs H . The semi-log scale is toaccommodate ∆ ≈ .
92 K is compatible with theCW temperature [29, 31] and shows no sign of closing be-low the QCP, implying that the AF phase of CeCd As evolves continuously with H . Magnetization data corrob-orate this assertion [29] showing no traces of the plateau-like phase transitions emblematic of TL magnets [39, 40].The field-dependence of ∆ demonstrates an essentialfeature. It shows a gradual increase to about 1.1 K at3.5 T followed by an abrupt closing upon approachingthe critical field. This non-monotonic behavior is an im-portant distinguishing hallmark that allows us to unam-biguously identify the ordered state of CeCd As . Model. —For Kramers ions in layered compounds, crys-tal field effects (CEF) lead to an energy splitting of the lo-cal Hilbert space of the rare-earth ion magnetic momentinto a series of doublets [41]. At temperatures much lowerthan the crystal field splitting, the lowest Kramers dou-blets, naturally parametrized as effective pseudo-spins
FIG. 3. ∆ vs H obtained using Eqs. (1) and (2) (symbols).The excitation spectrum gaps at the ordering vector Y ( E Y )and complimentary M point ( E M ) for the parameters of themodel (3) discussed in text. Inset: Brillouin zone with the Yand M points for the stripe phase in Fig. 1. S = , are responsible for the dominant magnetic prop-erties of insulating materials. Because of the entangle-ment with the orbital orientations that are tied to thelattice due to CEF, the pairwise interactions of thesepseudo-spins a priori retain no spin-rotational symme-tries [4, 23]. Instead, it is the discrete symmetries of thelattice that restrict possible forms of the bond-dependentinteractions. Together with the localized nature of f -orbitals that limits the ranges of interactions, these sym-metries lead to generic Hamiltonians that are expectedto adequately describe all rare-earth-based Kramers com-pounds on a given lattice [24].For a layered TL structure, the relevant point-groupsymmetry operations allow four terms in the Hamiltonianthat can be separated into bond-independent, H XXZ ,and bond-dependent, H bd , parts, see Ref. [27], H = X h ij i (cid:16) H XXZ h ij i + H bd h ij i (cid:17) + X h ij i H XXZ h ij i ,H XXZ h ij i m = J m (cid:16) S xi S xj + S yi S yj + ¯∆ S zi S zj (cid:17) (3) H bd h ij i = 2 J ±± h(cid:16) S xi S xj − S yi S yj (cid:17) ˜ c α − (cid:16) S xi S yj + S yi S xj (cid:17) ˜ s α i + J z ± h(cid:16) S yi S zj + S zi S yj (cid:17) ˜ c α − (cid:16) S xi S zj + S zi S xj (cid:17) ˜ s α i , where ˜ c (˜ s ) α = cos(sin) ˜ ϕ α , ˜ ϕ α are angles of the primi-tive vectors with the x axis, ˜ ϕ α = { , π/ , − π/ } , and { x, y, z } are the crystallographic axes, see Fig. 1. Thebond-independent exchange constants J m are J and J for the first- and second-neighbor couplings, respectively.Following prior works [26, 42], we use a minimal exten-sion of the model by the J term with the same XXZ anisotropy ¯∆. In an external field, Zeeman coupling H Z = − µ B X i h g ab (cid:16) H x S xi + H y S yi (cid:17) + g c H z S zi i (4)contains anisotropic g -factors of the pseudo-spins thatreflect the build-up of the ground-state doublets fromthe states of the J -multiplet of the rare-earth ions by a3ombined effect of SOC and CEF. The in-plane g -factoris uniform because of the TL three-fold symmetry [43]. Phase identification. —As the model (3) has no spin-rotational symmetries [23, 26], one expects gapped ex-citations throughout its phase diagram, but acciden-tal degeneracies render most of the phases, such aswell-known 120 ◦ phase and the nearby incommensuratephases, nearly gapless [27, 44]. Of the remaining phases,the CeCd As phenomenology of a single-phase field-evolution and a sizable spin-excitation gap strongly sug-gests so-called stripe phases as prime contenders for itsground state. In a stripe phase, ferromagnetic rows ofspins arrange themselves in an AF fashion, see Fig. 1. Inparticular, as we argue in this work, the non-monotonicfield-dependence of the gap is a hallmark of the stripephases. An alternative scenario in the Ising limit leadsto phases and transitions [40, 45] that are incompatiblewith the phenomenology of CeCd As , see SM [29] formore detail.Most importantly, the field-evolution of the spin-excitation spectrum in the stripe phase, at the QCP, andin the spin-polarized PM phase are all in accord with ourresults for CeCd As . Specifically, the spectrum minimain zero field are not associated with the ordering vector(identified as a Y-point), but are complementary to it(M-points in the Brillouin zone). This feature is charac-teristic of the systems with significant frustrating bond-dependent interactions [27, 28]. In an applied field, it isthe gap at the ordering vector that must close, leading toa rather abrupt switch of the minimal gap between theM and Y-points, as is demonstrated in Fig. 3 for a choiceof parameters in model (3) discussed next. Model parameters. —Assuming a stripe ground state,we obtain the field-evolution of spin excitations for model(3) in the AF and PM phases. There are two empiricalquantities that provide strong constraints on the modelparameters: the value of the critical field, H s , which wetake as 4.8 T, allowing for an ambiguity in the QCP forCeCd As , and the zero-field gap, ∆ , taken as 0.87 Kto account for an uncertainty of the fit. Qualitatively,with g ab ≈ H s strongly binds thecumulative exchange term, J + J , while ∆ restricts the J ±± anisotropic-exchange term, see SM [29] for details.We find the second anisotropic-exchange term, J z ± , tohave a minor effect on the spectrum [27] and neglect itin our consideration. This mild simplifying assumptionleaves two types of stripe states, stripe- x and stripe- y ,which correspond to spins along and perpendicular tothe bonds of the lattice, respectively, indistinguishableup to a change of the sign of the J ±± -term and a switchof the field direction from H k b to H k a . Because ex-periments in CeCd As have not discriminated betweenthe in-plane field directions, we take a minimal-modelapproach [29], by assuming J ±± <
0, which correspondsto stripe- x with H along the b -direction shown in Fig. 1. We have found that the excitation gaps are virtuallyinsensitive to the value of the XXZ anisotropy ¯∆, whichis loosely bound in the range 0.5–1.5 by the out-of-planesaturation field, extrapolated from M ( H ) data [29]. Thesituation is similar with the ratio J /J , which does notaffect observables at the fixed total J + J . Thus, bytaking XXZ anisotropy ¯∆ = 1 and making an ad hoc choice of J /J = 0 . f -orbitals,for the empirical H s and ∆ we obtain J ≈ .
19 K and J ±± ≈ − .
31 K [29]. With these model parameters, wederive the field-evolution of the gaps shown in Fig. 3.Our results show a gradual increase of the magnon gapenergy E M vs field at the complementary M-point andconcurrent decrease of the spin-excitation energy E Y atthe ordering vector [29] which inevitably takes the role ofthe global minimum of the spectrum upon approachingthe QCP, all in close accord with the data in Fig. 3. Atthe QCP, the asymptotic form of the spectrum adheres tothe expected relativistic form. Above the QCP, the gapat the Y point reopens, with the spectrum experiencinga roughly uniform, Zeeman-like shift vs H . Remarkably,the high-field data at 9 T deep in the PM phase is closelymatched by the theory with the same model parameters. Summary. —In summary, we have demonstrated that acombination of insights from the low-temperature specificheat data and theoretical modeling provide a comprehen-sive description of the ground state and excitations in aTL rare-earth anisotropic-exchange magnet, paving theway to a deeper understanding of a broad class of materi-als. The phenomenological constraints on the general mi-croscopic model have resulted in a precise identificationof CeCd As magnetic ground state as a stripe phase andwith a remarkable level of certainty regarding the part ofthe phase diagram where it belongs.This study is of immediate relevance to KCeS [46],KErSe [47], and isostructural CeCd P [48], where someof the same phenomenology has been observed. Futurestudies by thermodynamic and spectroscopic methods,such as low- T magnetization, nuclear and electronic mag-netic resonance, neutron and x-ray magnetic dichroismscattering, are expected to provide further insights intothe nature of the crossover region in the H – T phase dia-gram and into the role of structural disorder in the staticand dynamic properties of these materials. With moretheoretical input, they should yield more systematic con-straints on the model and elucidate the role of differ-ent terms in unusual magnetic states and excitations ofanisotropic-exchange magnets. Acknowledgments —Work at Los Alamos National Lab-oratory was performed under the auspices of the U.S.Department of Energy, Office of Basic Energy Sciences,Division of Materials Science and Engineering. P. F. S. R.acknowledges support from the Los Alamos LaboratoryDirected Research and Development program. Supportis acknowledged from the Northwestern-Fermilab Centerfor Applied Physics and Superconducting Technologies4K. E. A). Scanning electron microscope and energy dis-persive X-ray measurements were performed at the Cen-ter for Integrated Nanotechnologies, an Office of ScienceUser Facility operated for the U.S. Department of En-ergy (DOE) Office of Science. The work of A. L. C.was supported by the U.S. Department of Energy, Of-fice of Science, Basic Energy Sciences under Awards No.DE-FG02-04ER46174 and DE-SC0021221. P. A. M. ac-knowledges support from JINR Grant for young scien-tists 21-302-03. A. L. C. would like to thank Kavli Insti-tute for Theoretical Physics (KITP) where this work wasadvanced. KITP is supported by the National ScienceFoundation under Grant No. NSF PHY-1748958. [1] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Ba-lents, Correlated Quantum Phenomena in the StrongSpin-Orbit Regime, Annu. Rev. Cond. Mat. Phys. , 57(2014).[2] L. 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Hutchings, Point-Charge Calculations of EnergyLevels of Magentic Ions in Crystalline Electric Fields,Solid State Phys. , 227 (1964).[42] J. A. M. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu,M. B. Stone, H. Zhou, and M. Mourigal, Continuous Ex-citations of the Triangular-Lattice Quantum Spin LiquidYbMgGaO , Nature Phys. , 117 (2017).[43] Y. Li, G. Chen, W. Tong, L. Pi, J. Liu, Z. Yang, X. Wang,and Q. Zhang, Rare-Earth Triangular Lattice Spin Liq-uid: A Single-Crystal Study of YbMgGaO , Phys. Rev.Lett. , 167203 (2015).[44] J. G. Rau, P. A. McClarty, and R. Moessner, Pseudo-Goldstone Gaps and Order-by-Quantum-Disorder inFrustrated Magnets, Phys. Rev. Lett. , 237201(2018).[45] S. Miyashita, Magnetic Properties of Ising-Like Heisen-berg Antiferromagnets on the Triangular Lattice, J.Phys. Soc. Japan , 3605 (1986).[46] G. Bastien, B. Rubrecht, E. Haeussler, P. Schlender,Z. Zangeneh, S. Avdoshenko, R. Sarkar, A. Alfonsov,S. Luther, Y. A. Onykiienko, H. C. Walker, H. Kuhne,V. Grinenko, Z. Guguchia, V. Kataev, H.-H. Klauss,L. Hozoi, J. van den Brink, D. S. Inosov, B. Buchner,A. U. B. Wolter, and T. Doert, Long-Range MagneticOrder in the ˜ S = 1 / , SciPost Phys. , 041 (2020).[47] J. Xing, K. M. Taddei, L. D. Sanjeewa, R. S. Fishman,M. Daum, M. Mourigal, C. dela Cruz, and A. S. Sefat,Stripe Antiferromagnetic Ground State of Ideal Triangu-lar Lattice KErSe (2020), arXiv:2006.01602.[48] J. Lee, A. Rabus, N. Lee-Hone, D. Broun, and E. Mun,The Two-Dimensional Metallic Triangular Lattice Anti-ferromagnet CeCd P , Phys. Rev. B , 245159 (2019).[49] E. Gopal, Specific Heats at Low Temperatures, The In-ternational Cryogenics Monograph Series (Springer US,1966).[50] Y. T. Fan and W. H. Lee, Antiferromagnetic Spin Wavein Ce PdGe , Physical Review B , 132401 (2004).[51] W. P. Halperin, F. Rasmussen, C. Archie, andR. Richardson, Properties of Melting He: Specific Heat,Entropy, Latent Heat, and Temperature, J. Low Temp.Phys. , 617 (1978).[52] P. Pagliuso, D. Garcia, E. Miranda, E. Granado, R. Ser-rano, C. Giles, J. Duque, R. Urbano, C. Rettori,J. Thompson, M. Hundley, and J. Sarro, Evolution ofthe Magnetic Properties and Magnetic Structures Alongthe R m MIn m +2 (R=Ce, Nd, Gd, Tb; M=Rh, Ir; andm=1,2) Series of Intermetallic Compounds, J. Appl.Phys. , 08P703 (2006).[53] S. S. Stoyko and A. Mar, Ternary Rare-Earth ArsenidesREZn As (RE=La-Nd, Sm) and RECd As (RE=La-Pr), Inorg. Chem. , 11152 (2011).[54] Y. D. Li, Y. Shen, Y. Li, J. Zhao, and G. Chen, The Effectof Spin-Orbit Coupling on the Effective-Spin Correlationin YbMgGaO , Phys. Rev. B , 125105 (2018).[55] E. V. Komleva, V. Y. Irkhin, I. V. Solovyev, M. I.Katsnelson, and S. V. Streltsov, Unconventional Mag-netism and Electronic State in Frustrated Layered Sys-tem PdCrO , Phys. Rev. B , 174438 (2020). upplemental Material for Fingerprinting Triangular-Lattice Antiferromagnet byExcitation Gaps
I. EXPERIMENTAL DETAILS AND OTHERDATAA. Experimental methods
Bulk single crystals of CeCd As were grown bychemical vapor transport using iodine as a transportagent and recipe described in Ref. [1, 2]. PolycrystallineCeCd As was first synthesized by solid state reaction.A stoichiometric mixture of Ce, Cd and As pieces wassealed in a quartz tube under partial Ar atmosphereand heated to 800 ◦ C for one week. The resultingpolycrystal was sealed in a quartz tube with iodineand placed in the hot end of a zone furnace. Theother end of the tube was held at 700 ◦ C where singlecrystal platelets of typical size 1 mm were obtained. Thecrystallographic structure was verified by single-crystaldiffraction at room temperature, using Mo radiation ina Bruker D8 Venture diffractometer, and was consistentwith previous results [1, 2].Heat capacity measurements were performed in adilution refrigerator down to 70 mK using the heatpulse technique. A RuO thermometer was attacheddirectly to one side of the sample with GE varnish.The other side of the sample was attached to asapphire substrate with a heater on the opposite sideof the substrate. The weak thermal link wire wassilver painted directly to the sample. No evidencefor multiple timescale relaxation behavior was observedindicating good thermal contact between the sample,RuO thermometer, heater, and sapphire substrate, aswell as fast internal relaxation time of the sample.No subtraction of electron nor phonon heat capacitycontributions from the sample, heater, sapphire, orthermometer were performed because the magneticheat capacity of CeCd As is vastly dominant inthe temperature range of interest in this work. Itis estimated that the magnetic degrees of freedomcontribute more than 99 percent of the entropy changefrom 0 K to 2 K. The calorimeter was weakly thermallylinked to a copper temperature regulation block, anda temperature stabilized Lakeshore automatic bridgewith active feedback PID system was employed. Thesample RuO thermometer was previously calibrated inmagnetic field up to 9 T.Measurements of resistivity vs. temperature wereperformed in zero field down to 95 mK using a QuantumDesign physical property measurement system withan adiabatic demagnetization refrigeration attachment.The sample was taken from the same batch as thesample used for heat capacity measurements. It wasglued to a copper cold finger using GE varnish with cigarette paper providing electrical insulation. Four 25micron platinum wires were spot welded to the sample,and resistivity was measured by the 4-wire method.Multiple excitation currents were applied to ensureJoule heating was not significant.Magnetization measurements were performed usinga commercial Quantum Design MPMS SQUID-basedmagnetometer. B. Crystal structure
CeCd As crystallizes in the PrZn As -type structure(space group P /mmc ) with lattice parametersa=b=4.4051 Å and c=21.3511 Å, as shown in Fig.1. Magnetic rare-earth Ce +3 ions form planes of 2Dtriangular lattices that are separated from each otherby layers of As and Cd atoms with an aspect ratio ofinter-plane to intra-plane Ce spacing of approximately2.4. There is only one cerium site in this structure, butboth Cd and As atoms have two sites, one of which isonly 1/3 occupied. Though this partial occupancy doesnot distort the Ce triangular structure directly, the roleof disorder in this material remains poorly understood. FIG. 1. (left) Crystal structure of CeCd As , whichcrystallizes in space group P /mmc ). Orange andblue circles represent Cd and As atoms, respectively. (right)The two-dimensional triangular arrangement of the magneticCe atoms gives rise to the frustrated behavior in this work. C. Magnetization and susceptibility
The higher temperature investigation ofthermodynamic properties provide important cluesinto the nature of magnetism in CeCd As . Themagnetic susceptibility ( χ ) vs. temperature ( T ) is a r X i v : . [ c ond - m a t . s t r- e l ] F e b a)b) FIG. 2. a) Susceptibility ( χ = MH ) vs. Temperature ( T )taken with H=1 T along the c-axis, and within the ab-plane with the inverse susceptibility ( χ − ) in the insert. Theinfection points in χ − are due to the influence of exciteddoublets of the Ce moment. The light green curves are fitsto data as described in the text. b) Magnetization ( M ) vs.Field ( H ) at T =2 K to H =6 T along the c-axis and withinthe ab plane alongside the data of Liu et. al. [2]. The lightgreen curve is calculated by integrating the low temperaturesusceptibility. Even though the fit only reaches 1.2 µ b /Ce itis relatively good agreement. presented in Fig. 2a. The data were taken in an appliedfield of 1 T both along the c-axis (blue squares), andwithin the ab-plane (red circles). A simple Curie-Weisslaw does not describe the experimental results. χ ( T ) − ,shown in the insert, is visibly non-monotonic for thefield along the c-axis, with a local minimum near 140 K.Careful inspection of the susceptibility for the field inthe ab-plane reveals inflection points in its temperaturebehavior as well. Fig. 2b presents magnetization ( M )vs. field ( H ) results at 2 K for magnetic field up to6 T applied along the c-axis (open blue squares) andwithin the ab-plane (open red circles). We also plotdata at 1.9 K up to 16 T extracted from Ref. [2](filled circles and squares). The strong magnetization anisotropy of roughly 10 between the (easy) ab-planeand the (hard) c-axis was initially taken as evidence forstrong antiferromagnetic interactions of Ising-like spinsaligned along the c-axis [2]; however, inspection of thetemperature dependent susceptibility shows that thismay be an inaccurate description of the physics of thissystem.As recognized in Ref. [3], the high-temperature non-monotonic behavior in χ vs. T can be accounted for bya non-interacting model involving the Zeeman term andthe trigonal crystalline electric field (CEF) Hamiltonian H CEF = B O + B O + B O , in which B ni are theCEF parameters and O ni are the Stevens equivalentoperators [3]. The trigonal H CEF splits the j = sixfolddegenerate state of Ce into three doublets, two ofwhich are a mixture of | m j = ± / > and | ± / > states. In an attempt to capture the low-temperaturemagnetization of CeCd As , here we also include thespin Hamiltonian H spin = ˜ J P J i · J j , in which ˜ J > represents AF interactions between nearest neighborsthat mimic the RKKY interaction, and J i = J ( x ) i ˆ x + J ( y ) i ˆ y + J ( z ) i ˆ z is the total angular momentum operatoron site i. Following Ref. [4], we employ a mean-fieldapproximation ( J i · J j ∼ J · < J > ), which allows thespin Hamiltonian to be written simply as z ˜ JJ · < J > ,with z being the number of nearest neighbors. Ourresults give a CEF doublet hierarchy with the lowestenergy doublet dominated by the | ± / > states. Thisground state doublet gives rise to an g -factor anisotropythat causes the moments to lie in the ab-plane at lowtemperatures, which explains the aforementioned strongmagnetization anisotropy. The two excited doubletsthat would allow the magnetic moment to point outof the ab-plane are separated from the ground stateby 372 K, and 545 K, respectively, and hence are notpopulated at the low temperatures where the AF phaseemerges. They may be responsible for the observedinflection points in χ ( T ) .Considering a CEF plus single interactionHamiltonian does not describe the data well, butincluding two additional mean field antiferromagneticinteractions ( ˜ J = 0 . K and ˜ J = 0 . K) producesa reasonable, although not unique, fit shown aslight green curves in Fig 2(a) with CEF parameters( B = 11 . K, B = − . K, and B = 12 K) ingood agreement with Ref. [3]. The presence of twomean field antiferromagnetic interactions is indicativeof frustration. The fits reproduce the inflection pointsin χ ( T ) − , although suffer in accuracy for the c-axisdirection. This mismatch is likely because | χ ( T ) | isquite small along that direction leading to increasedmeasurement error. This demonstrates that theinflections are due to the CEF doublet hierarchy, andnot impurities as suggested in [2]. The inclusion of theseinteractions does indeed allow an accurate calculationof the M vs. H behavior matching Liu et. al. quitewell for low fields. The saturation of magnetizationand its magnitude at 6 T for the field in the ab-plane isslightly mismatched, but still qualitatively reproducesthe effect. The saturation magnetization of ∼ µ B / Ce in the ab-plane is consistent with the lowestenergy Ce doublet and the results in Ref. [3].Despite the measured susceptibility not beingdescribed by a Curie-Weiss law for the entiretemperature range, it is still possible to locally fit thedata to a Curie-Weiss law at low temperature in orderto extract the effective Weiss temperature. This willprovide an average measure and sign of the interactionstrength. We performed Curie-Weiss fits to the χ for T < K for c-axis, and for
T < K for the ab-plane, where the system is in the ground state doubletand far away from any inflection points. We obtaineffective Weiss temperatures ( Θ ) of -5.1 K and -4.5 K,respectively, which are consistent with previous results[2]. D. Crystal electric field levels
The crystal electric field levels of the j = Ce +3 are doubly degenerate due to Kramer’s theorem forhalf integer spin systems. The levels, | n = 0 , , > in increasing order of energy as expressed as the eigenfunctions of ˆ J z are listed in table I. Our results are alsosimilar to KErSe and CsErSe in which the CEF levelswere measured by powder neutron diffraction [5] TABLE I. The crystal electric field eigen functions, andenergy level relative to the ground state of the Ce +3 obtainedfrom the model as described in the main text. Take note ofthe ± , and ∓ that indicate each level is doubly degeneratein zero field. | > = 0 . | ± / > +0 . | ∓ / > E=0 K | > = 0 . | ± / > +0 . | ∓ / > E=372 K | > = | ± / > E=545 K
E. Zero field entropy
The effects of magnetic frustration become evidentin the heat capacity of CeCd As . The zero-field heatcapacity ( C / T ) and the associated change in entropy( ∆ S ) as a function of temperature are plotted in Fig.3 with a photograph of the sample in the inset. C / T data display a sharp lambda peak at T N = 412 mKindicating a phase transition from the high-temperatureparamagnetic state to a low-temperature AF state. Wealso observe the effect of frustration in the entropychange, ∆ S , plotted as the cyan curve in Fig. 3. By taking the integral of C / T , one obtains ∆ S from 0K to 2.2 K to be nearly 80% of Rln2, whereas thechange in entropy from 0 K to T N for CeCd As isonly approximately 25% of Rln2, which is a signatureof magnetic frustration. For an unfrustrated system,one should recover at least 50% Rln2 of entropy from T = 0 to T N , depending on the symmetry of the system,e.g. Heisenberg, Ising, or XY [6]. This is the case inCe PdGe [7], or the nuclear antiferromagnet He [8].
FIG. 3. Zero-field specific heat and integrated entropy dataas a function of temperature. At the antiferromagnetic phasetransition with T N =412 mK, the entropy is only 25% ofRln2. A photograph of the sample is in the inset. F. Entropy vs. field
The entropy change vs. field for relevant temperatureintegration ranges is shown in Fig. 4. The total entropychange from 0 K to 2.2 K monotonically decreases as afunction of field. In contrast, the entropy change from0 K to T N is nearly constant. Once the T U and T L feature emerge in the C/T the entropy change rapidlydecreases as field increases. The red and purple linesare linear fits to the 0 K to T U and 0 K to T L points,respectively. The entropy change trends towards zero at ∼ ∼ T U and T L , respectively. Thelinear fits interpolate to ∼ Rln(2) and ∼
80 % Rln(2) atzero field for T U and T L , respectively. G. Electrical resistivity
The resistivity ( ρ ) vs. temperature ( T ) is presentedin Fig. 5 and shows semiconducting behavior at mosttemperatures, although no satisfactory fit range for asingle gap semiconductor ( ∼ e − T/ − T ) could be found.Room temperature ρ is approximately 6.6 m Ω -cm. A D S ( R l n2 ) N L U FIG. 4. Entropy change as a function of field for variousintervals of temperature integration. small dip at 140 K can be observed. A similar anomalyin ρ is observed in CeCd P and non-magnetic LaCd P [9] ruling out a magnetic origin. It is likely related toeither a subtle change in phonon modes unique to thecrystal structure these compounds share or a structuraltransition. A log( ρ ) vs. T − / is shown in Fig. 6 withthe green dashed line indicating the location the zerofield T N =412 mK AF transition. The lack of any featureat T N suggests an extremely weak coupling betweencharge carries and the magnetic Ce +3 ion. The residualresistivity as T goes to zero is approximately 23 m Ω -cm. The somewhat linear log( ρ ) vs. T − / behaviorbetween 2.0 K and 0.3 K suggests that variable rangehoping ( ∼ e ( T /T ) . ) may be playing a role at thelowest temperatures, but again the lack of any sizable fitrange makes conclusions difficult. Multiple excitationcurrents were attempted, and no Joule heating effectcould be observed. H. Comparison with CeCd P Finally, we compare our results to those ofisostructural CeCd P , which has been synthesizedrecently in single crystal form [9] by the same groupthat reported the AF transition in CeCd As [10]. First,the magnetic susceptibility of CeCd P is qualitativelyand near quantitatively indistinguishable from both ourand their results on CeCd As , which suggests nearlyidentical local CEF environments. Second, CeCd P orders antiferromagnetically at T N = 410 mK at zerofield, and the transition also increases to 430 mK FIG. 5. Electrical resistivity of CeCd As as a function oftemperature.FIG. 6. Arrenhius plot of CeCd As . under a 1.5 T easy-plane field, although it is importantto emphasize that this field induced increase in T N for Ce triangular lattices can depend sensitively onprecise field direction within the ab-plane as shown inthe case of KCeS [11]. The authors were unable tomeasure heat capacity below 370 mK so no comparisonof the residual heat capacity, the emergence of anycrossover region, nor the extraction of the spin-wavegap could be done, although such a measurement wouldbe worthwhile. In both compounds they obverse amomentary field induced decrease of T N for fields below0.05 T. This is very likely to be domain selectionsince planer stripe magnetic structures in triangularlattices have three degenerate domains in zero field.There are some discrepancies that could play a rolein elucidating the behavior of these systems. TheCeCd As single crystals they made exhibit the samefield induced increase of T N we observed, however theirCeCd As made by flux growth demonstrate electricallyconductive behavior whereas ours made by chemicalvapor transport are semiconducting (see Fig. 5).However, the very low carrier density of their samplessuggests that both compounds likely sit close to a metal-insulator transition. It is self-evident at this pointthat exact nature of electrical conductivity in thesesystems is very sensitive to minor details of how theyare grown and will require further investigation. Thelack of any noticeable difference between our results interms of heat capacity, magnetization, and susceptibilitydemonstrates that the charge carries or lack thereof haveminimal impact on the magnetic degrees of freedom inthese compounds. There are also other closely relatedlanthanide compounds that deserve scrutiny [12]. II. THEORETICAL MODEL
For a typical layered triangular-lattice structure, therelevant point-group symmetry operations are the C (120 ◦ ) rotation around the z axis, C (180 ◦ ) rotationaround each bond, site inversion symmetry I , and twotranslations, T and T along δ and δ , respectively δ δ δ xyz FIG. 7. A sketch of the triangular-lattice layer of magneticions (empty circles) embedded in the octahedra of ligands(black dots) with the primitive vectors. Thick (blue) bondsare between magnetic ions and ion-ligand bonds are the thinsolid (dashed) lines for above (below) the plane. [13], see Fig. 7. These symmetries allow four terms inthe nearest-neighbor Hamiltonian that can be separatedinto bond-independent (
XXZ ) and bond-dependentparts, H = X h ij i (cid:16) H ¯∆ h ij i + H bd h ij i (cid:17) + X h ij i H ¯∆ h ij i ,H ¯∆ h ij i m = J m (cid:16) S xi S xj + S yi S yj + ¯∆ S zi S zj (cid:17) (1) H bd h ij i = 2 J ±± h(cid:16) S xi S xj − S yi S yj (cid:17) ˜ c α − (cid:16) S xi S yj + S yi S xj (cid:17) ˜ s α i + J z ± h(cid:16) S yi S zj + S zi S yj (cid:17) ˜ c α − (cid:16) S xi S zj + S zi S xj (cid:17) ˜ s α i , where ˜ c (˜ s ) α = cos(sin) ˜ ϕ α , the bond angles ˜ ϕ α arethat of the primitive vectors δ α with the x axis, ˜ ϕ α = { , π/ , − π/ } , and spin projections are incrystallographic axes that are tied to the lattice, seeFig. 7. The bond-independent exchange constants J m are J and J for the nearest- and second-nearest-neighbor couplings, respectively. Following previousconsiderations [14, 15], we use a minimal generalizationof the nearest-neighbor model by augmenting it withthe second-nearest-neighbor XXZ term with the sameanisotropy parameter ¯∆ .In an external field, the standard Zeeman coupling H Z = − µ B X i h g ab (cid:16) H x S x + H y S y (cid:17) + g z H z S z i , (2)contain anisotropic g -factors of the pseudo-spins thatreflect the build-up of the ground-state doublets fromthe states of the original J -multiplet of the rare-earthions due to a combined effect of spin-orbit coupling andCEF. The in-plane g -factor is uniform because of thethree-fold symmetry of the lattice [16]. III. PHASES
Fig. 8 shows a section of the classical J ±± – J z ± – J
3D phase diagram of the model (1) for a representativechoice of ¯∆ = 1 and antiferromagnetic J . All couplingsare in units of J and the Hamiltonian is invariantunder J z ± → − J z ± [17]. It is obtained by the energyminimization for the commensurate single- Q states and,thus, ignores more complicated multiple- Q states thatoccur near some of the phase boundaries as discussed inRef. [17]. Since they are unimportant for our presentconsideration we ignore them as well. This phasediagram is essentially the same for the other values ofthe easy-plane, or “XY-like” ≤ ¯∆ ≤ , aside from the ◦ phase extending to somewhat larger values of J z ± [15, 17].As is discussed in Refs. [15, 17], the ◦ phase isfavored by the XXZ part of the Hamiltonian whilethe stripe phases are favored by the bond-dependent J ±± and J z ± as well as by the J term. The stripe J ±± J z± J -0.05-0.1-0.15 0.050.10.2 0.05 0.1 0.150.30.4 0.150.1-0.2 stripe- yz stripe- x ∘ stripe- x stripe- yz -0.25-0.3-0.35 0.2 0.25 FIG. 8. Classical J ±± – J z ± – J phase diagram of the Hamiltonian from the main text with J term for representative ¯∆ = 1 obtained by the energy minimization for the commensurate single- Q states. All couplings in units of J > . phases differ by the mutual orientation of spins andbonds. In the stripe- x phase, favored by the negative J ±± , spins are fully in plane and along one of the bonds.In the stripe- yz phase, spins are perpendicular to oneof the bonds and are tilted out of plane for the non-zero J z ± . There is an obvious three-fold degeneracybetween the stripe states of different orientation of the“ferromagnetic” bonds along δ , , .The key message of Fig. 8 is that there are only threeordered antiferromagnetic phases in the phase diagramof the Hamiltonian for J > , with or without the J -term. For the “Ising-like” ¯∆ > and finite J ±± and J z ± ,stripe phases survive and continue to occupy much ofthe parameter space. In this limit, the XXZ and bond-dependent anisotropic terms also compete, resulting ina transition to a different stripe phase with the spinpointing out of the plane along the z axis. The fullphase diagram for ¯∆ > also contains ferrimagnetic “Y”phase in place of the ◦ phase [18] and, generally, hasa complicated cascade of the field-induced phases [19]. IV. CASE OF
CeCd As The essential empirical facts about CeCd As arethe following. It orders antiferromagnetically at T N ≈ . K. The zero-field specific heat shows activatedbehavior with a sizable excitation gap ∆ ≈ K, which is compatible to an estimate of a characteristicsuperexchange constant that one can infer from theextrapolated Curie-Weiss temperatures ( Θ CW ∼ K)[2].The magnetization field-dependence, M ( H ) , showsa monotonic increase for both in-plane and out-of-plane field direction before reaching a saturationwith some residual Van Vleck slope. Due to astrong easy-plane magnetic anisotropy [3], the actualsaturation in the c -direction is beyond the measuredfield range, which prevents an accurate determinationof the saturation field in that direction. The estimatesof the corresponding in-plane and out-of-plane g -factorsin (2) give a factor of ∼ between them, according toRef. [3].Most importantly, although taken above the orderingtemperature, magnetization curves in both principaldirections show no traces of the plateau-like featuresor of any other phase transitions that are emblematic ofthe triangular-lattice magnets [19, 20]. This lack of theother field-induced phases is also strongly corroboratedby the specific heat field-dependence in the in-planefield, which shows no sign of a closing of the excitationgap before the critical point to the paramagneticstate is reached at H abs ≈ . –4.8 T. Together, thesefacts strongly suggest that the H – T phase diagram ofCeCd As contains a single magnetically-ordered phasethat evolves continuously from the H = 0 state.To substantiate this statement, we show the specificheat vs temperature data collapse for the fields ≤ H . H abs using the rescaling of the temperature by the gap ∆ and an ad hoc multiplicative constant α for C ( T ) /T atdifferent fields, see Fig. 9. The single fit is by the leadingterm in the 2D activated behavior, Ax e − x , where x =∆ /T , see Sec. VIII for details.In addition, the field-dependence of the excitation gapextracted from the specific heat data in the main textdemonstrates a seemingly more subtle, but an essentialfeature. It has a noticeable gradual increase by 20%-60% (depending on the fit) from its zero-field value,followed by a rather abrupt closing upon approachingthe saturation field. This characteristic behavior turnedout to be an important distinguishing hallmark of thefield-induced transformations in the magnetic excitationspectrum.The other significant observations include acharacteristic T behavior of the specific heat atthe field-induced transition to the paramagneticstate and an initial moderate increase of the Néeltemperature vs field that is indicative of a suppressionof critical fluctuations.The key observations that are most important for thesubsequent discussion are the gapped ground state andthe single-phase character of the ordered phase. V. PHASE IDENTIFICATION
Because of the combined SOC and CEF effects,the Hamiltonian has no continuous spin-rotationalsymmetries. Therefore, one should generally expectthat all its ordered phases host gapped spin excitations.However, the 120 ◦ as well as the ferromagnetic statesof the classical model exhibit accidental continuous T /∆ α C / T ∆ = 0.97 K1 T, ∆ = 0.97
K3 T, ∆ = 1.13
K3.5 T, ∆ = 1.19
K4 T, ∆ = 1.13
K4.25 T, ∆ = 0.95
Kfit
FIG. 9. Data collapse of the C ( T ) /T vs T data, where C ( T ) is the specific heat at various fields below H abs , with T scaledwith the gap ∆ and α an ad hoc multiplicative constant.Solid line is the fit by the leading term in the 2D activatedbehavior, Ax e − x , where x = ∆ /T . degeneracies [17]. In simple terms, their ground stateenergies have no contribution from the bond-dependentterms, the orientation of their spin configurations isnot fixed beyond the one dictated by the XXZ term,and their spectra are gapless. The gaps open and spindirections get chosen as a result of a quantum order-by-disorder effect, but the gap magnitude is typically asmall fraction of the exchange constant [21].This consideration suggests the stripe phases asstrong contenders for the ground state of CeCd As as they do allow for a sizable spin-excitation gap.The second argument that will be explored later isthe simplicity of the field-evolution of the zero-fieldstripe phases for the most part of the phase diagram.That is, for a stripe state with a considerable gap,spins continuously tilt in a field until reaching aparamagnetic state without encountering intermediatestates. Lastly, the spectrum structure in the stripephases is both peculiar and characteristic of thesystems with significant frustrating or bond-dependentinteractions. Because of the presence of an accidentaldegeneracy elsewhere in the phase diagram, the spectralminima are not associated with the ordering vector, butare complementary to it [17, 22]. Such a structure has asignificant bearing on the field-induced transformationsin the magnetic excitation spectrum, which, as we willdemonstrate below, are in accord with the experimentalobservations in CeCd As .To be quantitative, we would like to demonstratewhich regions of the phase diagram of the HamiltonianCeCd As can possibly belong. For that, in Fig. 10 wepresent a 2D version of Fig. 8, the J ±± – J z ±
2D phasediagram of the Hamiltonian for a representative choiceof ¯∆ = 0 . , antiferromagnetic J , and J = 0 , whichshows the same three phases, the ◦ and two stripephases. The underlying intensity plot indicates the sizeof the gap in the spin excitation spectrum, all in unitsof J . The two solid white lines on each side show theboundaries on the range of J z ± and J ±± for the choiceof the zero-field gap E gap = 1 K for J varying between1 K and 1.2 K. The two sets of the thin black dashedlines for positive and negative J ±± show the projectionsof the outer boundaries of such ranges from the samephase diagrams but for different values of the XXZ anisotropy, ¯∆ = 0 . and ¯∆ = 1 . , altogether giving asense of how much the model parameters are constrainedby the gap value.Naively, an alternative scenario for the gapped groundstate that may seem to be a much more straightforwardoption is the Ising-like state. One can expect it to occurfor ¯∆ > in the absence of the bond-dependent terms.First, as was mentioned above, the XXZ and bond-dependent anisotropies may compete in this case, sothe stripe phases survive for larger values of | J ±± | ( J z ± ) ,leaving our consideration for them intact. Second, for ¯∆ > the gapless ◦ phase converts into a still gapless -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 J ±± /J J z ± / J ∆ = 0.8 ∘ stripe- yz stripe- x Δ =1 Δ =0.8 FIG. 10. The 2D J ±± – J z ± phase diagram of theHamiltonain for ¯∆ = 0 . , J = 0 , all in units of J > . Phasesare indicated. The intensity plot shows the gap in the spinexcitation spectrum. The white solid lines are for E gap = 1 Kfor J = 1 K and 1.2 K. The thin black dashed lines arethe projections of the outer boundaries of such regions for ¯∆ = 0 . and 1.5. ferrimagnetic “Y” phase, in which coplanar spins forma deformed “Y” with one spin pointing perpendicular tothe plane and two tilted away from it with the mutualangles that differ from ◦ , hence the ferrimagnetism[18]. Since the latter is not observed, and the stateis also gapless, this phase can be ruled out. The lastoption from this “Ising domain” is the “stripe- z ” phase,which occurs at finite J and ¯∆ > [19] and should bestable for a range of J ±± and J z ± . However, there aretwo arguments against CeCd As being in this phase.This phase should exhibit a complicated cascade of thefirst-order spin-flop-like transitions for the field in the c -direction, see Ref. [19], occurring already in the lowenough fields to be be visible in the available finite-temperature M ( H ) data. No traces of such transitionsare seen experimentally. Second, we have also calculatedthe field-dependence of the gap for a representativeset of parameters from this phase, see Sec. VI I. Asopposed to the stripe- x ( yz ) case, the gap behaviorvs field is monotonic and is not compatible with thephenomenology of the CeCd As . VI. STRIPE PHASE
Although the qualitative arguments presented abovestrongly point toward CeCd As being in a stripepart of the phase diagram, the model descriptionrequires defining, or restricting, of five parameters: twosuperexchange constants J and J , XXZ anisotropyconstant ¯∆ , and the bond-dependent terms J ±± and J z ± . A. A minimal model
The ground states and excitation spectra of thestripe phases of the Hamiltonian have been thoroughlyconsidered in zero field [17, 23] and above the saturationfield for both the out-of-plane [14, 23] and in-plane [24]field directions. The high-field description is reasonablystraightforward and gives clear insights into the roleof different terms of the Hamiltonian. In particular, J z ± does not contribute to the spin-wave spectrumin the polarized phase in the out-of-plane field andthe saturation fields H s in any of the three principaldirection, a ( x ) , b ( y ) , or c ( z ) , are also independent ofit. For the in-plane field, the J z ± term does contributeto the spectrum in the polarized state, but its kinematicform is such that it vanishes at all relevant highsymmetry points. For the zero-field spectrum, one canalso verify that adding a J z ± term that is comparableto J ±± does not provide any qualitative changes to it[17]. These insights give a significant reason to neglectthe J z ± term from the consideration altogether.A different qualitative incentive that makes such amove highly desirable is the significantly simpler spin-wave algebra that allows to reduce the rank of thematrices to diagonalize and to derive many expressionsin a simple analytical form. We, therefore, will not resistthis temptation and, in the following, will put J z ± = 0 .The second useful observation [23] is that the linearcombination of the superexchange constants J and J ,in which they enter in the expressions for observablessuch as saturation fields and energy gaps at highsymmetry points as well as into the classical energy ofthe state, is the same: J + J . This means that wecannot constrain them independently and they simplydefine an overall scale of the exchange matrix. Thus,for the following consideration we will simply fix theirratio to a reasonable value that is consistent withthe expectations for localized f-orbitals: J /J = 0 . .We note that completely neglecting J leads to someadditional subtle degeneracies in the spectrum both atzero and at the saturation field that are not generic andare useful to avoid by keeping J /J finite.These choices lead to significant simplifications andleave us with three terms to constrain, J , ¯∆ , and J ±± . B. Stripe domains
For J z ± = 0 , spin configurations in the stripe- x andstripe- yz phases are within the triangular lattice plane( x – y or ab plane), aligned with the bond ( J ±± < )and perpendicular to it ( J ±± > ), respectively, seeFig. 11(a) and (e). Their corresponding energies andexcitations in zero field are fully symmetric with respectto J ±± → − J ±± [23]. As was mentioned in Sec. III,there is a three-fold degeneracy of the stripe states M′ MY (a) b ( y ) z a ( x ) H φ (b) (c)(d) (e) FIG. 11. (a) One of the three domains of the stripe- x phase. (b) Spin configuration of that domain in the H k b field. (c)Brillouin zone with the ordering vectors of the stripe phases Y, M, and M . (d) Two domains of the stripe- x phase with theordering vectors at M and M . (e) Spin configuration of the stripe- yz phase. in zero field, illustrated in Fig. 11(d), which showstwo configurations that are degenerate with the one inFig. 11(a). The domains of all three states should bepresent in a material.There are two principal in-plane field directions, H k b and H k a . Theoretically, one of the three domainswith the spin configuration that is “most transverse” tothe external field will be selected by an infinitesimalfield. For instance, for the stripe- x phase in the H k b field, the domain in Fig. 11(a) is energetically favoredover the domains in Fig. 11(d). In the H k a field, two domains in Fig. 11(d) are preferred over the onein Fig. 11(a) and remain degenerate till the saturationfield. Experimentally, the domain selection often occursat some small but finite field because of the lowersymmetry of the spin system due to lattice distortions,domain surface energy, disorder pinning, and other real-life complications. The former reason is known to takeplace in case of α -RuCl [22] and a small-field transitionhas been observed in CeCd P [9], a material that isrelated to the present case.We also note that for J z ± = 0 , the finite-fieldconsideration of the stripe- x and stripe- yz phases isfully symmetric under the simultaneous change of signof J ±± and switching the roles of H k b with H k a . C. Saturation fields and magnetization
To be specific, we will focus on the case of J ±± < that corresponds to the stripe- x phase and select thefield direction H k b . In this case, the spins cantgradually toward the field as is shown in Fig. 11(b).The corresponding ordering vector for this domain isassociated with the Y-point in the Brillouin zone inFig. 11(c), while the M- and the M -points are referred to as complementary to it. We also remark that thischoice is also the simplest from the analytical point ofview as the canting angle in Fig. 11(b) is the samefor all the spins. For comparison, the choice of H k a for the same J ±± < case would not only lead to acoexistence of two domains from Fig. 11(d), but thereare four distinct spin tilt angles in this case that needto be found numerically.The saturation fields for the principal field directionscan be straightforwardly found from vanishing of thehigh-field spectrum gap at the ordering vector at thetransition [23, 24]. For the in-plane field directions forthe Hamiltonian with the Zeeman term, taking S = , H (T) M / M s H || a H || b ∆ = 1.0, g ab = 2.0 H ( a ) s H ( b ) s FIG. 12. The magnetization M ( H ) in the stripe- x phase at T = 0 for H k a and H k b . Parameters of the Hamiltonianare to match H exps ≈ . T in the b -direction, and g -factorsare as discussed in the text. h ( b ) s = g ab µ B H ( b ) s = 4 (cid:16) J + J − J ±± (cid:17) , (3) h ( a ) s = g ab µ B H ( a ) s = 4 (cid:18) J + J − J ±± (cid:19) , note that H ( b ) s > H ( a ) s since J ±± < . For the stripe- yz case, one needs to switch the sign of J ±± in (3) and H ( b ) s with H ( a ) s .Importantly, despite the uniform in-plane g -factor,the saturation fields are different for the two principalfield directions, with the difference ∆ h s = 2 | J ±± | .This is a general consequence of the bond-dependentinteractions in the anisotropic-exchange materials [22].Note that the recent work on a different Ce-basedtriangular-lattice material, KCeS , has found a clearindication of the splitting of the transition lines in the H – T phase diagram for the two in-plane field directions[11], providing an evidence of the same trend.Another important feature of Eq. (3) is that for thefixed J /J only two parameters of the Hamiltoniandefine H ( a/b ) s , J and J ±± , thus offering a strongconstraint on them from the empirical value of the in-plane H s . For the out-of-plane field, the saturation fieldis given by [23] h ( c ) s = g c µ B H ( c ) s = (cid:16) J + J (cid:17)(cid:16) (cid:17) + 4 | J ±± | , (4)that strongly depends on the XXZ anisotropy ¯∆ .To demonstrate the implications of these results,we complement them with the magnetization field-dependence M ( H ) in the stripe- x phase at T = 0 for the a - and b -directions in Fig. 12 and for the c -direction in Fig. 13, respectively. While for the b - and c -directions the spin tilting is simple and can be obtainedanalytically, for the a -direction the results are obtainedfrom the numerical energy minimization. In all threedirections, the stripe phase continuously deforms untilreaching spin saturation at the corresponding criticalfield H ( a/b/c ) s with no intermediate transitions.The parameters that are used to obtain M ( H ) inFig. 12 and Fig. 13 are discussed in more detail below.At this stage it suffices to say that their choice isdictated by matching experimental values of the zero-field gap and of the in-plane saturation field H exps ≈ . T, with the field assumed to be in the b -direction,and by varying the XXZ anisotropy ¯∆ in Fig. 13.The difference of the saturation fields H ( b ) s and H ( a ) s in Fig. 12 is to be expected from the discussion followingEq. (3), with XXZ anisotropy having no bearing on the M ( H ) curve for the field in the b -direction. Although M ( H ) curves in Fig. 13 are featureless, the strongdependence of H ( c ) s on ¯∆ , also in accord with Eq. (4), isobvious. Unfortunately, the experimental restrictionson the actual value of the saturation field in the c -direction in CeCd As are not tight enough, see [3]. H (T) M / M s ∆ = 0.5 ∆ = 1.5 ∆ = 1.0 H || c, g c = 0.49 H ( c ) s H ( c ) s H ( c ) s FIG. 13. Same as in Fig. 13 for H k c and for several ¯∆ . While the value as low as T as for ¯∆ = 0 . is probablya stretch, it is difficult to narrow down the shown rangeof ¯∆ = 0 . –1.5 more significantly with the available datagiven possible uncertainties in the g -factors. The goodnews is that other quantities of interest that do dependon ¯∆ do so rather insignificantly, as we show below.Lastly, our choices of the g -factors differ somewhat fromthe results suggested in Ref. [3], g ab = 2 . instead of 2.38and g c = 0 . instead of 0.46. These are to reflect slightlylower in-plane and higher out-of-plane initial slopes of M ( H ) in the present study. A smaller g ab -factor alsoseems to be supported by the value of the spin-excitationgap in the field 9 T, much above the saturation. D. Spin-wave theory
The spin-wave spectrum of the stripe phases in zerofield and in fields above the saturation have beenconsidered in the past [17, 23, 24]. Here we develop thelinear spin-wave theory (LSWT) for the field-inducedspin canted state in Fig. 11(b). The per-site classicalenergy is E cl S = (cid:16) J + J (cid:17)(cid:16) − ϕ (cid:17) + 2 J ±± (cid:16) ϕ (cid:17) − g ab µ B H sin ϕ/S, (5)with the canting angle ϕ . The energy minimization gives sin ϕ = H/H ( b ) s ≡ h, (6)see Eq. (3) for H ( b ) s .Some straightforward algebra with thetransformation of spins to a local reference frameand a Fourier transform give the LSWT Hamiltonianfor S = ˆ H (2) = 3 J X k (cid:18) A k a † k a k − B k (cid:16) a k a − k + H . c . (cid:17) (cid:19) , (7)1 (a) (b) E k ( K ) k x Γ MM′ Y Γ MM′ Y E k ( K ) k y k x k y ¯Δ = 1 ¯Δ = 0 FIG. 14. The 3D plots of the H = 0 magnon energy ε k (in Kelvins) from Eq. (10) throughout the Brillouin zone for tworepresentative sets of parameters. High-symmetry k -points are indicated. The XXZ anisotropy is (a) ¯∆ = 1 and (b) ¯∆ = 0 . where A k and B k are given by A k = 83 (1 + α − η ) h + 23 (1 + α )(1 − h ) −
83 (1 − h ) η + ¯∆ γ k + ¯ γ k + h ( γ k − ¯ γ k ) − η (cid:0) γ k − h ( γ k + ¯ γ k ) (cid:1) + α (cid:16) ¯∆ γ (2) k + ¯ γ (2) k + h (cid:16) γ (2) k − ¯ γ (2) k (cid:17)(cid:17) , (8) B k = − ¯∆ γ k + ¯ γ k + h ( γ k − ¯ γ k ) − η (cid:0) γ k − h ( γ k + ¯ γ k ) (cid:1) + α (cid:16) − ¯∆ γ (2) k + ¯ γ (2) k + h (cid:16) γ (2) k − ¯ γ (2) k (cid:17)(cid:17) , where α = J /J , η = J ±± /J , and γ k [¯ γ k ] = 13 (cid:18) cos k x ± k x √ k y (cid:19) ,γ (2) k h ¯ γ (2) k i = 13 (cid:18) cos √ k y ± k x √ k y (cid:19) , (9) γ k = 13 (cid:18) cos k x + cos k x √ k y (cid:19) . The standard Bogolyubov transformation of Eq. (7)yields the magnon energy for ≤ H ≤ H ( b ) s ε k = 3 J q A k − B k . (10)In Fig. 14, we show the 3D plots of the zero-field magnonenergy ε k in the stripe- x phase of Fig. 11(a), throughoutthe Brillouin zone, and for two representative sets ofparameters that are chosen to match the experimentalzero-field gap ∆ expH =0 ≈ K and the saturation fieldof H exps ≈ . T. The main message is that despitea rather drastic difference of the
XXZ anisotropyparameter between Fig. 14(a) and Fig. 14(b), theminima of the spectrum are at the M and M -pointsthat are complementary to the ordering vector of theground state, which is at the Y-point. In general,the structure of the low-energy part of the spectrum israther robust to the parameter choices and consists of a quasi-degenerate region in k -space connecting M andM -points, see also Refs. [17, 23] for unrelated choicesof parameters exhibiting the same pattern. The majordifference between Fig. 14(a) and Fig. 14(b) is that themaximum of the magnon band migrates from the Γ -point to Y-point upon reducing ¯∆ from 1 to 0. E. Gaps
From Eqs. (8) and (10), one can readily obtain theanalytic expressions for the magnon energy gaps at thehigh-symmetry k -points of interest E M = E M p h , with E M = q − J ±± (cid:0) − J ±± + ( J + J ) (cid:0) − ¯∆ (cid:1)(cid:1) , (11) E Y = E Y p − h , with E Y = q h ( b ) s (cid:0) − J ±± + ( J + J ) (cid:0) − ¯∆ (cid:1)(cid:1) , (12)where J ±± < , E M ( Y ) are zero-field gaps, h ( b ) s is fromEq. (3), and h = H/H ( b ) s . The “ordering gap” at the Y-point vanishes at the critical field for a transition toa paramagnetic state, as is expected. On the otherhand, the “accidental gap” at the M-point, which isthe spectrum minimum in zero field, increases with thefield. Clearly, one should expect their crossing at some H < H ( b ) s . F. Parameter sets
Before we proceed with the modeling of the CeCd As spectrum, we need to specify parameters of the modelthat meet the phenomenological criteria for it. Asis discussed above, we have already chosen J z ± = 0 ,fixed J = 0 . J , and assumed J ±± < . For theremaining parameters, J , ¯∆ , and J ±± , we have two2 H/H s E gap ( K ) ∆ = 0.5 ∆ = 1.5 ∆ = 1.0 E Y E M FIG. 15. Minimum of the spectrum ε k (solid lines) and E M ( Y ) energies (dashed lines) vs H/H s for the threeparameter sets. strong constraints, the zero-field gap ∆ expH =0 ≈ K andthe in-plane saturation field H exps ≈ . T, and one “soft”constraint for the out-of-plane saturation field H ( c ) s tobe within 15–30 T window.Since the theoretical value for H ( c ) s is the onlyquantity that strongly depends on ¯∆ , we use the latterconstraint to provide us with the broad bounds it. Then,we choose several reasonable values of ¯∆ and use the twostrong criteria to fix J and J ±± .Roughly speaking, the zero-field gap at the M-pointfrom Eq. (11) fixes J ±± and the in-plane saturation field H ( b ) s from Eq. (3) fixes J . Neglecting J ±± from Eq. (3)and setting ¯∆ = 1 in Eq. (11) yield the estimates J ≈ . K and J ±± ≈ . K. More precise calculations, using E M = ∆ expH =0 with Eq. (11), H ( b ) s = H exps with Eq. (3),and the in-plane g -factors g ab = 2 . and g c = 0 . asdiscussed above, produce three representative sets ¯∆ J J ±± H ( c ) s XXZ anisotropy parameter ¯∆ in the first column. Thelast column shows the out-of-plane saturation field H ( c ) s that corresponds to each set, giving a sense that thechoices of ¯∆ = 0 . and ¯∆ = 1 . are likely to be outliersand ¯∆ = 1 is a reasonable choice. G. Gap vs field results
We are now set to study the spectral properties ofCeCd As . Our Fig. 15 shows the field dependenceof the spectrum minimum for the three parameter setsintroduced above; the energies are in K and the fieldis in units of H ( b ) s . The solid line traces the true“minimal gap” of the spectrum ε k in Eq. (10), while Γ MM′ Y E k ( K ) k x k y ¯Δ = 1 H / H s = 0 . FIG. 16. ε k at H = 0 . H s for the ¯∆ = 1 parameter set. the dashed lines track the energies of the M and Y-points, Eqs. (11) and (12), respectively. The dashedcurves intersect below H s , as anticipated. The overallbehavior of the minimal gap is notable: a gradualincrease followed by a rather abrupt transition to asteep decrease and closing at the critical point, withthe results that only moderately depend on the XXZ anisotropy in the allowed range.The true minimal gap in Fig. 15 has some fieldregion where it is neither at M nor at Y-point, butat the k -points that are intermediate between them,the situation that is illustrated in Fig. 16 for the field H = 0 . H s and for the ¯∆ = 1 parameter set, for whichthe spectrum minimum is nearly degenerate along acontour that includes M, M , and Y-points.A complementary prospective is also offered by themagnon density of states (DoS) in Fig. 17, the quantitythat is directly related to the specific heat. Thisfigure explicitly shows the field-induced spectral weightredistribution due to the gap crossing and Van Hovesingularities associated with the spectrum degeneracies.It suggests that the higher density of states may lead toadditional features in the specific heat. H. Polarized phase
Above the saturation field, spins’ quantization axisaligns with the field direction and the spin-wave algebrasimplifies considerably [23, 24]. For the field H ≥ H ( b ) s in the b -direction, the A k and B k terms in the LSWT3 ω ( K ) H / H s ¯Δ = 1 FIG. 17. Magnon DoS vs
H/H s for the ¯∆ = 1 parameter set. Hamiltonian (7) are given by e A k = 83 (1 + α − η ) h − α )+ (cid:0) (cid:1) (cid:16) γ k + αγ (2) k (cid:17) + 2 ηγ k , (13) e B k = (cid:0) − ¯∆ (cid:1) (cid:16) γ k + αγ (2) k (cid:17) + 2 ηγ k , where α = J /J , η = J ±± /J , h = H/H ( b ) s as before, γ k and γ (2) k given in Eq. (9) and γ k = 13 (cid:18) cos k x − cos k x √ k y (cid:19) . (14)We note that in the case of H k a the expression is thesame up to the change J ±± → − J ±± .The energy spectrum has the same form as in (10) ε k = 3 J q e A k − e B k . (15)At the saturation field, Eqs. (10) and (15) yield the sameresult. The spectrum has a gapless mode at the Y-pointthat has an acoustic character, i.e., ε k ∝ | δ k | , where δ k = k − k Y . This is the behavior that is generallyexpected for the transitions in anisotropic systems, seemore discussion in Sec. VIII.Above the saturation field, the gap at the Y-point reopens, but the spectrum does not experienceany significant transformations aside from a roughlyuniform, Zeeman-like shift of the spectrum as a whole.That is, the Y-point remains a minimum for all H ≥ H ( b ) s with the gap e E Y = r(cid:16)e h − h ( b ) s (cid:17) (cid:16)e h − ( J + J ) (cid:0) (cid:1)(cid:17) , (16) H (T) E gap ( K ) ∆ = 1.0 ∆ = 0.5 ∆ = 1.5 E Y E M FIG. 18. Minimum of the spectrum ε k vs H for the threeparameter sets, and g ab = 2 . . where h ( b ) s is from Eq. (3) and e h = g ab µ B H .In Fig. 18, we present the field-dependence of thespectrum minimum in both regions, H ≤ H ( b ) s and H ≥ H ( b ) s , for the three parameter sets and g ab discussedabove.Our Fig. 19 reproduces Fig. 3 of the main text on thelinear scale. It shows the “minimal gap” together withthe gaps at the M- and Y-points throughout the entirefield regime for the ¯∆ = 1 parameter set and togetherwith experimental data. I. Ising phase gaps
As was discussed earlier, the only feasible alternativeto the gapped stripe state is an Ising-like state that doesnot need bond-dependent terms at all. However, on atriangular lattice, the simplest candidate, the nearest-neighbor-only phase, is a gapless and ferrimagnetic. Itis a relative of the ◦ phase with the plane containingthe triad of spins turned perpendicular to the basalplane and spins forming a deformed “Y” structure witha net magnetic moment [18]. H (T) E gap ( K ) E M E Y E min E Y E M E M E Y ¯ ∆ = 1.0 FIG. 19. Fig. 3 of the main text on the linear scale. H/H s E gap ( K ) ∆ = 1.0 Ising phase E Y E M FIG. 20. Same as Fig. 15 for the Ising-like stripe- z phase. The gapped state is reached with the help of a finite J and is a stripe-like phase with spins pointing alongthe z -axis, referred to as a stripe- z phase. It is knownto exhibit a staircase of the spin-flop transitions startingat low fields in the c -direction [19], the features that arenot observed in the M ( H ) data CeCd As .Nevertheless, we have volunteered to provide a studyof the field-dependence of the spectrum gap in thisphase for the in-plane field. Our Fig. 20 shows the theexcitation energies of the “accidental” and “ordering” Mand Y-points for a representative parameter set fromthis phase: J = 1 . K, J = 0 . J , J ±± = 0 , and ¯∆ = 1 . . This set of parameters matches the zero-field gap value of 1 K and the saturation field value of4.8 T with the same g -factor as before. Having J ±± = 0 does not change the field-dependence of the gaps in thestripe- z phase in any quantitative way. These resultsare also compared to the ones for the ¯∆ = 1 . stripe- x parameter set from Fig. 15.One can see a markedly different behavior of the gapsin the stripe- z phase. The gap at the M-point remainsan absolute minimum of the spectrum for the entire fieldrange, is monotonically decreasing vs field, and vanishesat the critical point together with the gap at theordering vector (Y-point). This is because for the stripe- z state at the full polarization point, all three stripedomains of different orientation are degenerate. There isno level-crossing in the spectrum and no abrupt changein the minimal gap behavior vs field. Needless to say,this is inconsistent with the CeCd As phenomenology. VII. NÉEL TEMPERATURE
We have modified the so-called self-consistentrandom-phase approximation (RPA) to calculate Néelordering temperature and its field-dependence for theselected parameter sets for the stripe state. The self-consistent RPA approach is based on the mean-fielddecoupling of the equations of motion for the spinGreen’s functions and has been recently employed in the context of the anisotropic-exchange systems, seeRefs. [17, 22] for details.In our case, the RPA approach for the orderingtemperature T N needed to be modified to account forthe order parameter corresponding to the component ofthe ordered moment that is transverse to the externalfield. The result is particularly simple T N = 3 J cos ϕN X µ, k A k ε k , (17)where ϕ is the spin canting angle in Fig. 11(b).Our Fig. 21 presents the results of such calculations.The overall shape of the H – T phase diagram is in ageneral accord with the data for CeCd As . However,there are two significant differences. While the RPAmethod offers a significant improvement over the “bare”mean-field values [17], the absolute values of T N are stillat least a factor of two larger than in the experiment.The second crucial difference is the lack of the notableinitial increase in the T N vs field in the RPA resultscompared with the experimental data, with the lattersuggesting a linear slope, δT N ∝ | H | . The observationsin a different Ce-based compound also indicate a similarincrease [11].Both discrepancies are of the same origin. Thequantitative successes and failures of the RPA methodare known [25], but the reason for them is not properlydiscussed. The key issue is that this method is basedon the picture of thermal reduction of the local orderparameter and, thus, is just a glorified Lindemanncriterion for melting of a given spin order. It isquite successful quantitatively in the 2D systems withcontinuous symmetries, in which the ordering happenswhen the 2D correlation length is very large and thefluctuating component into the 3D ordering is small.For all other cases, since it does not include criticalfluctuations into consideration, it fails. We can also H/H s T N ( K ) ∆ = 0.5 ∆ = 1.0 ∆ = 1.5 T NMF
FIG. 21. T N vs field by the self-consistent RPA approach forthe three parameter sets. Mean-field (MF) results for zerofield are shown for comparison. T N (17) are based onthe spectrum ε k from Eq. (10), they can only containterms ∝ H , but not the linear term. Therefore, theorigin of the latter is most likely in the field-inducedsuppression of the critical fluctuations. At larger fields,the order parameter reduction due to spin cantingdominates and the agreement improves. VIII. SPECIFIC HEAT AND VARIOUSASYMPTOTES
At low temperatures, population of spin excitationsis low and can be approximated by a bosonic statistics.This is also a description of such excitations within thespin-wave approach. Then, the specific heat is given by C ( T ) = ∂E∂T = X k (cid:16) ε k T (cid:17) e ε k /T ( e ε k /T − . (18)The results obviously depend on the spectrum ε k andon the dimensionality of the system. We use thisexpression to obtain the leading, asymptotically correctterms in the specific heat T -dependence in 2D forboth gapped and gapless spectra at low temperatures.Since the results are expected to be generic, the Debyeapproximation, that uses the low-energy form of thespectrum and a cut-off Debye momentum, suffices. A. Gapped spectrum
For the gapped spectrum near the minimum, spin-excitation energy can be approximated as ε k ≈ ∆ + J k , (19)with the “kinetic” energy that can be related tothe bandwidth W = Jk D , where k D is the Debyemomentum. In 2D, this yields C ( T ) in the Debyeapproximation C ( T ) = TW Z x m x dx x e x ( e x − , (20)where x = ∆ /T and x m = x + W/T . In the limit T (cid:28) ∆ , T (cid:28) W , a simple algebra gives C ( T ) ≈ TW (cid:18) ∆ T + 2∆ T + 2 (cid:19) e − ∆ /T (21)with the omitted terms of order O (cid:0) e − /T ; e − ∆ /T − W/T (cid:1) .A different, “relativistic” form of the gapped spectrumcan be of interest, especially close to the critical field ε k ≈ p ∆ + J k . (22)Following the same steps as above results in C ( T ) ≈ AT (cid:18) ∆ T + 3∆ T + 6 + 6 T ∆ (cid:19) e − ∆ /T (23) with the leading term coinciding with the one inEq. (21), both corresponding to an activated behaviorof the specific heat with the leading T − d +1 prefactor.These expressions allow to extract the value of thelowest gap in the spectrum from the activated behaviorof C ( T ) . B. Gapless spectrum at the critical point
The field-induced transitions of an antiferromagnetto a saturated, paramagnetic state are common. Forthe systems with the continuous spin symmetries,the magnetic field coupling is to a conserved totalmagnetization and the transition is of the Bose-Einsteincondensation type [26, 27]. In this case, the dispersionrelation of the bosonic excitations at the QCP is ε k ∝ k (dynamical exponent z = 2 ). Above the saturationfield the spectrum is Zeeman-shifted, but otherwiseunmodified.In our case, because of the symmetry-breakinganisotropic terms, magnetization is not a conservedquantity and the transition is of a different universalityclass, necessarily resulting in an acoustic-like ε k ∝ | k | (dynamical exponent z = 1 ), see Ref. [27]. Followingthe derivation that is analogous to the textbook one forphonons, the leading term in the specific heat is a powerlaw C ( T ) ≈ AT (24)with the power d = 2 and A is a constant.Tracking the field-dependent specific heat shouldallow to identify the field value that yields such abehavior at low temperatures. In practice, it may bedifficult to locate such a point exactly, so a different“double asymptotic” expansion may be useful. Oneneeds to consider specific heat (18) for the relativisticdispersion of Eq. (22) in the limit of T (cid:29) ∆ , butstill T (cid:28) W . A straightforward algebra yields a gap-dependent correction to (24) C ( T ) ≈ AT (cid:18) − ∆ αT (cid:19) (25)where α = 12 ζ (3) with ζ (3) ≈ . . This approximationshould be valid down to T ∼ ∆ / .In the main text we presented specific heat data forCeCd As for two fields near the QCP and their fitsusing the asymptotic expressions of Eqs. (25) and (24).The fits suggest that the 4.75 T field is at or veryclose to the QCP as it is well-fit by the T power-law,while the 4.6 T low-temperature data are well-fit by theasymptote in Eq. (25) with a small gap of ∆ = 0 . K.We would like to point out that the success of these fitsmay be fortuitous as the “real” LSWT dispersions fromEq. (10) also contain the non-linear terms that can affectthe asymptotic behavior in this temperature regime.6
C. Other anomalies in the specific heat
In the CeCd As , in the field region between ∼ Tand . T approaching the QCP, specific heat datademonstrates additional feature besides the one that isassociated with the proper phase transition, potentiallysuggesting a sliver of another phase.As was discussed in Sec. VI G, one possible scenarioof the origin of this feature is in the transformation ofthe spin-excitation spectrum, which occurs in a similarfield range and is responsible for additional Van Hovesingularities that may contribute to the specific heat.Unfortunately, the characteristic temperatures of thesefeatures are such that the bosonic approximation forthe specific heat may be unreliable. We, therefore,cannot substantiate this scenario as unbiased numericalmethods are needed. We point out that somefrustrated models on the triangular lattice consistentlydemonstrate a two-peak structure in their specific heatfor some parts of the phase diagram as obtained by exactdiagonalization [28]. 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