Cascade of magnetic-field-induced quantum phase transitions in a spin $\bm{1/2}$ triangular-lattice antiferromagnet
N. A. Fortune, S. T. Hannahs, Y. Yoshida, T. E. Sherline, T. Ono, H. Tanaka, Y. Takano
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Cascade of magnetic-field-induced quantum phase transitions in a spin triangular-lattice antiferromagnet N. A. Fortune, S. T. Hannahs, Y. Yoshida, ∗ T. E. Sherline, † T. Ono, H. Tanaka, and Y. Takano Department of Physics, Smith College, Northampton, MA 01063, USA National High Magnetic Field Laboratory, 1800 E. Paul Dirac Dr., Tallahassee, FL 32310, USA Department of Physics, University of Florida, P.O. Box 118440, Gainesville, Florida 32611-8440, USA Department of Physics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551, Japan (Dated: November 8, 2018)We report magnetocaloric and magnetic-torque evidence that in Cs CuBr — a geometricallyfrustrated Heisenberg S = triangular-lattice antiferromagnet — quantum fluctuations stabilize aseries of spin states at simple increasing fractions of the saturation magnetization M s . Only thefirst of these states — at M = M s — has been theoretically predicted. We discuss how the higherfraction quantum states might arise and propose model spin arrangements. We argue that thefirst-order nature of the transitions into those states is due to strong lowering of the energies byquantum fluctuations, with implications for the general character of quantum phase transitions ingeometrically frustrated systems. PACS numbers: 75.30.Kz, 75.40.Cx, 75.50.Ee
Geometric frustration appears in a wide variety ofphysical systems [1, 2, 3]. In a classical system, thisfrustration leads to a large number of states of identicalenergy. Quantum fluctuations can lift this degeneracy,creating classically unexpected ground states and excita-tions.For one of the simplest possible frustrated systems—aHeisenberg antiferromagnet with spins of quantum num-ber S = arranged on a triangular lattice—theory pre-dicts that quantum fluctuations should stabilize a novelup-up-down ( uud ) ground state [4, 5, 6]. Because thiscollinear state preserves the continuous rotation symme-try of the spin hamiltonian, low-energy excitations areseparated from the ground state by energy gaps, result-ing in the ground state of constant magnetization equalto of the saturation magnetization M s over a finitefield range. Experimentally, however, Cs CuBr is theonly known S = triangular-lattice antiferromagnet inwhich this up-up-down state occurs [7, 8, 9]. The sup-pression of this quantum stabilized state with increasingin-plane anisotropy prevents its formation in the isomor-phic compound Cs CuCl [5, 6].The spin hamiltonian for Cs CuBr is given by H = J X −→ S i · −→ S j + J X −→ S i · −→ S k , (1)where J = 11 . b axis and J = 8 . bc plane [10]. Not included in thehamiltonian are two small perturbations expected to bepresent: an antiferromagnetic interlayer coupling thatcauses the spins to order at 1.4 K in zero field, and ananisotropic superexchange interaction (Dzyaloshinskii-Moriya) that causes the spins to lie along the plane ofthe triangular lattice at zero field. The Dzyaloshinskii-Moriya interaction is also likely responsible for the sup- pression of the up-up-down transition in fields appliedalong the a axis (perpendicular to the triangular lattice)in Cs CuBr . In Cs CuCl , each of these is about 5% of J [11].Here we report the complete high-field phase dia-gram of Cs CuBr up to the saturation magnetic field H s = 28 . MM s = , we find a theoreticallyunexpected cascade of additional ordered antiferromag-netic phases at higher fractions of M s .The magnetocaloric experiment employed a minia-ture sample-in-vacuum calorimeter [12] inserted into themixing chamber of a dilution refrigerator. Inside thecalorimeter, the 5.35 mg sample was directly mounted ona 0.5 mm × × µ m ruthenium-oxide resistancethermometer with a minimum amount of nail polish. Thesample and thermometer were weakly thermally linkedvia 25 µ m diameter phosphor-bronze wires to a sapphirering embedded in a 7.0 mm diameter silver platform serv-ing as the thermal reservoir. These wires also served asthe electrical leads to the sample thermometer and me-chanical support for the sample and thermometer.When the magnetic field—produced by a 33 T resistivemagnet and applied along the crystallographic c axis [13]of the sample—is slowly swept up or down, the magne-tocaloric effect produces a temperature difference ∆ T be-tween the sample and the thermal reservoir that dependson the heat capacity C H , the temperature dependence ofthe magnetization ( ∂M/∂T ) H , the field sweep rate ˙ H ,and the weak link’s thermal conductance κ [14]:∆ T = − Tκ (cid:20)(cid:18) ∂M∂T (cid:19) H + C H T d (∆ T ) dH (cid:21) ˙ H. (2)Reversing the field sweep direction reverses the sign FIG. 1: (Color online) (a) Evolution of the temperature dif-ference between the sample and thermal reservoir due tothe magnetocaloric effect at 180 mK, with arrows indicatingthe field-sweep directions. (b) Derivative of magnetic torquewith respect to H at temperatures near 400 mK. To pro-duce a torque, the magnetic field was slightly tilted awayfrom the c axis toward the b axis, by the angle indicatedfor each curve. (c) Magnetic phase diagram deduced fromthe magnetocaloric-effect data taken at various temperatures.Circles indicate second-order phase boundaries, whereas othersymbols except the open diamonds indicate first-order bound-aries. Open diamonds are the positions of the large featuresnear H s and do not indicate a phase boundary. Lines areguides to the eye. Data for H ≤
18 T are from Ref. [10],where open circles are from specific heat. of the temperature difference, thereby revealing thesign and magnitude of ( ∂M/∂T ) H . Transitions be-tween phases appear as deviations from a smoothly vary-ing ∆ T . First-order phase transitions will also revealthe release/absorption of latent heat as the sample en-ters/leaves a lower entropy state. At sufficiently low tem-peratures, there will also be an additional heat release asa metastable state gives way to the lower energy stablestate for both field-sweep directions through a first-ordertransition. M/M =1 / a bdc s M/M =1 / s M/M =5 / s M/M =2 / s FIG. 2: (Color online) Collinear states on the triangular lat-tice at
M/Ms = , , , and . Arrows indicate down spinsantiparallel to the magnetic field. Vertices with no arrowsindicate up spins pointing in the direction of the field, withbroken lines marking rows containing both spins and solidlines marking rows of only up spins. The A phase may resem-ble the M/Ms = state, albeit not collinear. Magnetocaloric-effect measurements can be made us-ing swept fields [14], stepped fields [15], or modulatedfields [16]. The resolution and reproducibility of dc fieldmagnetocaloric measurements have traditionally beenlimited by temperature fluctuations, drift, and slow ther-mal response, all requiring high sweep rates producingadditional heating. In this experiment, we have overcomethese challenges to swept-field measurements through ac-tively stabilizing the temperature of the thermal reservoir(sapphire/silver platform), minimizing the heat capac-ity of the addenda, and reducing the thermal relaxationtime to less than 1 second. The reservoir temperaturewas maintained at a constant true temperature using thealgorithm outlined in Ref. [17] to correct for the magne-toresistance of the sensor.Magnetic phase transitions appear as anomalies in thesample temperature as shown in Fig. 1a. The phase di-agram deduced from our magnetocaloric-effect data isshown in Fig. 1c, along with phase boundaries for fields H ≤
18 T from Ref. [10]. Additional evidence for thisdiagram is provided by the magnetic-torque data shownin Fig. 1b. Even for S = spins, theory has long as-sumed that the field region above the uud phase containsonly one coplanar phase [4], at least for the isotropicHeisenberg hamiltonian. We find instead a remarkablecascade of phases in this field region. The boundariesbetween these ordered phases are nearly vertical, indi-cating that the phase diagram is primarily determinedby the zero-temperature energies, not the entropies, ofdifferent states. We are witnessing a cascade of quantumphase transitions.The uud “plateau” phase appears in the field range12.9 T–14.3 T [7, 8, 9, 10]. Below it is phase I, whichis known to be incommensurate [8, 9, 18]. Above it liesphase IIa, which is also incommensurate but distinct fromphase I [18]. The transitions between the uud phase andphases I and IIa are first-order [8, 9, 10, 19] and thelow-lying excitations in this phase are gapped [10, 19].In the field range 18.8 T–20.4 T, a new phase appears,the A phase. As seen in Figs. 1a and b, the transitions toit from phases IIa and IIb are second-order. Phase IIb,in the field range 20.4 T–22.1 T, may in fact be the sameas phase IIa.The magnetization of the A phase corresponds toroughly of the saturation magnetization but forms noplateau [8], suggesting that this phase is close to beingcollinear but is gapless. One likely arrangement for thenearby collinear state consists of alternating rows of up-down spins and only up spins, as depicted in Fig. 2b,an arrangement predicted to be the M/M s = groundstate of a triangular-lattice ring-exchange model for two-dimensional solid He [20].The most peculiar of all the new phases is the B phase,appearing at 22.1 T and only 70 mT wide. As seen in Fig.1a, the transitions between the B phase and phases IIband III are first-order. Like the A phase, the B phasecan be recognized in retrospect as a small feature in themagnetic induction measured in pulsed magnetic fields[8]. Unlike the A phase feature, however, the featureof the B phase is pointed, suggesting a magnetizationplateau and thus a collinear state with gapped low-lyingexcitations, at approximately of M s .Generalization of Lieb-Schultz-Mattis theorem [21]predicts that any gapped, ordered state must be com-mensurate [22, 23, 24]. Indeed, NMR of Cs showsthat the B phase is commensurate [25]. The collinearityand commensurateness suggest that the B phase may bethe state depicted in Fig. 2c, a repetition of two rows of uud spins and one row of all up spins. Quantum calcula-tions of the energy of this state have not yet been per-formed, but classically, this state is higher in energy thanthe coplanar and canted-spiral, three-sublattice states.Therefore, it is most likely that this new collinear, com-mensurate phase at of M s — like the previously knowncollinear, commensurate phase at of M s — owes its ex-istence to strong quantum fluctuations.Phase III, in the field region 22.1 T–23.1 T, is similar tophases IIa and IIb according to the magnetocaloric effect,implying that it is also incommensurate. The shapes ofthe boundaries between the B phase and phases IIb andIII indicate that phase IIb is higher, whereas phase III islower, in entropy than the B phase.Phase IV directly borders on phase III at a second-order transition line. The boundary between this phaseand the high-temperature, paramagnetic phase extrapo-lates to at most 26 T at zero temperature, well short of H s = 28 . H s .The -magnetization-plateau phase [8] is observed herein the field region 24.5 T–25.0 T. The boundaries be-tween this phase and phases IV and V are first-order.The requirement of collinearity and commensuratenessfor ordered magnetization-plateau states implies that theground state of this phase should be an arrangement suchas shown in Fig. 2d. Exact diagonalization for smallsystems shows that the ground state at M/M s = isindeed collinear, for 0 . . J /J . . state is, like the lower fractional states,higher in energy than the coplanar and canted-spiral,three-sublattice states. Stabilization of the commensu-rate, collinear state observed here appears to implythe existence of large quantum fluctuations capable ofsignificantly lowering the energy of this state below itsclassical expectation.Phase V covers the highest field region up to H s . Inthis phase, the magnetization increases steeply with in-creasing field [8], suggesting very rapid suppression ofquantum fluctuations by the increasing field. The shapeof the transition line between phase V and the param-agnetic phase is unlike all others in Fig. 1c, exhibitingslightly re-entrant behavior at about 25 T. These two fea-tures suggest that the phase is quite different from phasesI–IV. One possibility is that this is a canted spiral phase.Near H s , the sample temperature exhibits a large peakduring an upward field sweep and a deep dip during adownward sweep (Fig. 1a). They indicate a rapid changeof entropy with magnetic field, signifying the emergenceof a magnon energy gap at H s .The unexpected cascade of ordered phases within theantiferromagnetic phase boundary of Cs CuBr is quiteunlike the simple phase diagram of the semiclassical,spin- , triangular-lattice antiferromagnet RbFe(MoO ) [27]. The strong contrast demonstrates that even the sim-plest model of geometrically frustrated antiferromagneticinteractions is much richer than previously imagined,when it is governed by quantum mechanics, with impor-tant implications for many current theoretical models ofsuperconductivity and magnetism.One important implication is, of course, the prospectof new, as yet undiscovered quantum states in such mod-els, but a second is that the transitions to these states arecommonly first order. The transitions to all three gappedphases observed here—the uud phase at M s , the verynarrow B phase at M s , and the additional plateauphase at M s —are first-order. In contrast, theory usu-ally predicts second-order transitions to a magnetization-plateau-forming state with gapped low-lying excitations[4, 28]. As has been pointed out by Alicea et al. [6], how-ever, one possible explanation of their first order charac-ter could be due to the Dzyaloshinskii-Moriya interac- / ( N J S ) eee slope H c slope H c Classical P
M/M S gap gap slope H c slope H c DD slope H P M ee Q Q slope H c slope H c Classical P / ( N J S ) M/M S D gap gap slope H c slope H c D slope H P ee M ee a bdc FIG. 3: (Color online) Ground-state energy E of a frustratedquantum-mechanical Heisenberg antiferromagnet as a func-tion of magnetization M . (a) Macroscopic behavior of E ( M ),exhibiting a cusp at a gapped ground state, P. (b) Micro-scopic, extremely expanded view of the region near P, reveal-ing quantum-mechanically discrete ground states (dots). (c)Macroscopic behavior, when the transitions to state P arefirst-order. (d) Corresponding microscopic view. In a and c,the magnetic fields are in dimensionless units. tion, which introduces a cubic term in the free-energyfunctional.Here we suggest a new, alternative scenario for con-sideration. In general, as illustrated in Fig. 3a, whenquantum fluctuations select state P as a ground statewith energy gaps to the lowest-energy excitations, a cuspmust appear in the ground-state energy E as a functionof magnetization M [29]. These excitations are in factthe two ground states adjacent to P, as shown in Fig.3b. For second-order transitions, the critical fields H c and H c are the two derivatives ∂ E /∂M at P. Over thefield range between H c and H c (Fig. 3b), P remains thelowest state in the “total” energy E −
M H [30], manifest-ing itself as a magnetization plateau. When H is eitherat H c or H c , one of the excitations becomes gapless.We speculate, however, that preferential lowering of Pby quantum fluctuations might produce inflection pointsin the vicinity of P, as shown in Fig. 3c. In that case,the lowest total-energy state will change discontinuouslyat H c and H c from P to Q and Q (defined in Fig.3c). The transitions are now first-order, and are accom-panied by non-vanishing energy gaps, as depicted in Fig.3d. Because this scenario, if verified, relies only on thepresence of quantum fluctuations and not the particularsof the spin-orbit interactions in Cs CuBr , it would beapplicable to a broad range of quantum phase transitionsin geometrically frustrated systems.We thank A. Wilson-Muenchow, T. P. Murphy, J.-H.Park, and G. E. Jones for assistance, and J. Alicea, Y.Fujii, S. Miyahara, and O. Starykh for discussions. This research was supported by an award from Research Cor-poration, a Grant-in-Aid for Scientific Research from theJSPS, the Global CEO Program ‘Nanoscience and Quan-tum Physics’ at Tokyo Tech funded by Monkasho, andthe National High Magnetic Field Laboratory (NHMFL)UCGP program and travel grant. The experiments wereperformed at the NHMFL, which is supported by NSFand the State of Florida. Y.Y. is a JSPS postdoctoralfellow. ∗ Present address: Institute of Applied Physics and Mi-crostructure Research Center, University of Hamburg,Jungiusstrasse 11, D-20355 Hamburg, Germany. † Present address: Neutron Scattering Science Division,Oak Ridge National Laboratory, Oak Ridge, TN 37831,USA.[1] N. P. Ong and R. J. Cava, Science , 52 (2004).[2] D. Heidarian and K. Damle, Phys. Rev. Lett. , 127206(2005), and references therein.[3] J. E. Greedan, J. Mater. Chem. , 37 (2001).[4] A. V. Chubukov and D. I. Golosov, J. Phys.: Condens.Matter , 69 (1991).[5] S. Miyahara, K. Ogino, and H. Furukawa, Physica B , 587 (2006).[6] J. Alicea, A. V. Chubukov and O. A. Starykh, Phys. Rev.Lett. , 137201 (2009).[7] T. Ono et al ., Phys. Rev. B , 104431 (2003).[8] T. Ono et al ., J. Phys.: Condens. Matter , S773 (2004).[9] T. Ono et al ., J. Phys. Soc. Jpn. Suppl. , 135 (2005).[10] H. Tsujii et al ., Phys. Rev. B , 060406(R) (2007).[11] R. Coldea et al ., Phys. Rev. Lett. , 137203 (2002).[12] S. T. Hannahs and N. A. Fortune, Physica B ,1586 (2003).[13] B. Morosin and E. C. Lingafelter, Acta Crystallogr. ,807 (1960).[14] U. M. Scheven, S. T. Hannahs, C. Immer, and P. M.Chaikin, Phys. Rev. B , 7804 (1997).[15] B. Bogenberger et al ., Physica B , 248 (1993).[16] B. McCombe and G. Seidel, Phys. Rev. , 633 (1967).[17] N. Fortune et al ., Rev. Sci. Instrum. , 3825 (2000).[18] Y. Fujii et al ., Physica B , 45 (2004).[19] Y. Fujii et al ., J. Phys.: Condens. Matter , 145237(2007).[20] T. Momoi, H. Sakamoto, and K. Kubo, Phys. Rev. B ,9491 (1999).[21] E. H. Lieb, T. Schultz, and D. J. Mattis, Ann. Phys.(N.Y.) , 407 (1961).[22] M. Oshikawa, Phys. Rev. Lett. , 1535 (2000).[23] G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzin-gre, Eur. Phys. J. B , 167 (2002).[24] M. B. Hastings, Phys. Rev. B , 104431 (2004).[25] Y. Fujii et al ., unpublished.[26] S. Miyahara and N. Furukawa, unpublished.[27] A. I. Smirnov et al ., Phys. Rev. B , 134412 (2007).[28] K. Damle and T. Senthil, Phys. Rev. Lett. , 067202(2006).[29] C. Lhuillier and G. Misguich, in High Magnetic Fields:Applications in Condensed Matter Physics and Spec-troscopy , edited by C. Berthier, L. P. L´evy, and G. Mar-, edited by C. Berthier, L. P. L´evy, and G. Mar-