Quasi-one-dimensional Bose-Einstein Condensation in the Spin-1/2 Ferromagnetic-leg Ladder 3-I-V
Y. Kono, S. Kittaka, H. Yamaguchi, Y. Hosokoshi, T. Sakakibara
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Quasi-one-dimensional Bose-Einstein Condensationin the Spin-1/2 Ferromagnetic-leg Ladder 3-I-V
Y. Kono, ∗ S. Kittaka, H. Yamaguchi, Y. Hosokoshi, and T. Sakakibara Institute for Solid State Physics, the University of Tokyo, Kashiwa 277-8581, Japan Department of Physical Science, Osaka Prefecture University, Osaka 599-8531, Japan (Dated: November 9, 2018)Quantum criticality of the spin-1/2 ferromagnetic-leg ladder 3-I-V [=3-(3-iodophenyl)-1,5-diphenylverdazyl] has been examined with respect to the antiferromagnetic to paramagnetic phasetransition near the saturation field H c . The phase boundary T c ( H ) follows the power-law T c ( H ) ∝ H c − H for a wide temperature range. This characteristic behavior is discussed as a quasi-one-dimensional (quasi-1D) Bose-Einstein condensation, which is predicted theoretically for weakly cou-pled quasi-1D ferromagnets. Thus, 3-I-V provides the first promising candidate for this attractiveprediction. Low dimensionality—one or two dimensions (1D or2D)—plays an essential role in various exotic quantumphenomena such as a Tomonaga–Luttinger liquid (TLL),which is related to 1D spinon excitations [1], and aHaldane state [2] and Kosterlitz-Thouless transition [3],which are related to topological phases. However, inreal quantum magnets, the effects of three-dimensional(3D) interactions are inevitable so that they are always“quasi”low dimensional.Since the universality of a quantum phase tran-sition reflects the space dimensionality of the sys-tem [4, 5], the critical behavior near the quantum crit-ical point (QCP) in a quasi-low-dimensional magnetshould belong to the 3D universality class. A no-tion of Bose-Einstein condensation (BEC) has oftenbeen applied to describe a certain universality classof magnetic-field-induced QCP; 3D XY antiferromag-netic (AFM) ordering can be mapped onto the conden-sate of lattice-gas bosons [6, 7]. Realization of the 3DBEC QCP has been intensively studied in spin dimersystems such as TlCuCl [8] and BaCuSi O [9]. Asquasi-low-dimensional magnets, the spin-1/2 triangular-lattice antiferromagnet Cs CuCl [10], the spin-1/2AFM two-leg ladders (Cu H N) CuBr [11] and(Cu H N) CuBr [12], and the spin-1/2 alternatingAFM spin chain Cu(NO ) · O (copper nitrate) [13]are representative.Moving away from the QCP, the weaker 3D interac-tions are effectively masked, and low-dimensional char-acteristics can arise. One of the attractive issues of thistype of dimensional crossover is the theoretical proposalfor a BEC in quasi-low-dimensional magnets includingferromagnetic (FM) chains (quasi-1D) or planes (quasi-2D); the power law of the critical temperature near thesaturation field H c , T c ( H ) ∼ | H c − H | /φ , can exhibitcrossovers from φ = 3 / φ ≃ XXZ ferromagnet K CuF [15], whichis referred to as one of the candidates in Ref. [14]. Onthe other hand, candidates for the quasi-1D BEC caseremain to be explored.Recently, a new type of quasi-1D quantum magnet hasbeen synthesized using verdazyl radical-based molecules,each of which carries an S = spin—spin-1/2 FM-legladder [16–19]. The Hamiltonian of a typical two-leg spinladder in a magnetic field can be described as H = J || X i,α S i,α · S i +1 ,α + J ⊥ X i S i, · S i, − gµ B H X i,α S zi,α , (1)where J || is the interaction along each leg ( α = 1 , J ⊥ is the rung interaction between the legs, g is the g factor, and µ B is the Bohr magneton (see Fig. 1). The J ┴ J || J diag J J J ii +1 (cid:1) FIG. 1. Schematic of a part of the dominant intermolecularinteractions of 3-I-V, as represented in Ref. [16]. Each spherecorresponds to an S = spin (see the text). FM-leg case corresponds to J || < J ⊥ >
0. Pre-viously, we reported that the 3D BEC exponent wasobserved on one of the FM-leg ladders 3-Br-4-F-V [=3-(3-bromo-4-fluorophenyl)-1,5-diphenylverdazyl] near thelower critical and saturation field [20, 21]. 3-Br-4-F-V is a strong-rung ladder ( | J || /J ⊥ | < | J || /J ⊥ | > T c ( H ), near the saturationfield H c was precisely determined by several experimen-tal methods. The obtained T c ( H ) is in accordance withthe power law T c ( H ) ∝ ( H c − H ) over a wide tempera-ture range. The possibility of quasi-1D BEC is discussedbased on the dominant spin interactions of 3-I-V.Predominant intermolecular interactions of 3-I-V havebeen predicted as a FM-leg spin ladder along the a axisby ab initio molecular orbital (MO) calculations [16,17], and the intraladder interactions were estimated at J || /k B = − . J ⊥ /k B = 5 . J ⊥ = 0 [22], 3-I-V exhibits 3Dordering at zero field, and the 3D ordering phase reaches FIG. 2. Temperature dependence of the magnetic suscepti-bility χ = M/H in several magnetic fields (a) from 0.5 T to3.4 T and (b) from 3.5 T to 5.2 T with 0.1 T steps. Each curveis shifted by +0 .
002 emu/mol for clarity. The arrows indicatethe 3D ordering temperature T c at which χ has a cusp-likeminimum or maximum. the saturation field near 5.5 T [16]. The 3D ordering hasbeen attributed to the frustrated intra and interladdercouplings predicted by the MO calculations; the diago-nal interactions J diag < J > J <
0, and J <
0, which form triangles, as illustratedin Fig. 1. These interactions were estimated at approxi-mately 0 . J ⊥ [16]. Note that MO calculations have beenproven effective to evaluate predominant intermolecularinteractions of verdazyl radical compounds [17, 23–25].To extract the critical exponent φ of 3-I-V near the sat-uration field H c , we performed a precise determinationof the 3D ordering phase boundary using three meth-ods: magnetization, specific-heat, and magnetocaloric-effect (MCE) measurements. Single-crystal samples of3-I-V were synthesized as described in Ref. [17]. dc mag-netization measurements were performed by a force mag-netometer [26] on 2.52-mg randomly oriented samples.Specific-heat measurements were carried out by the stan-dard quasiadiabatic heat-pulse method on the same sam-ples. The MCE was measured by up and down magnetic-field sweepings at 50-80 mT/min with fixed bath temper-atures using a 0.42-mg crystal from the same samples.For all measurements, magnetic fields up to 6 T were ap-plied perpendicular to the a axis (perpendicular to theleg direction).Figure 2 shows the temperature dependence of themagnetic susceptibility χ = M/H in several magneticfields from 0.5 to 5.2 T. There exists a cusplike minimumor maximum in each curve as reported in Ref. [16]. Thiscusplike anomaly is typically observed in model materi-
FIG. 3. (a) Temperature dependence of the heat-capacity C and (b) its temperature derivative dC/dT in several magneticfields from 4.0 T to 5.4 T with 0.1-T steps. Each curve in(a) is shifted by +0 .
25 J mol − K − for clarity. The arrowsindicate the 3D ordering temperature T c at which C showsan inflection just beyond the peak. FIG. 4. (a), (c), and (e) show magnetocaloric-effect curvesat several fixed bath temperatures. (b), (d), and (f) show dT /d ( µ H ) obtained from (a), (c), and (e), respectively. THered or lighter gray (blue or darker gray) curves denote the up-(down-)sweep measurements. als discussed in the context of BEC [8, 15, 27] and ispredicted by theoretical calculations [28, 29]. In the caseof two-leg spin ladder systems, extrema associated witha crossover to the TLL regime have often been observedabove 3D ordering temperatures [13, 30–33], but 3-I-Vhas no such anomalies, which is attributed to the inter-ladder interactions. Thus, the cusplike anomaly indicatesthe position of the 3D ordering temperature T c .In Fig. 3(a), the temperature dependence of thespecific-heat C in several magnetic fields from 4.0 to 5.1 Texhibits a peak anomaly. The upturn behavior at lowtemperatures is attributed to nuclear Schottky contribu-tions from H, I, and N. Although the peak anomalyitself was defined as T c in a previous report [16], an inflec-tion point beyond the peak anomaly of C is a more plausi-ble way to define T c because it shows excellent agreementwith T c as determined from χ ( T ) and the MCE measure-ments. Therefore, we assign the sharp trough in dC/dT to T c as shown in Fig. 3(b).Figures 4(a), 4(c), and 4(e) show the MCE curvesat fixed bath temperatures ∼ ∼
60, and ∼
918 mK,respectively. The up-sweep and down-sweep curves oneach panel appear to be almost vertically symmetricwith each other. Such behavior indicates that the MCEmeasurements were performed under equilibrium condi-
FIG. 5. Phase boundary of the 3D ordering, T c ( H ), deter-mined from the present measurements. The open circles andsquares show T c obtained from χ ( T ) and C ( T ), respectively.Up and down triangles show T c obtained from dT /d ( µ H )of the up- and down-sweep MCE data, respectively. The in-set: magnetic-field dependence of C near the saturation fieldin the present (the closed diamonds) and previous (the opendiamonds, Ref. [16]) measurements. tions [6, 34]. Under these conditions, the first derivativeof T ( H ) exhibits a sharp peak (trough) in the up- (down-) sweep curves as shown in Figs. 4(b), 4(d), and 4(f). Thephase boundary T c ( H ) can be determined from the posi-tions of the peak (trough) anomalies in a similar mannerto that used in the MCE measurements of other systemsunder similar conditions [34–36]. Note that the temper-ature difference between the sample and the bath tem-perature ∆ T ( H ) at the equilibrium conditions behaveslike ∆ T ( H ) = − Tκ ˙ H (cid:18) ∂M∂T (cid:19) H , (2)where κ is the thermal conductivity between the sampleand the bath and ˙ H is the sweep rate [6]. This impliesthat the extrema in χ ( T )[ = M ( T ) /H ] should be associ-ated with the inflection points of the MCE curves. Thisfact supports the agreement of the definitions of T c in the χ ( T ) and MCE results.The 3D ordering temperatures T c ( H ), determinedfrom the present measurements are summarized in Fig. 5.All the definitions of T c are in excellent agreement witheach other, so that they give the exact phase boundaryof 3-I-V. We can also refer to the additional phase dis-cussed in the previous report [16]. In the inset of Fig. 5,the magnetic-field dependence of the specific heat for thepresent sample is compared with previous measurements.The present measurements show only a single sharp peak,different from the broad peak shown in the previous mea-surements. This implies that the broad peak in the pre- FIG. 6. (a) Enlarged plot of the main panel of Fig. 5 nearthe saturation field H c . Solid line is the linear-fitting lineof T c ( H ) from the χ ( T ) data below 1 K. Extrapolating thelinear-fitting line to zero temperature yields H c = 5.536(4) T.(b) Log-log plot of T c ( H ) vs µ ( H c − H ). The solid linescorrespond to T c ( H ) ∝ H c − H . The dotted line correspondsto T c ( H ) ∝ ( H c − H ) / for comparison. vious data may arise from a collapse of the sharp peakrather than an overlap of two phase transitions. Sucha collapse of a peak in specific-heat measurements indi-cates an effect of disorder [37] so that the sharper peakin the present results is attributed to the improvement ofthe sample quality. Moreover, there exist no additionalanomalies in χ ( T ) and C ( T ) near the additional phaseboundary defined previously. Thus, in fact, we concludethat the additional phase does not exist.The quantum criticality of the phase boundary nearthe saturation field is represented in Fig. 6. As indicatedin Fig. 6(a), T c ( H ) from the χ ( T ) data below 1 K is wellreproduced by the linear fitting of the data, yielding thecritical field µ H c = 5 . T c ( H ) vs µ ( H c − H ) for all the data of T c ( H ).It demonstrates that all the definitions of T c below 1 Kare consistent with the line corresponding to the criticalexponent φ = 1 (the solid lines), clearly distinguishedfrom φ = of the 3D BEC case (the dotted line). Thequasi-1D or quasi-2D BEC predicted in Ref. [14] is henceexpected to be realized in 3-I-V.The quasi-1D case is compatible with the spin interac-tions of 3-I-V predicted by the MO calculations. As dis-cussed in Ref. [14], the quasi-1D BEC exponent φ = 1can be found if effective interactions between magnonsare sufficiently small near QCP. In 3-I-V, quasiparticle(magnon) excitation near the QCP ( H c ) could be princi-pally derived from the transition from the triplet to sin-glet state on each rung because the rung interactions J ⊥ are antiferromagnetic. The magnons on each two-leg lad-der interact through the frustrated J - J - J interactionsas described above. Such frustration with the oppositesigns of interactions could suppress the effective interac-tions between those magnons, so that the predominantFM-leg interactions are relatively enhanced to cause thequasi-1D BEC exponent φ = 1. One can suspect thatthe ladder-type interactions cause the 2D characteris-tics, but it is required for the quasi-2D BEC that a 2Dplane consists of ferromagnetic interactions as describedin Ref. [14]. Thus, the quasi-2D BEC case is not suitablefor the present conditions. Crossover to the 3D BEC ex-ponent is not observed in the temperature range of thisstudy although the interladder interactions are on theorder of 0 . J ⊥ . This is also attributed to the effect ofthe frustration such that the crossover would be foundbelow 0.1 K. Since these arguments are only on the ba-sis of the MO calculations, inelastic neutron-scatteringmeasurements would be needed to estimate more accu-rate exchange parameters.To summarize, we have examined the quantum crit-ical phenomena near the saturation field ( H c ) on thespin-1/2 FM-leg ladder 3-I-V. The phase boundary of the3D ordering state near H c was precisely determined bymagnetization, specific-heat, and MCE measurements.All definitions of the 3D ordering temperatures T c arein excellent agreement with each other. The obtainedphase boundary shows the linearity of the power-law T c ( H ) ∝ H c − H below 1 K. The characteristic behav-ior would be caused by quasi-1D BEC with the predom-inant ferromagnetic interactions proposed by Ref. [14],which could be enhanced by the interladder frustration.Thus, spin-1/2 FM-leg ladders are promising for investi-gating the relationship between low dimensionality andBEC physics in quantum magnets.This work was supported in part by KAKENHI GrantsNo. 16J01784, No. 15K05158, No. 17H04850, No.15H03695, and No. 15H03682 from JSPS. The samplepreparation of 3-I-V was performed at Osaka PrefectureUniversity. 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