Various regimes of quantum behavior in S = 1/2 Heisenberg antiferromagnetic chain with fourfold periodicity
H. Yamaguchi, T. Okubo, K. Iwase, T. Ono, Y. Kono, S. Kittaka, T. Sakakibara, A. Matsuo, K. Kindo, Y. Hosokoshi
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Various regimes of quantum behavior in S = 1/2 Heisenbergantiferromagnetic chain with fourfold periodicity H. Yamaguchi , T. Okubo , K. Iwase , T. Ono , Y. Kono ,S. Kittaka , T. Sakakibara , A. Matsuo , K. Kindo , and Y. Hosokoshi Department of Physical Science,Osaka Prefecture University,Osaka 599-8531, Japan Institute for Solid State Physics,the University of Tokyo,Chiba 277-8581, Japan (Dated: September 29, 2018)
Abstract
We have succeeded in synthesizing single crystals of the verdazyl radical β -2,6-Cl -V [= β -3-(2,6-dichlorophenyl)-1,5-diphenylverdazyl]. The ab initio MO calculation indicated the formation of an S = 1 / PACS numbers: 75.10.Jm . INTRODUCTION Quantum spin systems have been investigated intensively over the last few decades, bothexperimentally and theoretically. They provide unique many-body phenomena caused bytheir strong quantum fluctuations. In previous studies, simple quantum spin systems con-sisting of one or two kinds of interactions have been fully understood. Among such systems,the S =1/2 Heisenberg antiferromagnetic chain (HAFC) is the simplest example, and itsground state is known to be described as a Tomonaga-Luttinger liquid (TLL), which is aquantum critical state with fermionic S = 1/2 spinon excitations [1, 2]. Haldane’s con-jecture in 1983 [3] stimulated studies on HAFC with higher spin values, and subsequentextensive investigations established the presence of the Haldane gap in HAFCs with integerspin values [4, 5]. The S =1/2 antiferromagnetic (AF) spin-ladders have also attracted muchattention as simple quantum spin systems in relation to quantum phase transitions and high- T c superconductors [6–9]. In such gapped spin systems, the field-induced quantum phasetransitions to the gapless TLL is the focus of intensive research towards the understandingof the quantum critical point [10, 11]. Although the ground states in those fundamentalsimple quantum spin systems have been fully understood, more complicated quantum spinsystems with multiple exchange interactions have not been sufficiently investigated, owingto a lack of experimental realizations. It is of great interest and importance to synthesizeand investigate unconventional quantum systems to further our understanding of quantumeffect in magnetic materials.A large number of experiments on inorganic materials forming quantum spin systemshave been reported previously. However, most of them are characterized by large valuesfor the exchange couplings, and these strong magnetic interactions make the observationof magnetic-field-dependent behavior in laboratory experiments difficult. Elemental sub-stitution or application of pressure is required to modulate the magnetic interactions insuch inorganic compounds. On the other hand, we can modify the magnetic interactionsby using simple chemical modifications in organic radical compounds. We focused on averdazyl radical with delocalized π -electron spins extending over the molecule. Owing to alack of high-quality single crystals, most previous studies on verdazyl-based materials havenot focused on their crystal structures, which are vital for gaining understanding of theirmagnetic states. Recently, we established synthetic techniques for preparing high-quality2erdazyl-based single crystals and demonstrated modulations of the magnetic interactionsby using simple chemical modifications [12–15]. This chemical modification method enabledus to synthesize varieties of unconventional magentic materials.In this paper, we report the first model compound of an S = 1 / β -2,6-Cl -V [=3-(2,6-dichlorophenyl)-1,5-diphenylverdazyl] and solved its crystal structure. An ab initio molecular orbital (MO) calculation indicates the formation of an S = 1 / II. EXPERIMENTAL
The synthesis of 2,6-Cl -V, whose molecular structure is shown in Fig. 1(a), was mainlyperformed through a conventional procedure [16]. The recrystallization in acetonitrileyielded crystals of α (green octahedron) and β (green needle) phases, which are definedbased on the values of unit cell volume per number of molecules ( V / Z ). X-ray intensitydata were collected at 296 K using a Rigaku CCD Mercury diffractometer. The structurewas solved by a direct method and was refined by full-matrix least squares techniques usingthe SHELX-97 [17].The magnetic susceptibility and magnetization curves were measured using a commercialSQUID magnetometer (MPMS-XL, Quantum Design) and a capacitive Faraday magnetome-ter with a dilution refrigerator. The experimental results were corrected for the diamagneticcontribution of -3.07 × − emu · mol − , which is determined based on the QMC analysis to bedescribed and close to that calculated by Pascal’s method. The specific heat was measuredwith a commercial calorimeter (PPMS, Quantum Design) by a thermal relaxation methodabove 2.0 K and by an adiabatic method down to 0.35 K. The high-field magnetizationat pulsed magnetic fields of up to about 75 T was measured using a non-destructive pulsemagnet at the Institute for Solid State Physics at the University of Tokyo. All experiments3ere performed using small randomly oriented single crystals with typical dimensions of2.0 × × .The ab initio MO calculations for real molecules are feasible and form a powerful ap-proach for investigating the electric properties of the molecules; these calculations wereperformed using the UB3LYP method as broken-symmetry (BS) hybrid density functionaltheory calculations. All the calculations were performed using the Gaussian 09 programpackage and 6-31G basis sets. The convergence criterion was 10 − hartree. To estimationthe intermolecular magnetic interaction, we applied our J evaluation scheme to multispinsystems that have been studied previously using the Ising approximation [19].Our QMC code is based on the directed loop algorithm in the stochastic series expansionrepresentation [20]. The calculation was performed for a system of 256 spins under theperiodic boundary condition. III. RESULTSA. Crystal structure and magnetic model
The crystallographic data are summarized in Table I [18]. The verdazyl ring ( whichincludes four nitrogen atoms), the upper two phenyl rings, and the bottom 2,6-dichlorophenylring are labeled as R , R , R , and R , respectively, as shown in Fig. 1(a). There arecrystallographically independent molecules, M and M , in the crystal, as shown in Fig. 1(b).It is significant that both molecules are no longer planar, and the dihedral angles of R -R ,R -R , R -R for M and M are about 34 ◦ , 38 ◦ , 89 ◦ and 64 ◦ , 33 ◦ , 77 ◦ , respectively. Weperformed ab initio MO calculations to evaluate quantitatively the dominant intermolecularmagnetic interactions on all molecular pairs within 4.5 ˚A. Consequently, we found that thereare three types of dominating interactions J , J , and J . They are evaluated to be J /k B = 5.6 K, J /k B = 18.7 K, and J /k B = 72.2 K, which are defined in the following eq. (1),and form an S = 1/2 HAFC with a four-fold magnetic periodicity consisting of J - J - J - J interactions along the a -axis, as shown in Fig. 1(c). The MO calculation indicated thatabout 64% of the total spin-density is present on R for both molecules. While R and R have about 15 % ∼
20 % of the relatively large spin-densities for each phenyl ring, the R has a spin-density less than about 1.3 % for both molecules. Therefore, the intermolecular4nteractions are mainly caused by short contacts of N or C related to the R ∼ R rings.The evaluated interactions J , J , and J correspond to interactions between the M -M ,M -M , and M -M molecular pairs, respectively, as shwon in Fig. 1(b). The C-C shortcontact between the M molecules related to J , which is doubled by an inversion symmetry,has a longer distance of 3.70 ˚A than the other two, and thus a relatively weak interactionexist there. Although the C-C contact between M and M molecules related to J has ashort distance of 3.40 ˚A, the small overlap of the π orbitals, which expand perpendicularto the planes, due to a relatively large dihedral angle between the related planes shouldmake the interaction weak. Conversely, the similarly C-C short contact of 3.40 ˚A betweenthe M molecules related to J , which is also doubled by an inversion symmetry, has aparallel stacking of the related planes, and thus the strongest interaction is expected there.Additionally, a weak ferromagnetic interchain interaction of J ′ /k B = -1.3 K was evaluatedbetween the M and M molecules from the MO calculation. Despite the N-C short contactof 3.54 ˚A, only a slight overlap of the π orbitals can be expected because of its almostside-by-side contact, as shown in Fig. 1(d). This interchain interaction forms square latticeperpendicular to the chain direction along the a axis. B. Magnetic susceptibility
Figure 2 shows the temperature dependence of the magnetic susceptibility ( χ = M/H ) at0.5 T. Above 200 K, it follows the Curie-Weiss law, and the Weiss temperature is estimatedto be θ W = -21.4(6) K. We observe a broad peak and shoulder at about 6.5 K and 30 K,respectively. Below 6.5 K, χ decreases with decreasing temperatures, which indicates theexistence of a nonmagnetic ground state separated from the excited states by an energy gap.The temperature dependence of χT shows a gradual change at the shoulder. The two-stepdecrease of χT with decreasing temperatures is expected to be associated with formationsof two types of singlet states originating from different types of exchange interactions. Wecalculated the magnetic susceptibility based on the S = 1/2 HAFC with four-fold magneticperiodicity by using the QMC method. The spin Hamiltonian is expressed as H = J X ij S i · S j + J X kl S k · S l + J X mn S m · S n − gµ B H X i S i , (1)5 ABLE I: Crystallographic data for β -2,6-Cl -VFormula C H Cl N Crystal system MonoclinicSpace group P / na/ ˚A 17.307(6) b/ ˚A 11.662(4) c/ ˚A 18.464(7) β /degrees 95.968(7) V /˚A Z D calc /g cm − α ( λ = 0.71075 ˚A )Total reflections 6469Reflection used 4167Parameters refined 469 R [ I > σ ( I )] 0.0670 R w [ I > σ ( I )] 0.1668Goodness of fit 1.052 where S is the spin operator, g the g -factor, g = 2.00, µ B the Bohr magneton, and H the external magnetic field. We obtained good agreement between the experiment andcalculation by considering the parameters J /k B = 8.6 K, J /k B = 34 K, and J /k B = 77K, as shown in Fig. 2. Comparing the experimental results and the ab initio values, theseparameters can be regarded as consistent with those evaluated from the MO calculation. C. Magnetization curve
Figure 3 shows the magnetization curves at various temperatures. We found a wide range1/2 magnetization plateau from 14 T to 70 T at 1.3 K. The magnetization curves at lower6 a) (d)(b)
ClNC (cid:973) b c RR R R Cl NNNN Cl a (cid:973) (cid:973) (c) J J J J J J J J J J J magnetic unit cellM M M M S =1/2 (cid:973) . M M abc FIG. 1: (color online) (a) The molecular structure of β -2,6-Cl -V. (b) Crystal structure forming achain along the a -axis, and (c) corresponding S = 1/2 HAFC with four-fold magnetic periodicity.(d) Molecular packing associated with the interchain interaction J ′ . Broken lines indicate C-C andN-C short contacts. Hydrogen atoms are omitted for clarity. temperatures exhibit a zero-field excitation gap of about 3.5 T, as shown in the lower insetof Fig. 3. The field derivative of the magnetization curve ( dM/dH ) indicates an asymmetricdouble peak structure associated with the phase boundaries, as shown in the upper inset ofFig. 3. Our QMC calculations at 0.60 and 1.3 K, using the parameters obtained from theanalysis of the magnetic susceptibility, reproduce the experimental results well, includingthe 1/2 magnetization plateau and the zero-field excitation gap, as shown in Fig. 3 and itslower inset. The ground state for this model is considered a singlet with the excitation gap.We discuss the origins of these exotic quantum behaviors of the magnetizations in depthlater. D. Magnetic susceptibility and magnetic specific heat in the low-temperature re-gion
Figures 4(a) and 4(b) show the low-temperature region of the magnetic susceptibility andthe magnetic specific heat in various magnetic fields, respectively. The magnetic specific7 .0120.0100.0080.0060.0040.0020.000 χ ( e m u / m o l ) Tempereture ( K ) 0.350.300.250.200.150.100.050.00 χ T ( e m u K / m o l ) B = 0.5 T experimentcalculation FIG. 2: (color online) Temperature dependence of magnetic susceptibility ( χ = M/H ) and χT of β -2,6-Cl -V at 0.5 T. The solid lines represent the calculated results for the S = 1/2 HAFC withfour-fold periodicity. heats C m are obtained by subtracting the lattice contribution assuming Debye’s T -law as0 . T (J/mol K), which is applicable below about 10 K in ordinary radical compounds.We calculated the magnetic specific heat at zero-field by using QMC method with theobtained parameters and reproduced the experimental result with the broad peak, as shownin Fig. 4(b). We found singular cusplike extremes in the magnetic susceptibility, as shownby the arrows in Fig. 4(a). Such a behavior originates from the Bose-Einstein condensationof magnons, which corresponds to a phase transition to a magnetic ordered state. Themagnetic specific heats consistently exhibit field-induced sharp peaks, which are conclusiveevidence for the phase transition to the ordered state, as shown in Fig. 4(b). We plottedthese specific temperatures in a T - H phase diagram with the double peak fields of dM/dH ,as shown in Fig. 4(c). A dome-like phase boundary appeared in the field-induced region,which is often observed in one-dimensional (1D) gapped spin systems. In the purely 1D case,the dome-like phase boundary is predicted to appear at the crossover temperature to theTLL phase as broad extremes of magnetic susceptibilities [11]. The present phase boundaryis interpreted as a phase transition to the ordered state with the aid of the expected weakferromagnetic interchain interaction. 8 .60 K0.30 K 0.15 K M a gn e ti za ti on ( µ B / f . u . ) d M / d H ( µ B / f . u . / T ) M a gn e ti za ti on ( µ B / f . u . ) experiment calculation FIG. 3: (color online) Magnetization curve of β -2,6-Cl -V at 1.3 K in high magnetic fields. Thelower inset shows those at 0.15, 0.30, and 0.60 K in low-field region, the upper inset shows theirfield derivative dM/dH . The arrows indicate phase transition fields. For clarity, the values ofthe vertical axes have been shifted arbitrarily. The solid lines with open squares represent thecalculated results for the S = 1/2 HAFC with four-fold periodicity. IV. DISCUSSION
Finally, we discuss the ground state of the present complicated spin model for the eval-uated parameters. We focus on two magnetic field regions, one below H / , which is thetransition field to the 1/2-plateau phase, and the other near the saturation field. First,we consider the ground state in the former case. For T ≪ J /k B , two spins connected bythe strongest AF J form a nonmagnetic singlet dimer. However, an effective interactionexists between the remaining spins connected by J through the triplet excited states of thesinglet dimer, as described in several studies on cuprates [21, 22]. The value of the effectiveinteraction J eff is given by J / J from the second-order perturbation treatment of the J term in the spin Hamiltonian. Consequently, the ground state below H / can be regardedas an effective S = 1 / J /k B = 8.6 K and J eff /k B = 7.5K, as shown in Fig. 5(a), where the stronger J /k B form a singlet dimer with the zero-fieldexcitation energy gap. The observed magnetic order at low temperatures is a long-rangeorder of this effective alternating chain with persistent quantum spin fluctuations originat-9 .0180.0160.0140.0120.0100.0080.006 χ ( e m u / m o l ) Temperature (K) T e m p e r a t u r e ( K ) Magnetic Field (T) specific heat magnetic susceptibility magnetization curve
Partial order(c)(b) (a) S m ( J / m o l K ) Temperature (K) C m / T ( J / m o l K ) Temperature (K) calculation
FIG. 4: (color online) Temperature dependence of (a) magnetic susceptibility χ and (b) C m /T of β -2,6-Cl -V at various magnetic fields. For clarity, C m /T for 4.0, 5.0, 6.0, 7.0, 8.0, and 8.5 T havebeen shifted up by 0.15, 0.25, 0.35, 0.45, 0.55, and 0.70 J/mol K , respectively. The arrows indicatephase transition temperatures. The inset shows magnetic entropy at 0 and 8.0 T. (c) Magneticfield versus temperature phase diagram of β -2,6-Cl -V. ing from the singlet dimmer associated with J . Correspondingly, the magnetic entropy S m ,obtained through the integration of C m /T , shows that the change associated with the phasetransition is only about 40% of the total entropy S totalm = R ln 2, as shown in the inset ofFig. 4(b). In the 1/2-plateau phase, the energy gap of the effective alternating chain disap-pears, and the spins connected by J are fully polarized. Therefore, if we express the singletand triplet eigenstates of two spins as | S> = 1 / √ | ↑↓ > − | ↓↑ > ) and | T > = | ↑↑ > , | T > = 1 / √ | ↑↓ > + | ↓↑ > ), | T − > = | ↓↓ > , respectively, we can only consider | T > forspins connected by J and | S> and | T > for spins connected by J near the saturation field10 a) J J J J J J J J J J J J J J J eff (b) J J J J J J J J J J J S eff J z J z J z S eff PolarizedPolarized Singlet J eff FIG. 5: (color online) Effective magnetic models in β -2,6-Cl -V at T ≪ J /k B . (a) The S = 1/2HAF alternating chain formed by J and J eff below H / . Broken ellipses indicate singlet pairsconnected by J . (b) The Ising ferromagnetic chain formed by J z with a weak XY AF interactionin terms of the effective spin S eff near the saturation field. The arrows indicate fully polarizedspins connected by J . at T ≪ J /k B . In such a case, considering the first- and second-order perturbations of the J term in the spin Hamiltonian, the effective spin Hamiltonian H eff is expressed using theeffective spin S eff , whose eigenstates are | ↑ > = | T > and | ↓ > = | S> , as following XXZmodel: H eff = J z X ij S z eff ,i S z eff ,j + J xy X ij ( S x eff ,i S x eff ,j + S y eff ,i S y eff ,j ) − h eff X i S z eff ,i , (2)where J z = − J / J , J xy = 2 J J / J ( J − J ), h eff = gµ B H − J − J / − J (2 J − J ) / J ( J − J ). The evaluated values of J z /k B = -1.9 K and J xy /k B = 0.24 K demonstratethat the ground state can be regarded as an effective Ising ferromagnetic chain with an weakXY AF interaction, as shown in Fig. 5(b). There is a first-order phase transition between | ↓ > and | ↑ > at h eff = 0, which corresponds to that between 1/2-plateau and saturatedphases in the magnetization. In this unique XXZ model, if we assume | J z | < | J xy | , an XY AFchain model becomes effective, and a TLL consisting of the effective spins expected to berealized. 11 . SUMMARY We have succeeded in synthesizing a verdazyl radical crystal of β -2,6-Cl -V. The abinitio MO calculation indicated the formation of an S = 1 / S = 1 / Acknowledgments
We thank T. Shimokawa, T. Tonegawa, and S. Todo for the valuable discussions. Thisresearch was partly supported by KAKENHI (Nos. 24740241, 24540347, and 24340075)DApart of this work was performed under the interuniversity cooperative research programof the joint-research program of ISSP, the University of Tokyo. This work was partiallysupported by the Strategic Programs for Innovative Research (SPIRE), MEXT, and theComputational Materials Science Initiative (CMSI), Japan. Some computations were per-formed using the facilities of the Supercomputer Center, ISSP, The University of Tokyo.Our QMC calculations were carried out using the ALPS application [24]. [1] S. Tomonaga, Prog. Theor. Phys. , 544-569 (1950).[2] J. M. Luttinger, J Math. Phys. , 1154-1162 (1963).[3] F. D. M. Haldane, Phys. Rev. Lett. , 1153 (1983).
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