Experimental Realization of a Quantum Pentagonal Lattice

Geometric frustration, in which competing interactions give rise to degenerate ground states, potentially induces various exotic quantum phenomena in magnetic materials. Minimal models comprising triangular units, such as triangular and Kagome lattices, have been investigated for decades to realize novel quantum phases, such as quantum spin liquid. A pentagon is the second-minimal elementary unit for geometric frustration. The realization of such systems is expected to provide a distinct platform for studying frustrated magnetism. Here, we present a spin-1/2 quantum pentagonal lattice in the new organic radical crystal α-2,6-Cl2-V [=α-3-(2,6-dichlorophenyl)-1,5-diphenylverdazyl]. Its unique molecular arrangement allows the formation of a partially corner-shared pentagonal lattice (PCPL). We find a clear 1/3 magnetization plateau and an anomalous change in magnetization in the vicinity of the saturation field, which originate from frustrated interactions in the PCPL.

nitroxide, the π-electron spin density of verdazyl radicals can be delocalized even in non-planar molecular structures. This makes the molecular orbitals (MOs) associated with exchange interactions flexible in shape and enables the design of lattice systems by chemical modification, thereby facilitating the synthesis of new materials forming unconventional lattice systems [18][19][20][21] . Here, we present the successful synthesis of a new verdazyl radical crystal, α-2,6-Cl 2 -V. Ab initio MO calculations indicate the formation of an S = 1/2 quantum pentagonal lattice consisting of one FM and two AFM interactions. Figure 1(b) shows the molecular structure of 2,6-Cl 2 -V. The crystallographic parameters at room temperature are as follows 22 : orthorhombic, space group Fdd2, a = 42.759(3) Å, b = 15.5551 (12) Å, c = 16.6127(13) Å, V = 11049.4(15) Å 3 , Z = 24, R = 0.0346, and R w = 0.0846. Furthermore, there is no indication of a structural phase transition down to around 23 K (see Supplementary Table S1). The central verdazyl ring with four nitrogen atoms and three phenyl rings are labeled R 1 , R 2 , R 3 , and R 4 , respectively [ Fig. 1(b)]. The crystals contain two crystallographically independent molecules, in which the large ionic radius of the Cl atom, introduced at the 2,6-position, induces a relatively large dihedral angle (> 80°) at R 1 -R 3 owing to electrostatic repulsion between the Cl and N atoms [ Fig. 1(b)]. This non-planar structure inhibits molecular stacking and the overlap of singly occupied MOs, resulting in the formation of the PCPL.

Results and Discussion
Ab initio MO calculations on the basis of the crystallographic arrangement at 23 K show that one FM and two AFM exchange interactions, labeled as J 1 , J 2 , and J 3 (see Supplementary Fig. S2), are dominant. They are evaluated as J 1 /k B = − 1.6 K, J 2 /k B = 3.9 K, and J 3 /k B = 3.3 K, which are defined in the Heisenberg spin Hamiltonian given by  = ∑ ⋅ , S S J n i j i j , where Σ , i j denotes the sum over the neighboring spin pairs. The three evaluated interactions form a twisted pentagonal unit consisting of J 1 -J 3 -J 2 -J 3 -J 1 , as shown in Fig. 1(c). The pentagonal units are connected to one another by sharing a corner, resulting in the formation of the S = 1/2 PCPL, as shown in Fig. 1(a). Figure 2(a) shows the temperature dependence of the magnetic susceptibilities (χ = M/H) at 0.1 T. At temperature above 50 K, the Curie-Weiss law is followed, χ = C/(T − θ W ). The estimated Curie constant is about C = 0.362 emu· K/mol, which is close to the expected value for noninteracting S = 1/2 spins, and the Weiss temperature is estimated to be θ W = − 0.85(5) K. Considering the mean-field approximation expressed as θ = − ( + )( + + )/ S S J J J k 2 1 2 2 9 W 1 2 3 B , this small absolute value of θ W indicates a weak internal field due to competition between the FM and AFM interactions. We observe a shoulder and corresponding two-step decrease in χΤ at about 1 K [inset of Fig. 2(a)] , which indicate the contribution of two or more types of AFM interactions. Figure 2(b) shows the temperature dependence of the total specific heat, C p , at zero-field. A clear broad peak appears at about 1.4 K, which is characteristic of a Schottky-like behavior associated with an energy gap between excited states. Consistent with such Schottky-like C p behavior, a clear 1/3 magnetization plateau is observed from 0.4 to 1.3 T, indicating an energy gap between the excited states, as shown in Fig. 4. The six spins in the unit cell and the 1/3 magnetization plateau suggest the existence of a nonmagnetic singlet state with an excitation energy gap formed by the AFM interactions J 2 and/or J 3 between the β sites. If such is the case, residual α-site spins interact with one another through the excited states of the singlet state as discussed later. The magnetization curve indeed increases less steeply than the Brillouin function for free S = 1/2 spins up to the plateau phase, highlighting the presence of non-negligible AFM internal fields. In this context, a sharp peak of the specific heat observed at about 0.07 K is explained by a zero-field phase transition to long-range AFM order of the α-site spins. In the 1/3 plateau phase, these spins are fully polarized along the external field direction such that the ordered state disappears. The phase transition actually disappears in the specific heat at 1.0 T, while the Schottky-like behavior is shifted to around 0.5 K in response to the decrease of the energy gap, as shown in Fig. 3.  In the case of the higher field region above the plateau phase, the field derivative of the magnetization curve (dM/dH) indicates linear behavior in H near the saturation field from 3.3 to 4.0 T, as shown in Fig. 4. This behavior is in clear contrast to the magnetization curve of conventional quantum spin systems, in which external field suppresses quantum fluctuations, which results in an upward curvature of the magnetization curve and a sharp dH/dH peak at the phase transition to fully polarized state 23,24 . The upturn observed in C p at 4.0 T should therefore be attributed to unconventional magnetic behavior near the saturation field.
In order to identify the contribution of abovementioned AFM interaction, we assumed a simple situation in which the fully polarized α-site spin works only as an internal field / J 2 1 for each β-site spin in the high-field region above the 1/3 magnetization plateau. Accordingly, we considered a lattice system consisting of only J 2 and J 3 with an effective internal field , where H ext is the external magnetic field. In the extreme case where /  J J 1 3 2 , the two spins connected by the J 2 interaction form a nonmagnetic singlet dimer with an excitation energy gap. In gapped cases such as this, the magnetization curve near the critical field-the end of the 1/3 plateau phase in the present case-increases with the square root of the applied field. For /  J J 1 3 2 in contrast, the AFM interaction J 3 forms a well-known S = 1/2 uniform AFM chain with a Tomonaga-Luttinger liquid ground state, and the energy gap disappears. The 1/3 plateau phase and sharp dM/dH peak that are observed here therefore indicate a singlet state arising from the AFM interaction J 2 with /  J J 1 3 2 , as described in Fig. 4. Accordingly, the effective interactions between the residual α-site spins are caused through the triplet excited states of the J 2 singlet dimer 22,25,26 . They are roughly evaluated from the second-order and third-order perturbation treatment of the J 2 term in the spin Hamiltonian (see Supplementary Information). We can consider that α-site spins form a uniform AFM chain with weak interchain interactions in the low-field region. These interactions should cause the phase transition to long-range AFM order of the α-site spins.
Finally, concerning the unconventional behavior near the saturation field, a plausible explanation is that this reflects a hidden order of spin multipoles caused by correlations between multi-magnon bound states, such as those in a spin-nematic phase 27 . The presence of such a phase is expected in the high-field region (near the saturation field) in S = 1/2 frustrated spin systems with FM interactions 27,28 , but has not been verified experimentally to date. In the PCPL, the three spins connected by the J 1 FM interaction may stabilize the three-magnon bound state in the vicinity of the saturation field, resulting in multipole spin order. Additional experimental techniques, notably neutron scattering using deuterated samples, should afford a more quantitative description of this field-induced phase and clarify the pentagonal frustration effect.

Methods
We synthesized 2,6-Cl 2 -V using a conventional procedure similar to that used for preparing the typical verdazyl radical 1,3,5-triphenylverdazyl 29 . The crystal structure was determined on the basis of intensity data collected using a Rigaku AFC-8R Mercury CCD RA-Micro7 diffractometer with Japan Thermal Engineering XR-HR10K. The magnetizations were measured using a commercial SQUID magnetometer (MPMS-XL, Quantum Design) and a capacitive Faraday magnetometer down to about 70 mK. The experimental results were corrected for diamagnetic contribution (− 2.57 × 10 −4 ), which is determined to become almost χΤ = const. above about 200 K, and close to the value calculated by Pascal's method. The specific heat was measured using a hand-made apparatus by a standard adiabatic heat-pulse method down to about 50 mK. Considering the isotropic nature of organic radical systems, all experiments were performed using small randomly oriented single crystals. The ab initio MO calculations were performed using the UB3LYP method as broken-symmetry hybrid-density functional theory calculations. All the calculations were performed using the Gaussian 09 program package, and the basis functions used were 6-31G. To estimate the intermolecular magnetic interaction of the molecular pairs within 4.0 Å, we applied our previously presented evaluation scheme 30 .