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Discontinuous Quantum and Classical Magnetic Response of the Pentakis Dodecahedron

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Discontinuous Quantum and Classical Magnetic Response of the Pentakis Dodecahedron Discontinuous quantum and classical magnetic response of the pentakis dodecahedron N. P. Konstantinidis Department of Mathematics and Natural Sciences, The American University of Iraq, Sulaimani, Kirkuk Main Road, Sulaymaniyah, Kurdistan Region, Iraq (Dated: July 6, 2021) The pentakis dodecahedron, the dual of the truncated icosahedron, consists of 60 edge-sharing triangles. It has 20 six-fold and 12 five-fold coordinated vertices, with the former forming a dodec- ahedron, and each of the latter connected to the vertices of one of the 12 pentagons of the dodec- ahedron. When spins mounted on the vertices of the pentakis dodecahedron interact according to the nearest-neighbor antiferromagnetic Heisenberg model, the two different vertex types necessitate the introduction of two exchange constants. As the relative strength of the two constants is varied the molecule interpolates between the dodecahedron and a molecule consisting only of quadran- gles. The competition between the two exchange constants, frustration, and an external magnetic field results in a multitude of ground-state magnetization and susceptibility discontinuities. At the classical level the maximum is ten magnetization and one susceptibility discontinuities when the 12 five-fold vertices interact with the dodecahedron spins with approximately one-half the strength of their interaction. When the two interactions are approximately equal in strength the number of discontinuities is also maximized, with three of the magnetization and eight of the susceptibility. At 1 the full quantum limit, where the magnitude of the spins equals 2 , there can be up to three ground- state magnetization jumps that have the z-component of the total spin changing by ∆Sz = 2, even though quantum fluctuations rarely allow discontinuities of the magnetization. The full quantum case also supports a ∆Sz = 3 discontinuity. Frustration also results in nonmagnetic states inside the singlet-triplet gap. These results make the pentakis dodecahedron the molecule with the largest number of magnetization and susceptibility discontinuities from the quantum to the classical level, taking its size also into account. PACS numbers: 75.10.Hk Classical Spin Models, 75.10.Jm Quantized Spin Models, 75.50.Ee Antiferromag- netics, 75.50.Xx Molecular Magnets I. INTRODUCTION is the smallest molecule associated with noncontinuous magnetic response [17, 18]. Magnetization discontinuities have also been found for extended systems [19{21]. The The antiferromagnetic Heisenberg model (AHM) is addition of higher order isotropic exchange-interaction a prototype of strongly correlated electronic behavior, terms in the Hamiltonian enriches the discontinuous LEC describing interactions between localized spins [1, 2]. magnetic response [22, 23]. Recently theories for the cal- Of special interest are lattices and molecules where in culation of the ground states of classical Heisenberg spin the classical lowest-energy configuration (LEC) antifer- systems have been developed [24{30]. romagnetic interactions are not associated with parallel spins due to competing interactions, something known as The fullerene molecules form a family very closely frustration [3, 4]. Frustration originates in the topology connected to noncontinuous magnetic response, as al- of the lattice or molecule in question, which defines the ready mentioned. They are made of 12 pentagons and n geometry of spin interactions. It has important conse- 2 −10 hexagons, where n is the number of vertices of the quences both at the classical and quantum level includ- molecule, which are three-fold coordinated [31]. Their ing phases without conventional order such as the spin- dual molecules consist of triangles, and have twelve five- liquid phase, singlet excitations inside the singlet-triplet fold coordinated vertices that originate in the fullerene gap, and magnetization plateaus in an external magnetic pentagons, while the rest of the vertices originate in the field added to the AHM [5{9]. fullerene hexagons and have six nearest neighbors. The total number of vertices of the dual fullerene molecules Another important consequence of frustration are dis- n arXiv:2101.06739v2 [cond-mat.str-el] 4 Jul 2021 is N = + 2, while their spatial symmetry is the one of continuities in the LEC magnetization and susceptibility 2 as a function of an external field. This is unexpected as the molecule they are derived from. the AHM lacks any anisotropy in spin space, which is typ- The fullerene molecules associated with ground-state ically the origin of discontinuous magnetic behavior, as quantum magnetization discontinuities belong to the for example in the Ising model. Instead, the origin of the icosahedral Ih spatial symmetry group [11{13], which has jumps is the frustrated connectivity of the interactions the highest number of operations among point symmetry between spins. Ground-state magnetization discontinu- groups, equal to 120 [32]. When placed in an increasing ities are a generic feature of fullerene molecules at the magnetic field the projection of the total spin S along the classical and the full quantum limit, as are discontinu- field axis Sz, except from the standard changes between ities in the magnetic susceptibility at the classical level successive sectors with ∆Sz = 1, presents discontinuities [10{15]. The icosahedron, which is a Platonic solid [16], with ∆Sz = 2 where the lowest-energy state of specific 2 Sz sectors never becomes the ground state in the field. one of the susceptibility for J ∼ 0:5. For J ∼ 1 the For the smallest Ih fullerene molecule, the dodecahedron higher-magnetic-field dodecahedron discontinuities dis- 1 (n = 20), this occurs once for spins with magnitude s = 2 appear, and for a window extending up to J ∼ 1:1 the and twice for s = 1. For bigger Ih fullerenes there is at discontinuities change their character. The most discon- 1 least one discontinuity for s = 2 . tinuous response in this J-range includes three magneti- Molecular ground-state magnetization discontinuities zation and eight susceptibility jumps. For J & 1:1 the in the absence of anisotropy are much more often found effect of frustration diminishes and eventually only a sin- for classical spins, generating an increased interest to gle susceptibility discontinuity survives. The large num- find molecules with jumps when the spins mounted at ber of discontinuities for J . 1:1 is a dramatic change 1 from the single magnetization discontinuity of the small- their vertices have s = 2 . The quantum discontinuities of the Ih-symmetry molecules provide the motivation to est member of the family, the icosahedron. The LEC consider their equally high-symmetric duals. The icosa- magnetic response of the pentakis dodecahedron is the hedron, the dual of the dodecahedron and the smallest most discontinuous to have been found taking also the Ih-symmetry fullerene dual, has N = 12 equivalent ver- size of the molecule into account. tices and no discontinuity for small s [17, 18, 33{41]. The The ground-state magnetization response is also rich next-larger Ih fullerene is perhaps the most well-known in magnetization discontinuities at the full quantum limit 1 among this family of molecules, the truncated icosahe- s = 2 , something which is also in contrast with the icosa- dron (n = 60) [42], which has also been studied within hedron which has no jumps. The discontinuity of the the context of the AHM [10, 12, 43]. Its dual is the pen- dodecahedron (J = 0) [11{13] survives again for J . 1, takis dodecahedron with N = 32 (Fig. 1). It has 20 similarly to the classical case. Further similarities with six-fold coordinated vertices that form a dodecahedron, the classical case are the new discontinuities for lower and 12 five-fold coordinated vertices with each connected magnetic fields. There is a number of them for J . 1, to the vertices of one of the 12 pentagon faces of the and they even involve the Sz = 0 state, in which case the dodecahedron. The pentakis dodecahedron consists of discontinuity leads directly to the Sz = 3 sector, having 60 edge-sharing triangles, the smallest frustrated unit. a strength of ∆Sz = 3. In accordance with the classical Since the five-fold have only six-fold coordinated vertices case, discontinuities occur up to J ∼ 1:1, while for higher as nearest neighbors, while the six-fold coordinated ver- J values they disappear due to the dwindling frustration. tices also have nearest neighbors of their own kind, there The maximum number of discontinuities for a single J is are two geometrically different types of nearest-neighbor three, including the case J = 1 where the two exchange bonds, which are taken to be nonnegative and are asso- constants are equal. Their nature resembles the one of ciated with two different exchange interaction constants the Ih-symmetry fullerenes, where the two lowest-energy in the AHM. states on the sides of the discontinuity are nondegenerate. In this paper, in order to investigate correlations be- However, for small J states with multiple degeneracy are z tween ground-state magnetic response and spatial sym- connected with a magnetization step of ∆S = 2. Frus- tration also results in nonmagnetic excitations inside the metry for dual fullerene molecules of Ih symmetry, and also the existence of classical and quantum magnetic re- singlet-triplet gap when the two exchange constants are sponse discontinuities, the AHM on the pentakis dodec- comparable in strength. ahedron is considered. The ground-state magnetic re- The plan of this paper is as follows: Sec. II introduces sponse is calculated as a function of the relative strength the AHM with an added magnetic field, and Sec. III of the two types of inequivalent nearest-neighbor bonds. discusses the LEC and magnetic response for classical This allows the calculation of the magnetic properties spins. Sec. IV presents the low-energy spectrum and between two well-defined limits. When the exchange in- magnetization response in the quantum-mechanical case. teraction between the five-fold and the six-fold coordi- Finally Sec.
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