Discontinuous quantum and classical magnetic response of the pentakis dodecahedron.

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 dodecahedron, and each of the latter connected to the vertices of one of the 12 pentagons of the dodecahedron. 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 quadrangles. 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 the full quantum limit, where the magnitude of the spins equals12, there can be up to three ground-state magnetization jumps that have thez-component of the total spin changing by ΔSz= 2, even though quantum fluctuations rarely allow discontinuities of the magnetization. The full quantum case also supports a ΔSz= 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.

Capped carbon nanotubes with a number of ground state magnetization discontinuities increasing with their size.

The classical ground state magnetic response of fullerene molecules that resemble capped carbon nanotubes is calculated within the framework of the antiferromagnetic Heisenberg model. It is found that the magnetic response depends subtly on spatial symmetry and chirality. Clusters based on armchair carbon nanotubes which are capped with non-neighboring pentagons and have D 5d spatial symmetry have a number of magnetization discontinuities which increases with their size. This occurs even though the model completely lacks magnetic anisotropy, and even though the only source of frustration are the two groups of six pentagons located at the ends of the molecules, which become more strongly outnumbered as the clusters are filled in the middle with more unfrustrated hexagons with increasing size. For the cluster with 180 vertices there are already seven magnetization and one susceptibility discontinuities. Contrary to that, similar molecules which have D 5h spatial symmetry reach a limit of one magnetization and two susceptibility ground state discontinuities, while fullerene molecules based on zigzag carbon nanotubes and capped by neighboring pentagons also reach a fixed number of discontinuities with increasing size.

Influence of the biquadratic exchange interaction in the classical ground state magnetic response of the antiferromagnetic icosahedron.

The icosahedron has a ground state magnetization discontinuity in an external magnetic field when classical spins mounted on its vertices are coupled according to the antiferromagnetic Heisenberg model. This is so even if there is no magnetic anisotropy in the Hamiltonian. The discontinuity is a consequence of the frustrated nature of the interactions, which originates in the topology of the cluster. Here it is found that the addition of the next order isotropic spin exchange interaction term in the Hamiltonian, the biquadratic exchange interaction, significantly enriches the classical ground state magnetic response. For relatively weak biquadratic interaction new discontinuities emerge, while for even stronger the number of discontinuities for this small molecule can go up to seven, accompanied by a susceptibility discontinuity. These results demonstrate the possibility of using a small entity like the icosahedron as a magnetic unit whose ground state spin configuration and magnetization can be tuned between many different non-overlapping regimes with the application of an external field.

Thermalization of a dimerized antiferromagnetic spin chain.

Thermalization is investigated for the one-dimensional anisotropic antiferromagnetic Heisenberg model with dimerized nearest-neighbor interactions that break integrability. For this purpose the time evolution of local operator expectation values after an interacting quench is calculated directly with the Chebyshev polynomial expansion, and the deviation of the diagonal from the canonical thermal ensemble value is calculated for increasing system size for these operators. The spatial and spin symmetries of the Hamiltonian are taken into account to divide it into symmetry subsectors. The rate of thermalization is found to weaken with the dimerization parameter as the Hamiltonian evolves between two integrable limits, the non-dimerized and the fully dimerized where the chain breaks up into isolated dimers. This conclusion is supported by the distribution of the local operator off-diagonal elements between the eigenstates of the Hamiltonian with respect to their energy difference, which determines the strength of temporal fluctuations. The off-diagonal elements have a low-energy peak for small dimerization which facilitates thermalization, and originates in the reduction of spatial symmetry with respect to the non-dimerized limit. For increasing dimerization their distribution changes and develops a single low-energy maximum that relates to the fully dimerized limit and slows down thermalization.

Ground state magnetic response of two coupled dodecahedra.

The antiferromagnetic Heisenberg model on the dodecahedron possesses a number of ground state magnetization discontinuities in a field at the classical and quantum level, even though it lacks magnetic anisotropy. Here the model is considered for two dodecahedra coupled antiferromagnetically along one of their faces, as a first step to determine the magnetic response of collections of fullerene molecules. The magnetic response is determined from the competition among the intra-, interdodecahedral exchange and magnetic field energies. At the classical level the discontinuities of the isolated dodecahedron are renormalized by the interdodecahedral coupling, while new ones show up, with the maximum number of ground state discontinuities being six for a specific range of the coupling. In the full quantum limit where the individual spin magnitude [Formula: see text], there are two ground state discontinuities originating in the single discontinuity of the isolated dodecahedron, and another one due to the intermolecular coupling, generating a total of three discontinuities which come one right after the other. These results show that the magnetic response of more than one dodecahedra interacting together is quite richer than the one of a single dodecahedron.

Thermalization away from integrability and the role of operator off-diagonal elements.

We investigate the rate of thermalization of local operators in the one-dimensional anisotropic antiferromagnetic Heisenberg model with next-nearest neighbor interactions that break integrability. This is done by calculating the scaling of the difference of the diagonal and canonical thermal ensemble values as a function of system size, and by directly calculating the time evolution of the expectation values of the operators with the Chebyshev polynomial expansion. Spatial and spin symmetry is exploited and the Hamiltonian is divided into subsectors according to their symmetry. The rate of thermalization depends on the proximity to the integrable limit. When integrability is weakly broken thermalization is slow, and becomes faster the stronger the next-nearest neighbor interaction is. Three different regimes for the rate of thermalization with respect to the strength of the integrability breaking parameter are identified. These are shown to be directly connected with the relative strength of the low and higher energy difference off-diagonal operator matrix elements in the symmetry eigenbasis of the Hamiltonian. Close to the integrable limit the off-diagonal matrix elements peak at higher energies and high-frequency fluctuations are important and slow down thermalization. Away from the integrable limit a strong low-energy peak gradually develops that takes over the higher frequency fluctuations and leads to quicker thermalization.

Antiferromagnetic Heisenberg model on the icosahedron: influence of connectivity and the transition from the classical to the quantum limit.

The antiferromagnetic Heisenberg model on the icosahedron presents unconventional properties at the classical and quantum level, which originate in the frustrated nature of the interactions between the spins. Here we examine the importance of the connectivity of the icosahedron for the appearance of a magnetization discontinuity as a function of an external field which separates two families of lowest energy configurations. We also investigate the transition from the classical to the quantum limit. The influence of connectivity on the magnetic properties is revealed by considering the cluster as being made up of a closed strip of a triangular lattice with two additional spins attached. The classical magnetization discontinuity is shown to evolve continuously from the discontinuity effected by these two spins when they are uncoupled to the cluster. In the second part the transition from the classical to the quantum limit is examined by focusing on the low energy spectrum, taking fully into account the spatial and the spin symmetry of the model in the characterization of the states. A symmetry analysis of the highly degenerate lowest energy classical manifold identifies as its direct fingerprint the low energy quantum states for spin magnitude as low as s = 1, with the latter following a tower of states behavior which relates to the icosahedron having a structure reminiscent of a depleted triangular lattice. The classical character of the AHM for small s is also detected on the ground state energy and correlation functions. On the other hand the classical magnetization discontinuity in a field eventually disappears for small s, after a weak reentrant behavior.