We solve the quantum-mechanical antiferromagnetic Heisenberg model with spins positioned on vertices of the truncated icosahedron using the density-matrix renormalization group (DMRG). This describes magnetic properties of the undoped C_{60}60 fullerene at half filling in the limit of strong on-site interaction UU. We calculate the ground state and correlation functions for all possible distances, the lowest singlet and triplet excited states, as well as thermodynamic properties, namely the specific heat and spin susceptibility. We find that unlike smaller C_{20}20 or C_{32}32 that are solvable by exact diagonalization, the lowest excited state is a triplet rather than a singlet, indicating a reduced frustration due to the presence of many hexagon faces and the separation of the pentagonal faces, similar to what is found for the truncated tetrahedron. This implies that frustration may be tuneable within the fullerenes by changing their size. The spin-spin correlations are much stronger along the hexagon bonds and exponentially decrease with distance, so that the molecule is large enough not to be correlated across its whole extent. The specific heat shows a high-temperature peak and a low-temperature shoulder reminiscent of the kagomé lattice, while the spin susceptibility shows a single broad peak and is very close to the one of C_{20}20.

The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic nearest-neighbour interactions JJ and antiferromagnetic next-nearest-neighbour interactions J'J′ (within the isotropic Heisenberg model) was previously characterized as a phase with commensurate order. In this paper, we demonstrate that the system in fact exhibits an incommensurate quantum spin spiral. Even though the ground state is translationally invariant in terms of the local spin expectations ‹S_i›‹Si#8250;, the spiral can be detected via the connected spin-spin correlations ‹{S_i\cdot S_j}#8250;‹Si⋅Sj#8250;-‹{S_i}#8250;\cdot‹{S_j}#8250;‹Si#8250;⋅‹Sj#8250; between the apical spins. It has a long wavelength that grows with J'J′ and that soon exceeds finite-system sizes typically employed in numerical simulations. A faithful treatment thus requires the use of state-of-the-art simulations for large, periodic systems. In this work, we are able to accurately treat up to L=400L=400 sites (200 unit cells) with periodic boundary conditions using the density-matrix renormaliztion group (DMRG). Exploiting the SU(2) symmetry allows us to directly compute the lowest-energy state for a given total spin. Our results are corroborated by variational uniform matrix product state (VUMPS) calculations, which work directly in the thermodynamic limit at the cost of a lower accuracy.

We compute ground-state properties of the isotropic, antiferromagnetic Heisenberg model on the sodalite cage geometry. This is a 60-spin spherical molecule with 24 vertex-sharing tetrahedra which can be regarded as a molecular analogue of a capped kagome lattice and which has been synthesized with high-spin rare-earth atoms. Here, we focus on the S=1/2S=1/2 case where quantum effects are strongest. We employ the SU(2)-symmetric density-matrix renormalization group (DMRG). We find a threefold degenerate ground state that breaks the spatial symmetry and that splits up the molecule into three large parts which are almost decoupled from each other. This stands in sharp contrast to the behaviour of most known spherical molecules. On a methodological level, the disconnection leads to ``glassy dynamics’’ within the DMRG that cannot be targeted via standard techniques. In the presence of finite magnetic fields, we find broad magnetization plateaus at 4/5, 3/5, and 1/5 of the saturation, which one can understand in terms of localized magnons, singlets, and doublets which are again nearly decoupled from each other. At the saturation field, the zero-point entropy is S=\ln(182)\approx 5.2S=ln(182)≈5.2 in units of the Boltzmann constant.

The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic nearest-neighbour interactions J and antiferromagnetic next-nearest-neighbour interactions J (within the isotropic Heisenberg model) was previously characterized as a phase with commensurate order. In this paper, we demonstrate that the system in fact exhibits an incommensurate quantum spin spiral. Even though the ground state is translationally invariant in terms of the local spin expectations Si , the spiral can be detected via the connected spin-spin correlations Si • Sj − Si • Sj between the apical spins. It has a long wavelength that grows with J and that soon exceeds finite-system sizes typically employed in numerical simulations. A faithful treatment thus requires the use of state-of-the-art simulations for large, periodic systems.In this work, we are able to accurately treat L = 400 sites (200 unit cells) with periodic boundary conditions using the density-matrix renormaliztion group (DMRG). This is possible due to the exploitation of the SU(2) symmetry which results in effective bond dimensions of more than 1,000,000. Our results are corroborated by variational uniform matrix product state (VUMPS) calculations, which work directly in the thermodynamic limit at the cost of a lower accuracy.