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arXiv:2011.12083 · SciPost Physics

The antiferromagnetic $S=1/2$ Heisenberg model on the C$_{60}$ fullerene geometry

Roman Rausch, Cassian Plorin, Matthias Peschke

Published Apr 23, 2021• 24 citations
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Abstract

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 U U . 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 .

DOI
10.21468/scipostphys.10.4.087

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The antiferromagnetic S=1/2S=1/2S=1/2 Heisenberg model on the C60_{60}60​ fullerene geometry
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 U U . 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 nearest-neighbor antiferromagnetic Heisenberg model is considered for spins si=12{s}_{i}=\frac{1}{2}si​=21​ and 1 located on the vertices of the dodecahedron and the icosahedron, which belong to the point symmetry group Ih{I}_{h}Ih​. Taking into account the permutational and spin-inversion symmetries of the Hamiltonian results in a drastic reduction of the dimensionality of the problem, leading to full diagonalization for both clusters. There is a strong signature of the frustration present in the systems in the low-energy spectrum for si=12{s}_{i}=\frac{1}{2}si​=21​, where the first excited states are singlets. Frustration also results in a doubly peaked specific heat as a function of temperature for the dodecahedron. Furthermore, there is a discontinuity in the magnetization as a function of magnetic field for the dodecahedron, where a specific total spin sector never becomes the ground state in a field. This discontinuity is accompanied by a magnetization plateau. For si=1{s}_{i}=1si​=1 both systems again have singlet excitations. The magnetization of the dodecahedron has now two discontinuities in an external field and also magnetization plateaux, and the specific heat of the icosahedron, a two-peak structure as a function of temperature. The similarities between the two systems suggest that the antiferromagnetic Heisenberg model on a larger cluster with the same symmetry, the 60-site cluster, will have similar properties.
Using quantum Monte Carlo techniques, we study the one-band Hubbard model at half filling, on a C60{\mathrm{C}}_{60}C60​ molecule. Spin-spin correlations are measured at couplings U/t=4 and 8, and results are compared with similar studies on square lattices. At the lowest temperatures accessible to our simulations, we find that the spin correlations on the C60{\mathrm{C}}_{60}C60​ molecule tend to vanish at large distances, with a correlation length of 3--4 lattice spacings. Spin-wave calculations in the Heisenberg limit are also reported, in qualitative agreement with Monte Carlo simulations. These results suggest that the ground state of the half-filled Hubbard model on the C60{\mathrm{C}}_{60}C60​ fullerene is spin disordered (defined as having a correlation length smaller than the cluster size), in contrast with the spin-density-wave state that exists on the two-dimensional square lattice. We believe that a ``dimer'' state may properly describe the undoped C60{\mathrm{C}}_{60}C60​ molecule.
We study the spin-1/2 Heisenberg antiferromagnet on a series of finite-size clusters with features inspired by the fullerenes. Frustration due to the presence of pentagonal rings makes such structures challenging in the context of quantum Monte Carlo methods. We use an exact diagonalization approach combined with a truncation method in which only the most important basis states of the Hilbert space are retained. We describe an efficient variational method for finding an optimal truncation of a given size which minimizes the error in the ground-state energy. Ground-state energies and spin-spin correlations are obtained for clusters with up to 32 sites without the need to restrict the symmetry of the structures. The results are compared to full-space calculations and to unfrustrated structures based on the honeycomb lattice. \textcopyright{} 1996 The American Physical Society.
The magnetic and thermodynamic properties of spin-1∕21∕21∕2 Heisenberg diamond chains are investigated in three different cases: (a) J1{J}_{1}J1​, J2{J}_{2}J2​, J3>0{J}_{3}>0J3​>0 (frustrated), (b) J1{J}_{1}J1​, J3<0{J}_{3}<0J3​<0, J2>0{J}_{2}>0J2​>0 (frustrated), and (c) J1{J}_{1}J1​, J2>0{J}_{2}>0J2​>0, J3<0{J}_{3}<0J3​<0 (nonfrustrated), where the bond coupling Ji{J}_{i}Ji​ (i=1,2,3)>0(i=1,2,3)>0(i=1,2,3)>0 stands for an antiferromagnetic (AF) interaction, and <0<0<0 for a ferromagnetic (F) interaction. The density-matrix renormalization-group (DMRG) technique is invoked to study the properties of the system in the ground state, while the transfer-matrix renormalization-group (TMRG) technique is applied to explore the thermodynamic properties. The local magnetic moments, spin-correlation functions, and static structure factors are discussed in the ground state for the three cases. It is shown that the static structure factor S(q)S(q)S(q) shows peaks at wave vectors $q=a\ensuremath{\pi}∕3$ (a=0,1,2,3,4,5)(a=0,1,2,3,4,5)(a=0,1,2,3,4,5) for different couplings in a zero magnetic field, which, however, in the magnetic fields where the magnetization plateau with m=1∕6m=1∕6m=1∕6 pertains, exhibits the peaks only at q=0q=0q=0, $2\ensuremath{\pi}∕3$, and $4\ensuremath{\pi}∕3$, which are found to be couplings independent. The DMRG results of the zero-field static structure factor can be nicely fitted by a linear superposition of six modes, where two fitting equations are proposed. It is observed that the six modes are closely related to the low-lying excitations of the system. At finite temperatures, the magnetization, susceptibility, and specific heat show various behaviors for different couplings. The double-peak structures of the susceptibility, and specific heat against temperature are obtained, where the peak positions and heights are found to depend on the competition of the couplings. It is also uncovered that the XXZXXZXXZ anisotropy of F and AF couplings leads the system of case (c) to display quite different behaviors. In addition, the experimental data of the susceptibility, specific heat, and magnetization for the compound Cu3(CO3)2(OH)2{\mathrm{Cu}}_{3}{(\mathrm{C}{\mathrm{O}}_{3})}_{2}{(\mathrm{O}\mathrm{H})}_{2}Cu3​(CO3​)2​(OH)2​ are fairly compared with our TMRG results.
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We present an exact diagonalization study on layered J1−J2J_{1}-J_{2}J1​−J2​ antiferromagnetic Heisenberg spin model to examine the role of frustration induced by inplane next-nearest neighbor coupling J2J_{2}J2​, in presence of interlayer antiferromagnetic coupling J⊥J_{\perp}J⊥​. A finite lattice of 24 spins in layered geometry of (4×3)⊕(4×3)(4\times 3)\oplus (4\times 3)(4×3)⊕(4×3) is considered and the resulting Hamiltonian matrix diagonalized using Davidson iterative algorithm to obtain the ground and a few low-lying excited states. The lattice size (242424 spins with Hilbert space dimensionality of 270415627041562704156 in Sztot=0S_{z}^{tot}=0Sztot​=0 subspace) has been kept relatively small because of the large number of runs required to sample J1−J2−J⊥J_{1}-J_{2}-J_{\perp}J1​−J2​−J⊥​ parameter space. Quantities like spin-gap, Shannon entropy, spin-spin correlation, static spin structure-factor, magnetic specific-heat and magnetic susceptibility are calculated for various values of spin-spin coupling parameters. With increase in interlayer coupling, the system is driven to states with long range order and the interval of quantum paramagnetically disordered state, sandwiched between Néel and collinear ordered states, narrows on the scale of inplane frustration parameter.