The antiferromagnetic $S=1/2$ Heisenberg model on the ${\text{C}}_{60}$ 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$_{60}$ fullerene at half filling in the limit of strong on-site interaction $U$. 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$_{20}$ or C$_{32}$ 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 kagome lattice, while the spin susceptibility shows a single broad peak and is very close to the one of C$_{20}$.

wavefunction and the matrix-product operator (MPO) representation of the Hamiltonian. 117 The latter can be further reduced using the lossless compression algorithm of Ref. [42]. It 118 gives only a small benefit of 8% reduction for H itself, with the resulting maximal MPO 119 bond dimension of χ (H) = 35 × 32 (from 38 × 35). The benefit for H 2 is larger, yielding 120 χ H 2 = 564 × 468 (reduced from 1444 × 1225, hence by 55%). With these optimizations, 121 the ground state can be found quite efficiently and we can take the variance per site 122 as a global error measure that is immune to local minima. 123 Since DMRG requires a linear chain of sites, we must map the C 60 vertices onto a chain, Thus we conclude that as long as the numbering of the sites is reasonable and more or 140 less minimizes the hopping distances, the dependence on the numbering itself is small 141 and an inaccuracy that results from a suboptimal numbering can simply be compensated 142 by moderately increasing the bond dimension. This is in line with the conclusions of 143 Ummethum, Schnack and Läuchli [29]. Finally, we note that by checking the energy 144 variance (Eq. 2) and the distribution of spin-spin correlations at a given distance (see  Table 1: Properties of the ground state and the lowest eigenstates: total energy E, energy density E/L, the gap to the ground state, the total spin S tot , the full bond dimension χ SU(2) with spin-SU(2) symmetry, the maximal bond dimension of the largest subspace χ sub,SU (2) , the effective bond dimension χ eff that would be required when not exploiting the symmetry, the energy variance per site (Eq. 2), and the overlap with the ground state.

Correlation functions 163
The truncated icosahedron is an Archimedean solid, so that all of its sites (vertices) are 164 equivalent; but since two hexagons and one pentagon come together at a vertex, there 165 are two different nearest-neighbour bonds: one that is shared between the two hexagons 166 and two that run between a pentagon and a hexagon (with the total count of 30 and 167 60, respectively, see Fig. 3 and Fig. 4). We shall call them "hexagon bonds" (H-bonds) 168 and "pentagon bonds" (P-bonds). The wavefunction must respect this geometry, but 169 as the mapping to a chain introduces a bias, this only happens for a sufficiently large 170 bond dimension. Thus, we can average over the respective bonds and take the resulting 171 distribution width as a measure of error, with a δ-distribution expected in the limit of 172 χ → ∞. Figure 1 shows the result for distances up to d = 4, from which we see that for 173 the given bond dimension, the distributions have already become sufficiently sharp.

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Similarly, we have up to five distinct types of bonds for the remaining distances d =   Finally, we also note that for d = 5, 6, 7 the correlations acquire mixed signs and for  gap is actually about as large as for the dodecahedron, but the maximal distance is d = 9 206 and the drop-off across the whole molecule is larger. In this sense, the C 60 spin state is 207 disordered.

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The fullerenes C n have a kind of thermodynamic limit n → ∞, where we expect that 209 the magnetic properties should approach the properties of the hexagonal lattice with Néel 210 order [47], which should be detectable by large spin-spin correlations in a finite system.

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Clearly, we are still far away from that limit: The pentagons disrupt the bipartiteness and 212 lead to a disordered state instead.     . The icosidodecahedron values are according to our own DMRG calculation. The C 60 alternating HP bonds are formed by alternating jumps along H and P (cf. Fig. 3), starting with H; and link two sites within a hexagon up to d = 3. Note that the icosidodecahedron has two inequivalent bonds for d = 2, 3, 4, but the correlation along the second-type bond is very small and is omitted. The weak bond for C 12 at d = 3 is +0.0003 and thus barely visible. The inset shows an exponential fit for the distance dependence of the C 60 spin-spin correlations, either by taking the maximal values for each d or by taking bond-averaged values.
The symmetry should be restored when averaging over the whole degenerate subspace.

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The icosahedral group has the irreducible representations A (1), where a (i) indicates the ancilla site attached to the physical site i. 279 We then apply a propagation in β using the TDVP (time-dependent variational prin-  The TDVP algorithm is known to get stuck in a product state without being able to 290 build up the initial entanglement [21]. We find that this happens whenever the sites i and To strike a balance between accuracy and running time, we can limit the bond dimen-295 sion per subspace to χ sub,SU(2) ∼ 300 − 600, rather than limiting the total bond dimension.

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This ensures that the largest matrix is at most χ sub,SU(2) × χ sub,SU (2) and the duration of 297 the remaining propagation can be estimated. The downside is that the resulting χ SU(2) at 298 each site does not in general correspond to the χ SU (2) lowest singular values and has to be 299 seen as an order-of-magnitude estimate. A benchmark of this approach for the numerically 300 solvable C 20 is given in Appendix A. Table 3 shows the parameters that were used in the 301 thermodynamic calculations.

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The relevant quantities are the partition function the specific heat per site (or per spin): and the zero-field uniform magnetic susceptibility χ sub,SU (2)   where M is the magnetization at a given external field strength B and the Hamiltonian 307 is changed to H → H − B · S, with the total spin S: While the specific heat could be exactly calculated using the squared Hamiltonian average  The result for c (T ) is shown in Fig. 8 and is compared to smaller molecules that exhibit a 313 two-peak structure: For the truncated tetrahedron they are so close to each other that they 314 cannot be resolved, while being distinct for the dodecahedron. For C 60 , we find instead in the region of low-energy states which is absent in the other systems. 321 We recall that for a two-level system given by the Hamiltonian H = diag (0, ∆), the 322 specific heat has a Schottky peak at T /∆ ≈ 0.417. In other words, a maximum appears 323 when the temperature is tuned to the middle of the gap ∆. This is roughly consistent 324 with the gap values given in Tab. 1. The fact that we have a shoulder rather than a clear 325 peak implies that several states of close energy contribute to c (T ), i.e. a comparatively 326 high density of states close to the first excited state. In fact, we can see that as the bond 327 dimension in the DMRG calculation is increased, we are able to better describe the low-328 lying states, leading to a flattening of a very shallow peak to a shoulder. Furthermore, we 329 can say that the states in this vicinity must be singlets or triplets, since the quintet gap

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In the case of a molecule, a fair comparison should in any case be to a finite kagomé 337 plaquette that has a finite-size gap.

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The low-temperature behaviour of the specific heat is consequently also very difficult to is found in a finite system which moves up to merge with the shoulder as the system 342 size is increased [63], while tensor-network calculations directly in the thermodynamic 343 limit show no such peak in the first place [64]. The shoulder below the main peak is 344 remarkably similar to the shape that we obtain for C 60 . We also note that the main 345 peak lies below T = 1 for both C 60 and the kagomé lattice, while it is above T = 1 for   Figure 9 shows the result for the susceptibility χ (T ). It can be interpreted in a similar way, 354 the difference being that singlet states do not contribute anymore. Moreover, it is easy to 355 show that for high temperatures, χ (T ) follows a universal Curie law χ (T ) ∼ 3/4 · T −1 , 356 while for T → 0 we expect χ → 0, since the ground state is a spin singlet and not 357 susceptible to small fields. In between, χ (T ) should have at least one peak. We observe 358 that it is positioned at a higher temperature for the truncated tetrahedron due to the 359 larger singlet-triplet gap (see Tab. 2). The dodecahedron and C 60 , on the other hand, are 360 remarkably close, though χ (T ) tends to be slightly larger for C 60 and does not go to zero 361 as fast for very small temperatures, which we ascribe to the smaller singlet-triplet gap. Néel-like state is prevented by the perturbing pentagonal faces, and one would need larger 381 fullerenes to approach the honeycomb lattice limit.

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In terms of thermodynamics, we find a two-peak structure of the specific heat, similar 383 to what is found for the dodecahedron or the kagomé lattice down to T ∼ 0.1 − 0.2.

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The low-temperature feature is very shallow for C 60 , forming a shoulder, which indicates 385 relatively densely lying singlet and triplet excited states. The spin susceptibility shows a 386 broad peak very similar to the dodecahedron, but approaches zero less rapidly for T → 0.

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All the properties of C 60 are quite different from the icosidodecahedron, which is highly 388 frustrated, with many low-energy singlet states, a non-degenerate first excited singlet and 389 triplet, as well as a three-peak structure in the specific heat. On the other hand, we cannot claim that the finite-temperature results are numerically exact, since a much higher 436 bond dimension may be required to achieve such precision, we expect that the qualitative 437 behaviour should be captured as well.