# C

###### Abstract

The antiferromagnetic Heisenberg Hamiltonian is investigated on a truncated tetrahedron, which is a closed 12 site system. We find that the ground state has many similarities to that of . We study 2- and 4-spin correlations in the classical ground state of the truncated tetrahedron and calculate the same correlations in the exact S=1/2 ground state. We find that the classical correlations survive for a range of bond strengths in the Heisenberg Hamiltonian and that one can construct a good trial wavefunction based on the classical ground state. This suggests that the correlations present in the classical ground state of also survive in the exact ground state of that system, for a range of bond strengths about the physically relevant . We calculate the momentum-space correlation function , which is measurable by neutron scattering, for both and . We also calculate correlations at finite temperature.

###### pacs:

75.10-b, 75.25+z, 75.30.Kz and 75.30.-morrelations for the S=1/2 Antiferromagnet on a Truncated Tetrahedron {instit} Center for Materials Science, MS-K765, Los Alamos National Laboratory, Los Alamos, NM 87545 {instit} Theoretical Division, MS-B262, Los Alamos National Laboratory, Los Alamos, NM 87545 \narrowtext

## 1 Introduction

We have found that an antiferromagnetic Heisenberg Hamiltonian on has an exotic ground state with nontrivial topology.[1] The Heisenberg Hamiltonian arises from a tight-binding Hamiltonian describing the half-filled orbitals at the carbon sites, assuming that there is strong Coulomb repulsion between electrons in these orbitals. The repulsion may be either long or short range. The exotic classical ground state has nontrivial topology, with a skyrmion number 7. (The vector field that is normal to the spin plane of the pentagons covers the sphere 7 times.) is, however, a spin 1/2 system, and quantum corrections are expected to be important. Estimates using perturbation theory indicate that the corrections are about 60% to the ground state energy and about 40% to the magnetization per site. Given the size of these corrections, there is concern that the perturbative results may be misleading.

The truncated tetrahedron is system that is similar to , but is small enough that the problem can be solved exactly.[2] The classical ground state for also has nontrivial topology, with a skyrmion number 1. Here we investigate 2- and 4-spin correlations in the exact ground state of the Heisenberg antiferromagnet on , and compare to correlations in the classical ground state. We find that the classical correlations survive in the exact ground state for a wide range of the parameter , the ratio of the two bond strengths in the system. We also construct an variational wavefunction based on a coherent state representation of the classical ground state spin configuration, and show that it has a large overlap with the exact ground state.

These results suggest that the correlations found in the classical ground state of also survive for for a range of bond strengths. There is expected to be no phase transition in the exact ground state as a function of on-site repulsion in the Hubbard model on the lattice.[1, 3] The spin correlations for the large limit Heisenberg antiferromagnet are therefore expected to be present, although somewhat reduced in magnitude, for the physically relevant case of intermediate .

## 2 Classical Approximation

As in the case of , all carbon atoms on are equivalent, but there are two inequivalent types of bonds. The magnetic exchange constant for two neighboring sites on the same triangle is where is the hopping matrix element and is the Hubbard interaction. The constant connecting a site on one triangle with a nearest neighbor on another triangle is . The Hamiltonian is

(1) |

where the first sum is over the 12 triangle bonds, and the second over the 6 non-triangle bonds. We take =1 and investigate the properties of this Heisenberg Hamiltonian on as a function of . We first treat the Hamiltonian classically, so that in Eq. (1) is a classical unit vector. This is the same as the limit. We discuss the properties of the exact quantum ground state in the next section.

The truncated tetrahedron has 4 triangular faces and 4 hexagonal faces. An antiferromagnet is frustrated on the triangular faces. For an isolated triangle, the ground state energy is per bond, which is smaller than the per bond that an unfrustrated system would have. If the 12 spins on the truncated tetrahedron are able to find an arrangement so that the different triangles do not further frustrate each other, the ground state energy will reach the lower bound . The lower bound is an energy per triangle bond, and per non-triangle bond. The classical ground state configuration was found numerically by minimizing the energy over the spin variables (see Fig. 1). We find that does attain the lower bound , but only in a topologically nontrivial spin arrangement. In comparison, also attains its lower bound, but other fullerenes , , and do not.[1]

The ground state configuration for is the same for all positive and . In addition to the global rotational symmetry of the ground state, there is a discrete parity symmetry whereby each spin . The state related to a ground state by parity cannot be reached by a global rotation on . Although the ground state spin configuration is non co-planar, it may be easily visualized. There is a global rotation such that the spins on each triangle lie in the plane of that triangle. The skyrmion number for a set of vectors defined at points on the surface of a sphere is given by

(2) |

where is the solid angle subtended by three vectors and the sum is over a set of fundamental triangles that cover the sphere. The skyrmion number for and is calculated by introducing “twist” bond variables on the triangle bonds for the vectors , with

(3) |

and one finds that both of the parity-related ground states of have skyrmion number 1. (The skyrmion number cannot be defined directly on the spin variables because the solid angle in Eq. (2) is undefined for many sets of 3 neighboring spins.) The defined by any pair of spins on the same triangle are equal. The spin configuration may be chosen so that the point along the normals to the triangles as shown in Fig. (1). The sum of all is zero. The net magnetic moment in the ground state is also zero.

In the presence of an applied magnetic field, has a second order metamagnetic transition at which is discontinuous. The skyrmion number is 1 both above and below the transition. In contrast, and , both of whose ground states have skyrmion number 7, have a first order metamagnetic transition at which is discontinuous.

## 3 Spin 1/2

For , the matrices in Eq. (1) represent 2 x 2 Pauli spin matrices. (Our convention is to use matrices, which are a factor of 2 larger than matrices.) The Heisenberg Hamiltonian on the truncated tetrahedron was solved in each sector separately.[4] The dimension of the sector is 924 and that of the sector is 792, so that the total spin subspace has dimension . The lowest energy eigenvalue is present in the sector and absent from the sector, so the ground state is a singlet. We calculate various correlation functions as a function of . In , is expected to be nearly equal to, but slightly larger than , because the corresponding non-pentagon bond is about 3% shorter than the pentagon bond.[5] has not been synthesized, but we consider to be the region of physical interest.

First we consider the quantum two-spin correlations . There are five inequivalent ’s on the truncated tetrahedron, which are represented by , , , and , where the subscripts refer to the site labeling in Fig. (1). The ’s, with the exception of , are plotted in Fig. (2). (The classical is zero, and the quantum for all .) The functions and are the triangle and non-triangle bond energies divided by the bond strengths. One sees that as , approaches -1, the value on an isolated triangle. Surprisingly, as , the correlation function remains large and negative. When , there is no correlation between spins on different triangles. Then is averaged over a manifold of degenerate ground states and is equal to zero. For small, however, the expectation is over the unique ground state and is nonzero. In the opposite limit of large , approaches -3 and all other fall to zero. This reflects the fact that in this limit, the wavefunction becomes a product of singlet valence bonds on the 6 non-triangle bonds, denoted . The values of and , the third and fourth nearest neighbor correlation functions, both peak at .

We now calculate correlations between the spin plane normals in the exact ground state. This will involve four-spin correlation functions. As we pointed out in the discussion of the classical ground state, the sum of the spin plane normals for the four triangles is zero, so that classically

(4) |

These correlations may be measured by

(5) |

where

(6) |

The spins on triangle are numbered counterclockwise . (The in Eq. (4) are normalized to unit length, but the in Eqs. (5-6) are not.)

The function measures the relative orientation of the spin-plane normals, and involves correlations between spins over the whole system. It is plotted in Fig. (3). As increases from zero, increases slowly to its maximum at . For large , approaches zero as the wavefunction approaches and all correlations other than those for the singlet on nearest neighbor non-triangle bonds vanish. The four-point function falls off at somewhat smaller values of than the two-point functions in Fig. (2). The quantum correlation functions will be compared with classical and other correlation functions below. Note that the skyrmion number itself is not a good quantum number for the quantum system.

Singh and Narayanan[6] have emphasised that in identifying a nonvanishing expectation value of an operator in the ground state as evidence for magnetic order, it is important to show that the expectation value is considerably larger at zero than at infinite temperature. Note in this regard that and all of the are traceless operators and thus have zero expectation value at .

A classical ground state has each spin pointing in a fixed direction. The most straightforward translation of a classical ground state into a spin 1/2 wavefunction is to have each of the quantum spins point in the same fixed directions. The wavefunction for the first spin is the two-component spinor that is the ground state of , where points in the classical spin direction. These are coherent states. The full wavefunction is

(7) |

Since the classical ground state configuration is independent of , so is the trial wavefunction (and other trial wavefunctions discussed below). The expectation of the spin correlation functions in the trial wavefunction and in the exact groundstate for are shown in Table 1. The exact ground state has large 2-spin and 4-spin correlations, even at large separations. The correlation functions and are identical in and in the classical ground state. The state does reasonably well with all of the correlation functions, including . The largest error is the underestimate of , which has some singlet character in , but can have only Ising character in .

Any global rotation of a classical ground state gives another ground state. Thus a better trial state would be to sum over all global rotations, or equivalently to project it onto the singlet subspace. As mentioned in Section II, the parity operation results in another set of ground states that cannot be reached by rotations. We therefore define states and as the (normalized) singlet projections of the parity related coherent states. An additional related state is found by taking a linear combination of these,

(8) |

where and are complex numbers with . We determine and in Eq. (8) by maximizing . (Virtually identical results are obtained by minimizing the energy.) The amplitudes and are found to have equal magnitudes and a phase difference that is almost independent of . In Fig. (4) we plot the norm of the overlap squared of these states and of with the exact ground state . One sees that and are both peaked near , and that is a good approximation to the ground state when . becomes exact for , but is already failing in the region of physical interest, .

The correlation functions of the trial states , , and are given in Table 1. ( has the same correlation functions as , and is not shown.) The state gives no indication of either the magnitude or sign of any spin correlation function other than . As the variational wavefunction is improved in the sequence , the correlation functions generally approach those of the exact ground state. Convergence is best at short distances and worst for the long distance . By examining Figs. (2-3) and Table 1, one sees that the trial state gives a reasonable approximation to all of the correlation functions not just for , but also as . The agreement is remarkably good considering that zero-point spin-wave fluctuations have not been added to any of the trial states.

We now compare the energy of trial states and that obtained from perturbation theory to the exact ground state energy. High energy contributions from non-singlet states are removed by projecting onto the singlet subspace. The energy of is thus lower than that of . The energy of is -8.803 -12.325 , which is an upper bound on the exact ground state energy. For , this energy is -21.128, which is to be compared with an exact ground state energy of -22.804. From perturbation theory, the leading correction to the ground state energy is

(9) |

where and the sum is over intermediate states with two adjacent spin flips. (The ground state is determined by stability against single spin flips.) For this is

(10) |

The perturbed ground state energy is with . This expression clearly overestimates the correction to the classical ground state energy when , but it does fairly well for . For , , which should be compared with the exact ground state energy of -22.804.

For , we expect the trial state to be less successful than for in competing with the other trial states. The reason is that contains highly frustrated triangle bonds, which can be sacrificed at relatively low cost by the formation of singlets on the non-triangle bonds. A pentagon is less frustrated than a triangle, with a classical energy per bond of -.809, as compared to a triangle’s -0.5 and an unfrustrated bond’s -1. It is therefore reaonable to expect that the correlations found in the classical ground state of , which with skyrmion number 7 are more exotic than those in the truncated tetrahedron, should survive for a larger range of values.

Neutron scattering experiments can measure , the Fourier transform of the spin-spin correlation function. We calculate , which is averaged over all relative orientations of and the molecule. This would be appropriate for a sample containing molecules of random orientations. Neglecting form factors,

(11) |

where is the cartesian distance between sites and . We assume that the triangle and non-triangle bonds lengths are equal. The calculated S(q) for , , , and the coherent state are shown in Fig. (5). (S(q) for is not zero at q=0 because is not a singlet.) If the triangle and non-triangle bond lengths differ by very little, as is the case for the pentagon and non-pentagon bonds in , it is difficult to separate the contributions to S(q) from and . The longer distance spin correlations, present in all of the wavefunctions except , cause higher frequency oscillations in . This leads, for example, to a shoulder at , which is strongest in but still visible in the exact ground state.

For , we do not know and show for the coherent state obtained from the classical ground state, and for the state with singlets on the non-pentagon bonds. There is a great deal more structure in S(q) for than for . These features will, however, be smoothed out somewhat for the exact ground state wavefunction for as they were for .

Finite temperature: We have calculated the correlation functions at finite temperature, using the exact eigenfunctions with Boltzmann weight. As expected, all for distinct spins approach zero as . Two of the correlation functions, however, are nonmonotonic. The correlation between spins on the same triangle increases in magnitude by 6% with increasing before decreasing. The correlation , which is quite small at zero temperature, actually changes sign and increases in magnitude by a factor of 16 before decreasing to zero.

The finite temperature properties can help to interpret recent Quantum Monte Carlo studies on the Hubbard model with 60 electrons by Scalettar et al.[7] They find that for , the spin correlations on resemble those of the classical ground state at short and intermediate distances, but decay to near zero at the largest separations. A crucial question is whether this long distance decay is a property of the quantum ground state, or whether it is caused by the finite temperature of the simulation. For , we calculated the exact Heisenberg partition function for corresponding to and the lowest temperature reached. The temperature , where is the gap to the first excited state. The first excited state is, however, 9-fold degenerate counting spatial and spin degrees of freedom, and higher excited states are fairly close. At , the system has a probability 0.19 to be in the quantum ground state, and the correlation function is 48% of its zero temperature value. should require a lower temperature than for the long distance spin correlations to reach their zero temperature values. If the ground state had spin correlations extending across the molecule, one would expect the excitation energies, which correspond to magnons at allowed nonzero wavevector, to be smaller than those of by roughly a factor of two (). By this estimate, at a given temperature would have similar properties to at twice that temperature. At , has a probability 0.028 to be in the ground state, and a long distance correlation function that is reduced by a factor of 6 from its zero temperature value. This suggests that the decay of spin correlations at large distances observed in the Quantum Monte Carlo studies may be a finite temperature effect.

## 4 Summary and Conclusion

The classical Heisenberg antiferromagnet on the truncated tetrahedron has an exotic ground state with nontrivial topology and nonzero spin correlations up to the longest distances in the system. To see whether quantum fluctuations for destroy these correlations, we exactly diagonalized the problem. We find that the classical ground state gives a very good indication of the exact quantum 2-spin and 4-spin correlation functions for . The classical ground state is also used to generate trial wavefunctions with good overlap against the ground state for . Although the wavefunction with singlet valence bonds on the non-triangle bonds is the correct one for , this state gives no inkling of the long range correlations that develop for the region of physical interest.

This work has implications for the magnetic properties of , in which an exotic classical ground state has been found. The results for suggest that the quantum fluctuations will not destroy the classical spin correlations in . The quantum fluctuations may play a smaller role in , because the spin arrangement is less frustrated than that of . There is also concern that spin correlations in can be destroyed because is not large enough. We have argued that the exact quantum ground state of the Hubbard model in is unlikely to have a phase transition as a function of , and thus that the large spin correlations, somewhat reduced in magnitude, should be observable in real .[1] Evidence for these magnetic correlations may be found in inelastic neutron scattering experiments. It may also be possible to infer the spin correlations by doing NMR on a molecule with two nuclei replaced by . It remains to investigate how doping affects the spin correlations, and to what extent spin correlations affect the properties of doped crystals, such as .[8]

The authors would like to thank A. Balatsky, E. Dagotto, S. Doniach, D. Guo, M. Inui, D. Rokhsar, and J. R. Schrieffer for useful conversations. DC acknowledges the support of the Center for Materials Science through the Program in Correlated Electron Theory. This work was supported by the US Department of Energy.

## References

- [1] D. Coffey and S. A. Trugman, preprint LA-UR-91-3302.
- [2] The ground state energy of the Hubbard model on the truncated tetrahedron as a function of doping has been studied by S. R. White, S. Chakravarty, M. P. Gelfand, and S. A. Kivelson, Phys. Rev. B 45, 5062 (1992).
- [3] For a finite quantum system at zero temperature, a phase transition occurs as a function of the parameters only if the quantum numbers change. For and , the only quantum numbers are the spin and the angular momentum (actually what is left of under the finite rotation group of the tetrahedron or icosahedron). For both and , for . For , we have determined from the exact spectrum that in the Heisenberg limit . For , in the large limit, we know only that the classical ground state has =0. The simplest hypothesis consistent with these facts is that the Hubbard model has no phase transition as a function of for either or .
- [4] There is a more efficient valence bond basis for calculating eigenfunctions, but this system is small enough that we did not need to use that method. See K. Chang, I. Affleck, G. W. Hayden, and Z. G. Soos, J. Phys. Condens. Matter 1, 153 (1989).
- [5] C. S. Yannoni et al., J. Am. Chem. Soc. 113, 3190 (1991).
- [6] R. R. Singh and R. Narayanan, Phys. Rev. Lett. 65, 1072 (1990).
- [7] R. T. Scalettar, E. Dagotto, L. Bergomi, T. Jolicoeur, and H. Monien, Florida State University preprint.
- [8] A. F. Hebard et al., Nature 350, 600 (1991).