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arXiv:cond-mat/0202216 · Physical review. B, Condensed matter

Antiferromagnetism in the exact ground state of the half-filled Hubbard model on the complete bipartite graph

Gianluca Stefanucci, Michele Cini

Published Sep 10, 2002• 3 citations
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Abstract

As a prototype model of antiferromagnetism, we propose a repulsive Hubbard Hamiltonian defined on a graph $\ensuremath{\Lambda}=\mathcal{A}\ensuremath{\cup}\mathcal{B}$ with $\mathcal{A}\ensuremath{\cap}\mathcal{B}=\ensuremath{\emptyset},$ and bonds connecting any element of $\mathcal{A}$ with all the elements of $\mathcal{B}.$ Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the $\mathcal{A}$ sites and/or of the $\mathcal{B}$ sites. This is the Hubbard model defined on the so-called ${(N}_{A}{,N}_{B})$ complete bipartite graph, ${N}_{A}{(N}_{B})$ being the number of elements in $\mathcal{A}(\mathcal{B}).$ In this paper, we analytically find the exact ground state for ${N}_{A}{=N}_{B}=N$ at half filling for any N; the repulsion has a maximum at a critical N-dependent value of the on-site Hubbard U. The wave function and the energy of the unique, singlet ground state assume a particularly elegant form for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ We also calculate the spin-spin correlation function and show that the ground state exhibits an antiferromagnetic order for any nonzero U even in the thermodynamic limit. We are aware of no previous explicit analytic example of an antiferromagnetic ground state in a Hubbard-like model of itinerant electrons. The kinetic term induces nontrivial correlations among the particles, and an antiparallel spin configuration in the two sublattices becomes energetically favored at zero temperature. On the other hand, if the thermodynamic limit is taken and then zero temperature is approached, a paramagnetic behavior results. The thermodynamic limit does not commute with the zero-temperature limit, and this fact can be made explicit by the analytic solutions.

DOI
10.1103/physrevb.66.115108

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We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.
As an extension of our previous rigorous investigation on the spin correlations in the ground states of the half-filled Hubbard model and the periodic Anderson model [G. S. Tian, Phys. Rev. B 50, 6246 (1994)], in the present paper we study the behavior of these correlation functions at finite temperature. We show rigorously that, at any $T\ensuremath{\ne}0,$ the predominant spin correlations in these systems are antiferromagnetic. Furthermore, based on this result, we also show that a quasi-one (or two)-dimensional itinerant electron ferrimagnet must have a gapless branch of ferromagnetic excitations. This conclusion is consistent with the previous results derived by the spin-wave theory.
The half-filled Hubbard model on the Bethe lattice with coordination number z=3z=3z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy EEE, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength U/tU/tU/t. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion U/tU/tU/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large U/tU/tU/t, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (∣∣≃0.35| |\simeq 0.35∣∣≃0.35 for U/t→+∞U/t\to +\inftyU/t→+∞). $ - ^2$ shows a maximum for U/t=2.4U/t=2.4U/t=2.4--2.9 which results from the interplay between the usual increase of $ $ with increasing U/tU/tU/t and the formation of important permanent moments $ $ at large U/tU/tU/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of U/tU/tU/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.