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arXiv:cond-mat/9604121

Ferromagnetism in an itinerant electron system: Hubbard model on complete graph

D. F. Wang

Published Apr 18, 1996• 1 citation
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Abstract

In this work, the ground states of the Hubbard model on complete graph are studied, for a finite lattice size $L$ and arbitrary on-site energy $U$. We construct explicitly the ground states of the system when the number of the electrons $N_e \ge L+1$. In particular, for $N_e=L+1$, the ground state is ferromagnetic with total spin $s_g=(N_e-2)/2$.

DOI
10.48550/arxiv.cond-mat/9604121

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In this work, the ground states of the Hubbard model on complete graph are studied, for a finite lattice size LLL and arbitrary on-site energy UUU. We construct explicitly the ground states of the system when the number of the electrons Ne≥L+1N_e \ge L+1Ne​≥L+1. In particular, for Ne=L+1N_e=L+1Ne​=L+1, the ground state is ferromagnetic with total spin sg=(Ne−2)/2s_g=(N_e-2)/2sg​=(Ne​−2)/2.
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Let L(G) be the line graph of a graph G=(V,E). The Hubbard model on L(G) has ferromagnetic ground states with a saturated spin if the interaction is repulsive (U>0) and if the number of electrons N satisfies N>or=M. M= mod E mod + mod V mod -1 if G is bipartite and M= mod E mod + mod V mod otherwise. The author shows that the ferromagnetic ground state is unique if N=M. Further he gives a sufficient condition for the existence of other ground states if N>M. The results are valid also for a multi-band Hubbard model on a bipartite graph. In the case of a periodic lattice, the results are related to the existence of a flat energy band.
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 A\mathcal{A}A with all the elements of B.\mathcal{B}.B. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the A\mathcal{A}A sites and/or of the B\mathcal{B}B sites. This is the Hubbard model defined on the so-called (NA,NB){(N}_{A}{,N}_{B})(NA​,NB​) complete bipartite graph, NA(NB){N}_{A}{(N}_{B})NA​(NB​) being the number of elements in A(B).\mathcal{A}(\mathcal{B}).A(B). In this paper, we analytically find the exact ground state for NA=NB=N{N}_{A}{=N}_{B}=NNA​=NB​=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.