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We study the single-band Hubbard model under the action of an external magnetic field using the cumulant Green's functions method (CGFM). The starting point of the method is to diagonalize a cluster containing N correlated sites (seed) and employ the cumulants calculated from the cluster solution to obtain the full Green's functions for the lattice. All calculations are done directly, and no self-consistent process is needed. We benchmark the one-dimensional results for the gap, the ground-state energy, and the double occupancy obtained from the CGFM against the corresponding exact results of the thermodynamic Bethe ansatz and the quantum transfer matrix methods. The results for the CGFM tend systematically to the exact one as the cluster size increases. The particle-hole symmetry of the density of states is fulfilled. The method can be applied to any parameter space for one, two, or three-dimensional Hubbard Hamiltonians and can also be extended to other strongly correlated models, like the Anderson Hamiltonian, the t-J, Kondo, and Coqblin-Schrieffer models. We also calculate the effects of positive magnetic fields as a function of the chemical potential, and we identify a finite cluster effect (Phase VI) characterized by a partially filled band and negative magnetization (n_up < n_down). This phase survives for clusters containing up to N=8 sites but tends to disappear as the size of the cluster increases. We include a simple application to spintronics, where we used these clusters as correlated quantum dots to realize a single-electron transistor when connected to Hubbard leads. We calculate the phase diagram, including the new cluster phase, using the magnetic field and chemical potential as parameters for N=7 and N=8.