======Seminário de Matéria Condensada - 19/04/2018, 11:00, Sala 426 (Torre Nova) ======

==== Ferromagnetism beyond Lieb's theorem  ====

=== Raimundo Rocha dos Santos (UFRJ)  ===

Modelling itinerant ferromagnetism still poses major challenges to theoreticians. In 1963 John Hubbard proposed [1] a single-band model, but, as it turned out, ferromagnetism only appears within mean-field approximations. Since then, distinct routes to ferromagnetism have been proposed, some of which are based on multi-band models. The development of this route was boosted by a theorem proved by Elliott Lieb [2], according to which the Hubbard model on bipartite lattices with unequal number of sites on each sublattice, and at half filling, should have a non-zero spin in the ground state. While a total non-zero spin is suggestive of long-range order (LRO), a systematic investigation of LRO had not been carried out so far. Another issue of interest is whether Lieb’s theorem can be extended to lattices in which the on-site repulsion is inhomogeneous. An example of a lattice falling under the conditions of the theorem is the ‘CuO2 lattice (also known as ‘Lieb lattice’, or as a decorated square lattice), in which ‘d-orbitals’ occupy the vertices of the squares, while ‘p-orbitals’ lie halfway between two d-orbitals; both d and p orbitals can accommodate only up to two electrons. In this talk we report on Determinant Quantum Monte Carlo (DQMC) simulations for the Lieb lattice [3]. We quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of U and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, U<sub>d</sub> = U<sub>p</sub>, originally considered by Lieb, and the inhomogeneous (IH) case, U<sub>d</sub> not equal to U<sub>p</sub>. For the H case at half filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all U. For the IH system at half filling, we argue that the case where U<sub>p</sub> and U<sub>d</sub> differ falls under Lieb’s theorem, provided they are positive definite, so we used DQMC to probe the cases U<sub>p</sub> =0,U<sub>d</sub> = U and U<sub>p</sub> =U,U<sub>d</sub> =0. We found that the different environments of d and p sites lead to a ferromagnetic insulator when U<sub>d</sub> = 0; by contrast, U<sub>p</sub>= 0 leads to a metal without any magnetic ordering. In addition, we have also established that at density ρ = 1/3, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each d site; this filling corresponds to a van Hove singularity in the density of states, and the Fermi surface is nested, similarly to what happens in the Hubbard model in the simple square lattice.

1. J. Hubbard, Proc. R. Soc. London. Ser. A **276**, 238 (1963).\\
2. E. H. Lieb, Phys Rev Lett **62**, 1201 (1989); (E) 62, 1927 (1989).\\
3. N. C. Costa, T. Mendes-Santos, T. Paiva, R. R. dos Santos, and R. T. Scalettar, Phys. Rev. B **94**, 155107 (2016).