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 noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction
U is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``
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. We use exact determinant quantum Monte Carlo (DQMC) simulations to 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,
Ud=Up, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. 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 ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases
Up=0,Ud=U and
Up=U,Ud=0. We found that the different environments of
d and
p sites lead to a ferromagnetic insulator when
Ud=0; by contrast,
Up=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each
d site.
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.