We review recent theoretical investigations of the thermodynamics of two-dimensional quantum magnets on bipartite lattices. Despite the simple, semiclassical nature of the ground state, different types of low temperature scaling can occur in these systems. None of these scaling proposals allows for a conclusive description of experimental and numerical data. However, these are well reproduced assuming a wide nonuniversal crossover region smoothly connected to the high temperature limit. This provides evidence that for a quantitative description of experimental results and numerical data at finite temperatures nonuniversal effects are much more important then previously thought.
We study the properties of a linear chain of spins governed by the Hamiltonian H=J1 ∑ j=1NSj·Sj+1+J2 ∑ j=1NSj·Sj+2and derive equations for an upper bound of the free energy by means of a temperature dependent Hartree-Fock approximation. These equations can be solved at zero temperature, yielding an approximate wavefunction for the ground state, of which the energy is an upper bound for the exact ground state energy. The upper bounds obtained are improvements of the results of Majumdar and Ghosh. There are several different cases to be considered, depending on the relative values of J1 and J2. In none of these is there a gap between the ground state and the first excited state in the thermodynamic limit. Finally, correlation functions are shortly discussed in connection with spiral structures in classical spin systems.
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 with all the elements of
B. Since all the hopping matrix elements associated with each bond are equal, the model is invariant under an arbitrary permutation of the
A sites and/or of the
B sites. This is the Hubbard model defined on the so-called
(NA,NB) complete bipartite graph,
NA(NB) being the number of elements in
A(B). In this paper, we analytically find the exact ground state for
NA=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.