Analysis of a family of Heisenberg systems with simple eigenfunctions for the ground state and low lying excitations

Abstract

We analyze a family of one and bi-dimensional frustrated Heisenberg systems, whose ground state ͑GS͒ and low lying excitation spectrum can be exactly described as a product. The magnetic lattice is composed by clusters of spin S ions. These clusters are connected to each other by intermediate, spin ions ͑here called "connectors"͒. The value of S and are arbitrary. Three major properties of these systems are: ͑i͒ The GS is exponentially degenerate. ͑ii͒ The low lying excitations are separated from the GS by a gap, and they are localized. However, neighbor excitations interact, exchanging energy; two "bottom" excitations repel each other. ͑iii͒ The magnetization curve has a staircaselike form. On introducing a direct connector-connector coupling, a break in the lattice translational symmetry can occur, due to the uniform magnetic field. Finally, we study the effect of a weak departure from the hypothesis of the model, with the aim to make it more adaptable to fit different actual magnetic systems. We obtain bands of delocalized excitations. Bounded states between excitations also appear.

Ferrimagnetism in the Hubbard, dimmer-connector frustrated chain

“…We note that F 2 = F (F + 1) is a constant of motion for each . The spin Hamiltonian H H was studied by Niggemann et al [10], and other authors [9][10][11]. Introducing the ratio…”

“…(ii) For 0.5 < R < 1.10, it holds that F 2 = 1, F 2 −1 = 0, and the system is in the phase of a period two (labeled as P 2 in the literature [9]). However, the antiferromagnetic exchange J couples the (non-zero) dimmer spin F 2 to its neighbors, S A, 2 and S A, 2 +1 , thus yielding a singlet (non-magnetic) state.…”

“…Here the system is in the Kolezhuk "Ferrimagnetic Phase" [12] (labeled as P ∞ in the literature [9]), where spins 1 (dimmers) and 1/2 (connectors) alternate, and the antiferromagnetic coupling yields a non-zero global spin S = N/2 (in spite of quantum fluctuations).…”

“…We illustrate former concepts by means of the "connector-dimmer" AB 2 chain (see figure 1); there the B sublattice corresponds to the dimmers, while the A sublattice contains the "connectors" of reference [9]. This chain transforms into the bipartite "lozenge" lattice when the hopping amplitude between the B sites on the same dimmer vanishes (vertical lines in figure 1).…”

“…In this case, Lieb's theorem [4] ensures that the net spin of the chain is S = The AB 2 chain has been studied in the context of Heisenberg [9][10][11] and half-filled band Hubbard [6][7][8] models. While [9][10][11] include two antiferromagnetic exchanges, J (connecting A-B sites) and J 0 (connecting both B sites on the same dimmer), [6][7][8] only consider a non-vanishing electronic hopping between A and B sites (say, a "lozenge" Hubbard lattice), in order to satisfy Lieb's hypothesis of a bipartite lattice [4]. Lieb's results were generalized, showing that the ferrimagnetic order persists in the "lozenge" and in other one-dimensional lattices, when site dependent Hubbard parameters are considered [7].…”