Presence or absence of order by disorder in a highly frustrated region of the spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices

The geometrically frustrated spin-1/2 Ising-Heisenberg model on triangulated Husimi lattices is exactly solved by combining the generalized star-triangle transformation with the method of exact recursion relations. The ground-state and finite-temperature phase diagrams are rigorously calculated along with both sublattice magnetizations of the Ising and Heisenberg spins. It is evidenced that the Ising-Heisenberg model on triangulated Husimi lattices with two or three interconnected triangles-in-triangles units displays in a highly frustrated region a quantum disorder irrespective of temperature, whereas the same model on triangulated Husimi lattices with a greater connectivity of triangles-in-triangles units exhibits at low enough temperatures an outstanding quantum order due to the order-by-disorder mechanism. The quantum reduction of both sublattice magnetizations in the peculiar quantum ordered state gradually diminishes upon increasing the coordination number of the underlying Husimi lattice.

Figure 1

The spin-1/2 Ising-Heisenberg model on the triangulated Husimi lattices (figures on the left) and its rigorous mapping equivalence with the effective spin-1/2 Ising model on the pure Husimi lattices (figures on the right) established by means of the generalized star-triangle transformation (figure at the bottom). The triangulated Husimi lattices, which include two different lattice sites occupied by the Ising (empty circles) and Heisenberg (full circles) spins, are drawn for two particular values of the coordination number $q=3$ and 4.

Figure 2

The ground-state phase diagram of the spin-1/2 Ising-Heisenberg model on the triangulated Husimi lattices in the $\mathrm{\Delta }\text{−}{J}_{\mathrm{H}}/J_{\mathrm{I}}$ plane.

Figure 3

The critical temperature as a function of the ratio between the Heisenberg and Ising interaction for several values of the coordination number $q$ and three different values of the exchange anisotropy: (a) $\mathrm{\Delta }=0$, (b) $\mathrm{\Delta }=1$, and (c) $\mathrm{\Delta }=2$.

Figure 4

Thermal variations of the sublattice magnetizations of the Ising spins (solid lines) and the Heisenberg spins (broken lines) for several values of the interaction ratio $J_{\mathrm{H}}/J_{\mathrm{I}}$ and a few values of the coordination number and the exchange anisotropy: (a) $q=3$, $\mathrm{\Delta }=1$; (b) $q=3$, $\mathrm{\Delta }=2$; (c) $q=4$, $\mathrm{\Delta }=1$.