Inhomogeneous quantum antiferromagnetism on periodic lattices

We study quantum antiferromagnets on two-dimensional bipartite lattices. We focus on local variations in the properties of the ordered phase which arise due to the presence of inequivalent sites or bonds in the lattice structure, using linear spin wave theory and quantum Monte Carlo methods. Our primary finding is that sites with a high coordination tend to have a low ordered moment, at odds with the simple intuition of high coordination, favoring more robust Néel ordering. The lattices considered are the dice lattice, which is dual to the kagome, the CaVO lattice, an Archimedean lattice with two inequivalent bonds, and finally the crown lattice, a tiling of squares and rhombuses with a greater variety of local environments. We first present results for the on-site magnetizations and local bond expectation values for the $S=1∕2$ Heisenberg model on these lattices, and then discuss two exactly soluble cases which provide a simple analytical framework for understanding our lattice studies.

Figure 1

The dice lattice, showing a localized excitation.

Figure 2

The three dice lattice energy bands along the main directions in $\mathbf{k}$ space. (Special points in the hexagonal Brillouin zone are $O$, the origin, $A$, the vertex, and $B$, the midpoint of an edge.)

Figure 3

The CaVO lattice, showing a state localized on a strip (open circles correspond to bigger amplitudes than filled circles).

Figure 4

The CaVO energy bands along the main symmetry directions. (Special points in the square Brillouin zone are $O$, the origin, $X$, the midpoint of the edge, and $\Gamma$, the vertex).

Figure 5

A unit cell of the crown lattice, with sites numbered from 1 to 7.

Figure 6

The seven crown lattice energy bands along some of the main symmetry directions. Special points in the square Brillouin zone are $O$, the origin, $X$, the zone boundary along the $k_x$ axis, and $\Gamma$, the $\pi ,\pi$ vertex.

Figure 7

States on strips (see text). Amplitudes are different on the sites with open (filled) circles.

Figure 8

(Color online) Dependence of the local staggered magnetization $m_i$ on the coordination number z for the inequivalent sites of the crown lattice. Quantum Monte Carlo data (open) are compared to linear spin wave theory results (full).

Figure 9

(Color online) Dependence of the local averaged bond strength ${\overline{b}}_i$ on the coordination number $z$ for the inequivalent sites of the crown lattice. Quantum Monte Carlo data (open circles) are compared to linear spin wave theory results (full circles).

Figure 10

A Heisenberg star: a central spin is coupled to $\zeta$ mutually uncoupled neighbors.