Ground state of a two-dimensional quasiperiodic quantum antiferromagnet

We consider the antiferromagnetic spin-$1∕2$ Heisenberg model on a two-dimensional bipartite quasiperiodic tiling. The broken symmetry ground state in this model is inhomogeneous, reflecting the fact that there are a variety of local environments in such a structure. An important symmetry of the quasicrystal, namely that of invariance under discrete scale transformations is used to define an approximate real space renormalization scheme for the octagonal tiling. We solve for some of the fixed point properties of this quasiperiodic antiferromagnet. The ground state energy and local order parameters are calculated, and the results compare favorably with numerical values obtained by quantum Monte Carlo calculation. Despite the novel features of the ground state in this type of antiferromagnet, there are some interesting similarities with the well-known square lattice antiferromagnet. The most striking of these is the proximity of the values of the ground state energies of these two paradigms for two very dissimilar classes of solids.

Figure 1

Inhomogeneous ground state structure on the tiling. The circles have sizes that depend on the strength of the local order parameter.

Figure 2

A portion of the octagonal tiling showing the six different nearest neighbor environments $A,B,\dots ,F$.

Figure 3

Portion of original (black) tiling, showing sites of the $\alpha$ class (black dots) which become sites of the new inflated (grey) tiling.

Figure 4

The six different nearest neighbor environments of the octagonal tiling.

Figure 5

The $\alpha$ site clusters defined on the original (left) and once inflated (right) tilings.

Figure 6

Second generation $A$ cluster.

Figure 7

Five-spin units (surrounded by circles) on the square lattice. The new $\sqrt{5}\times \sqrt{5}$ unit cell is shown.

Figure 8

Tiling showing block centers (black dots). The grey lines connect pairs of sites that are shared between two blocks.

Figure 9

Block spin centers (filled circles) showing the central and all peripheral blocks for three cases: top, a $z=5$, $z^{\prime }=3$ site; middle a $z=6$, $z^{\prime }=4$ site; bottom, a $z=z^{\prime }=8$ site.

Figure 10

$(z+1)$ spin cluster (Heisenberg star).

Figure 11

$m_s\left(z\right)$ values versus $z$ obtained for increasing orders of RG. The zero order analytical curve is indicated by a dashed line in each figure. (a) First (circles) and second (rectangles) order RG. (b) Third (circles) and fourth (rectangles) order RG and QMC data for the full octagonal tiling. (c) Adjusted fourth order data (grey squares) and QMC data (grey circles).

Figure 12

Two segments of the Fibonacci chain.

Figure 13

Cut-and-project method. Selected edges of a square lattice are projected onto the parallel space $\left(E_1\right)$. The edges that are selected have perpendicular space $\left(E_2\right)$ projections that fall within the segment marked $W$.

Figure 14

Projection into perpendicular space of vertices of the octagonal tiling. Domains corresponding to the six families are labeled (eightfold symmetry determines labels of unmarked domains).

Figure 15

Superposition of two successive approximants.