Exact quantum ground state of a two-dimensional quasicrystalline antiferromagnet

We present the exact dimer ground state of a quantum antiferromagnet defined on a quasicrystal constructed from the bronze-mean hexagonal quasicrystal. A coupling isotropy on the first- and second-neighbor bonds is sufficient to stabilize a product state of singlets on the third-neighbor bonds. We also provide a systematic approach for constructing additional crystals, quasicrystals, and amorphous structures that can sustain an exact dimer ground state.

Figure 1

The construction of model-X from bronze-mean hexagonal quasicrystal: First, we deplete all vertices that are shared by six big triangles to generate the hexagonal tiles. Next, we deplete the connections shared by two triangles, constructing the two rhombus tiles. On the resulting tiling, we place a spin on each vertex, and the connection between the vertices, which now connects a pair of spins, acts as the bonds. The bonds are of two different lengths, namely $s$ and $l$, shown by dashed orange and dotted blue lines. We assume an equal Heisenberg coupling of strength $J^{\prime }$ on both these bonds. We further add a subset of the third-neighbor interactions, shown by thick green lines, with an exchange interaction of strength of $J$. We call the resulting spin model “model-X.” The singlets (light purple ellipses) reside on $J$ bonds. As the $J^{\prime }$ couplings do not contribute to the energy, this product state of singlets is an eigenstate of (2).

Figure 2

Model-Y, defined on a lattice, is made out of the same tiles as model-X. In model-Y, the dashed orange bonds have an exchange interaction of strength $J^{\prime }$, and the green bonds bear an exchange interaction of strength $J$. Similar to model-X, model-Y can also admit an exact singlet eigenstate with singlets on the $J$ bonds, which becomes a ground state of the system for $J\gtrsim 1.45J$.

Figure 3

A general scheme for obtaining a model with an exact dimer ground state from a graph where each vertex has four connections. First, we decorate the connections with new vertices (marked by solid circles), which is followed by a generalized star-triangle-type transformation to decimate all the old vertices. From there, one can generate two possible graphs by connecting a subset of the diagonals of the newly generated quadrangles (keeping in mind that no site can be part of two diagonals), both of which can host an exact dimer state.

Figure 4

(a) The 144-site model-X cluster used to perform the DMRG calculations mentioned in the main text. (b) The 144-site model-Y cluster used to perform the DMRG calculations mentioned in the main text.

Figure 5

The DMRG results for model-X: (a) the ground state energy per site and (b) the spin-spin correlations on selected bonds. (c) and (d) are respectively the same for model-Y. The spin systems used for our calculations are depicted in Figs. 4 and 4. In both calculations we have used a open boundary condition for a fair comparison.

Figure 6

(a) An example of a amorphous system with an exact ground state constructed via the prescription put forward in the main text. The dashed lines are placed randomly on the plane to create an amorphous system with coordination number 4. (b) An example of a four-coordinated quasicrystal (foreground) generated from Penrose rhomb tiling (background). The method explained in the main text now can be directly used on this system to produce a model with an exact dimer eigenstate.