AKLT models on decorated square lattices are gapped

The nonzero spectral gap of the original two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) models has remained unproven for more than three decades. Recently, Abdul-Rahman et al. (arXiv:1901.09297) provided an elegant approach and proved analytically the existence of a nonzero spectral gap for the AKLT models on the decorated honeycomb lattice (for the number $n$ of spin-1 decorated sites on each original edge no less than 3). We perform calculations for the decorated square lattice and show that the corresponding AKLT models are gapped if $n\ge 4$. Combining both results, we also show that a family of decorated hybrid AKLT models, whose underlying lattice is of mixed vertex degrees 3 and 4, are also gapped for $n\ge 4$. We develop a numerical approach that extends beyond what was accessible previously. Our numerical results further improve the nonzero gap to $n\ge 2$, including the establishment of the gap for $n=2$ in the decorated triangular and cubic lattices. The latter case is interesting, as this shows that the AKLT states on the decorated cubic lattices are not Néel ordered, in contrast to the state on the undecorated cubic lattice.

Figure 1

Illustration of lattices. Some are decorated, namely (a), (d), (e), (f), (g), and (i), with additional sites added to underlying lattices. Underlying lattices of (a)–(d) are trivalent; underlying lattices of (e)–(h) are four-valent; the underlying lattice of (i) is of mixed vertex degrees of 3 and 4.

Figure 2

Illustration of local structure of decorated lattices: (a) the decorated honeycomb and (b) the decorated square lattice, both with $n=2$.

Figure 3

Illustration of local lattice structure and tensors for (a) the honeycomb or any trivalent lattices and (b) square lattice or any four-valent lattices. The solid purple dots represent virtual qubits. The solid line between two neighboring qubits represents a maximally entangled state of the form $|{\varphi }^{+}\rangle \equiv \left(\right|\uparrow \uparrow \rangle +|\downarrow \downarrow \rangle )/\sqrt{2}$. The solid vertical bar denotes the operator $\widehat{K}=\left(\right|\uparrow \rangle \langle \downarrow |-|\downarrow \rangle \langle \uparrow |)/\sqrt{2}$, which maps $|{\varphi }^{+}\rangle$ to a singlet, up to normalization. The tensors $T_l^L$ consist of $W^L$ and $V$'s inside the dotted square labeled by the channel's symbol $E_L$, and similarly the tensors $T_r^R$ consist of $W^R$ and $V$'s inside the dotted square labeled by $E_R$.

Figure 4

The function $1-b_L\left(n\right)$ vs $n$, which is an indicator of injectivity the mapping ${\mathrm{\Gamma }}_{L-C_n}$. Since $b_R\left(n\right)=b_L\left(n\right)$, this also applies to ${\mathrm{\Gamma }}_{R-C_n}$.

Figure 5

The function $1-b_G\left(n\right)$ vs $n$, which is an indicator of injectivity for the mapping ${\mathrm{\Gamma }}_G$.

Figure 6

The function $d\left(n\right)-1/4$ vs $n$. It is an indicator of a nonzero spectral gap if negative for the decorated square lattice.

Figure 7

The function $d^{\mathrm{mix}}\left(n\right)-1/4$ vs $n$. It is an indicator of a nonzero spectral gap if negative for a decorated lattice, whose underlying lattice has mixed degrees 3 and 4. One should evaluate $d^{\mathrm{mix}}\left(n\right)-1/4$ instead of $d^{\mathrm{mix}}\left(n\right)-1/3$ to check the gappedness.