Classifying local fractal subsystem symmetry-protected topological phases

We study symmetry-protected topological (SPT) phases of matter in $2\mathrm{D}$ protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases, no nontrivial SPT phases are possible at all. In cases where nontrivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.

Figure 1

Example of a symmetry generator (top) $S\left(g_i^{(a,j_0)}\right)$ or (bottom) $S\left(g_i^{(b,j_0)}\right)$ for the fractal generated by $f\left(x\right)=\overline{x}+1+x$ with $p=2$. Sites with blue or red squares are acted on by $u_{ij}\left({\mathit{g}}^{\left(a\right)}\right)$ or $u_{ij}\left({\mathit{g}}^{\left(b\right)}\right)$, respectively, and form a valid space-time trajectory of a cellular automaton.

Figure 2

Measurement of the twist phases for (left) $\mathrm{\Omega }(g_{i^{\prime }}^{(a,j_0-l_y)},g_i^{(a,j_0)})$ and (right) $\mathrm{\Omega }(g_{i^{\prime }}^{(b,j_0+l_y)},g_i^{(a,j_0)})$. Due to locality, the twist phase may only be nontrivial if the support of (left) $S(g_{i^{\prime }}^{(a,j_0-l_y)}$ or (right) $S\left(g_{i^{\prime }}^{(b,j_0+l_y)}\right)$ has some overlap with the yellow box of size $(2l_x+1)\times (2l_y+1)$ about $(i,j_0)$. This implies that the twist phase must be trivial for $i^{\prime }$ outside of a small region around $i$, a property which we call locality. However, this must be true for all choices of $j_0$, which greatly constrains the allowed twist phases. In the case of twist phases between the same type of symmetry (left), only the trivial set of twist phases, all $\mathrm{\Omega }(g_{i^{\prime }}^{(a,j_0-l_y)},g_i^{(a,j_0)})=1$ is allowed. Between an $a$ and a $b$ type symmetry (right), we show that only a finite number of solutions exist.

Figure 3

Visualization of the matrices ${\mathit{K}}^{(k,m)}$ for the example of $f\left(x\right)=1+x+x^2$ and $p=2$, for $k=0$ and $m=0,1,2,3$ (other $k$ can be obtain by shifting every element $k$ steps to the left). Each blue cell $(i^{\prime },i)$, counting from the top left, represents a nonzero matrix element $K_{i^{\prime }i}^{(0,m)}=1$. The arrows indicate evolution by $\mathit{K}\to \mathit{F}\mathit{K}{\mathit{F}}^{-1}$, under which they exhibit cycles of period $2^{\lceil {log}_2(m+1)\rceil }$, as can be seen. Each of them are only nonzero about a small diagonal band (nongray squares) of width given by $D_m=1+2m$. A ${\mathit{K}}^{(k,m)}$ is local if this white band fits inside some diagonal band [Eq. (56)]. If a ${\mathit{K}}^{(k,m)}$ is local, then it can be seen that under evolution it retains locality (the white band never gets larger), a property which we call consistent locality. The main result of this paper is that any consistently local matrix can be written as a linear sum of local ${\mathit{K}}^{(k,m)}$. Since there are only a finite number of local ${\mathit{K}}^{(k,m)}$, there are only a finite number of consistently local matrices that can be written, and therefore a finite number of distinct phases in the thermodynamic limit. The number will depend on the choice of $(l_x,l_y)$, i.e., how local the model is. Notice that consistent locality is nongeneric: if we just pick a local matrix by filling in elements along the diagonal band at random, it will generically quickly become highly nonlocal after a few steps of evolution.

Figure 4

A visualization of the process to defining a twist phase for pseudosymmetries $\tilde{S}$. Twisting with respect to ${\tilde{S}}_i^{(a,j_0,j_0+R)}$ can be thought of as acting via a unitary $U$, which has some region $\mathcal{A}\left(U\right)$ shown as the yellow square. To measure the change in charge of another symmetry, we take the expectation value of the commutator [Eq. (94)] of $U$ with ${\tilde{S}}_{i^{\prime }+k_a{p}^n}^{(b,j_0+l_y+p^n,j_0-p^n)}$, as shown, for large $n$. This may be nontrivial for small $|i^{\prime }-i|$ when they overlap, and is the sign of a nontrivial pseudo-SPT.

Figure 5

Explicit examples of some possible phases for the case of $G={\mathbb{Z}}_2^{(f,y)}\times {\mathbb{Z}}_2^{(\overline{f},\overline{y})}$ with $f\left(x\right)=1+x$: the Sierpinski triangle symmetry. The models are constructed following the procedure of Sec 5, and are simply cluster models defined on some underlying graph. The models all have $l_y=0$ and ${\mathit{W}}_0={\mathit{W}}^{(0,0)}$ given by $\left(a\right){\mathit{K}}^{(0,0)}, \left(b\right){\mathit{K}}^{(-1,1)}, \left(c\right){\mathit{K}}^{(-1,2)}$, and $\left(d\right){\mathit{K}}^{(-2,3)}$. Each site [gray circle in (a)] consists of an $a$ and a $b$ qubit, which are represented by blue and red vertices. Example of fractal subsystems on which the symmetries act are also shown in (a): green highlighted vertices for the $a$ type subsystems, and orange for the $b$. The reader may verify that $\prod X$ on these subsystems is indeed a symmetry of the cluster model defined on all these graphs. The case (a) is simply the previously studied honeycomb lattice cluster model, which is translation invariant. The other three are previously undiscovered phases, and are only translation invariant with a period of (b) 2 or [(c) and (d)] 4 along the $x$ and $y$ directions. The graphs for phases with ${\mathit{K}}^{(k,m)}$ for $k$ other than the ones chosen here may be obtained simply by shifting each blue $a$ vertex left/right by a number of sites, while maintaining connectivity of the graph. For each case, we also show one cycle of the matrices ${\mathit{W}}_j={\mathit{F}}^j{\mathit{W}}_0{\mathit{F}}^{-j}$, the matrix characterizing the twist phases for symmetries defined with respect to row $j$, presented in the same manner as in Fig. 3. The lower-leftmost $a$ and $b$ vertices of each graph are defined to be at coordinate $(x,y)=(0,0)$. Although we have only shown examples on an $8\times 8$ torus, these may be tiled onto any commensurate system size.