Fractal Symmetric Phases of Matter

Abstract

We study spin systems which exhibit symmetries that act on a fractal subset of sites, with fractal structures generated by linear cellular automata. In addition to the trivial symmetric paramagnet and spontaneously symmetry broken phases, we construct additional fractal symmetry protected topological (FSPT) phases via a decorated defect approach. Such phases have edges along which fractal symmetries are realized projectively, leading to a symmetry protected degeneracy along the edge. Isolated excitations above the ground state are symmetry protected fractons, which cannot be moved without breaking the symmetry. In 3D, our construction leads additionally to FSPT phases protected by higher form fractal symmetries and fracton topologically ordered phases enriched by the additional fractal symmetries.

Figures (8)

• Figure 1: Fractal structures generated by (left,blue) the Sierpinski rule $a_i^{(t+1)} = a_{i-1}^{(t)} + a_{i}^{(t)}$ and (right,red) the Fibonacci rule $a_i^{(t+1)} = a_{i-1}^{(t)} + a_{i}^{(t)} + a_{i+1}^{(t)}$, starting from the initial state $a_i^{(0)}=\delta_{i,0}$. In the polynomial representation, the row $t$ is given by ${f(x)}^t$, with (blue) $f(x)=1+x$ and (red) $f(x)=x^{-1}+1+x$ over $\mathbb{F}_2$. Notice that self-similarity at every row $t=2^l$ (here, we show evolution up to $t=40$).
• Figure 2: The number of independent symmetry generators $k(L)$ and a choice of $q_\alpha(x)$ for the Sierpinski model on an $L\times L$ torus for few particular $L$. Here, $m,n$ are positive integers, $m>n$, and $0\leq \alpha<k$ labels the symmetry polynomials $q_\alpha^{(L)}(x)$, and $\lfloor\cdot\rfloor$ denotes the floor function.
• Figure 3: A valid history for the state $s_0=1$ for the Fibonacci rule CA. The forward evolution (red) is fully deterministic, and here an unambiguous choice has been made for states leading up to it (orange). Lattice points are labeled by $(i,j)$ corresponding to $x^i y^j$ in the polynomial representation.
• Figure 4: In $(a)$, we show how to place the Sierpinski FSPT on to the honeycomb lattice naturally. The orange circle is the unit cell, and blue/red sites correspond to the $a$/$b$ sublattice sites. The interactions involve four spins on the highlighted triangles triangles. In $(b)$, we show the sites affected by a choice of symmetry operations on an infinite plane. The large circles are those affected by a particular $\mathbb{Z}_2^{(a/b)}$ type symmetry (Eq \ref{['eq:fsptsym']}). In $(c)$, we perform a symmetry twist on the Sierpinski FSPT on a $7\times 7$ torus. The chosen symmetries $g_1$ ($g_2)$ corresponds to operations on all spins highlighted by a large blue (red) circle. The green triangles correspond to terms in the twisted Hamiltonian $H_\text{twist}(g_1)$ that have flipped sign. The charge response $T(g_1,g_2)=-1$ is given by the parity of red circles that also lie in the green triangles, and is independent of where we make the cut $j_0$.
• Figure 5: $(a)$ We illustrate the terms in the Hamiltonian for the Fibonacci FSPT (Eq \ref{['eq:fspt']} with $f=x^{-1}+1+x$). The model is defined on a square lattice, with a two-site unit cell (circled), $a$ (blue) and $b$ (red). The two terms in the Hamiltonian at $h=0$ are illustrated in the two triangles. Also shown are the edge Pauli operators along the left edge. $(b)$ We show a family of symmetry elements on a $10\times 10$ slab. The black outlined circles represent the band of $R=2$ unit cells on which we fix the action of the symmetry so that it acts only as $\hat{\mathcal{X}}^{(b)}_{0,7}$ on the left edge in this case (with $(0,0)$ being the top left unit cell). This fixes how the symmetry must act on the top and some of the right edge (gray outlined circles), but there is still some freedom along the remaining sites on the right edge (yellow question marks), which will determine how it acts on the remaining sites (transparent orange circles). There are $2^{L_x-R}=2^8$ symmetry elements (corresponding to the 8 question marks) satisfying our constraint. $(c)$ We also show the family of symmetry elements which act only as $\hat{\mathcal{Z}}^{(b)}_{0,7}$, and therefore forms a projective representation with the symmetry element shown in $(b)$ on the left edge. Note that these symmetries will generally have some non-trivial action along the other edges.