Strong planar subsystem symmetry-protected topological phases and their dual fracton orders

We classify subsystem symmetry-protected topological (SSPT) phases in $3+1$ dimensions $(3+1\text{D}$) protected by planar subsystem symmetries: short-range entangled phases which are dual to long-range entangled Abelian fracton topological orders via a generalized “gauging” duality. We distinguish between weak SSPTs, which can be constructed by stacking $2+1\text{D}$ SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite Abelian group. Finally, we show that fracton orders realizable via $p$-string condensation are dual to weak SSPTs, while those dual to strong SSPTs exhibit statistical interactions prohibiting such a realization.

Figure 1

Left: Examples of our construction of 1-foliated or weak 2- or 3-foliated models, for $G={\mathbb{Z}}_N\times {\mathbb{Z}}_N$, in the graphical notation. 2D SPTs to be stacked are shown in the blue boxes, and the large arrow points to the resulting SSPT after stacking. The color of the edges connecting two vertices indicate its weight modulo $N$. Right: Examples of $\mathbf{M}$ matrices that cannot be obtained by stacking 2D phases onto 2- or 3-foliated models. The type-1 phase is only strong for even $N$, and type-2 strong phases can only be realized for 2-foliated symmetries.

Figure 2

The commuting (but nonprojector) Hamiltonian describing the fracton dual of the ${\mathbb{Z}}_2$ strong model is shown. Qubit degrees of freedom live on the links. The Hamiltonian is a sum over all cubes $c$ of the term $B_c$, shown, which consists of Pauli $X,Z,S=\mathrm{diag}(1,i)$, and $C{Z}_{12}={(-1)}^{(Z_1-1)(Z_2-1)/4}$ operators between qubits on links of different orientations as shown by the colored lines. In addition, the Hamiltonian also has the usual cross terms $A_v^{\mu \nu }$ from the X-cube which is the product of four $Z$ operators lying in the $\mu \nu$ plane touching a vertex $v$. Thus, $H=-{\sum }_c{B}_c-{\sum }_v\left(A_v^{xy}+A_v^{yz}+A_v^{zx}\right)$.