$\mathbb{Z}_N$ generalizations of three-dimensional stabilizer codes

Abstract

In this work, we generalize several three-dimensional Z2 stabilizer models--including the X-cube model, the three-dimensional toric code, and Haah's code--to their ZN counterparts. Under periodic boundary conditions, we analyze their ground state degeneracies and topological excitations, and uncover behaviors that strongly depend on system size. For the X-cube model, we identify excitations with mobility restricted under local operations but relaxed under nonlocal ones derived from global topology. These excitations, previously confined to open boundaries in the Z2 model, now appear even under periodic boundaries. In the toric code, we observe nontrivial braiding between string and point excitations despite the absence of ground state degeneracy, indicating long-range entanglement independent of topological degeneracy. Again, this effect extends from open to periodic boundaries in the generalized models. For Haah's code, we find new excitations--fracton tripoles and monopoles--that remain globally constrained, along with a relaxation of immobility giving rise to lineons and planons. These results reveal new forms of topological order and suggest a broader framework for understanding fracton phases beyond the conventional Z2 setting.

Figures (25)

• Figure 1: $\mathbb{Z}_N$ generalization of the $2d$ toric code. Each link hosts an $N$-dimensional spin. (a) Vertex and plaquette stabilizer terms. (b) A Wilson loop operator for a $\mathbb{Z}_N$ electric charge (blue circle labeled “e”) encircling a magnetic excitation (red circle labeled “m”). The Wilson loop consists of Pauli $Z$ operators with non-uniform exponents $1, a_x, a_x^2, \dots$. The yellow arrow marks the intersection of the electric and magnetic Wilson lines, where their braiding phase is generated. (c) A single electric charge excitation can be created by an electric Wilson loop operator winding around the system in the $\hat{x}$-direction, when $a_x^{L_x} \not\equiv 1 \pmod{N}$. (d) A single magnetic excitation can be created by a magnetic Wilson line operator in the $\hat{x}$-direction when $a_x^r \equiv 0 \pmod{N}$.
• Figure 2: $\mathbb{Z}_2$ X-cube model. A two-dimensional spin lives on each link. (a) Cube stabilizer $S_c$. (b–d) Vertex stabilizers $V_v^{xy}$, $V_v^{yz}$, and $V_v^{zx}$, defined in the corresponding planes. (e) A pair of $\hat{x}$-directional lineons is created by a straight line operator $W_{\mathcal{C}}$, which flips $V_v^{zx} = 1$ and $V_v^{xy} = 1$ at the endpoints of the string $\mathcal{C}$. (f) A lineon dipole along the $\hat{z}$ direction becomes a planon that moves freely in the $xy$-plane. (g) A fracton quadrupole created by a membrane operator $O_{\mathcal{M}}$ in the $zx$-plane. A fracton dipole along $\hat{z}$ becomes a planon in the $xy$-plane when acted upon by membrane operators in $yz$ and $zx$ planes. (h) Top view of braiding a $\hat{z}$-oriented fracton dipole around a $\hat{x}$-oriented lineon. The braiding phase $(-1)$ arises from the intersection of a $Z$ membrane along the dipole's path and the line operator $W_{\mathcal{C}}$, as indicated by the arrow.
• Figure 3: Pictorial representations of the $\mathbb{Z}_N$ generalization of the X-cube model [Eq. (\ref{['eqn : H_gXC']})]. (a) Cube term $S_c$. (b–d) Vertex terms $V_v^{xy}$, $V_v^{yz}$, and $V_v^{zx}$, defined in their respective planes.
• Figure 4: Vertex excitations of $\mathbb{Z}_N$ generalized X-cube model via local string operator $W_\mathcal{C}$ Eq. (\ref{['eqn : app_gXC_line']}). The charge vectors of each excitation are indicated by arrows. (a) A pair of $x$-directional lineons can be created by the straight line operator $W_\mathcal{C}$ with the exponents $\{a_x^{\ell-1}b, a_x^{\ell-2}b,\cdots,a_x b, b\}$. (b) A single vertex excitation can be created by the $x$-directional local line operator $W_\mathcal{C}$ when $b=0$ mod $N_x$. (c) A single vertex excitation can be created by the $z$-directional local line operator $W_\mathcal{C}$ when $b=0$ mod $N_z$. (d) A single vertex excitation can be created by the $y$-directional local line operator $W_\mathcal{C}$ when $b=0$ mod $N_y$.
• Figure 5: Promotion of vertex excitations from lineon to planon. The charge vectors of each excitation are indicated by arrows. (a) The pictorial representation of bending the path of lineon dipole along $x$-axis indicated by dashed loop , from $\hat{z}$-direction to $\hat{y}$-direction. When $b_x=b_y=-a_z^{\ell_z}$ and $b_y'=-a_x^{\ell_x}b_x=-a_z^{\ell_z}b_z'$, lineon pair at bending points vanishes, then the lineon dipole becomes planon which is mobile on $yz$-plane. (b) A single vertex exciation can be looked as a planon instead of along $yz$-plane a lineon when $b_z=0$ mod $N_x$ in Eq. \ref{['eqn : app_gXC_lineon2Planon']}.