Symmetry-enriched topological order and quasi-fractonic behavior in $\mathbb{Z}_N$ stabilizer codes

TL;DR

This paper develops a unifying framework for symmetry-enriched topological order in $ abla_N$ BB stabilizer codes by using a polynomial (Laurent) representation over $\mathbb{Z}_N[x^{\pm},y^{\pm}]$. A central result is that topological order and fusion rules for general $N$ can be determined from the corresponding codes with prime qudit dimensions, enabling a direct extension of the Bernstein–Khovanskii–Kushnirenko (BKK) approach to fusion rules for qudits. The authors also show that these BB codes exhibit SET-induced quasi-fractonic mobility and size-dependent ground-state degeneracy, resolved via a mobility group and an exact GSD formula on a torus. A computational algebraic method based on Gröbner bases over the ring of integers is introduced to efficiently compute topological order and SET properties for explicit models, with concrete results for the WCF and DCY models and a toric BB code. Together, these results provide a practical roadmap for analyzing and constructing SET-enabled qudit stabilizer codes across arbitrary $N$.

Abstract

We study a broad class of qudit stabilizer codes, termed $\mathbb{Z}_N$ bivariate-bicycle (BB) codes, arising either as two-dimensional realizations of modulated gauge theories or as $\mathbb{Z}_N$ generalizations of binary BB codes. Our central finding, derived from the polynomial representation, is that the essential topological properties of these $\mathbb{Z}_N$ codes can be determined by the properties of their $\mathbb{Z}_p$ counterparts, where $p$ are the prime factors of $N$, even when $N$ contains prime powers ($N = \prod_i p_i^{k_i}$). This result yields a significant simplification by leveraging the well-studied framework of codes with prime qudit dimensions. In particular, this insight directly enables the generalization of the algebraic-geometric methods (e.g., the Bernstein-Khovanskii-Kushnirenko theorem) to determine anyon fusion rules in the general qudit situation. Moreover, we analyze the model's symmetry-enriched topological order (SET) to reveal a quasi-fractonic behavior, resolving the anyon mobility puzzle in this class of models. We also present a computational algebraic method using Gröbner bases over the ring of integers to efficiently calculate the topological order and its SET properties.

Figures (7)

• Figure 1: Illustration of the WCF model. (a) The $X$-type stabilizer $A_{(i,j)}^{\operatorname{WCF}}$. (b) The $Z$-type stabilizer $B_{(i,j)}^{\operatorname{WCF}}$. (c) The minimal coupling terms between gauge and matter qudits. (d) The zero flux terms (yellow) and the gauging symmetry operators (light blue). The dark blue and the orange dots correspond to the gauge-qudit Pauli $X$ and $Z$ operators on the edges, respectively. Similarly, the red and purple squares indicate the matter-qudit operators $\sigma_x$ and $\sigma_z$ at the vertices. The explicit forms of each operator are shown next to their symbols. We omit the coordinates of operators in (d) for brevity.
• Figure 2: Illustration of the DCY model. (a) the $X$-type stabilizer $A_{(i,j)}^{\operatorname{DCY}}$. (b) the $Z$-type stabilizer $B_{(i,j)}^{\operatorname{DCY}}$. In (a) and (b), the location of the stabilizers is marked by the coordinate $(i,j)$ in each figure, while their constituent Pauli operators are shown without explicit coordinates. (c) The minimal coupling terms between gauge and matter qudits. (d) The zero flux condition (yellow) and the gauging symmetry operators (light blue) with all coordinate labels omitted. Throughout the figures, deep blue dots, orange dots, red squares, and purple squares represent the $X$, $Z$, $\sigma^z$, and $\sigma^x$ operators, respectively.
• Figure 3: The quasi-fractonic behavior of the DCY model. Here $N=4$ and $a=1$. This figure depicts the process of moving excitations in the DCY model. The gray dots represent $Z$-type stabilizers at each position, where we only draw the stabilizers along the $x$-directions in each step. The yellow dots represent the excitations, denoted by the monomials next to the dots. In Step 1, the anyonic excitations are split by moving. After Step 3, the original excitation is moved from $(0,0)$ to $(4,0)$.
• Figure 4: Illustration of the $X$-type (a) and the $Z$-type (b) stabilizers of the bivariate-bicycle code specified by $f=c^{(0)}+c^{(1)}x+c^{(2)}x^{\bar{\alpha}}y^{b}$ and $g=d^{(0)}+d^{(1)}y+d^{(2)}x^{a}y^{\bar{\beta}}$. On the lattice, a unit cell at the vertex $(i,j)$ consists of two qudits on the edges $(i,j+\frac{1}{2})$ and $(i+\frac{1}{2},j)$. The deep blue and orange circles on the lattice edges represent the constituent $X$ and $Z$ Pauli operators, respectively. Each operator is labeled by a monomial in which the exponents of $x$ and $y$ denote the unit cell coordinates, and the coefficient gives the operator's power.
• Figure 5: The topological index of the $\mathbf{BB}_p(f,g)$ code with $f$ and $g$ from Eq. \ref{['eq:tc_layout']} for two special cases: (a) all coefficients are non-zero modulo $p$. (b) only $d^{(1)}$ modulo $p$ is zero. For both cases, the parameter space is divided into five regimes by four hyperbolas, and the corresponding expression of the topological index in each regime is indicated. Parameters are restricted to $\beta\ge\alpha\ge0$ for (a), and to $\alpha,\beta\ge 0$ and $a\le 0$ for (b), as all other values can be reduced to this range by suitable origin shifts and axis reflections.

Theorems & Definitions (20)

• Definition 1
• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3: McCoy's Theorem, Mccoy1942Mccoy1957
• Lemma 1
• proof
• Lemma 2
• proof
• proof : Proof of Theorem \ref{['lem:topo_prime_factors']}