Fragmented Topological Excitations in Generalized Hypergraph Product Codes

TL;DR

This work constructs a family of exactly solvable spin models, termed orthoplex models, from generalized hypergraph product codes and reveals novel fracton physics across dimensions. In 3D, the model exhibits a Type-I fracton phase with a non-monotonic ground state degeneracy and non-Abelian lattice defects tied to dislocations. In 4D, it uncovers fragmented loop excitations where microscopic point-like constituents assemble into a topologically connected loop upon projection, highlighting a new intermediate class of excitations. The framework extends to arbitrary dimensions via a generalized HGP protocol, offering a versatile platform to study fracton orders and potential implications for quantum error correction and hidden topological structures.

Abstract

Product code construction is a powerful tool for constructing quantum stabilizer codes, which serve as a promising paradigm for realizing fault-tolerant quantum computation. Furthermore, the natural mapping between stabilizer codes and the ground states of exactly solvable spin models also motivates the exploration of many-body orders in the stabilizer codes. In this work, we investigate the fracton topological orders in a family of codes obtained by a recently proposed general construction. More specifically, this code family can be regarded as a class of generalized hypergraph product (HGP) codes. We term the corresponding exactly solvable spin models \textit{orthoplex models}, based on the geometry of the stabilizers. In the 3D orthoplex model, we identify a series of intriguing properties within this model family, including non-monotonic ground state degeneracy (GSD) as a function of system size and non-Abelian lattice defects. Most remarkably, in 4D we discover \textit{fragmented topological excitations}: while such excitations manifest as discrete, isolated points in real space, their projections onto lower-dimensional subsystems form connected objects such as loops, revealing the intrinsic topological nature of these excitations. Therefore, fragmented excitations constitute an intriguing intermediate class between point-like and spatially extended topological excitations. In addition, these rich features establish the generalized HGP codes as a versatile and analytically tractable platform for studying the physics of fracton orders.

Figures (10)

• Figure 1: A schematic picture of fragmented topological excitations. In (a), for comparison, we show a standard loop excitations with orange color, where excitation energy is distributed along the loop. In (b), we show a fragmented loop excitation composed of point-like excitations scattered in a 3D space, while actually its projection onto the 2D plane $P$ forms a connected loop (the dashed orange circle), revealing its topological nature.
• Figure 2: Cells in a 3D cubic lattice. In (a), we show a $0$-cell, a $1$-cell and a $2$-cell, all colored blue. In (b), we show a $1$-cell $\gamma_1$ (colored red) and the set of nearest cells of it $\Delta \gamma_1$, that is composed of two vertices (colored orange) and four plaquettes (colored blue).
• Figure 3: Hamiltonian and excitations in 3D orthoplex model. In (a), we show two representative Hamiltonian terms (i.e., stabilizers). Qubits involved in the $A$ term ($X$-stabilizer) and $B$ term are respectively colored red and blue, and the two qubits shared by the two terms are colored purple. In (b), we demonstrate a lineon $\mathbf{l}^e$ and a planon $\mathbf{p}^m$, each is generated by a string composed of Pauli $X$ operators denoted by the gray dotted line, with red filled circles specifying the qubits acted by the Pauli $X$ operators. For clarity, the mobile regions of the two excitations are schematically highlighted by the transparent green regions.
• Figure 4: Non-Abelian line defect in 3D orthoplex model. In (a) and (b), we demonstrate modified stabilizers around the removed qubits far from and on the dislocation line, respectively, where the modified stabilizers are obtained by taking the products of the truncated $A$ and $B$ stabilizers. Note that in (a), the modified stabilizer preserve the original orthoplex geometry but is "stretched". Qubits involved in $A$, $B$ and both of them are respectively colored red, blue and purple. In (c) and (d), we demonstrate how a planon changes its type (i.e., $\mathbf{p}^e \leftrightarrow \mathbf{p}^m$) when it circles the dislocation line, as it must have crossed the half plane of removed qubits for an odd number of times.
• Figure 5: Hamiltonian of 4D orthoplex model. In (a) we show a representative $A_{\gamma^{yz}_2}$ term that involves eight qubits on nearest $\gamma_1$ and $\gamma_3$. In (b), we show a representative $B_{\gamma_0}$ term that involves eight qubits on nearest $\gamma_1$. In both cases, the qubits form vertices of a 4D orthoplex. For clarity, we draw links along direction $w$ by gray lines.