We investigate two-dimensional compactifications of three-dimensional fractonic stabilizer models. We find the two-dimensional topological phases produced as a function of compactification radius for the $X$-cube model and Haah's cubic code. Furthermore, we uncover translation symmetry enrichment in the compactified cubic code that leads to twisted boundary conditions.
We investigate two-dimensional compactifications of three-dimensional fractonic stabilizer models. We find the two-dimensional topological phases produced as a function of compactification radius for the X-cube model and Haah's cubic code. Furthermore, we uncover translation symmetry enrichment in the compactified cubic code that leads to twisted boundary conditions.
The authors generalize here three-dimensional \ensuremath{\mathbb{Z}}2 stabilizer models---the X-cube, toric code, and Haah's code---to their \ensuremath{\mathbb{Z}}N counterparts, revealing size-dependent ground-state degeneracy and novel excitations. They demonstrate that the generalized X-cube hosts quasi-fractons that remain immobile under local operations but become mobile via global ones. In the toric code, both closed-loop and open-string excitations emerge, while Haah's code features new fracton tripole and monopole excitations. These findings uncover unconventional topological orders and broaden the understanding of fracton phases beyond the conventional \ensuremath{\mathbb{Z}}2 framework.
ZN generalizations of three-dimensional stabilizer codes
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.
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.
Topological holography (TH), or SymTFT, realizes symmetries and dualities of a quantum system as boundary data of a topological bulk in one higher dimension. We formulate fracton topological holography (FTH), extending this mechanism from liquid topological orders to fracton stabilizer codes. The construction is organized as a general four-stage framework: prepare the bulk model and compute its excitations, determine boundary data and admissible gapped top boundaries, identify the low-energy preserving operator algebra together with its symmetry, relation, and twist data, and then switch among top boundaries to compare the induced boundary descriptions. As a type-I example, we develop FTH for the X-cube model with smooth and rough top boundaries; for a minimal effective Hamiltonian, both yield transverse-field plaquette Ising models, with exchanged subsystem symmetry and twist data, and the boundary switch is implemented by a linear-depth local unitary sequential quantum circuit (SQC). As a type-II example, we formulate FTH for Haah's cubic code in the Laurent-polynomial stabilizer formalism and analyze the natural (Z) and (X) top boundaries, which induce two two-dimensional qubit systems related locally by exchanging generalized plaquette Ising and transverse-field terms and nonlocally by a symmetry--relation duality. These results show that FTH is a genuine extension of TH to both type-I and type-II fracton orders. FTH therefore provides a concrete framework for organizing and understanding duality, with the prospect of offering a systematic route to new dualities.
We study several exotic systems, including the X-cube model, on a flat three-torus with a twist in the xy plane. The ground-state degeneracy turns out to be a sensitive function of various geometrical parameters. Starting from a lattice, depending on how we take the continuum limit, we find different values of the ground-state degeneracy. Yet, there is a natural continuum limit with a well-defined (though infinite) value of that degeneracy. We also uncover a surprising global symmetry in 2+1 and 3+1 dimensional systems. It originates from the underlying subsystem symmetry but the way it is realized depends on the twist. In particular, in a preferred coordinate frame, the modular parameter of the twisted two-torus $\ensuremath{\tau}={\ensuremath{\tau}}_{1}+i{\ensuremath{\tau}}_{2}$ has rational ${\ensuremath{\tau}}_{1}=k/m$. Then, in systems based on $\text{U}(1)\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ subsystem symmetries, such as momentum and winding symmetries or electric and magnetic symmetries, this symmetry is a projectively realized ${\mathbb{Z}}_{m}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{m}$, which leads to an m-fold ground state degeneracy. In systems based on ZN symmetries, like the X-cube model, each of these two Zm factors is replaced by Zgcd(N,m).
In this work, we develop a coupled layer construction of fracton topological orders in d=3 spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple d=2 topological phases and from stacks of d=3 fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.