Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1806.08633 · Physical review. B./Physical review. B

Foliated fracton order in the checkerboard model

Wilbur E. Shirley, Kevin Slagle, Xie Chen

Published Mar 18, 2019• 56 citations
arXivHTMLPDF

Abstract

In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization-group transformation for the model that utilizes toric code bilayers as an entanglement resource and show how to extend the model to general three-dimensional manifolds. Furthermore, we use universal properties distilled from the structure of fractional excitations and ground-state entanglement to characterize the foliated fracton phase and find that it is the same as two copies of the X-cube model. Indeed, we demonstrate that the checkerboard model can be transformed into two copies of the X-cube model via an adiabatic deformation.

DOI
10.1103/physrevb.99.115123

Related papers

Foliated fracton order in the Majorana checkerboard model

Taige Wang, Wilbur E. Shirley, Xie Chen

We establish the presence of foliated fracton order in the Majorana checkerboard model. In particular, we describe an entanglement renormalization group transformation which utilizes toric code layers as resources of entanglement and furthermore discuss entanglement signatures and fractional excitations of the model. In fact, we give an exact local unitary equivalence between the Majorana checkerboard model and the semionic X-cube model augmented with decoupled fermionic modes. This mapping demonstrates that the model lies within the X-cube foliated fracton phase.

arXiv:1904.01111•Physical review. B./Physical review. B•Aug 16, 2019•35 citations

Planon-modular fracton orders

Evan Wickenden, Marvin Qi, Arpit Dua +1 more

There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.

arXiv:2412.14320•Physical review. B./Physical review. B•Jun 2, 2025•0 citations

Twisted foliated fracton phases

Wilbur E. Shirley, Kevin Slagle, Xie Chen

In the study of three-dimensional gapped models, two-dimensional gapped states should be considered as a free resource. This is the basic idea underlying the notion of ``foliated fracton order'' proposed in Shirley et al. [Phys. Rev. X 8, 031051 (2018)]. We have found that many of the known type-I fracton models, although they appear very different, have the same foliated fracton order, known as ``X-cube'' order. In this paper, we identify three-dimensional fracton models with different kinds of foliated fracton order. Whereas the X-cube order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry, the different orders correspond to twisted versions of the gauge theory for which the system prior to gauging has nontrivial order protected by the planar subsystem symmetry. We present constructions of the twisted models and demonstrate that they possess nontrivial order by studying their fractional excitation contents.

arXiv:1907.09048•Physical review. B./Physical review. B•Sep 2, 2020•73 citations

Universal entanglement signatures of foliated fracton phases

Wilbur Shirley, Kevin Slagle, Xie Chen

Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In , we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-partite entanglement quantities, generalizing the proposal of topological entanglement entropy designed for conventional topological phases. We present arguments for the universality of these quantities and show that they attain non-zero constant value in non-trivial foliated fracton phases.

arXiv:1803.10426•SciPost Physics•Jan 31, 2019•86 citations

Fracton and topological order in the XY checkerboard toric code

Maximilian Vieweg, Kai Phillip Schmidt

We introduce the XY checkerboard toric code. It represents a generalization of the Z 2 toric code with two types of star operators with x and y flavor and two anisotropic star sublattices forming a checkerboard lattice. The quantum phase diagram is deduced exactly by a duality transformation to two copies of self-dual Xu-Moore models, which builds on the existence of a subextensive number of Z 2 conserved parities. For any spatial anisotropy of the star sublattices, the XY checkerboard toric code realizes two quantum phases with Z 2 topological order and an intermediate phase with type-I fracton order. The properties of the fracton phase such as subextensive ground-state degeneracy can be analytically deduced from the degenerate limit of isolated stars. For the spatially isotropic case the extension of the fracton phase vanishes. The topological phase displays anyonic excitations with restricted mobility and dimensional reduction. All phase transitions are first order.

arXiv:2411.19051•Physical Review Research•Jun 23, 2025•4 citations

Renormalization of Ising cage-net model and generalized foliation

Zongyuan Wang, Xiuqi Ma, David T. Stephen +2 more

A large class of type-I fracton models, including the X-cube model, have been found to be fixed points of the foliated renormalization group (RG). The system size of such foliated models can be changed by adding or removing decoupled layers of 2D topological states and continuous deformation of the Hamiltonian. In this paper, we study a closely related model---the Ising cage-net model---and find that this model is not foliated in the same sense. In fact, we point out certain unnatural restrictions in the foliated RG, and find that removing these restrictions leads to a generalized foliated RG under which the Ising cage-net model is a fixed point, and which includes the original foliated RG as a special case. The Ising cage-net model thus gives a prototypical example of the generalized foliated RG, and its system size can be changed either by condensing/uncondensing bosonic planon excitations near a 2D plane or through a linear-depth quantum circuit in the same plane. We show that these two apparently different RG procedures are closely related, as they lead to the same gapped boundary when implemented in part of a plane. Finally, we briefly discuss the implications for foliated fracton phases, whose universal properties will need to be reexamined in light of the generalized foliated RG.

arXiv:2301.00103•Physical review. B./Physical review. B•Jul 26, 2023•9 citations
In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization-group transformation for the model that utilizes toric code bilayers as an entanglement resource and show how to extend the model to general three-dimensional manifolds. Furthermore, we use universal properties distilled from the structure of fractional excitations and ground-state entanglement to characterize the foliated fracton phase and find that it is the same as two copies of the X-cube model. Indeed, we demonstrate that the checkerboard model can be transformed into two copies of the X-cube model via an adiabatic deformation.
We establish the presence of foliated fracton order in the Majorana checkerboard model. In particular, we describe an entanglement renormalization group transformation which utilizes toric code layers as resources of entanglement and furthermore discuss entanglement signatures and fractional excitations of the model. In fact, we give an exact local unitary equivalence between the Majorana checkerboard model and the semionic X-cube model augmented with decoupled fermionic modes. This mapping demonstrates that the model lies within the X-cube foliated fracton phase.
There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.
In the study of three-dimensional gapped models, two-dimensional gapped states should be considered as a free resource. This is the basic idea underlying the notion of ``foliated fracton order'' proposed in Shirley et al. [Phys. Rev. X 8, 031051 (2018)]. We have found that many of the known type-I fracton models, although they appear very different, have the same foliated fracton order, known as ``X-cube'' order. In this paper, we identify three-dimensional fracton models with different kinds of foliated fracton order. Whereas the X-cube order corresponds to the gauge theory of a simple paramagnet with subsystem planar symmetry, the different orders correspond to twisted versions of the gauge theory for which the system prior to gauging has nontrivial order protected by the planar subsystem symmetry. We present constructions of the twisted models and demonstrate that they possess nontrivial order by studying their fractional excitation contents.
Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In , we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-partite entanglement quantities, generalizing the proposal of topological entanglement entropy designed for conventional topological phases. We present arguments for the universality of these quantities and show that they attain non-zero constant value in non-trivial foliated fracton phases.
We introduce the XY checkerboard toric code. It represents a generalization of the Z 2 toric code with two types of star operators with x and y flavor and two anisotropic star sublattices forming a checkerboard lattice. The quantum phase diagram is deduced exactly by a duality transformation to two copies of self-dual Xu-Moore models, which builds on the existence of a subextensive number of Z 2 conserved parities. For any spatial anisotropy of the star sublattices, the XY checkerboard toric code realizes two quantum phases with Z 2 topological order and an intermediate phase with type-I fracton order. The properties of the fracton phase such as subextensive ground-state degeneracy can be analytically deduced from the degenerate limit of isolated stars. For the spatially isotropic case the extension of the fracton phase vanishes. The topological phase displays anyonic excitations with restricted mobility and dimensional reduction. All phase transitions are first order.
A large class of type-I fracton models, including the X-cube model, have been found to be fixed points of the foliated renormalization group (RG). The system size of such foliated models can be changed by adding or removing decoupled layers of 2D topological states and continuous deformation of the Hamiltonian. In this paper, we study a closely related model---the Ising cage-net model---and find that this model is not foliated in the same sense. In fact, we point out certain unnatural restrictions in the foliated RG, and find that removing these restrictions leads to a generalized foliated RG under which the Ising cage-net model is a fixed point, and which includes the original foliated RG as a special case. The Ising cage-net model thus gives a prototypical example of the generalized foliated RG, and its system size can be changed either by condensing/uncondensing bosonic planon excitations near a 2D plane or through a linear-depth quantum circuit in the same plane. We show that these two apparently different RG procedures are closely related, as they lead to the same gapped boundary when implemented in part of a plane. Finally, we briefly discuss the implications for foliated fracton phases, whose universal properties will need to be reexamined in light of the generalized foliated RG.