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arXiv:2301.00103 · Physical review. B./Physical review. B

Renormalization of Ising cage-net model and generalized foliation

Zongyuan Wang, Xiuqi Ma, David T. Stephen, Michael Hermele, Xie Chen

Published Jul 26, 2023• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.108.035148

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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.
The Ising cage-net model, first proposed by Prem et al. [Phys. Rev. X 9, 021010 (2019)], is a representative type I fracton model with nontrivial non-Abelian features. In this paper, we calculate the ground state degeneracy of this model and find that, even though it follows a similar coupled layer structure as the X-cube model, the Ising cage-net model cannot be ``foliated'' in the same sense as X-cube as defined in Shirley et al. [Phys. Rev. X 8, 031051 (2018)]. A more generalized notion of ``foliation'' is hence needed to understand the renormalization group transformation of the Ising cage-net model. The calculation is done using an operator algebra approach that we develop in this paper, and we demonstrate its validity through a series of examples.
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.
We introduce a class of gapped three-dimensional models, dubbed "cage-net fracton models," which host immobile fracton excitations in addition to non-Abelian particles with restricted mobility.Starting from layers of two-dimensional string-net models, whose spectrum includes non-Abelian anyons, we condense extended one-dimensional "flux strings" built out of pointlike excitations.Flux-string condensation generalizes the concept of anyon condensation familiar from conventional topological order and allows us to establish properties of the fracton-ordered (equivalently, flux-string-condensed) phase, such as its groundstate wave function and spectrum of excitations.Through the examples of doubled-Ising and SUð2Þ k cagenet models, we demonstrate the existence of strictly immobile Abelian fractons and of non-Abelian particles restricted to move only along one dimension.In the doubled-Ising cage-net model, we show that these restricted-mobility non-Abelian excitations are a fundamentally three-dimensional phenomenon, as they cannot be understood as bound states among two-dimensional non-Abelian anyons and Abelian particles.We further show that the ground-state wave function of such phases can be understood as a fluctuating network of extended objects-cages-and strings, which we dub a cage-net condensate.Besides having implications for topological quantum computation in three dimensions, our work may also point the way towards more general insights into quantum phases of matter with fracton order.
We present a framework for non-Abelian p-loop, fracton, and planon condensation in 3+1 dimensions by constructing extended cage-net fracton models using decoupled layers of the Hu-Geer-Wu (HGW) string-net model. These cage-net models extend the conventional cage-net models based on the Levin-Wen (LW) string-net model in the sense that they inherit the tail degrees of freedom of the HGW models, which are essential for completely describing the internal spaces of quasiparticles. This approach allows us to explicitly derive the quasiparticle spectra of the cage-net models by projecting those of the parent 2D HGW layers. Utilizing this framework, we can condense the p-loops formed by non-Abelian anyons within a fracton phase. Specifically, we construct the condensation projector for (σσˉ,1)(σ\barσ, 1)(σσˉ,1)-loops within the extended Ising Cage-Net (ICN) model. We demonstrate that condensing these non-Abelian loops drives a phase transition that maps the ICN model to the X-cube (XC) model defined on a truncated cubic lattice, a process that explicitly reveals the splitting of non-Abelian planons into distinct sub-dimensional excitations. Furthermore, our framework extends to the condensation of fractons and planons: we demonstrate that in the ICN model fracton condensation drives the decoupling of the 3D fracton order back into isolated 2D topological order layers, while planon condensation collapses the system entirely into a trivial phase. Our results establish a concrete Hamiltonian mechanism for phase transitions between distinct fracton orders and provide a generalizable method for analyzing the evolution of sub-dimensional excitations.
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.
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.