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arXiv:1709.10094 · Physical review. B./Physical review. B

Simple anisotropic three-dimensional quantum spin liquid with fractonlike topological order

Olga Petrova, Nicolas Regnault

Published Dec 26, 2017• 48 citations
arXivHTMLPDF

Abstract

We present a three-dimensional cubic lattice spin model, anisotropic in the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction, that exhibits fractonlike order. This order can be thought of as the result of interplay between two-dimensional ${\mathbb{Z}}_{2}$ topological order and spontaneous symmetry breaking along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction. Fracton order is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground-state degeneracy: On an ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}\ifmmode\times\else\texttimes\fi{}{L}_{z}$ three-torus, it has a ${2}^{2{L}_{z}}$ topological degeneracy and an additional symmetry-breaking nontopological degeneracy equal to ${2}^{{L}_{x}{L}_{y}\ensuremath{-}2}$. The fractons can be combined into composite excitations that move either in a straight line along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ direction or freely in the $xy$ plane at a given height $z$. While our model draws inspiration from the toric code, we demonstrate that it cannot be adiabatically connected to a layered toric code construction. Additionally, we investigate the effects of imposing open boundary conditions on our system. We find zero energy modes on the surfaces perpendicular to either the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}$ or $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}$ directions and their absence on the surfaces normal to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$. This result can be explained using the properties of the two kinds of composite two-fracton mobile excitations.

DOI
10.1103/physrevb.96.224429

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We present a three-dimensional cubic lattice spin model, anisotropic in the z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi direction, that exhibits fractonlike order. This order can be thought of as the result of interplay between two-dimensional Z2{\mathbb{Z}}_{2}Z2​ topological order and spontaneous symmetry breaking along the z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi direction. Fracton order is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground-state degeneracy: On an Lx\ifmmode×\else\texttimes\fiLy\ifmmode×\else\texttimes\fiLz{L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}\ifmmode\times\else\texttimes\fi{}{L}_{z}Lx​\ifmmode×\else\texttimes\fiLy​\ifmmode×\else\texttimes\fiLz​ three-torus, it has a 22Lz{2}^{2{L}_{z}}22Lz​ topological degeneracy and an additional symmetry-breaking nontopological degeneracy equal to 2LxLy\ensuremath−2{2}^{{L}_{x}{L}_{y}\ensuremath{-}2}2Lx​Ly​\ensuremath−2. The fractons can be combined into composite excitations that move either in a straight line along the z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi direction or freely in the xyxyxy plane at a given height zzz. While our model draws inspiration from the toric code, we demonstrate that it cannot be adiabatically connected to a layered toric code construction. Additionally, we investigate the effects of imposing open boundary conditions on our system. We find zero energy modes on the surfaces perpendicular to either the x\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}x\ifmmode^\elseˆ\fi or y\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}y\ifmmode^\elseˆ\fi​ directions and their absence on the surfaces normal to z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi. This result can be explained using the properties of the two kinds of composite two-fracton mobile excitations.
We propose a coupled-layer construction of a class of fracton topological orders in three spatial dimensions, which has no immobile excitations but is characterized by single quasiparticle excitations constrained in one-dimensional subspaces and dipole excitations mobile in two-dimensional subspaces. The simplest model is obtained by stacking and coupling layers of the two-dimensional toric codes on the square lattice and can be exactly solved in the strong-coupling limit. The resulting subdimensional excitations are understood as a consequence of anyon pair condensation induced by the coupling between layers. We also present generalizations of the construction for layers of the Kitaev-honeycomb models, the ZN{Z}_{N}ZN​ toric codes, and the toric codes and the doubled semion models on the honeycomb lattice.
We study models with fractonlike order based on Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy SA{S}_{A}SA​ of these models in a subregion AAA of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to AAA.
Starting from an isotropic configuration of intersecting, two-dimensional toric codes, we construct a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point- like topological excitations ("fractons"), and degenerate ground-states on the torus that are locally indistinguishable. Our proposal leads to a simple description of the fracton excitations and of the ground-state as a "loop" condensate, and provides a basis for building new 3D topological orders such as a natural, ZNZ_{N}ZN​ generalization of this fracton phase, which we introduce. We describe the rich phase structure of our layered ZNZ_{N}ZN​ system. By invoking a lattice duality, we demonstrate that when N≥5N \ge 5N≥5, there is an intermediate phase that appears between the decoupled, layered system and the fracton topologically-ordered state, which opens the possibility of a continuous transition into the fracton topological phase. We conclude by presenting a solvable model, that interpolates between the fracton phase and a confined phase in which the phase transition is first-order.
In three dimensions, gapped phases can support "fractonic" quasiparticle excitations, which are either completely immobile or can only move within a low-dimensional submanifold, a peculiar topological phenomenon going beyond the conventional framework of topological quantum field theory.In this work we explore fractonic topological phases using three-dimensional coupled wire constructions, which have proven to be a successful tool to realize and characterize topological phases in two dimensions.We find that both gapped and gapless phases with fractonic excitations can emerge from the models.In the gapped case, we argue that fractonic excitations are mobile along the wire direction, but their mobility in the transverse plane is generally reduced.We show that the excitations in general have infinite-order fusion structure, distinct from previously known gapped fracton models.Like the two-dimensional coupled-wire constructions, many models exhibit gapless (or even chiral) surface states, which can be described by infinite-component Luttinger liquids.However, the universality class of the surface theory strongly depends on the surface orientation, thus revealing a different type of bulk-boundary correspondence unique to 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.
In this work, we develop a coupled layer construction of fracton topological orders in d=3d=3d=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=2d=2d=2 topological phases and from stacks of d=3d=3d=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.