Fracton topological order via coupled layers

In this work, we develop a coupled layer construction of fracton topological orders in $d=3$ spatial dimensions. These topological phases have sub-extensive 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.Comment: 20 pages, 18 figures, published versio

Symmetry-protected self-correcting quantum memories

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D "cluster-state" model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally.Comment: 39 pages, 16 figures, comments welcome; v2 includes much more explicit detail on the main example model, including boundary conditions and implementations of logical operators through local moves; v3 published versio

Strong planar subsystem symmetry-protected topological phases and their dual fracton orders

We classify subsystem symmetry-protected topological (SSPT) phases in 3 + 1 dimensions (3 + 1D) protected by planar subsystem symmetries: short-range entangled phases which are dual to long-range entangled Abelian fracton topological orders via a generalized “gauging” duality. We distinguish between weak SSPTs, which can be constructed by stacking 2 + 1D SPTs, and strong SSPTs, which cannot. We identify signatures of strong phases, and show by explicit construction that such phases exist. A classification of strong phases is presented for an arbitrary finite Abelian group. Finally, we show that fracton orders realizable via p-string condensation are dual to weak SSPTs, while those dual to strong SSPTs exhibit statistical interactions prohibiting such a realization

Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$u_t = u_{xx} + f(x,u,u_x)\,,$$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria $v_t=0$ are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds. An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries $x=0$ and $x=1$, respectively, and the overlapping reach of polar serpents in the shooting meander. In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors $\mathcal{A}$ with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.Comment: 73+(ii) pages, 40 figures, 14 table; see also parts 1 and 2 under arxiv:1611.02003 and arxiv:1704.0034

Measurement-Based Quantum Computation on Symmetry Breaking Thermal States

We consider measurement-based quantum computation (MBQC) on thermal states of the interacting cluster Hamiltonian containing interactions between the cluster stabilizers that undergoes thermal phase transitions. We show that the long-range order of the symmetry breaking thermal states below a critical temperature drastically enhance the robustness of MBQC against thermal excitations. Specifically, we show the enhancement in two-dimensional cases and prove that MBQC is topologically protected below the critical temperature in three-dimensional cases. The interacting cluster Hamiltonian allows us to perform MBQC even at a temperature an order of magnitude higher than that of the free cluster Hamiltonian.Comment: 8 pages, 7 figure

Entanglement entropy of (3+1)D topological orders with excitations

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group $G$ and its group 4-cocycle $\omega\in\mathcal{H}^4[G;U(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant $\ln d_i$ to the entanglement entropy, where $d_i$ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\,\omega)$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\,\omega)$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\omega$ from the others.Comment: 12 pages, 4 figures; v2: minor changes, published versio

The decomposition of the hypermetric cone into L-domains

The hypermetric cone \HYP_{n+1} is the parameter space of basic Delaunay polytopes in n-dimensional lattice. The cone \HYP_{n+1} is polyhedral; one way of seeing this is that modulo image by the covariance map \HYP_{n+1} is a finite union of L-domains, i.e., of parameter space of full Delaunay tessellations. In this paper, we study this partition of the hypermetric cone into L-domains. In particular, it is proved that the cone \HYP_{n+1} of hypermetrics on n+1 points contains exactly {1/2}n! principal L-domains. We give a detailed description of the decomposition of \HYP_{n+1} for n=2,3,4 and a computer result for n=5 (see Table \ref{TableDataHYPn}). Remarkable properties of the root system $\mathsf{D}_4$ are key for the decomposition of \HYP_5.Comment: 20 pages 2 figures, 2 table