The authors present a generic lattice construction for a generalized X-cube model of fracton topological order and propose a coarser definition of phase of matter for fracton orders. Similar to other fracton models, this model hosts emergent subdimensional excitations, which are topologically robust excitations that are confined to only move along certain lines or surfaces. The model is constructed from stacks of intersecting surfaces, and the subdimensional excitations are confined to move only along these (possibly curved) surfaces or their intersections. Surprisingly, for certain curved intersecting surfaces, a robust ground-state degeneracy on a manifold with trivial topology can result.
As an unconventional realization of topological orders with an exotic interplay of topology and geometry, fracton (topological) orders feature subextensive topological ground-state degeneracy and subdimensional excitations that are movable only within a certain subspace. It has been known, in the exactly solvable three-dimensional X-cube model that universally represents the type-I fracton orders, that mobility constraints on subdimensional excitations originate from the absence of spatially deformable stringlike operators. To unveil the interplay of topology and geometry, in this paper, we study the dynamical signature in the X-cube model in the presence of external Zeeman fields via large-scale quantum Monte Carlo simulation and stochastic analytic continuation. We compute both real-space correlation functions and dynamic structure factors of subdimensional excitations (i.e., fractons, lineons, and planons) in the fracton phase and their evolution into the trivial paramagnetic phase by increasing external fields. We find, in the fracton phase, that the correlation functions and the spectral functions show clear anisotropy exactly caused by the underlying mobility constraints. On the other hand, the external fields successfully induce quantum fluctuations and offer mobility to excitations along the subspace allowed by mobility constraints. These numerical results provide the evolution of a dynamical signature of subdimensional particles in fracton orders, indicating that the mobility constraints on local dynamical properties of subdimensional excitations are deeply related to the existence of fracton topological order. The results will also be helpful in potential experimental identifications in spectroscopy measurements such as neutron scattering and nuclear magnetic resonance.
We study quantum phase transitions out of the fracton ordered phase of the
ZN X-cube model. These phase transitions occur when various types of subdimensional excitations and their composites are condensed. The condensed phases are either trivial paramagnets, or are built from stacks of
D=2 or 3 deconfined gauge theories, where
D is the spatial dimension. The nature of the phase transitions depends on the excitations being condensed. Upon condensing dipolar bound states of fractons or lineons, for $N\ensuremath{\ge}4$ we find stable critical points described by decoupled stacks of
D=2 conformal field theories. Upon condensing lineon excitations, when
N>4 we find a gapless phase intermediate between the X-cube and condensed phases, described as an array of
D=1 conformal field theories. In all these cases, effective subsystem symmetries arise from the mobility constraints on the excitations of the X-cube phase and play an important role in the analysis of the phase transitions.
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.
We generalize the Hyperbolic Fracton Model from the
{5,4} tessellation to generic tessellations, and investigate its core properties: subsystem symmetries, fracton mobility, and holographic correspondence. While the model on the original tessellation has features reminiscent of the flat-space lattice cases, the generalized tessellations exhibit a far richer and more intricate structure. The ground-state degeneracy and subsystem symmetries are generated recursively layer-by-layer, through the inflation rule, but without a simple, uniform pattern. The fracton excitations follow exponential-in-distance and algebraic-in-lattice-size growing patterns when moving outward, and depend sensitively to the tessellation geometry, differing qualitatively from both type-I or type-II fracton model on flat lattices. Despite this increased complexity, the hallmark holographic features -- subregion duality via Rindler reconstruction, the Ryu-Takayanagi formula for mutual information, and effective black hole entropy scaling with horizon area -- remain valid. These results demonstrate that the holographic correspondence in fracton models persists in generic tessellations, and provide a natural platform to explore more intricate subsystem symmetries and fracton physics.
Fracton topological order hosts fractionalized point-like excitations (e.g., fractons) that have restricted mobility. In this paper, we explore even more bizarre realization of fracton phases that admit spatially extended excitations with restricted mobility. We construct exactly solvable lattice models in all spatial dimensions
D and divide excitations into and ones. At infrared scales, simple excitations are geometrically manifold-like, e.g., points, strings, membranes, and so on. We introduce a useful notation---a pair of integers
(n,m)---to label simple excitations.
n=0,1,2,⋯ represent particles, strings, membranes,
⋯, respectively;
m is the dimension of the submanifold where mobility of excitations is allowed. With this notation, fractons are labeled by
(0,0) and the subdimensional mobility of spatially extended excitations can be discussed systematically. On the other hand, complex excitations at infrared scales are geometrically non-manifold-like, whose physical characterization (e.g., creation operators, excitation energy, fusion rules, mobility) is far more intricate than that of simple excitations. Several concrete models are analyzed in details. For example, complex excitations dubbed chairons and yuons are constructed in
D=4. In
D=5, except chairons, cloverions and xuons are found. Finally, we construct a family tree in which parent models and descendent models share excitation DNA. Several implications in, e.g., condensed matter physics and gravity are briefly discussed.