Fractonic phases of matter are novel quantum ground states supporting sub-dimensional emergent excitations with mobility restrictions and/or immobile fractons. The ground state degeneracy of such phases is sub-extensive and depends on the geometry of the underlying lattice. Due to these unusual properties, fractonic phases are considered as models for quantum memory or as examples of quantum glassy behaviors. While there exist a number of exactly solvable models with interactions between multiple particles/spins (twelve or more), the realization of such models in real materials is extremely challenging. In this work, we provide a realistic quantum model of quadratic spin interactions on the breathing pyrochlore lattice, inspired by a classical spin model studied earlier. We show that the emergent excitations in this model are immobile when they are present alone. They can only move as a cluster or when they reside at the corners of a membrane excitation. Using the membrane operators acting on the ground state manifold, we construct degenerate ground states with periodic boundary conditions. It is shown that the ground state degeneracy explicitly depends on the lattice geometry. We discuss the implications of these results in light of the rank-2 tensor gauge theory.
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.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.
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.
For nearly three decades, research on frustrated magnetism in three dimensions (3D) has centered on the pyrochlore lattice of corner-sharing tetrahedra and the classical spin liquid (CSL) known as spin ice. We propose that a lattice of corner-sharing octahedra -- the octochlore lattice -- may provide a next-generation platform for 3D frustrated magnetism, with realizations in anti-perovskite and alkali-rare-earth fluoride compounds. We study the phase diagram of Ising moments on the octochlore lattice, finding a variety of frustrated phases including CSLs and phases with subextensive ground state degeneracy intermediate between spin liquids and long-range order. Utilizing a cluster multipole framework, we present a unified treatment of this variety of frustrated behaviors. In addition to a spin ice CSL, we identify a fracton CSL with excitations restricted to move along one-dimensional (1D) lines, a classical U(1) equivalent of the paradigmatic X-cube model harboring fracton topological order. These "lineon" quasiparticles carry magnetic quadrupole moments, contrasting the famous magnetic monopoles of spin ice. These two CSLs lie at the boundaries of a parent "frustrated chains" phase with subextensive degeneracy. Each CSL corresponds to a condensate of different bound states of 1D ferro-spinons, giving rise to quasi-critical dimensional crossovers near the ends of the frustrated chains phase associated to avoided Kasteleyn-like transitions. We also find a spin nematic phase whose ground states may be viewed as fracton crystals, exhibiting both uniaxial and biaxial orders. The latter is caused by spontaneous dimensional reduction owing to accidental symmetries of the subextensive ground state manifold. This work paves the way for the realization of fracton CSLs and the exploration of other exotic states in underexplored octochlore magnetic materials.
Fracton emerges from strongly interacting many-body systems whose excitations, referred to as sub-dimensional particles, have restricted mobility or kinetic motions. These entities have garnered significant interest due to their interdisciplinary implications spanning topological quantum codes, quantum field theory, emergent gravity, quantum information, and more, revealing unique nonequilibrium behaviors such as nonergodicity and glassy dynamics. This review presents a structured and educational overview of fracton phenomena, specifically focusing on gapless fracton liquids. Noteworthy for their compressibility and gapless excitations, fracton liquids facilitate collective modes reminiscent of those gauge fluctuations found in Maxwell's electromagnetic framework. However, they are distinct due to an additional higher-moment conservation law that restricts the mobility of individual charges and monopoles. Our exploration begins with the theoretical foundation of 3D fracton liquids, presenting a variety of emergent symmetric tensor gauge theories and reviewing their equilibrium and dynamic properties. Following this theoretical groundwork, we discuss the material realization of various fracton liquids in Yb-based breathing pyrochlore lattices, close-packed tiling structures and other synthetic quantum matter platforms. Additionally, we introduce a general protocol to foster and manipulate emergent fracton spin liquids in the realm of frustrated magnetism.