Topological glass states

In connection with recent discussion of topological order and topological phase transitions in quantum systems, we reexamine circumstances that lead to the appearance of a topological glass in certain classical lattice spin models. Local bonding enforces constraints on low energy states which organize themselves into topologically distinct classes that break ergodicity but not any apparent symmetry as in the usual Landau theory of phase transitions. Various properties of such a topological glass are demonstrated using two classical Ising-like models. Quantum field theories with non-trivial topological structures have attracted a great deal of interest in recent years [1–5]. Although the fundamental variables of these systems are topologically simple (e.g., ordinary spins or boson/fermion fields on a periodic lattice), their low energy properties are characterized by a non-trivial topological order and order parameters that are non-local functions of the original microscopic variables. The effective variables may take the form of fermions or bosons dressed by Chern-Simon field [1] or stringlike objects [2, 3]. Topological excitations with fractionalized charge (or spin) quantum number often appear in these systems as a result of the topological order. The study of this class of problems is important for a better understanding of some of the more exotic states of matter such as the FQH states. Many existing examples of topological order are from quantum systems with a non-trivial Berry phase structure. Non-trivial topology may also arise, however, from the spatial bonding pattern in the ground state which can be understood from a purely classical viewpoint. Using simple and clear examples from statistical mechanics, we examine in this paper how various aspects of topological order, such as ground state degeneracy, reduced phase space, topological excitations and topological phase transitions, come together in such a scenario. Whereas topological order is often characterized by topological excitations with fractionalized charge (or spin) quantum number in quantum systems, it manifests itself as breaking of ergodicity in classical systems. These related but also distinct features are noteworthy as they contribute to a more complete identification and classification of topologically ordered many-body quantum states. The classical systems we consider have an extensive ground state entropy and no apparent spatial order in the usual sense of the word at any temperature. Nevertheless, the lowenergy part of the phase space decomposes into topologically distinct sectors, leading to the breaking of ergodicity at sufficiently low temperatures. In this respect, they resemble the