Sufficient symmetry conditions for Topological Quantum Order

Abstract

We prove sufficient conditions for Topological Quantum Order at zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries, thus providing a unifying framework based on a symmetry principle. These symmetries may be actual invariances of the system, or may emerge in the low-energy sector. Prominent examples of Topological Quantum Order display Gauge-Like Symmetries. New systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. We analyze the physical consequences of Gauge-Like Symmetries (including topological terms and charges) and show the insufficiency of the energy spectrum, topological entanglement entropy, maximal string correlators, and fractionalization in establishing Topological Quantum Order. General symmetry considerations illustrate that not withstanding spectral gaps, thermal fluctuations may impose restrictions on suggested quantum computing schemes. Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.T he role of invariance (symmetry) principles in accounting for observed regularities is well known (1). Invariances become particularly effective in quantum mechanics where the linear character of Hilbert space enables us to construct superpositions of states that transform as irreducible representations of various symmetry groups. First discovered were invariance principles of a geometric character relating to space-time displacements and uniform motion. The best known local invariance operations (i) relate different coordinate systems to one other by such geometric deformations that leave the (local) metric invariant and (ii) appear as local transformations that link different gauge theory representations. It was recently realized that probing nonlocal (topological) structures uncovers new invariance principles with physical (and experimental) consequences. Particle-wave type dualities link the seemingly different local and nonlocal structures.Understanding the thermodynamic phases of matter via symmetry principles enables characterization by universal behaviors as in Landau's theory of phase transitions (2). In this theory, the order parameter(s) of the system relate to probing its local structure. A non-vanishing order parameter, encoding the breaking of a symmetry, defines the ordered state while the restoration of that symmetry signals the transition to a disordered state. A new paradigm, Topological Quantum Order (TQO), extends the Landau symmetry-breaking framework (3). In essence, this new order is associated with robustness against local perturbations, and hence cannot be described in principle by local order parameters. Thus, the underlying order remains hidden to ordinary local probes. Indeed, this new order exhibits nonlocal correlations that potentially lead to novel physical consequenc...

Fractionalization and Topological Order

“…From the behavior of the non-local operators one might mistakenly conclude that this fragility to thermal fluctuations is intrinsic to topological order [24]. On the other hand, we can compare this to the well-known example of a classical Z 2 lattice gauge theory in three dimensions [3,30], which also lacks a local order parameter in the zero temperature limit.…”

“…This phenomenon was first discussed in the context of topological phases in Ref. [24] and termed "thermal fragility". ‡ This is quite distinct from conventionally ordered phases, where the thermal uncertainty in the expectation value of a local order parameter scales with the density of defects, and becomes negligibly small at low temperature.…”

Hubbard Model on the Pyrochlore Lattice: A 3D Quantum Spin Liquid

“…At finite temperatures, the expectation value of any quantity not invariant under all local symmetries must vanish by Elitzur's theorem. In this case, in contrast to the T = 0 case, it is the local gaugetype symmetries corresponding to the presence of local loop processes that determine the topological quantum order [18]. Further, when ∆J ≪ T ≪ E g , where E g denotes the spectral gap between states in the Klein-point manifold and all other states, then the local symmetries determined by the ice rules emerge.…”

“…These states are connected only by planar processes, loops with lengths of order L 2 in a finite system, which [following Eqs. (6) and (8)] have energies of order −∆J/2 L 2 , and this type of exponentially small spectral gap is associated with topological order [18,31]; similar exponentially small differences arise also in measurements for other quasi-local operators. In the topologically ordered system, local loops are processes within the same topological sector while system-scale loops are topological operations that change this sector.…”

“…In systems of finite size, this symmetry is broken but the accompanying spectral gaps are exponentially small in L 2 . The associated topological degeneracy in this and other models with d ≥ 1 processes [18] is analogous to 2D quantum dimer models where the ground states are also equal-amplitude superpositions of dimer coverings. These models have non-local d = 1 symmetries associated with the parity of the even or odd number of dimers cut by 1D loops around the entire system, and are understood as gapped quantum liquids with Z 2 topological order [8].…”