Symmetry and Topological Order

Abstract

We prove sufficient conditions for Topological Quantum Order at both zero and finite temperatures. The crux of the proof hinges on the existence of low-dimensional Gauge-Like Symmetries (that notably extend and differ from standard local gauge symmetries) and their associated defects, 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), discuss associated braiding, 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 certain suggested quantum computing schemes and lead to "thermal fragility". Our results allow us to go beyond standard topological field theories and engineer systems with Topological Quantum Order.

Figures (2)

• Figure 1: Schematics of the interactions and symmetries involved in three $D=2$ examples. See (a-c) of text. The left panel represents (a) an Ising gauge theory with local ($d=0$) symmetries. The middle panel represents (b) an orbital compass model with $d=1$ symmetries; here the symmetry operations span lines. The right panel depicts (c) an XY model with $d=2$ symmetries; the symmetry here spans the entire $D=2$ dimensional plane.
• Figure 2: (Color online.) The physical engine behind our theorem are topological defects. For instance, introducing a (one-dimensional) soliton into a general (anisotropic) orbital compass model state such as depicted in panel (b) of Fig. \ref{['examples']} leads to only a finite energy cost. This penalty is depicted here by a single energetic bond (dashed line). The energy-entropy balance associated with such $d=1$ Ising type domain walls is the same as that in a $D=1$ Ising system. At finite temperature, entropic contributions overwhelm energy penalties and no local order is possible. Order is manifest in non-local quantities associated with topological defects. Similar results occur in other systems with low-dimensional Gauge-Like Symmetries.