Topological order and emergence

Abstract

Topologically ordered systems play a prominent role in current research in condensed matter physics; examples include systems that exhibit the quantum Hall effect, topological insulators, and topological superconductors. These systems possess properties that are characterized by topological invariants, exhibit phase transitions that cannot be characterized by spontaneous symmetry breaking, and exhibit order that cannot be characterized in terms of a local order parameter. They thus fall outside the scope of the Landau–Ginsburg theory of phase transitions, which, arguably, has informed much of the discussion, in both the physics and philosophy literature, of emergence in condensed matter systems. Nevertheless, some authors have claimed that topologically ordered systems exhibit emergence. This essay offers a critical assessment of this claim. In particular, it identifies two types of topological order and observes that, whereas the alleged mechanisms underwriting these types differ, they nevertheless share certain features; in particular, the lowenergy behavior of such systems can be described by effective topological quantum field theories. This suggests that a unified account of the emergence of topological order should look to a law-centric, as opposed to a mechanismcentric, view of emergence.