Speaker
Description
The classification of phases of matter at finite temperature has traditionally rested on Landau's paradigm of spontaneous symmetry breaking and local order parameters. At absolute zero, this paradigm has been decisively enlarged over the past decades: topologically ordered and symmetry-protected topological phases sharply distinguish ground states that share exactly the same symmetries. Most of these topological distinctions, however, are defined strictly at zero temperature, and whether equally sharp, symmetry-free distinctions can persist at finite temperature remains far less understood.
In this talk, I address this question through two contrasting spin models. The first is the spin-1 spin ice on the pyrochlore lattice, which also serves as an effective model for the hydrogen-bond network of high-pressure water ice. Combining exact duality mappings—onto 3D $XY$ and Ising loop-gas models—with Monte Carlo simulations, I show that the topological phase transitions present at zero temperature are immediately rounded into continuous crossovers at any finite temperature, because thermally excited point-like monopoles screen the emergent gauge field. This naturally explains why molecular ice-VII and symmetric ice-X, which share identical crystal symmetry, are connected by a crossover rather than a genuine phase transition.
The second model is the 3D $\mathbb{Z}_2$ toric code in a generic magnetic field, in which every higher-form symmetry is explicitly broken. In sharp contrast to the spin ice, here the topological order survives up to a genuine finite-temperature phase transition. The protection is purely geometric: the Bianchi identity forbids point-like magnetic monopoles and forces flux to proliferate only through closed loops. Using large-scale quantum Monte Carlo, I show that the topological entanglement entropy stays quantized at $\ln 2$ throughout this phase. This quantity, however, is notinvariant under quasi-local channels—a constant-depth circuit can manufacture the same $\ln 2$ from a trivial product state—and therefore cannot certify the phase on its own. I then propose the decoded Wilson-loop correlation, a channel-invariant order parameter that equals $1$ in the topological phase and $0$ in the trivial phase, furnishing a genuine topological invariant of the finite-temperature, field-driven 3D toric code as a mixed (Gibbs) state.
Taken together, these examples reveal that the survival of topological order at finite temperature is dictated by the geometry of its topological excitations: point-like defects offer no protection and yield only crossovers, whereas loop-like defects can sustain a sharp phase distinction that breaks no symmetry—one captured by an appropriately constructed channel invariant.