Thermodynamic signatures of topological phases

Abstract

For a long time, it was believed that the Ginzburg-Landau formalism was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although topological insulators are well understood by now, a thermodynamic description of their behavior remained elusive, firstly because the edges are lost in the thermodynamic limit, when the system size goes to infinity, and secondly because topological quantum field theories involve non-local order parameters, and hence cannot be captured by the conventional Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the paradigmatic Bernevig-Hughes-Zhang model in two dimensions, it has been shown that at the topological phase change the thermodynamic potential does not scale extensively due to the boundary effects.

Here, we extend this thermodynamic approach firstly to the Kitaev chain in one-dimension and then to different topological models in various dimensions (the Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero temperature. Surprisingly, we find that all these models exhibit the same universal behavior in the order of the topological-phase transition, which depends on the dimensionality of the system. Next, we use this thermodynamic approach to describe topological phases at finite-temperatures. We calculate the entropy and heat capacity for the Kitaev chain and verify its behavior in comparison with conformal field theory. Furthermore, we extend the topological phase transition to finite-temperatures, by numerically deriving the topological phase diagram for the bulk of the system, and show that it displays a good agreement with the one calculated analytically from the Uhlmann phase. Finally, we calculate measurable quantities such as the density of states and the heat capacity for different topological insulators. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter, provided that nonlinear terms are appropriately taken into account for the thermodynamic extensive variables.