Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although they are well understood, a thermodynamic description of these materials remained elusive, firstly because the edges yield a non-extensive contribution to the thermodynamic potential, and secondly because topological field theories involve non-local order parameters, and cannot be captured by the Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the Bernevig-Hughes-Zhang model in two dimensions, it was shown that at the topological phase transition the thermodynamic potential does not scale extensively due to boundary effects. Here, we extend this approach to different topological models in various dimensions (the Kitaev chain and 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, all models exhibit the same universal behavior in the order of the topological-phase transition, depending on the dimension. Moreover, we derive the topological phase diagram at finite temperature using this thermodynamic description, and show that it displays a good agreement with the one calculated from the Uhlmann phase. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter.

Topological insulators are states of matter distinguished by the presence of symmetry-protected metallic boundary states. These edge modes have been characterized in terms of transport and spectroscopic measurements, but a thermodynamic description has been lacking. The challenge arises because in conventional thermodynamics the potentials are required to scale linearly with extensive variables such as volume, which does not allow for a general treatment of boundary effects. In this paper, we overcome this challenge with Hill thermodynamics. In this extension of the thermodynamic formalism, the grand potential is split into an extensive, conventional contribution, and the subdivision potential, which is the central construct of Hill's theory. For topologically nontrivial electronic matter, the subdivision potential captures measurable contributions to the density of states and the heat capacity: it is the thermodynamic manifestation of the topological edge structure. Furthermore, the subdivision potential reveals phase transitions of the edge even when they are not manifested in the bulk, thus opening a variety of possibilities for investigating, manipulating, and characterizing topological quantum matter solely in terms of equilibrium boundary physics.

In the field of topological insulators, the topological properties of quantum states in samples with simple geometries, such as a cylinder or a ribbon, have been classified and understood during the past decade. Here we extend these studies to a Möbius band and argue that its lack of orientability prevents a smooth global definition of parity-odd quantities such as pseudovectors. In particular, the Chern number, the topological invariant for the quantum Hall effect, lies in this class. The definition of spin on the Möbius band translates into the idea of the orientable double cover, an analogy used to explain the possibility of having the quantum spin Hall effect on the Möbius band. We also provide symmetry arguments to show the possible lattice structures and Hamiltonian terms for which topological states may exist in a Möbius band, and we locate our systems in the classification of topological states. Then, we propose a method to calculate Möbius dispersions from those of the cylinder, and we show the results for a honeycomb and a kagome Möbius band with different types of edge termination. Although the quantum spin Hall effect may occur in these systems when intrinsic spin-orbit coupling is present, the quantum Hall effect is more intricate and requires the presence of a domain wall in the sample. We propose an experimental setup which could allow for the realization of the elusive quantum Hall effect in a Möbius band.

We find a series of topological phase transitions of increasing order, beyond the more standard second-order phase transition in a one-dimensional topological superconductor. The jumps in the order of the transitions depend on the range of the pairing interaction, which is parametrized by an algebraic decay with exponent α. Remarkably, in the limit α = 1 the order of the topological transition becomes infinite. We compute the critical exponents for the series of higher-order transitions in exact form and find that they fulfill the hyperscaling relation. We also study the critical behaviour at the boundary of the system and discuss potential experimental platforms of magnetic atoms in superconductors. arXiv:1710.05691v2 [cond-mat.stat-mech]

Periodically driven systems play a prominent role in optical lattices. In these ultracold atomic systems, driving is used to create a variety of interesting behaviours, of which an important example is provided by topological states of matter. Such Floquet topological phases have a richer classification than their equilibrium counterparts. Although there exist analogues of the equilibrium topological phases that are characterised by a Chern number, the corresponding Hall conductivity, and protected edge states, there is an additional possibility. This is a phase that has a vanishing Chern number and no Hall conductivity, but nevertheless hosts anomalous topological edge states (Rudner et al (2013 Phys. Rev. X 3 031005)). Due to experimental difficulties associated with the observation of such a phase, it has not been experimentally realised in optical lattices so far. In this paper, we show that optical lattices prove to be a good candidate for its realisation and observation, because they can be driven in a controlled manner. Specifically, we present a simple shaking protocol that serves to realise this special Floquet phase, discuss the specific properties that it has, and propose a method to experimentally detect this fascinating topological phase that has no counterpart in equilibrium systems.