A. Quelle

Universalities of thermodynamic signatures in topological phases

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.

Dynamical competition between quantum Hall and quantum spin Hall effects

In this paper we investigate the occurrence of quantum phase transitions in topological systems out of equi- librium. More specifically, we consider graphene with a sizable spin-orbit coupling, irradiated by circularly- polarized light. In the absence of light, the spin-orbit coupling drives a quantum spin Hall phase where edge currents with opposite spins counter-propagate. On the other hand, the light generates a time-dependent vector potential, which leads to a hopping parameter with staggered time-dependent phases around the benzene ring. The model is a dynamical version of the Haldane model, which considers a static staggered flux with zero total flux through each plaquette. Since the light breaks time-reversal symmetry, a quantum Hall phase protected by an integer topological invariant arises. By numerically solving the complete problem, with spin-orbit coupling and light, and investigating different values of the driving frequency {\omega}, we show that the spectrum exhibits non-trivial gaps not only at zero energy, but also at {\omega}/2. This additional gap is created by photon resonances between the valence and conduction band of graphene, and the symmetry of the spectrum forces it to lie at {\omega}/2. By increasing the intensity of the irradiation, the topological state in the zero energy gap undergoes a dynamical phase transition from a quantum spin Hall to a quantum Hall phase, whereas the gap around {\omega}/2 remains in the quantum Hall regime.

Driving protocol for a Floquet topological phase without static counterpart

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 that their equilibrium counterparts. Although analogues of the equilibrium topological phases exist, which 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 vanishing Chern number and no Hall conductivity, but nevertheless hosts anomalous topological edge states. Due to experimental difficulties associated with the observation of such a phase, it has not been experimentally realised so far. In this paper, we show that optical lattices prove to be a good candidate for both its realisation and subsequent 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.

Thermodynamic signatures of edge states in topological insulators

Topological insulators are states of matter distinguished by the presence of symmetry protected metallic boundary states. These edge modes have been characterised 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 like 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 Hills theory. For topologically non-trivial 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 new possibilities for investigating, manipulating, and characterizing topological quantum matter solely in terms of equilibrium boundary physics.

Bandwidth-resonant Floquet states in honeycomb optical lattices

We investigate, within Floquet theory, topological phases in the out-of-equilibrium system that consists of fermions in a circularly shaken honeycomb optical lattice. We concentrate on the intermediate regime, in which the shaking frequency is of the same order of magnitude as the band width, such that adjacent Floquet bands start to overlap, creating a hierarchy of band inversions. It is shown that two-phonon resonances provide a topological phase that can be described within the Bernevig-Hughes-Zhang model of HgTe quantum wells. This allows for an understanding of out-of-equilibrium topological phases in terms of simple band inversions, similar to equilibrium systems.

Nontrivial topological states on a Möbius band

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 Mobius 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 Mobius 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 Mobius band. We also provide symmetry arguments to show the possible lattice structures and Hamiltonian terms for which topological states may exist in a Mobius band, and we locate our systems in the classification of topological states. Then, we propose a method to calculate Mobius dispersions from those of the cylinder, and we show the results for a honeycomb and a kagome Mobius 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 Mobius band.

Resonances in a periodically driven bosonic system

Periodically driven systems are a common topic in modern physics. In optical lattices specifically, driving is at the origin of many interesting phenomena. However, energy is not conserved in driven systems, and under periodic driving, heating of a system is a real concern. In an effort to better understand this phenomenon, the heating of single-band systems has been studied, with a focus on disorder- and interaction-induced effects, such as many-body localization. Nevertheless, driven systems occur in a much wider context than this, leaving room for further research. Here, we fill this gap by studying a noninteracting model, characterized by discrete, periodically spaced energy levels that are unbounded from above. We couple these energy levels resonantly through a periodic drive, and discuss the heating dynamics of this system as a function of the driving protocol. In this way, we show that a combination of stimulated emission and absorption causes the presence of resonant stable states. This will serve to elucidate the conditions under which resonant driving causes heating in quantum systems.

Topological Floquet states on a Möbius band irradiated by circularly polarised light

Abstract Topological states of matter in equilibrium, as well as out of equilibrium, have been thoroughly investigated during the last years in condensed-matter and cold-atom systems. However, the geometric topology of the studied samples is usually trivial, such as a ribbon or a cylinder. In this paper, we consider a graphene Mobius band irradiated with circularly polarised light. Interestingly, due to the non-orientability of the Mobius band, a homogeneous quantum Hall effect cannot exist in this system, but the quantum spin Hall effect can. To avoid this restriction, the irradiation is applied in a longitudinal-domain-wall configuration. In this way, the periodic time-dependent driving term tends to generate the quantum anomalous Hall effect. On the other hand, due to the bent geometry of the Mobius band, we expect a strong spin–orbit coupling, which may lead to quantum spin Hall-like topological states. Here, we investigate the competition between these two phenomena upon varying the amplitude and the frequency of the light, for a fixed value of the spin–orbit coupling strength. The topological properties are analysed by identifying the edge states in the Floquet spectrum at intermediate frequencies, when there are resonances between the light frequency and the energy difference between the conduction and valence bands of the graphene system.