Topological Phase Transitions in Finite-size Systems Across Boundary Conditions

Ge Yang, Marcos Rigol

In finite systems driven unitarily across topological phase transitions, the Chern number and the Bott index have been found to exhibit different behaviors depending on the boundary conditions and on the commensurability of the lattice. For periodic boundary conditions, the Chern number does not change for finite commensurate lattices (or in the thermodynamic limit). On the other hand, the Chern number can change for incommensurate lattices with periodic boundary conditions and the Bott index can change for lattices with open boundary conditions. Here we show that the scalings of the fields at which those two indices change exhibit Landau-Zener and near-adiabatic regimes depending on the speed at which the strength of the drive is ramped up and on the system size. Those regimes are preceded by a regime in which the topological indices do not change. The latter is the only regime that, for nonvanishing ramp speeds, survives in the thermodynamic limit. We then show that the dc Hall response can be used to detect topological phase transitions independently of the behavior of the topological indices.

M. A. Contínentino, S. Rufo, Griffith M. Rufo

M. A. Contínentino

Christopher N. Varney, Kai Sun, Marcos Rigol +1 more

In this paper, we investigate signatures of topological phase transitions in interacting systems. We show that the key signature is the existence of a topologically protected level crossing, which is robust and sharply defines the topological transition, even in finite-size systems. Spatial symmetries are argued to play a fundamental role in the selection of the boundary conditions to be used to locate topological transitions in finite systems. We discuss the theoretical implications of these results, and utilize exact diagonalization to demonstrate its manifestations in the Haldane-Fermi-Hubbard model. Our findings provide an efficient way to detect topological transitions in experiments and in numerical calculations that cannot access the ground-state wave function.

Jakub Mildner

Philipp Jaeger

Dynamical phase transitions (DPT) are receiving a rising interest. They are known to behave analogously to equilibrium phase transitions (EPT) to a large extend. However, it is easy to see that DPT can occur in finite systems, while EPT are only possible in the thermodynamic limit. So far it is not clear how far the analogy of DPT and EPT goes. It was suggested, that there is a relation between topological phase transitions (TPT) and DPT, but many open questions remain. Typically, to study DPT, the Loschmidt echo (LE) after a quench is investigated, where DPT are visible as singularities. For one-dimensional systems, each singularity is connected to a certain critical time scale, which is given by the dispersion in the chain. In topological free-fermion models with winding numbers 0 or 1, only the LE in periodic boundary conditions (PBC) has been investigated. In open boundary conditions (OBC), these models are characterized by symmetry protected edge modes in the topologically non-trivial phase. It is completely unclear how these modes affect DPT. We investigate systems with PBC governed by multiple time scales with a Z topological invariant. In OBC, we provide numerical evidence for the presence of bulk-boundary correspondence in DPT in quenches across a TPT.