We focus on the dynamical characters for a topological superconductor model. Considering a chiral p-wave superconductor with open and periodic boundary conditions in orthogonal directions, we show the bulk-boundary correspondence comparing to the topological phase diagram by Pfaffian. The expectations of Pauli matrixes are used to reflect the topological phase for this model. The dynamical quantum phase transition revealed by quench does occur in the parameter region consistent with the topological phase diagram. Besides quenching, temporal dynamical non-analyticities occur until the parameter crossing the topological critical point slowly. And the dynamics of the excitation behave differently in different topological phases. The dynamical phase transition behaves robustly against noise during evolution until the amplitude of noise reaching a threshold with the time-Anderson localization dominates. This work manifests the topological phase can be detected in a dynamical way.
We focus on the dynamical characters for a topological superconductor model. Considering a chiral p-wave superconductor with open and periodic boundary conditions in orthogonal directions, we show the bulk-boundary correspondence comparing to the topological phase diagram by Pfaffian. The expectations of Pauli matrixes are used to reflect the topological phase for this model. The dynamical quantum phase transition revealed by quench does occur in the parameter region consistent with the topological phase diagram. Besides quenching, temporal dynamical non-analyticities occur until the parameter crossing the topological critical point slowly. And the dynamics of the excitation behave differently in different topological phases. The dynamical phase transition behaves robustly against noise during evolution until the amplitude of noise reaching a threshold with the time-Anderson localization dominates. This work manifests the topological phase can be detected in a dynamical way.
Dynamical quantum phase transitions occur in dynamically evolving quantum systems when nonanalyticities occur at critical times in the return rate, a dynamical analog of the free energy. This extension of the concept of phase transitions can be brought into contact with another, namely, that of topological phase transitions in which the phase transition is marked by a change in a topological invariant. Following a quantum quench dynamical quantum phase transitions can happen in topological matter, a fact which has already been explored in one-dimensional topological insulators and in two-dimensional Chern insulators. Additionally, in one-dimensional systems a dynamical bulk boundary correspondence has been seen, related to the periodic appearance of zero modes of the Loschmidt echo itself. Here we extend both of these concepts to two-dimensional higher-order topological matter in which the topologically protected boundary modes are corner modes. We consider a minimal model which encompasses all possible forms of higher-order topology in two-dimensional topological band structures. We find that dynamical quantum phase transitions can still occur, and can occur for quenches which cross both bulk and boundary gap closings. Furthermore, a dynamical bulk boundary correspondence is also found, which takes a different form to that in one dimension.
A dynamical equivalent of the bulk-boundary correspondence has been suggested to occur in one and two dimensional topological models following sudden quenches. Depending on the topological invariant of the time evolving and initial phases involved large boundary contributions to a dynamical free energy occur. Moreover they occur periodically between the critical times at which this dynamical free energy becomes non-analytic, \emph{i.e.}~at dynamical quantum phase transitions. At these critical times the eigenvalue spectrum of the Loschmidt matrix which underlies the dynamical free energy closes its gap. The boundary contributions are understood to be due to zero-modes or in-gap bands of this matrix, forming a close analogy with equilibrium topological models and their edge modes. The exact cause of this phenomena and its generality remain unknown. In this article we test the dynamical bulk-boundary correspondence for a complicated two dimensional topological superconductor with a rich phase diagram, allowing quenches between many different Chern numbers. We show that there is no straightforward correspondence between the equilibrium phases quenched between and the dynamical bulk boundary correspondence. Furthermore the correspondence can depend on the orientation of the edges, suggesting a possible weak topological variant.
We investigate the nonequilibrium dynamics of two multiband boundary-obstructed topological systems (BOTSs) and a higher-order topological superconductor (HOTSC). The singular behavior exhibited by the dynamical free energy indicates the presence of dynamical quantum phase transitions (DQPTs) within these systems. By studying the nonequilibrium dynamics in the BOTS featuring even-parity Cooper pairing, we find that there are DQPTs in this BOTS via quenching from different topological phase regions. The analysis of the energy spectra shows that two previously considered identical topological phases are actually different. Based on this result, we identify that all DQPTs in the BOTS featuring even-parity Cooper pairing result from topology-changing quenches. Moreover, we also study DQPTs in another BOTS featuring odd-parity Cooper pairing, and we obtain similar results to those from the BOTS featuring even-parity Cooper pairing. The appearance of DQPTs in different BOTSs reveals that the occurrence of DQPTs is related to the topology of BOTSs. In addition, we adopt the same method to investigate DQPT in HOTSC for comparison. The numerical results show that DQPT always takes place for quenches between topological and trivial phases. This is completely different from the results of BOTSs, for which not all topology-changing quenches are accompanied by DQPTs. The study of DQPTs unveils the apparent difference between BOTS and HOTSC, and thus our study provides an alternative approach to revealing the topological nature of novel higher-order topological systems, which does not rely on the specific definitions of topological invariants.
Dynamical quantum phase transitions (DQPTs) manifested in the nonanalyticities in the temporal evolution of a closed quantum system generated by the time-independent final Hamiltonian, following a quench (or ramping) of a parameter of the Hamiltonian, is an emerging frontier of nonequilibrium quantum dynamics. We, here, introduce the notion of a dynamical topological order parameter (DTOP) that characterizes these DQPTs occurring in quenched (or ramped) two-dimensional closed quantum systems; this is quite a nontrivial generalization of the notion of DTOP introduced in Budich and Heyl [Phys. Rev. B 93, 085416 (2016)] for one-dimensional situations. This DTOP is obtained from the ``gauge-invariant'' Pancharatnam phase extracted from the Loschmidt overlap, i.e., the modulus of the overlap between the initially prepared state and its time-evolved counterpart reached following a temporal evolution generated by the time-independent final Hamiltonian. This generic proposal is illustrated considering DQPTs occurring in the subsequent temporal evolution following a sudden quench of the staggered mass of the topological Haldane model on a hexagonal lattice where it stays fixed to zero or unity and makes a discontinuous jump between these two values at critical times at which DQPTs occur. What is remarkable is that while the topology of the equilibrium model is characterized by the Chern number, the emergent topology associated with the DQPTs is characterized by a generalized winding number.
We study the competition of disorder and superconductivity for a one-dimensional p-wave superconductor in incommensurate potentials. With the increase in the strength of the incommensurate potential, the system undergoes a transition from a topological superconducting phase to a topologically trivial localized phase. The phase boundary is determined both numerically and analytically from various aspects and the topological superconducting phase is characterized by the presence of Majorana edge fermions in the system with open boundary conditions. We also calculate the topological Z2 invariant of the bulk system and find it can be used to distinguish the different topological phases even for a disordered system.
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.