Survival probability in a quenched Majorana chain with an impurity

Abstract

We investigate the dynamics of a one-dimensional p-wave superconductor with next-nearest-neighbor hopping and superconducting interaction derived from a three-spin interacting Ising model in transverse field by mapping to Majorana fermions. The next-nearest-neighbor hopping term leads to a new topological phase containing two zero-energy Majorana modes at each end of an open chain, compared to a nearest-neighbor p-wave superconducting chain. We study the Majorana survival probability (MSP) of a particular Majorana edge state when the initial Hamiltonian (H_{i}) is changed to the quantum critical as well as off-critical final Hamiltonian (H_{f}), which additionally contains an impurity term (H_{imp}) that breaks the time-reversal invariance. For the off-critical quenching inside the new topological phase with H_{f}=H_{i}+H_{imp}, and small impurity strength (λ_{d}), we observe a perfect oscillation of the MSP as a function of time with a single frequency (determined by the impurity strength λ_{d}) that can be analyzed from an equivalent two-level problem. On the other hand, the MSP shows a beating like structure with time for quenching to the phase boundary separating the topological phase (with two edge Majoranas at each edge) and the nontopological phase where the additional frequency is given by inverse of the system size. We attribute this behavior of the MSP to the modification of the energy levels of the final Hamiltonian due to the application of the impurity term.

The Lieb–Mattis theorem revisited

“…Examples of other problems discussed with the use of the LMT or the Lieb-Schultz-Mattis theorem can be found in Refs. [7][8][9][10][11][12][13][14]. On the other hand, there are many attempts to extend these theorems (e.g.…”

Disordered Kitaev chain with long-range pairing: Loschmidt echo revivals and dynamical phase transitions

“…The behavior of edge states under a sudden quantum quench has been investigated in two-dimensional topological insulator [36], where it was shown that, in the abrupt transition from the topological insulator to the trivial insulator phase, there is a collapse and revival of the edge states [30,34]. Similar results have been obtained for the one-dimensional Kitaev model [31,32,37,38].…”

“…Very recently, the studies of the connection between quenching dynamics and quantum phase transition [5,12,20], the topological order [26][27][28] and also the dynamics of an edge state [29][30][31][32], have attracted the attention of the scientists. Specifically, searching the robustness and response of the topological edge states to quantum quenches [28,31,[33][34][35].…”

“…Very recently, the studies of the connection between quenching dynamics and quantum phase transition [5,12,20], the topological order [26][27][28] and also the dynamics of an edge state [29][30][31][32], have attracted the attention of the scientists. Specifically, searching the robustness and response of the topological edge states to quantum quenches [28,31,[33][34][35]. The behavior of edge states under a sudden quantum quench has been investigated in two-dimensional topological insulator [36], where it was shown that, in the abrupt transition from the topological insulator to the trivial insulator phase, there is a collapse and revival of the edge states [30,34].…”

“…Current experimental progresses in realization of longrange interacting quantum models with tunable long range interactions [39] has renewed the interest in studying the non-equilibrium dynamics of quantum systems with infinite-range interactions [40][41][42][43][44]. Motivated by the short-range one dimensional Kitaev chain [31,32], a long-range pairing version of an integrable p-wave superconducting chain of fermions has been proposed, where strength of super conducting pairing between two sites separated by a distance r falling off in a power-law fashion as r α [44][45][46][47][48][49][50][51][52][53][54][55][56]. Despite numerous attempts to link the significant features of quantum phase transition (QPT) to the quench dynamics (LE), the general principle has not been established to connect the QPT to dynamics of the systems, specifically in the disordered and long range quantum systems.…”