We analyze the multiple timescales that appear in the dissipative dynamics of an open Kitaev chain. Starting from the unified quantum master equation (UQME), we reduce the problem to a singularly perturbed three‐dimensional system that describes the nearly degenerate subspace spanned by a pair of Majorana zero modes (MZMs). Application of Chiba’s renormalization‐group (RG) method yields an explicit β ‐function for the slow amplitude and the asymptotic expansions of the slow and fast eigenvalues, together with rigorous error bounds on the timescale t = O (1/ ɛ 2 ). A weak longitudinal field, which renders the underlying closed system nonintegrable, produces a quadratic correction of structural type to the same β ‐function. The sign of the correction implies a shortening of the slow timescale; the prefactor k is not evaluated numerically. The analytic predictions are confirmed by numerical integration of both the reduced master equation and a short chain. The resulting framework provides a mathematically controlled description of how dissipation and the loss of integrability jointly determine the long‐time fate of topological edge modes. MSC2020 Classification: 81T17, 34E13, 82C10, 81Q80, 37A60
We analyze the multiple timescales that appear in the dissipative dynamics of an open Kitaev chain. Starting from the unified quantum master equation (UQME), we reduce the problem to a singularly perturbed three‐dimensional system that describes the nearly degenerate subspace spanned by a pair of Majorana zero modes (MZMs). Application of Chiba’s renormalization‐group (RG) method yields an explicit β ‐function for the slow amplitude and the asymptotic expansions of the slow and fast eigenvalues, together with rigorous error bounds on the timescale t = O (1/ ɛ 2 ). A weak longitudinal field, which renders the underlying closed system nonintegrable, produces a quadratic correction of structural type to the same β ‐function. The sign of the correction implies a shortening of the slow timescale; the prefactor k is not evaluated numerically. The analytic predictions are confirmed by numerical integration of both the reduced master equation and a short chain. The resulting framework provides a mathematically controlled description of how dissipation and the loss of integrability jointly determine the long‐time fate of topological edge modes. MSC2020 Classification: 81T17, 34E13, 82C10, 81Q80, 37A60
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system at zero chemical potential as well as in the symmetric case of Δ = t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion interaction. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
Understanding the dynamics of open quantum systems is intrinsically important to control and utilize quantum hardware in various applications. The generic behavior of purely dissipative open quantum many-body systems with k-local, normal dissipation processes can be investigated using random matrix theory, revealing different decay timescales of observables determined by their complexity, as shown by Wang et al. [K. Wang et al., Phys. Rev. Lett. 124, 100604 (2020)]. The time evolution of an open quantum system is entirely described by the eigenvalue spectrum of the Lindbladian, in which the hierarchy of decay timescales is reflected in the formation of well-separated eigenvalue clusters. Here, we extend the analysis of this spectrum to the presence of unitary dynamics using a random matrix model. In the case of strong dissipation, the unitary dynamics can be treated perturbatively and the k locality of the Hamiltonian determines how susceptible the spectrum is to such a perturbation. For the physically most relevant case of (dissipative) two-body interactions, the correction in the first order of the perturbation vanishes, leading to the relative robustness of the spectral features, which in turn implies the same robustness of the corresponding separation of decay timescales. We find the perturbative approach for the spectrum to yield an excellent agreement with exact-diagonalization results. For weak dissipation, the spectrum flows into clusters except for outlier eigenmodes, which we identify to be the local symmetries of the Hamiltonian. We support our analytic findings by numerical simulations and show that our generic model applies to a nonrandom Heisenberg chain.
Majorana zero modes, spontaneous Z2f symmetry breaking, and long range edge correlation in Kitaev chains: analytic solutions and density-matrix-renormalization-group study
We study Kitaev model in one-dimension with open boundary condition by using exact analytic methods for non-interacting system in the half filled limit as well as in the symmetric case of Δ=t, and by using density-matrix-renormalization-group method for interacting system with nearest neighbor repulsion. We show explicitly that the emergence of Majorana zero mode is associated with the spontaneous symmetry broken of fermion number parity operator Z2f, which is +1 for even number of fermions and −1 for odd number of fermions. Our results are in support of recent proposal by Klassen and Wen\cite{Klassen}, who proposed such spontaneous symmetry broken in systems with Majorana zero modes. We suggest and examine an edge correlation function of Majorana fermions to characterize the long range order in the topological superconducting states and study the phase diagram of the interating Kitaev chain.
The relaxation of a many-body system is still a challenging problem that has not been well understood. In this work we exactly calculate the dynamics of the quantum XY model with boundary dissipation, in which the density matrix in terms of Majorana operators can be decoupled into independent subspaces Ki represented by different number of Majorana fermions. The relaxation is characterized by multiple timescales, and in the long-time limit it is determined by the single-particle relaxation process in a typical timescale T∗. For the bulk bands, we find ${T}^{*}\ensuremath{\propto}{N}^{3}/\ensuremath{\gamma}{n}^{2}$ in the weak dissipation limit and ${T}^{*}\ensuremath{\propto}\ensuremath{\gamma}{N}^{3}/{n}^{2}$ in the strong dissipation limit, where N is the chain length, $\ensuremath{\gamma}$ is the dissipation rate, and n is the band index. For the edge modes, ${T}^{*}\ensuremath{\propto}1/\ensuremath{\gamma}$ indicates the most vulnerable to dissipation in the long-chain limit. These results are counterintuitive because it means any weak dissipation can induce strong relaxation, while strong dissipation can induce weak relaxation. We find that these two limits correspond to two different physics, which are explained based on the first- and second-order perturbation theory in an equivalent non-Hermitian model. Finally, we show that even in the long-chain limit with weak dissipation, the relaxation may exhibit strong odd-even effect, which can be washed out by the strong dissipation. These results shed new insight into the dynamics of topological qubits in the environment, and the mechanism for many-body dissipation may also have universal significance in understanding the many-body dissipation in the other models.
Open quantum systems with nearly degenerate energy levels have been shown to exhibit long-lived metastable states in the approach to equilibrium, even when modelled with certain Lindblad-form quantum master equations. This is a result of dramatic separation of timescales due to differences between Liouvillian eigenvalues. These metastable states often have nonzero coherences that die off only in the long-time limit once the system reaches thermal equilibrium. We examine two distinct situations that give rise to this effect: one in which dissipative dynamics couple together states only within a nearly degenerate subspace, and one in which they give rise to jumps over finite-energy splittings, between nearly degenerate subspaces. We find, in each case, that a change of basis can often lead to a representation that more naturally captures the impact of the system-bath interaction than does the energy eigenbasis, revealing that separate timescales are associated with separate processes (e.g., decoherence into a nonenergy eigenbasis, decay of population correlations to the initial state). This approach is paired with the inspection of quantum trajectories, which further provide intuition as to how open system evolution is characterized when coherent oscillations, thermal relaxation, and decoherence all occur simultaneously. Published by the American Physical Society 2024
We investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between φ=π, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case φ=0. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number ν=1 and 2, giving rise to distinct topological phases that support two and four MZMs. The ν=0 gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.