Dissipation-Induced Steady States in Topological Superconductors: Mechanisms and Design Principles

The search for conditions supporting degenerate steady states in nonequilibrium topological superconductors is important for advancing dissipative quantum engineering, a field that has attracted significant research attention over the past decade. In this study, we address this problem by investigating topological superconductors hosting unpaired Majorana modes under the influence of environmental dissipative fields. Within the Gorini-Kossakowski-Sudarshan-Lindblad framework and the third quantization formalism, we establish a correspondence between equilibrium Majorana zero modes and non-equilibrium kinetic zero modes. We further derive a simple algebraic relation between the numbers of these excitations expressed in terms of hybridization between the single-particle wavefunctions and linear dissipative fields. Based on these findings, we propose a practical recipes how to stabilize degenerate steady states in topological superconductors through controlled dissipation engineering. To demonstrate their applicability, we implement our general framework in the BDI-class Kitaev chain with long-range hopping and pairing terms – a system known to host a robust edge-localized Majorana modes.

💡 Research Summary

The manuscript investigates how environmental dissipation can be harnessed to create and control degenerate steady states in topological superconductors (TS) that host unpaired Majorana modes. Using the most general Gorini‑Kossakowski‑Sudarshan‑Lindblad (GKSL) master equation together with the third‑quantization formalism, the authors establish a precise correspondence between equilibrium Majorana zero modes (MZMs) and non‑equilibrium kinetic zero modes (ZKMs) that appear in the spectrum of the Liouvillian super‑operator.

The model assumes a quadratic fermionic Hamiltonian written in terms of 2N Majorana operators (\hat w_j) and a set of (N_B) linear jump operators (\hat L_v = \sum_j l_{v,j}\hat w_j). In the third‑quantization picture the GKSL dynamics is mapped onto a non‑Hermitian single‑particle problem with an effective matrix (X = A - iM), where (A) encodes the Hamiltonian and (M) encodes the dissipative couplings. The eigenvalues of (X) determine the Liouvillian spectrum; eigenvalues equal to zero correspond to ZKMs, which are the open‑system analogues of MZMs.

A central result is the algebraic relation

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