Researchers Map Hidden Edge States in Dissipative Superconductor

Algebraic decay occurs in the gapless regime and an exponential falloff is observed in the gapped phase, enabling local observables to directly diagnose the topology. The correspondence breaks down in a dissipative topological superconductor, where local observables become unable to decipher the expected topology.

Shaina Gandhi of the Indian Institute of Technology Guwahati and colleagues from Bilkent University, using a bond-dissipative dimerized Kitaev chain within a third-quantized rapidity-matrix formulation, find that at zero chemical potential the Majorana rapidity matrix decomposes into two independent non-Hermitian sectors. Each sector possesses its own generalised Brillouin zone and non-Bloch winding number, thereby revealing a sector-resolved non-Bloch bulk-boundary correspondence.

Non-Hermitian topology and boundary sensitivity in open quantum systems

Non-Hermitian lattices are strongly boundary sensitive, with open boundaries reshaping the spectrum and causing bulk eigenstates to accumulate at the edges, a phenomenon known as the non-Hermitian skin effect (NHSE). This failure of Bloch bulk-boundary correspondence motivates a non-Bloch formulation based on generalised Brillouin zones (GBZs).

Non-Hermitian topological phases are classified by the point-gap and line-gap structure of the complex spectrum, the former capturing spectral winding and closely associated with the NHSE, while the latter permits Hermitian-like classifications. Open quantum systems provide a natural setting for broader manifestations of this boundary-sensitive physics. For Markovian dynamics generated by a quadratic Lindblad master equation, two-point relaxation is governed by a finite-dimensional rapidity (damping) matrix obtained from third quantization.

The Liouvillian damping gap is defined by the slowest rapidity decay rate, and the rapidity matrix itself can display a skin effect, meaning its open boundary condition (OBC) spectrum is not obtained from the Bloch unit circle. This gap-relaxation correspondence is well established in the dissipative Su-Schrieffer-Heeger (SSH) chain. In this system, the periodic boundary condition (PBC) damping gap vanishes for t1 ≤t2, and the particle density inherits the decay of the gapless channel, relaxing algebraically.

Conversely, under OBC the non-Bloch spectrum is gapped, and the skin effect produces a directional damping front, known as chiral damping, beyond which the decay becomes exponential. The scenario becomes more complex when considering a topological superconductor, which hosts Majorana zero modes protected by particle-hole symmetry. Superconducting pairing couples normal and anomalous correlations, so the closed dynamical object is the full Majorana covariance matrix.

Distinct Majorana rapidity channels can then coexist, and a physical observable may mix them rather than resolve them independently. Consequently, a gapless sector need not govern every observable. In the present model, the local density, constructed from cross-sector covariances, can relax exponentially even when one sector is gapless, an insensitivity termed sector-blindness. This phenomenon is established by a bond-dissipative modified dimerized Kitaev chain, whose Hermitian limit hosts trivial, topological superconducting, and SSH-like regimes.

Each sector forms a non-Hermitian SSH-like rapidity chain with generally distinct effective couplings, GBZs, damping gaps, and non-Bloch windings. Although the total damping gap does not capture this direct-sum structure, the sector windings predict the number of isolated OBC edge rapidities and establish a sector-resolved non-Bloch bulk-boundary correspondence. For finite μ, the sectors hybridize, necessitating the full rapidity matrix.

Since the Majorana density is sector-blind, resolving the slow channel requires same-sector covariance elements, which are not directly accessible. Therefore, attention turns to the spatial entanglement spectrum (ES), which for a Gaussian state is determined entirely by the covariance matrix restricted to a spatial subsystem. Within this spectrum, finite-time entanglement zero events, corresponding to crossings or touchings of the entanglement eigenvalues, can diagnose post-quench topology.

Recent work has shown that PBC ES dynamics can detect topology defined from the OBC spectrum in a Lindbladian SSH model with a Liouvillian skin effect. The findings demonstrate that, for balanced gain and loss, finite-time zero events in the total spatial ES signal the presence of nontrivial OBC sector topology even though the physical evolution remains periodic. Together, the sector edge-count rule, the covariance-selection rule, and the entanglement response establish a sector-resolved dynamical correspondence between PBC evolution and hidden OBC topology.

i − Σj h t2b† jaj+1 + ∆2b† ja† j+1 + H.c. Each sector is thus a non-Hermitian SSH chain.

This sector decomposition is exact only at μ = 0. For finite μ, the two Majorana sectors hybridize, so the sector GBZs and windings are no longer separately defined; the corresponding full-matrix crossover is analysed in Appendix E. Before defining the sector invariants, it is noted why a non-Bloch description is required for the present model.

Direct diagonalization of the Bloch rapidity matrix X(k) under PBC and XOBC under OBC gives qualitatively different spectra. Finite bond dissipation causes the Hermitian bulk gap closings to broaden into finite real-gapless rapidity windows within the periodic boundary condition spectrum.

The boundaries of these windows correspond to second-order exceptional points, identified by the vanishing of the biorthogonal phase rigidity. Detailed spectra for both periodic and open boundary conditions, alongside exceptional-point diagnostics, are presented in Appendix B. At zero chemical potential, the Majorana rapidity matrix decomposes into two independent non-Hermitian sectors, each possessing a generalised Brillouin zone and non-Bloch winding number, revealing a sector-resolved non-Bloch bulk-boundary correspondence. The open boundary condition bulk rapidity continuum is not derived from the Bloch unit circle.

Instead, it is obtained from a sector-dependent non-Bloch continuation. The sector-dependent generalised Brillouin zone construction and the resulting non-Bloch bulk continua are detailed in Appendix C. Excluding a uniform shift, each sector block is purely offdiagonal and therefore chiral symmetric, which quantizes the non-Bloch winding.

The sector winding number then reduces to a value determined by the sector parameters. The local density is a cross-sector covariance and relaxes at the sum of the two sector rates, so it remains sector-blind even when one sector is gapless and topological.

Sector decomposition reveals hidden topology in dissipative superconductors

At zero chemical potential, the Majorana rapidity matrix in a dissipative topological superconductor decomposes into two independent sectors, allowing for the identification of a sector-resolved non-Bloch bulk-boundary correspondence. Previously, discerning topological properties via local observables in such systems was thought to be straightforward, relying on established relationships between relaxation dynamics and topology.

However, local observables become sector-blind, unable to detect the underlying topology despite its presence. This sector-blindness arises because the local density relaxes at the sum of the two sector rates, even when one sector exhibits gapless and topological behaviour, necessitating a new approach to characterise these complex systems.

Dissipation-induced topology requires entanglement spectrum analysis for reliable detection

The pursuit of strong quantum technologies hinges on identifying and controlling topological states of matter, materials exhibiting properties protected from local disturbances. These systems promise stable quantum bits, essential for fault-tolerant computation, and scientists have increasingly turned to dissipation, the loss of energy, as a means to engineer and detect these states. However, the findings reveal a surprising fragility in this approach; while dissipation can create topological behaviour, standard methods for diagnosing it via local measurements become unreliable. Analysing the entanglement spectrum under specific conditions offers a reliable alternative for detecting these hidden characteristics.

Researchers found that in a dissipative topological superconductor, local measurements cannot reliably detect the system’s topology despite its presence. This means conventional methods that link energy relaxation to topological properties fail in these materials, as local density relaxes based on combined sector rates. The study demonstrates that analysing the entanglement spectrum with periodic-boundary Lindblad evolution can instead reveal hidden topological content sector by sector, offering a new way to characterise these complex systems. The authors identified this behaviour using a bond-dissipative dimerized Kitaev chain model.