Steady-state skin effect in bosonic topological edge states under parametric driving

Technical Summary: Steady-state skin effect in bosonic topological edge states under parametric driving

Problem Statement
The non-Hermitian skin effect (NHSE), characterized by the extreme sensitivity of lattice spectra to boundary conditions and the accumulation of eigenmodes at boundaries, has been extensively studied mathematically and in classical or artificial setups. However, realizing this phenomenon in realistic quantum condensed matter systems remains challenging. Conventional approaches often rely on explicit dissipation or open quantum systems, which can obscure the intrinsic quantum nature of the effect. The authors address the need to identify a platform where the NHSE arises from the intrinsic quantum properties of bosonic systems—specifically the non-Hermiticity inherent in bosonic Bogoliubov–de Gennes (BdG) Hamiltonians—rather than from external dissipation.

Methodology
The study proposes a theoretical framework utilizing bosonic topological edge states subjected to parametric driving. The core methodology involves:

1. Model Construction: The authors construct a model based on a bosonic Chern insulator (specifically the QWZ model) with added parametric pumping terms. In the rotating frame, these pumping terms introduce "anomalous" pairing terms ($a_i a_i$ and $a_i^\dagger a_i^\dagger$) that break particle-number conservation. This renders the effective Hamiltonian non-Hermitian in the BdG formalism without requiring explicit dissipation.
2. Spectral Analysis: The spectral properties are analyzed under two boundary conditions: Cylindrical Boundary Conditions (CBC) and Fully Open Boundary Conditions (FOBC). The authors examine the complex energy spectra and the winding numbers associated with the edge states to identify the presence of the skin effect.
3. Steady-State Calculation: To investigate the physical realization of the skin effect, the authors compute the one-particle reduced density matrix using non-equilibrium Green's functions. They introduce a constant, $\omega$-independent dissipation term in the self-energy to stabilize the system and define a non-equilibrium steady state (NESS). This approximation is justified within the rotating frame under parametric driving, where the self-energy shift allows for a consistent description of dissipation that would be impossible in thermal equilibrium.
4. Observables: The study calculates the spatial distribution of particle accumulation and the quadrature fluctuations ($\langle x_{i,\theta}^2 \rangle$) to characterize the nature of the steady state.

Key Contributions and Results

• Realization of NHSE via BdG Non-Hermiticity: The paper demonstrates that the NHSE can be realized in a bosonic system driven by parametric pumping. The non-Hermiticity arises intrinsically from the bosonic commutation relations and the anomalous pairing terms, rather than from coupling to an external bath.
• Boundary-Dependent Spectral Behavior: Under CBC, the system exhibits complex edge energies with positive imaginary parts, leading to dynamical instability. However, under FOBC, the skin effect manifests as a transition where the edge states acquire real energies (or negligible imaginary parts), effectively suppressing the dynamical instability associated with edge amplification. This spectral difference between CBC and FOBC is a hallmark of the NHSE.
• Role of Unitary Symmetry: The study identifies a specific onsite unitary symmetry ($U = \hat{1} \otimes \tau_x \otimes \sigma_x$) that protects the skin effect. In the presence of this symmetry, the system exhibits distinct spectral windings in different symmetry sectors. The authors note that breaking this symmetry (e.g., via a perturbation $\epsilon$) can cause the FOBC spectrum to revert to the CBC spectrum, leading to the disappearance of the skin effect or triggering "infinitesimal instability."
• Steady-State Corner Accumulation: In the steady state, the system exhibits pronounced particle accumulation localized at the corners of the lattice. This accumulation is driven by the transient amplification of wavepackets due to the strong non-normality of the effective Hamiltonian.
• Quadrature Anisotropy and Squeezing: A significant finding is the manifestation of the bosonic quantum nature through "quadrature anisotropy." While the bulk remains near the vacuum state (standard quantum limit), the driven edges and corners show enhanced deformation of the quadrature ellipse. Notably, the fluctuations at the driven edge are squeezed (below the standard quantum limit), while the corners exhibit large fluctuations with strong anisotropy. This distinguishes the bosonic skin effect from its fermionic counterpart, where such transient amplification and squeezing do not occur.

Significance
The paper claims to bridge the gap between the mathematical theory of non-Hermitian matrices and practical quantum physical systems. By proposing a mechanism that utilizes intrinsic BdG non-Hermiticity, the authors provide a platform to study the NHSE in quantum condensed matter that is largely independent of microscopic model details. The results suggest that the bosonic skin effect is not merely a spectral curiosity but leads to observable steady-state phenomena, such as corner particle accumulation and quadrature squeezing, which are direct consequences of the interplay between non-equilibrium dynamics, non-Hermiticity, and bosonic quantum statistics. The work highlights that these effects are robust in systems like photonic, magnonic, and phononic platforms where bosonic BdG descriptions are applicable.