[Paper Review] Interplay between non-Hermiticity and non-Abelian gauge potential in topological photonics

Jiaqi Cai, Qingyun Yang+4 more|arXiv (Cornell University)|Dec 6, 2018

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper proposes a superconducting circuit platform to study the interplay between non-Hermitian loss/gain and non-Abelian gauge potentials in topological photonics. It demonstrates that non-Abelian gauge fields induce a 'flying' Hofstadter butterfly and robust edge modes, while non-Hermiticity breaks bulk-edge correspondence, enabling a gapless quantum spin Hall phase with stable counter-propagating edge modes despite complex-energy spectra and avoided level mixing.

ABSTRACT

Topological phases in spinless non-Hermitian models have been widely studied both theoretically and experimentally in some artificial materials using photonics, phononics and magnon. In this work, we investigate the interplay between non-Hermitian loss and gain and non-Abelian gauge potential realized in a two-component superconducting circuit. In our model, the non-Hermiticity along only gives rise to trivial gain and loss to the states; while the non-Abelian gauge along gives rise to flying butterfly spectra and associated edge modes, which in photonics can be directly measured by the intensity of photons at the boundaries. These two terms do not commute, and their interplay can give rise to several intriguing non-Hermitian phases, including the fully gapped quantum spin Hall (QSH) phase, gapless QSH phase, trivial gapped phase and gapless metallic phase. The bulk-edge correspondence is absent and we find that during the closing of energy gap in the gapped QSH phase, the system enters the gapless QSH phase regime which still supports two counter-propagating edge modes. We have also unveiled the intriguing role of non-Hermiticity on the chiral symmetry and time-reversal symmetry of the Hermitian models, which can be applied to other physical models.

Motivation & Objective

• To investigate the interplay between non-Hermitian loss/gain and non-Abelian gauge potentials in a controllable superconducting circuit platform.
• To understand how non-commuting non-Hermitian and non-Abelian terms affect topological phases in a two-component system.
• To examine the breakdown of bulk-edge correspondence in non-Hermitian topological systems and the survival of edge modes in gapless regimes.
• To explore the role of non-Hermiticity in breaking or preserving symmetries such as chiral and time-reversal symmetry in topological models.

Proposed method

• Realizes a two-component superconducting circuit model with tunable non-Abelian gauge potential via dynamic flux modulation.
• Introduces non-Hermitian loss and gain through engineered dissipation in the system, controlled via coupling to external reservoirs.
• Uses a cylindrical geometry with momentum $k_y$ as a synthetic flux to simulate threaded flux and observe the 'flying' Hofstadter butterfly effect.
• Employs exact diagonalization and spectral analysis in finite and infinite systems to study eigenvalue distributions in the complex plane.
• Applies symmetry analysis, including a novel anti-unitary symmetry $Q$, to characterize spectral symmetries and edge mode robustness.
• Uses a staggered potential $V_{\text{stag}}$ to reopen the band gap and stabilize the quantum spin Hall phase in the presence of non-Abelian coupling.

Experimental results

Research questions

• RQ1 How does non-Abelian gauge potential induce a 'flying' Hofstadter butterfly in a non-Hermitian system?
• RQ2 What happens to topological edge modes when non-Hermitian loss and gain are introduced alongside non-Abelian gauge fields?
• RQ3 Can robust edge modes survive in a gapless quantum spin Hall phase when bulk and edge states have different complex-energy eigenvalues?
• RQ4 How does the non-commutativity of non-Hermitian and non-Abelian terms alter the topological phase diagram?
• RQ5 To what extent is bulk-edge correspondence preserved or broken in non-Hermitian topological systems with non-Abelian gauge fields?

Key findings

• The non-Abelian gauge potential alone generates a 'flying' Hofstadter butterfly and supports topologically protected edge modes that obey bulk-edge correspondence.
• Non-Hermitian loss and gain alone induce trivial gain and loss without topological protection, but do not break the edge mode robustness when combined with non-Abelian terms.
• The non-commuting interplay between non-Hermitian and non-Abelian terms leads to four distinct non-Hermitian phases: fully gapped QSH, trivial gapped, gapless QSH, and metallic phases.
• Bulk-edge correspondence is explicitly broken, yet robust counter-propagating edge modes persist in the gapless QSH phase due to energy-level separation in the complex plane.
• The resonant coupling between edge modes and bulk states is forbidden because their complex eigenvalues do not overlap, ensuring edge mode stability under perturbations.
• A novel anti-unitary symmetry $Q$ is identified that protects spectral symmetry and enables identification of edge modes via their non-symmetric complex-energy distribution.