Non-Hermitian Aharonov-Bohm Cage in Bosonic Bogoliubov-de Gennes Systems

TL;DR

This work addresses how non-Hermitian Aharonov-Bohm cages create degenerate flat bands (DFBs) in bosonic Bogoliubov-de Gennes systems and how to classify their degeneracy types. The authors develop a transfer-matrix formalism for a ladder of coupled bosonic Kitaev-Majorana chains, derive the annihilating polynomial of the dynamical matrix, and show that flat bands appear under the conditions $t=J$ and $t_1\cos heta_1 = t_2\\cos heta_2$, yielding a momentum-independent spectrum $E=\, mmlsqrt{t_1^2-t_2^2}$. By linking the transfer-matrix structure to the minimal polynomial of the BdG Hamiltonian, they classify DFBs into DP2s, 1stEP2s, 2ndEP2s, and EP4s, with corresponding local ranges up to 4. The paper further generalizes to $N$ coupled chains to realize $2N$-order EPs (EP2Ns), showing that appropriate tuning of inter-chain transfer terms yields $ ilde{H}^{2N}=0$ and a local range of $2N$, i.e., a highly degenerate AB cage. This framework offers a design principle for highly degenerate flat bands and provides a path toward experimental observation in platforms such as superconducting circuits, with potential extensions to higher dimensions. All mathematical relations are expressed with explicit polynomials and transfer-matrix constraints that connect spectral degeneracy to real-space localization and dynamical response.

Abstract

The non-Hermitian Aharonov-Bohm (AB) cage is a unique localization phenomenon that confines all possible excitations. This confinement leads to fully flat spectra in momentum space, which are typically accompanied with the degeneracy with various types. Classifying the degeneracy type is crucial for studying the dynamical properties of the non-Hermitian AB cage, but the methods for such classification and their physical connections remain not very clear. Here, we construct a non-Hermitian AB cage in a bosonic Bogoliubov-de Gennes (BdG) system with various types of degenerate flat bands (DFBs). Using the transfer matrix, we demonstrate the localization mechanism for the formation of AB cage and derive the minimal polynomial in mathematics for classifying the degeneracy types of DFBs, thus providing comprehensive understanding of the correspondence among the degeneracy type of DFBs, the minimal polynomial, and the transfer matrix. With such correspondence, we propose a scheme to realize highly degenerate flat bands.

Figures (6)

• Figure 1: (a) Schematic diagram of the ladder model, consisting of coupled two Kitaev-Majorana chains (right side), where the conjugated coupling and two-boson creation/annihilation processes are denoted by the black lines, red solid/dashed lines, respectively. Corresponding to these couplings, the hopping strengths of the particle and hole degrees of freedom between $a_n$ and $b_n$ is shown in the left side. Under a transformation that exchanges the particle and hole [i.e. $\alpha(\beta)\rightarrow \alpha^{\dagger}(\beta^{\dagger})$, $\alpha^{\dagger}(\beta^{\dagger})\rightarrow \alpha(\beta)$], the coupling strength acquires an anti-conjugated form thereby manifesting the particle hole symmetry in BdG systems 10.1063/5.0035358. (b) The $\theta_1$-$\theta_2$ phase diagram of degeneracy type of DFBs, where the red lines, yellow lines, blue regions and black dots correspond to EP4s type, DPs type, 1stEP2s type and 2ndEP2s type in the energy band. (c) The dependence of the absolute value of eigenvalues variation on the perturbation $\delta J$ for different types of DFBs, when $J = t = 2$ and the wave vector is chosen as $k=0$. The parameter choices are $\left\{ t_1=t_2=2, \theta_1=-\theta_2=\pi/3 \right\}$ for the EP4s type, $\left\{t_1=2\sqrt{3}, t_2=2, \theta_1=\pi/3,\theta_2=\pi/6 \right\}$ for the 2ndEP2s type, $\left\{t_1=t_2=2, \theta_1=\theta_2=0\right\}$ for the 1stEP2s type, and $\left\{t_1=1, t_2=2, \theta_1=0, \theta_2=\pi/3\right\}$ for the DPs type. The $\theta_1,\theta_2$ for the 1stEP2s type are located at the origin point, while those for the other types are marked with pentagrams in their corresponding colors in (b).
• Figure 2: (a) The transfer matrix $U$ between different sites, where $U_l$ ($U_r$, $U_{\uparrow}$, $U_{\downarrow}$) denotes the leftward (rightward, upward, downward) transition. (b) The dynamical matrix $\tilde{H}$ in real space involving all possibilities of propagation, which is exemplified by a specific transfer matrix $U_{a_n,b_n}$ from $b_n$ to $a_n$.
• Figure 3: Schematic diagram of the localization mechanism for the formation of DFBs, which can be simplified as the excitation at the $n$-th column cannot hop to the $(n\pm 2)$-th column along the red and green propagation paths. In each path, the destructive interference occurs within the transfer matrix.
• Figure 4: The realization of AB cage with $2N$-order EPs in $N$ coupled bosonic Kitaev-Majorana chains. Left part: the transfer matrices along the corresponding directions are denoted by the black arrows. Middle part: the conditions for the formation of AB cage match those for the prohibited red propagation paths, which cause the excitation at the $\left(n-4\right)$-th column to be localized between the $\left(n-5\right)$-th column and $\left(n-3\right)$-th column. Right part: the largest local range among possible excitations, illustrated by the allowed transitions (green lines).
• Figure 5: The illustration of gauge-invariant Wilson loop and the sufficient and necessary conditions for the forming of AB cage. Left part: the blue path $l_{loop}$ is a closed loop for excitation transition, and the trace of the link along such loop corresponds to a gauge-invariant Wilson loop. Right part: the excitations transition along the red path $l_1$ and green path $l_2$ to further sites when the transition along the chain is forbidden by the condition $J=t$. To form the AB cage, the excitation should only transition along such paths for finite periods.