Nontrivial topological phases in "Zig-Zag" arrays of polarization transmons

TL;DR

Problem addressed: realize and characterize topological phases in a multi-orbital, long-range coupled 1D system. Approach: implement an extended Zig-Zag chain using polarization transmons with degenerate dipole orbitals, and analyze with topological invariants (winding number $\\mathcal{W}$) and inverse participation ratio (IPR). Findings: the model supports gapped topological and trivial phases, with in-gap edge states tied to nonzero $\\mathcal{W}$ and, in the trivial regime, Tamm-like defect modes; sufficiently strong next-nearest-neighbor coupling $g$ yields pairs of edge states in the gap. Methods validation: finite-element electromagnetic modeling confirms that the linearized circuit reproduces the extended Zig-Zag band structure and edge-mode profiles, indicating experimental feasibility. Significance: this platform enables experimental exploration of topological and many-body phenomena in multi-orbital superconducting systems.

Abstract

In recent years, quantum simulators of topological models have been extensively studied across a variety of platforms and regimes. A new promising research direction makes use of meta-atoms with multiple intrinsic degrees of freedom, which to date have been predominantly studied in the classical regime. Here, we propose a superconducting quantum simulator to study an extension of the well-known "Zig-Zag" model with long-range cross-polarization couplings using polarization transmons hosting degenerate dipole orbitals. We map the phase transitions of the extended "Zig-Zag" model both numerically and analytically using inverse participation ratios and topological invariants. We demonstrate the existence of in-gap localized trivial and Tamm edge states. With linearized meta-atoms, we show via electromagnetic modeling that the proposed arrangement closely reproduces the extended "Zig-Zag" model. This work paves the way towards experimental investigation of the previously inaccessible topological quantum many-body phenomena.

Figures (4)

• Figure 1: Polarization transmon as an elementary cell for the quantum topological "Zig-Zag" model simulator. a) A realistic geometry for an electrical circuit of the polarization transmon. Four closely positioned superconducting islands form axial and diagonal capacitors ($C, C^\prime \approx 33.9, 7.7$ fF assuming $\epsilon_\text{eff}\approx6$). The nearest neighbors are connected by identical Josephson junctions (characterized by the $E_J/h \approx 14.9$ GHz constant) forming the quantum potential of the meta-atom. b) The doubly degenerate energy spectrum of a 7-site "Zig-Zag" chain with $\theta = 90 \degree$ and only nearest-neighbour couplings ($g=0$). Two pairs of degenerate bulk bands and two edge state distributions are also shown. c) The nearest-neighbour $\pi$-type coupling ($t_{\bot}$) and $\sigma$-type coupling ($t_{||}$). Additional NNN cross-polarization coupling $g$ couples the $p_x$ and $p_y$ subspaces. d) Superconducting "Zig-Zag" model visualized as coupled $p_x$ and $p_y$ bosonic subspaces (orthogonal polarizations in the classical limit), which are SSH-chains with alternating couplings $t_{||}$ and $t_{\bot}$ coupled via $g$ interaction.
• Figure 2: Extended "Zig-Zag" band structure $E_{1-4}(K)$ (left column) and winding diagram (right column) for $t_{||} = -0.1$, $t_{\bot} = 0.037$ and varying NNN coupling strength. a)$g = 0.016$: the system is gapped and $\mathcal{W} = 1$. b)$g = 0.0315$: the system is gapless and $\mathcal{W}$ is undefined. c)$g = 0.048$: the system is gapped and $\mathcal{W} = 0$.
• Figure 3: Phase (IPR) and transition frequency diagrams for extended "Zig-Zag" chains of length $N=9$a,c) and 101 b,d). In a,b), the white dashed line (left) indicates $t_\bot = t_{||}$, the red lines are analytic phase transition points and the orange dashed line (left) indicates $t_\bot = 0.037$, where the spectra c,d) are calculated. e,f) The mid-spectrum eigenstates for $g = 0.016$ and $g = 0.1$ respectively. As one can see in the top panel, both polarizations contribute to the topological edge state due to the NNN interaction.
• Figure 4: A quantitative comparison between three classical electromagnetic eigenmodes and single-excitation wavefunctions calculated from the second quantization formalism for the proposed architecture, $N=9$. a) Energy spectrum obtained from the second quantization Hamiltonian. b) Lowest mode of the higher bulk band. c) Lower frequency edge mode. d) Higher frequency edge mode. In the top parts of the b-d) panels, the data from the FEM modeling are shown; the vertical component of the electric field directly above the electrode plane is plotted with color. In the bottom, extracted amplitudes of perpendicular polarization components $p_{x,y}$ are shown with bars, and the amplitudes obtained from second quantization Hamiltonian are shown in black outline.