Topological Zak phase in strongly coupled LC circuits

We show the emergence of topological Bogoliubov bosonic excitations in the relatively strong coupling limit of an LC (inductance-capacitance) one-dimensional quantum circuit. This dimerized chain model reveals a ${\mathbb{Z}}_2$ local symmetry as a result of the counter-rotating wave (pairing) terms. The topology is protected by the sublattice symmetry, represented by an antiunitary transformation. We present a method to measure the winding of the topological Zak phase across the Brillouin zone by a reflection measurement of (microwave) light. Our method probes bulk quantities and can be implemented even in small systems. We study the robustness of edge modes toward disorder.

Figure 1

(a) Schematic of the dimerized lattice with two sublattices $A$ and $B$ and alternating couplings $(v,w)$. (b) Dimer chain of LC oscillators described by the Hamiltonian (1). (c) Bulk band structure in the strong (solid blue) and weak (dashed red) coupling regimes.

Figure 2

(a) The circle defined by $h\left(k\right)e^{i\phi \left(k\right)}$ in the two phases: For $v<w$ it winds once around the origin, while for $v>w$ the winding number equals to zero. The topological phase transition occurs at $v=w$. (b) $cos\phi \left(k\right)$ in the two phases. In the topological phase, it is a surjective function onto $[-1,1]$ while in the trivial phase it is limited to [0,1]. [(c), (d)] The correlation function ${\chi }_n\left(\omega \right)$ in Eq. (12) in the trivial (c) and topological (d) phases for a ring composed of 20 unit cells. The envelope of the resonances reveals the winding of the Zak phase as explaned in the text.

Figure 3

(a) Energy spectrum of a finite chain with $N=100$ unit cells (200 sites) as a function of the hopping matrix elements $v/w$. Midgap energy states appear in the topological phase $v<w$. (b) Bogoliubov coefficients of the two midgap edge states and energy spectrum for the same chain. [(c), (d)] The correlation function ${\chi }_{n=1}\left(\omega \right)$ in a system of 20 unit cells with open boundary conditions.

Figure 4

Band structure for different disorder realizations. (a) Disorder in the hopping terms $v, w$. (b) Disorder in the chemical potential ${\epsilon }_0$. The disorder is introduced by adding a noise term uniformly distributed between $[-\mathit{dis},\mathit{dis}]$.