Topological phases in frustrated synthetic ladders with an odd number of legs

The realization of the Hofstadter model in a strongly anisotropic ladder geometry has now become possible in one-dimensional optical lattices with a synthetic dimension. In this work, we show how the Hofstadter Hamiltonian in such ladder configurations hosts a topological phase of matter which is radically different from its two-dimensional counterpart. This topological phase stems directly from the hybrid nature of the ladder geometry and is protected by a properly defined inversion symmetry. We start our analysis by considering the paradigmatic case of a three-leg ladder which supports a topological phase exhibiting the typical features of topological states in one dimension: robust fermionic edge modes, a degenerate entanglement spectrum, and a nonzero Zak phase; then, we generalize our findings—addressable in the state-of-the-art cold-atom experiments—to ladders with a higher number of legs.

Figure 1

(a) Schematic representation of the Hofstadter Hamiltonian on a three-leg ladder. In the presence of a nonvanishing coupling between the extremal states $m=+1$ and $m=-1$ (${\tilde{\mathrm{\Omega }}}_j\ne 0$), the ladder is equivalent to a cylinder. (b) Phase diagram for a three leg-ladder.

Figure 2

The spectrum $E\left(k\right)$ of the Hamiltonian $\widehat{H}$ for a three-leg ladder with periodic boundary conditions along the real direction pierced by a flux $\gamma =2\pi /3$ for different values of the coupling $\tilde{\mathrm{\Omega }}$; $\mathrm{\Omega }=1$ and $L_x\to +\infty$.

Figure 3

(a) The eigenvalues $E_{\alpha }$ of the Hamiltonian $\widehat{H}$ with open boundary conditions along the real direction. (b) The wave functions corresponding to the two states between the lower and the intermediate band. (c) The Zak phase of the lower band as a function of $\tilde{\mathrm{\Omega }}$. (d) The 60 largest eigenvalues of the entanglement spectrum for $\ell =4$; $n$ is an index which labels the eigenvalues. (a–d) $L_x=96$, $L_y=3$, $\gamma =2\pi /3$. (a–b, d) $\tilde{\mathrm{\Omega }}=\mathrm{\Omega }$.

Figure 4

Schematic plot of a three-leg ladder with ${\widehat{H}}_2^{\mathrm{osc}}$ in the presence (a) and absence (b) of inversion symmetry. Eigenvalues with open boundary conditions along the real direction in the presence (c) and absence (d) of inversion symmetry. Schematic plot of a three-leg ladder with ${\widehat{H}}_2^{\mathrm{osc}}$ in the presence (e) and absence (f) of the generalized antiunitary inversion symmetry.

Figure 5

The eigenvalues $E_{\alpha }$ of the Hamiltonian $\widehat{H}+{\widehat{H}}_{\mathrm{noise}}$ in the presence of inversion symmetric disorder (a) and completely random disorder (b). Here $L_x=96$, $\mathrm{\Omega }=\tilde{\mathrm{\Omega }}=1$, and $\gamma =2\pi /3$.

Figure 6

The density profile for a three-leg ladder of length $L_x=96$ sites and $N=97$ particles. Here $\mathrm{\Omega }=\tilde{\mathrm{\Omega }}=1$ and $\gamma =2\pi /3$. The inset displays the expectation values of ${\widehat{n}}_j^{*}$.

Figure 7

Density profile and the expectation value of the operator ${\widehat{n}}_j^{**}$ in the presence of harmonic confinement in the trivial region (a,b) and in the topological region (b,d). Here $L_x=96$, $\mathrm{\Omega }=1$, $\omega =0.006$, and $N=22$ particles; and $\tilde{\mathrm{\Omega }}=0$ in the trivial region and $\tilde{\mathrm{\Omega }}=\mathrm{\Omega }$ in topological region.

Figure 8

The correlator $|\langle {\widehat{d}}_{j,m}^{†}{\widehat{d}}_{j,-I}\rangle |$ as a function of $m$ for a $L_y=81$-leg ladder and different fluxes $\gamma$. Here $\mathrm{\Omega }=\tilde{\mathrm{\Omega }}=1$. Periodic boundary conditions both along the real and the synthetic directions are used and $L_x\gg L_y$.

Figure 9

(a) The eigenvalues $E_{\alpha }$ of the Hamiltonian $\widehat{H}$ with open boundary conditions along the real direction and $\tilde{\mathrm{\Omega }}=\mathrm{\Omega }$. (b) The Zak phase of the lower band as a function of $\tilde{\mathrm{\Omega }}$; here $L_x=120$, $L_y=5$, $\gamma =2\pi /5$, and $\mathrm{\Omega }=2$.