Reduction of Topological $\mathbb{Z}$ Classification in Cold-Atom Systems

One of the most challenging problems in correlated topological systems is a realization of the reduction of topological classification, but very few experimental platforms have been proposed so far. We here demonstrate that ultracold dipolar fermions (e.g., ${}^{167}\mathrm{Er}$, ${}^{161}\mathrm{Dy}$, and ${}^{53}\mathrm{Cr}$) loaded in an optical lattice of two-leg ladder geometry can be the first promising test bed for the reduction $\mathbb{Z}\to {\mathbb{Z}}_4$, where solid evidence for the reduction is available thanks to their high controllability. We further give a detailed account of how to experimentally access this phenomenon; around the edges, the destruction of one-particle gapless excitations can be observed by the local radio frequency spectroscopy, while that of gapless spin excitations can be observed by a time-dependent spin expectation value of a superposed state of the ground state and the first excited state. We clarify that even when the reduction occurs, a gapless edge mode is recovered around a dislocation, which can be another piece of evidence for the reduction.

Figure 1

(a) Sketch of the model (1). Blue (red) circles denote $A$ ($B$) sublattice, respectively. Brown (gray) lines represent hopping $V$ ($t$), respectively. (b) Phase diagram of the intraladder interaction $U$ vs the interladder interaction $J$. Red dots are data points. (c) Highest 20 Schmidt eigenvalues (${\lambda }_{\alpha }$) for each case of parameters. Each eigenvalue is normalized with ${\lambda }_0$. These data are obtained under the OBC and for $L=30$. For the calculation of the Schmidt eigenvalues (${\lambda }_{\alpha }$), we consider a virtual cut dividing the system at the strong bond (gray line) with $i=L/2$. We have observed the same structure of ES for larger $L$.

Figure 2

(a) [(b)] The charge gap ${\mathrm{\Delta }}_c$ and the spin gap ${\mathrm{\Delta }}_s$ as functions of the interaction strength under the OBC for $J=0$ ($U=5t$), respectively. In panel (b), the data of ${\mathrm{\Delta }}_c$ are multiplied by 0.1.

Figure 3

(a) Sketch of a dislocation with $L_d=1$. (b) The spin gap ${\mathrm{\Delta }}_s$ for $V=0.2t$, $0.4t$, $0.6t$, $0.8t$, and $0.9t$ in the presence of dislocations. The data are obtained for $(U,J)=(5t,0.018t)$.