Two-leg Su-Schrieffer-Heeger chain with glide reflection symmetry

The Su-Schrieffer-Heeger (SSH) model lays the foundation of many important concepts in quantum topological matters. Here, we show that a spin-dependent double-well optical lattice allows one to couple two topologically distinct SSH chains in the bulk and realize a glided-two-leg SSH model that respects the glide reflection symmetry. Such a model gives rise to intriguing quantum phenomena beyond the paradigm of a traditional SSH model. It is characterized by Wilson lines that require non-Abelian Berry connections, and the interplay between the glide symmetry and interaction automatically leads to charge fractionalization without jointing two lattice potentials at an interface. Our work demonstrates the versatility of ultracold atoms to create new theoretical models for studying topological matters.

Figure 1

Glided-two-leg SSH model in a spin-dependent optical lattice. (a) Spin-up (red color) and spin-down atoms (green color) are loaded in two different double-well lattices, each of which is shifted from the other by half of the lattice spacing $d/2$. A microwave field provides an interleg tunneling $t$ (wiggles). The single (purple) and double (orange) lines represent two tunnelings $t_1$ and $t_2$, respectively. (b) Tight-binding model. Red (dark gray) and blue (black) dots represent different sublattice sites, respectively.

Figure 2

Band structure, Bloch oscillation, and Wilson line. (a) Band structure and Bloch oscillation. The light blue and yellow regions correspond to the first and second BZ, respectively. The first, second, third, and fourth bands are represented by solid, dotted, dotted, and solid curves, respectively. The $-$ (red) and $+$ (blue) branches of the eigenstates of the glide operator are distinguished by colors. In the presence of an external force, a particle starting from $A$ in the first band crosses the zone boundary and enters the second band in the second BZ (point $B^{\prime }$, which is identical to $B$). After traveling for one reciprocal lattice vector $2\pi /d$, the particle ends at $C$, a state orthogonal to the initial one at $A$. After another reciprocal lattice vector $2\pi /d$, the particle returns to $A$. The primes in symbols $A^{\prime }$–$D^{\prime }$ represent the different signs of $\eta$ in the second BZ from the first BZ. (b) Wilson line. When a particle initially stays at the first band, ${\left|W_{0\to k}^{11}\right|}^2$ corresponds to the probability of the particle to remain at this band. Since the electric field cannot couple the $+$ and $-$ branches, ${\left|W_{0\to k}^{11}\right|}^2$ remains as 1 or 0 unless crossing the zone boundary.

Figure 3

Charge fractionalization. (a) At half filling, a repulsive interaction leads to spontaneous symmetry breaking and a ferromagnet emerges. The red (dark gray) and green (light gray) clouds represent the Wannier wave functions of the spin-up and spin-down particles, respectively. (b) Doping an extra particle forms two domain walls (blue ovals). If $t_2=0$, the two domain walls are deconfined because of interchain tunneling $t$ (black wiggle). (c) If $t=0$, the two domain walls are confined for any finite intrachain tunneling $t_2$ (purple arrow).

Figure 4

The distribution of the extra particle. The parameters used in the TEBD simulation are $t_1=-0.4E_R, t_2=-4\times {10}^{-4}{E}_R, t=8\times {10}^{-4}{E}_R$, and $U=0.012E_R$. The lattice site is $N=30$ with an open boundary condition. (a) For deconfined domain walls, the result of the TEBD simulation agrees with the density distribution of two hard-core particles (blue). (b) The additional local potential $V_L=V_R=-0.002E_R$ (yellow curve). When two deconfined domain walls are localized at the right well of the lattice site $j_L=7$ and the left well of the lattice site $j_R=24$, the extra particle density is centered around these two lattice sites.