Spin Chern phases and persistent spin texture in a quasi 2D SSH model

Abstract

We construct a quasi-two-dimensional Su Schrieffer-Heeger model (SSH) like model and uncover a rich set of topological phases with nontrivial spin textures in the presence of complex hopping and spin orbit coupling. Despite its simple structure, the combined effect of complex hopping and spin orbit interaction gives rise not only to the conventional quantum anomalous Hall insulating (QAHI) phase, but also to distinct combinations of spin Chern phases, namely quantum anomalous spin Hall insulating (QASHI) phase. Furthermore, we demonstrate that the bulk bands of this model can host persistent spin textures, whose formation and stability are governed by the relative strengths of nearest and next nearest neighbor complex hopping. To elucidate the underlying mechanisms, we develop a low energy continuum theory that captures the emergence of these topological phases and clarifies the origin of the persistent spin textures. Interestingly, the resulting spin textures closely resemble those typically observed in conventional semiconductor systems with topologically trivial band structures. However, in our case, they emerge within a nontrivial topological framework, enabled by carefully engineered hopping patterns that intertwine lattice geometry, complex hopping, and spin orbit coupling

Figures (9)

• Figure 1: Spin texture of Rashba SOC with effective momentum-dependent vector field, ${\bf h}_{\rm RSOC}=(k_y,-k_x)$ (a), Dresselhauss SOC with effective vector field, ${\bf h}_{\rm DSOC}=(k_x, -k_y)$, (b) and both the SOC with equal strength leading to an effective field, ${\bf h}_{\rm DSOC+\rm RSOC}=(k_y+kx, -k_x-k_y)$ (c). Evidently, we see unidirectional spin texture when both SOCs are of equal strength.
• Figure 2: Schematic illustration of the quasi two-dimensional spinful SSH model. One-dimensional SSH chains extend along the $x$ direction and are coupled along the $y$ direction to form a ladder-like 2D lattice geometry. The hopping $\gamma$ represents anisotropy between different sublattices. The interlayer hopping $\delta$ in (c) shows staggered pattern.
• Figure 3: a) The topological phase diagram from the full lattice Hamiltonian (Eq. \ref{['eqn:main_ham']}) in the plane of model parameters $v$ and $w$ of the standard 1D SSH. Here we set $\delta = 0.5,\tau = 0.2$. The black solid line represent the phase boundary obtained from the expression of Dirac points $k_{D_y}$.b) Same as (a) once we introduce long-range hopping $\gamma=0.5,\eta_{1,2}=0$. Fig (c-f) shows dispersion relation with $w = 2, v = 1$. For Fig (c,d) $\gamma = 0, \eta = 0$ , Fig (e,f) $\gamma = 0.5, \eta = 1$
• Figure 4: The topological phase diagram from the full lattice Hamiltonian in the plane of model parameters $v$ and $w$ standard 1D SSH with spin conserving SOI term $\zeta = 1$ . Here we have $\delta = 0.5, \tau = 0.2, \gamma = 0.5$.Fig (c-f) shows dispersion relation with $w = 2, v = 1$. For Fig (a,c,d) $\eta_1= \eta_2 = 1$ , Fig (b,e,f) $\eta_1 = 1, \eta_2 = 2$
• Figure 5: The topological phase diagram from the full lattice Hamiltonian in $v-w$ plane for a set of four parameters $(\eta_1,\eta_2)=(0,0)$ (a), $(1,0)$ (b), $(1,1)$ (c) and $(4,1)$ (d) with spin conserving SOI term $\zeta = 1$ and RSOI term $\alpha = 1$ . Here we have $\delta = 0.5, \tau = 0.2, \gamma = 0.5$.