Reentrant topological phases and spin density wave induced by 1D moiré potentials

TL;DR

This work addresses how a commensurate 1D moiré potential reshapes topology and correlations in a spin-1/2 fermionic lattice. It combines analytical winding-number analysis for the single-particle sector with DMRG for the interacting many-body regime to reveal a sequence of trivial–topological–trivial–topological–trivial transitions driven by moiré strength, accompanied by edge states and PM-SDW order. The study shows that moiré geometry renormalizes effective fields, producing reentrant topology with universal scaling exponents, and that PM-SDW persists in interacting regimes while being tunable by on-site and nearest-neighbor interactions. These findings broaden the landscape of moiré topological physics in 1D, offer a route to tunable topological and spin-density orders in ultracold-atom experiments, and motivate extensions to higher dimensions and incommensurate moiré systems.

Abstract

Recent studies of 2D moiré materials have opened opportunities for advancing condensed matter physics. However, the effect of 1D moiré potentials on topological and correlated phases remains largely unexplored. Here we reveal a sequence of trivial-to-topological transitions and periodic-moiré-spin density waves induced by the 1D commensurate moiré potentials for spin-1/2 fermionic atoms. Such reentrant topology from a trivial phase is absent without the moiré potential and can be understood as the renormalization of topological parameters by the moiré strength. We then unveil the critical exponent and localization properties of the single-particle eigenstates. The periodic spin density wave of many-body ground states is contributed by the moiré potential, and is enhanced by on-site interactions but suppressed by nearest-neighbor interactions. Our results enrich the topological physics with multiple transitions and spin-density orders in 1D moiré systems, and the realization of the proposed model is promising in near-future ultracold atom setups.

Figures (9)

• Figure 1: Topological phase diagram and related properties in the single particle region. (a) Real-space winding number $\nu$ and (b) inverse of the zero-mode localization length $\Lambda^{-1}$ as functions of $m_o$ and $m_z$. Red dashed curve in (a) denotes the topological phase boundary revealed by the momentum-space winding number $\nu_k$. Horizontal and vertical black dashed lines in (a) correspond to the cuttings for $m_z=2.4$ in (c) and $m_o=2.8$ in (d), respectively. Real-space winding number $\nu$ and energy gap $\Delta E$ as functions of (c) $m_o$ for $m_z=2.4$ and (d) $m_z$ for $m_o=2.8$ under the OBC. (e) Energy spectrum with respect to $m_o$ under the OBC. The zero-energy modes in the topological regions are highlighted in red. The zoom in shows a detailed view of the energy spectrum in the second topological region. (f) Density distributions of the two zero-energy edge modes for $m_z=2.4$ and $m_o=1.7$. Other parameters are $t_s=0.95$, and $A=32$.
• Figure 2: Finite-size scaling of the topological invariant. (a) $\partial \nu / \partial m_o$ as a function of $m_o$ with $m_z=2.4$ and various system sizes. The finite-size critical point at $m_o=m_o^{(L)}$ is given by the peak of each curve indicated by the arrow with $m_o^{(L)}$. (b) Finite-site scaling of the distance from $m_o^{(L)}$ to the ideal transition point $m_{oc}=1.223$. (c) $\partial \nu / \partial m_z$ as a function of $m_z$ with $m_o=2.8$ and various system sizes. The peak is indicated by the arrow with $m_z^{(L)}$. (d) Finite-site scaling of the distance from $m_z^{(L)}$ to $m_{zc}=1.678$. Other parameter is $t_s=0.95$.
• Figure 3: Influence of disorders on the reentrant topological transition. Disorder averaged real-space winding number $\overline{\nu}$ and energy gap $\overline{\Delta E}$ are plotted as functions of $m_o$ under the OBC. Disorder is added on the spin-dependent hopping $t_j=t+W_j$ (a), spin-flip hopping $t_{sj}=t_s+W_j$ (b), Zeeman potential $m_{zj}=m_z+W_j$ (c), and all these three components (d). Other parameters are the same as Fig. \ref{['fig1']} (c), $W_j\in[-W,W]$ with $W=0.2$, and $20$ different disorder realizations are used. Error bars indicate the standard deviation of the sampled data between different disorder realizations.
• Figure 4: Flat band structure and the flatness parameter. The energy spectrum in momentum space for (a) $m_o=0.1$, (b) $m_o=1$ and (c) $m_o=10$ under PBC. (d) Inverse flatness of ${s}$-th energy band $f_{s}^{-1}$ for ${s}=1,10$ as a function of $m_o$. Other parameters are $t_s=0.95$ and $m_z=2.4$.
• Figure 5: Localization properties of single particle eigenstates. (a) Real-space fractal dimension (FD) and (b) momentum-space FD of eigenstates versus $m_o$ with $A=32$. The three colored dots indicate the chosen parameter of the three scaling lines in (d). (c) Density distributions for typical extended (blue), critical (black), and localized (red) states for $A=20$. (d) Finite-size scaling of the real-space FD for three typical states labeled in the legend with $L=21A$, $N=2L$, and $A=\{32,48,64,80,96\}$. Other parameters are $t_s=0.95$ and $m_z=2.4$.