Reentrant topological phases and entanglement scalings in moiré-modulated extended Su-Schrieffer-Heeger Model

TL;DR

The paper analyzes moiré-pattern-induced reentrant topological transitions in a one-dimensional extended SSH model, establishing that the universality class remains invariant (with $ ext{μ} o 1$ and $z o 1$) across reentrant sequences. It provides an analytic phase boundary for the solvable case $w=0$ via renormalization of the intracell hopping by moiré strength $m_0$, and demonstrates numerically that the general case $w eq 0$ sustains the same critical behavior. A bulk-boundary correspondence is established between OBC zero-energy edge modes and the PBC entanglement spectrum, and the central charge extracted from entanglement entropy matches the number of emergent zero-energy pairs, reflecting the winding-number changes during transitions. The results extend to complex moiré patterns with winding numbers exceeding 1 and suggest feasible experimental realizations in ultracold-atom quantum simulators, providing a unified framework for moiré-modulated topological phase transitions in 1D systems.

Abstract

Recent studies of moiré physics have unveiled a wealth of opportunities for significantly advancing the field of quantum phase transitions. However, properties of reentrant phase transitions driven by moiré strength are poorly understood. Here, we investigate the reentrant sequence of phase transitions and the invariant of universality class in moiré-modulated extended Su-Schrieffer-Heeger (SSH) model. For the simplified case with intercell hopping $w=0$, we analytically derive renormalization relations of Hamiltonian parameters to explain the reentrant phenomenon. For the general case, numerical phase boundaries are calculated in the thermodynamic limit. The bulk boundary correspondence between zero-energy edge modes and entanglement spectrum is revealed from the degeneracy of both quantities. We also address the correspondence between the central charge obtained from entanglement entropy and the change in winding number during the phase transition. Our results shed light on the understanding of universal characteristics and bulk-boundary correspondence for moiré induced reentrant phase transitions in 1D condensed-matter systems.

Figures (7)

• Figure 1: (Color online) Illustration and phase diagram of the extended SSH model. (a) Illustration of the investigated extended SSH model with the intracell hopping $v_j$ modulated by moiré pattern $m_j=m_o[\cos(2\pi j/a_1)+\cos(2\pi j/a_2)]$. (b) The phase diagram of the extended SSH model with $m_o=0$. The phase diagram consists of three different regions, the topological trivial region with winding number $\nu=0$, the topological non-trivial regions with $\nu=1$ and $\nu=2$.
• Figure 2: (Color online) Characterization of reentrant physics with intercell hopping $w=0$. (a) The topological phase diagram of moiré-modulated SSH model. Red dashed curve is the analytical phase boundary obtained from the divergence of localization length. The vertical black dashed line corresponds to the cutting line in (c), and the black pentagram mark denotes the chosen parameters in (e,f). (b) The second energy gap $\Delta E_2$ plotted as functions of $m_o$ and $J_2$. (c) Real-space winding number $\nu$ and entanglement entropy $\mathcal{S}$ plotted as functions of moiré strength $m_o$ for $J_2=1.6$. (d) The energy spectrum versus $m_o$ for $J_2=1.6$, with zero-energy modes drawn in red. (e) Density distributions $\braket{\hat{n}_j}$ of four localized edge states for $J_2=1.6$ and $m_o=1.5$. (f) Entanglement spectrum $\zeta_i$ for $J_2=1.6$ and $m_o=1.5$. Other parameters are $\varepsilon=1$, $A=32$, $a_1=3$ and $a_2=7$.
• Figure 3: (Color online) Critical behaviors with intercell hopping $w=0$. (a) $\partial \nu / \partial m_o$ as a function of $m_o$ for various system sizes near the first trivial-to-topological transition point $m_{oc1}=1.1283(3)$. The finite-size critical point $m_{oc}^{(L)}$ is revealed by the peak position of each curve. Inset panel shows the $\log$-$\log$ plot of system size $L$ and $\Delta m_{oc}(L)=|m_{oc}(L)-m_{oc1}|$. (b) $\partial \nu / \partial m_o$ near the second trivial-to-topological transition point $m_{oc2}=2.5685(7)$. Inset is the $\log$-$\log$ plot of system size $L$ and $\Delta m_{oc}(L)=|m_{oc}(L)-m_{oc2}|$. (c) $\mathrm{Log}$-$\log$ plot of the second energy gap versus system size $L$ near the first (c) and second (d) trivial-to-topological transition points, respectively. Other parameter are $\varepsilon=1$, $a_1=3$, $a_2=7$, and $J_2=1.6$.
• Figure 4: (Color online) Entanglement scalings with intercell hopping $w=0$. Entanglement entropy $\mathcal{S}(L)$ scales as a function of rescaled system size $L$ in the first (a) and second (c) trivial-to-topological phase transition points. Entanglement entropy $\mathcal{S}(l)$ scales as a function of rescaled bipartition length $l$ or the first (b) and second (d) phase transition points. Inset panels are $\mathcal{S}(l)$ against unscaled $l$. All solid lines are linear fittings which indicate the central charge $c\approx 2$. $A=512$ for (b,d), other parameter are $\varepsilon=1$, $a_1=3$, $a_2=7$, and $J_2=1.6$.
• Figure 5: (Color online) Characterization of reentrant physics with intercell hopping $w\neq0$. (a) The topological phase diagram for $\varepsilon=0.3$. Red and blue dashed curves are the thermodynamic limit phase boundary obtained from condition equation $q(k)=0$. The vertical black dashed line correspond to the cutting line in (d), and black pentagram marks denote the chosen parameters in (e,f) and (g,h), respectively. (b) The first energy gap $\Delta E_1$, and (c) the second energy gap $\Delta E_2$ plotted as functions of $m_o$ and $J_2$. (d) $\nu$ and $\mathcal{S}$ plotted as functions of $m_o$ for $J_2=1.27$. (e,f) Density distributions $\braket{\hat{n}_j}$, and entanglement spectrum $\zeta_i$ for $\nu=2$ with $J_2=1.27$ and $m_o=0.7$. (g,h) $\braket{\hat{n}_j}$, and $\zeta_i$ for $\nu=1$ with $J_2=1.27$ and $m_o=1.2$. Other parameters are $\varepsilon=0.3$, $A=32$, $a_1=3$ and $a_2=7$.