Real-space topology and charge order in the Haldane-Holstein Model

TL;DR

We address how retarded electron-phonon coupling destabilizes Chern topology in the half-filled Haldane-Holstein model. Using unbiased determinant quantum Monte Carlo, topology is diagnosed with the many-body Bott index and the local Chern marker derived from equal-time Green's functions, complemented by static mean-field and exact-diagonalization analyses in respective limits. The main result is a robust first-order transition from a Chern insulator to a staggered charge-density-wave at moderate electron-phonon coupling (around $g/t_1\approx 1.6$ for $t_2/t_1=0.2$), accompanied by gap closing and loss of boundary spectral weight; the transition boundary tracks the sign problem minimum in DQMC simulations. In the antiadiabatic limit, the Holstein coupling maps to an attractive Hubbard model with coexisting CDW and superconductivity, while in the adiabatic limit static mean-field theory captures the transition boundary. These findings demonstrate a concrete mechanism by which electron-phonon coupling can induce an abrupt destruction of Chern topology via CDW formation and offer experimentally accessible signatures for correlated topological materials such as layered quantum Hall insulators and transition-metal dichalcogenide systems.

Abstract

We study the half-filled Haldane-Holstein model, where a paradigmatic Chern insulator is coupled to fully dynamical phonons, and provide an unbiased characterization of how retarded electron-phonon interactions destabilize Chern topology. Using determinant quantum Monte Carlo, we find that increasing the coupling drives an abrupt, first-order transition from a Chern insulator to a staggered charge-density wave that acts as a dynamical sublattice (Semenoff) mass. The transition is simultaneously signaled by a nearly quantized many-body Bott index and a real-space local Chern marker constructed from the interacting Green's function, both of which collapse as the charge order parameter becomes extensive. Spectral and open-boundary calculations reveal concomitant gap closing and the loss of boundary spectral weight at the critical coupling. Despite the generic phase problem induced by broken time-reversal symmetry, we show that it remains mild in the low-frequency regime and that the average phase factor sharply tracks the CI-CDW boundary. Our results establish a concrete route by which electron-phonon coupling can trigger a discontinuous collapse of Chern topology and provide experimentally relevant signatures for correlated topological platforms.

Figures (12)

• Figure 1: Schematic picture of the Haldane model hopping terms (a) and the charge-ordered phase (b) induced by Holstein phonons. Contour plots of the Bott index (c) and charge structure factor (d) are shown as functions of $g/t_1$ and $t_2/t_1$, for $L=6$, $\omega_0/t_1=1$, and $T/t_1=1/12$. The combination of both quantities allows us to map the semimetal (SM), charge density wave (CDW), and Chern insulator (CI) phases. Markers represent the critical values of $g$ obtained through finite-size scaling, and dashed lines serve as a guide; the star marker on the vertical axis is from Ref. Zhang20192.
• Figure 2: (a) Chern marker $C$, normalized charge structure factor and Bott index as functions of $g/t_{1}$. (b) Average determinant sign, $\langle \mathrm{sign}\rangle$ versus $g/t_{1}$; inset: finite-size scaling of $S_c/N$ near the critical region. (c) Colormap of $\langle \mathrm{sign}\rangle$ as a function of $g/t_{1}$ and $t_{2}/t_{1}$ for $L=6$ and $T/t_{1}=1/12$. (d)–(f) Reweighted histograms of $S_{c}$ at $g/t_{1}=1.6$, $1.622$, and $1.7$, respectively, computed for $L=12$ and $T/t_{1}=1/10$; the color of each histogram bar encodes the average sign for that reweighted bin. All data for $\omega_0/t_1 = 1$.
• Figure 3: Single-particle spectral function for $L=15$, $t_2/t_1=0.2$, and $T/t_1=1/30$. (a)-(c) show the CI phase at $g/t_1=1.5$: spectral weight resolved on sublattice $A$ [(a)] and $B$ [(b)], and the total spectral function [(c)]. (d)–(f) correspond to the critical coupling $g/t_1=1.6$, and (g)–(i) show the CDW phase ($g/t_1=1.7$).
• Figure 4: Phase diagram of the Haldane-Holstein model for $t_{2}/t_{1} = 0.2$. The Chern insulator (CI) phase appears at small values of $\lambda\! =\!g^{2}/\omega_{0}^{2}$, while the charge-density-wave (CDW) phase emerges at stronger couplings. In the $\omega_{0} \!\to\! \infty$ limit, superconductivity (SC) coexists with CDW, as confirmed by exact diagonalization (red marker). The solid black marker denotes the critical point obtained from density matrix renormalization group (DMRG) calculations He2024. Open black markers represent the determinant quantum Monte Carlo (DQMC) critical points, and the solid line is a guide to the eye. The blue dotted line indicates the mean-field critical $\lambda$, which becomes exact in the $\omega_{0} \!\to\! 0$ limit.
• Figure 5: Results for open boundary conditions. Local density of states near the Fermi level for (a) $g/t_{1}=1.0$, (b) $g/t_{1}=1.5$, and (c) $g/t_{1}=2.0$. Panels (d)–(f) show the corresponding real-space local Chern marker $C(\mathbf{R})$. Data are for $L=15$, $\omega_{0}/t_{1}=1$, $t_{2}/t_{1}=0.2$, and $T/t_{1}=1/30$.