Charge Order in the half-filled bond-Holstein Model

TL;DR

We study the half-filled bond-Holstein model on a square lattice to characterize a charge-density-wave (CDW) transition using numerically exact determinant quantum Monte Carlo (DQMC). A momentum-dependent electron-phonon coupling on bonds yields a phonon-mediated nearest-neighbor repulsion that enhances CDW and raises $T_{\rm cdw}$ compared with the site-Holstein model. Finite-size scaling of the charge structure factor $S_{\rm cdw}$ yields $T_{\rm cdw}$ with high accuracy and machine-learning analyses corroborate the phase boundary and reveal a high-temperature metal-to-bipolaron-liquid crossover. In the atomic limit $t=0$, CDW persists for bond-Holstein due to intersite interactions, while site-Holstein lacks such order, underscoring distinct strong-coupling physics and potential relevance to oxide materials with bond-stretching phonons.

Abstract

We use determinant quantum Monte Carlo to study the half-filled `bond-Holstein' model on a square lattice. We find that the model exhibits a charge-density-wave (CDW) phase transition with a critical temperature $T_\mathrm{cdw}$ considerably higher than that of the canonical `site-Holstein' model. Using a finite-size scaling analysis of the charge structure factor $S_{\rm cdw}$, we obtain $T_\mathrm{cdw}$ to greater than one percent accuracy. At the same time, local observables also show clear signatures consistent with the transition temperatures inferred from our scaling analysis. We attribute the enhanced CDW tendencies to a phonon-mediated nearest-neighbor electron repulsion that is directly proportional to the dimensionless electron-phonon coupling $λ$ in the atomic ($t\rightarrow 0$) limit. This behavior contrasts with the site-Holstein case, where the same limit yields only an on-site attraction. We supplement our analysis with results from several unsupervised machine learning methods, which not only confirm our estimates of $T_\mathrm{cdw}$ but also provide insight into the high-temperature crossover between a metallic and bipolaron liquid regime.

Figures (7)

• Figure 1: A cartoon sketch of the (a) site- and (b) bond-Holstein models (see also Sec. \ref{['subsec:Holstein']}), shown here in one dimension for simplicity. In the site-Holstein model, local lattice displacements couple to the total local electron charge density, leading to a momentum independent eph interaction. In the bond-Holstein model, generalized oscillators are defined on each of the system's bonds and couple to neighboring electron densities with opposite signs. This mechanism results in a momentum dependent eph coupling $g(\boldsymbol{q})$.
• Figure 2: Position space snapshots of the charge density correlation function $\langle \hat{n}_i \hat{n}_0 \rangle$ in the bond-Holstein model as a function of $\beta$, obtained on an $L=8$ lattice with dimensionless coupling $\lambda_{\rm bond}=0.4$. The strength of the correlations at each lattice site relative to the reference site in the bottom left is indicated by the common color scale at the bottom, where red (blue) corresponds to high (low) correlation.
• Figure 3: (a) & (b) Structure factor $S(\pi,\pi)$ and $S(\pi,\pi)/L^{\gamma/\nu}$ for the site-Holstein model with the Ising universality class exponents ${\gamma}/{\nu} = {7}/{4}$ and different lattice sizes $L$. Panels (c) & (d) show the analogous data for the bond-Holstein model. The vertical dashed line in the lower panels indicates the transition temperature $T_{\rm cdw}$ determined from the crossing of the curves for different lattice sizes (see text). In both cases, the dimensionless coupling has been fixed to $\lambda_{\rm bond}=0.4$. The apparent difference in abruptness of the two CDW transitions is a consequence of the more 'zoomed-in' range of $\beta$ in the site case.
• Figure 4: Average (a) electron kinetic energy $\mathcal{K}$, (b) $e$-ph potential energy $\mathcal{V}$, and (c) double occupancy $\mathcal{D}$ as functions of temperature for the bond-Holstein model with dimensionless coupling $\lambda_{\rm bond}=0.4$ and lattice size $L=8$. The insets plot the first derivative of each quantity with respect to temperature. The vertical dashed lines indicate the CDW transition temperature, as determined in Fig. \ref{['fig:ScaledScdw']}.
• Figure 5: (a) Specific heat $C$ of the bond-Holstein model as a function of temperature. (b) The Binder ratio $B$ (see text) as a function of temperature. The vertical dashed lines in both panels denote the position of the Binder ratio crossing derived from panel (b). The legend in panel (a) is common to both panels.