Disorder Suppression of Charge Density Wave in the Honeycomb Holstein Model

TL;DR

This work investigates how random hopping disorder impacts charge-density-wave (CDW) order in the Holstein model on a honeycomb lattice of Dirac-like electrons. Using determinant quantum Monte Carlo, the authors map CDW transitions via the CDW structure factor $S_{CDW}$ and real-space correlations, and assess transport through the current-current conductivity, while varying the electron-phonon coupling and disorder strength. They find that disorder suppresses CDW correlations and lowers long-range order, shifting the critical temperature in a way that signals enhanced localization; moderate to strong disorder also reduces kinetic energy and $\sigma_{dc}$, and narrows or closes the CDW gap in $A(\omega)$. The results illuminate the interplay between electron-phonon coupling and disorder in 2D Dirac systems, revealing how Anderson localization can destabilize correlated phases and modify critical behavior in disordered lattice models.

Abstract

The formation of charge-density-wave order in Dirac fermion systems via electron-phonon coupling represents a significant topic in condensed matter physics. In this work, we investigate this phenomenon within the Holstein model on the honeycomb lattice, with a specific focus on the effect of disorder. While the interplay between electron-electron interactions and disorder has long been a central theme in the field, recent attention has increasingly turned to the combined influence of disorder and electron-phonon coupling. Using determinant quantum Monte Carlo simulations, we concentrate on the phase transitions of charge-density-wave order on the honeycomb lattice. Disorder is introduced through the random hopping of electrons in the system, which can localize electrons via the Anderson effect. Our primary result is that disorder suppresses the charge-density-wave phase, and the interplay between disorder and electron-phonon interactions extends the phase area. We also determine the transition temperature \(β_c\) to the ordered phase as a function of the electron-phonon coupling. Additionally, we observed a suppression of electron kinetic energy and dc conductivity under disorder, highlighting the role of Anderson localization in the degradation of electronic transport. These findings offer significant theoretical insight into the stability and critical phenomena of correlated phases in disordered two-dimensional systems.

Figures (6)

• Figure 1: (a) A honeycomb lattice structure with $L = 6$. $\vec{a}_{1}$ and $\vec{a}_{2}$ are the two primitive vectors of the lattice, which define the periodic structure of the honeycomb lattice. The black solid dots and white hollow dots represent the two sub-lattices of the honeycomb lattice. The trajectory (the red dashed line) corresponds to the horizontal axis of plot (b). (b)The influence of different disorder strengths $\Delta$ (taking $\Delta =0.0$, $\Delta =0.5$, $\Delta =1.0$, $\Delta =1.5$ respectively, distinguished by markers of different colors and shapes) on the charge-correlation function $c(\boldsymbol{r})$. Under the conditions of a system size $L = 6$ and fixed parameters $\omega_{0}/t=1$, $\lambda_{D}=2/3$, $\beta=6$, $\mu=-4$.
• Figure 2: The charge-correlation function $c(\boldsymbol{r})$ along a specific real-space path shown as the red line in FIG.\ref{['fig:01']}(a). Under the conditions of a system size $L = 6$ and fixed parameters $\omega_{0}/t= 1$, $\lambda_{D}=2 / 3$, with different temperature parameters $\beta$ and different disorder strengths are considered: (a) $\Delta = 0.0$, (b) $\Delta = 0.5$, (c) $\Delta = 1.0$, (d) $\Delta = 1.5$.
• Figure 3: The variation of the CDW structure factor $S_{CDW}$ with the inverse temperature $\beta$ for different lattice sizes, with $\lambda_{D} = 2/3$, $\omega_{0} = 1$, $\rho = 1$, and disorder strength $\Delta$ being (a) $\Delta = 0$, (b) $\Delta = 0.5$, (c) $\Delta = 1.0$, (d) $\Delta = 1.5$ respectively.
• Figure 4: The crossing plot indicating the critical temperature for different disorder strengths $\Delta$, which yield (a) $\beta_{c}=5.8$, (a) $\beta_{c}=5.8$, (a) $\beta_{c}=6.7$, (a) $\beta_{c}=7.9$. Here, $\lambda_{D} = 2/3$, $\omega_{0} = 1$, and $\rho = 1$, and the critical exponents $\gamma = 7/4$ and $\nu = 1$ of the two-dimensional Ising model are used.
• Figure 5: (a) Kinetic energy of electrons and (b) dc conductivity as functions of the inverse temperature, and for different disorder strengths, at fixed $L = 6$, $\omega_{0} = 1$, and $\lambda_{D} = 2/3$.