Effect of strain on charge density wave order in the Holstein model

We investigate charge ordering in the Holstein model in the presence of anisotropic hopping, $t_x,t_y=1-\delta ,1+\delta$, as a model of the effect of strain on charge-density-wave (CDW) materials. Using quantum Monte Carlo simulations, we show that the CDW transition temperature is relatively insensitive to moderate anisotropy $\delta \lesssim 0.3$, but begins to decrease more rapidly at $\delta \gtrsim 0.4$. However, the density correlations, as well as the kinetic energies parallel and perpendicular to the compressional axis, change significantly for moderate $\delta$. Accompanying mean-field theory calculations show a similar qualitative structure, with the transition temperature relatively constant at small $\delta$, and a more rapid decrease for larger strains. We also obtain the density of states $N\left(\omega \right)$, which provides clear signal of the charge ordering transition at large strain, where finite size scaling of the charge structure factor is extremely difficult because of the small value of the order parameter.

Figure 1

The mean-field order parameter $x_1$ versus $\beta$ for four different values of strain $\delta$ at ${\lambda }_D=0.25$. The critical transition temperature $T_{\mathrm{cdw}}={\beta }_c^{-1}$ decreases with increasing strain. See Fig. 2.

Figure 2

The mean-field critical temperature ${\beta }_c/{\beta }_{c_0}$ versus the strain $\delta$ for ${\lambda }_D=0.25$, where ${\beta }_{C0}=1.353$ in the isotropic $\delta =0$ case. The inset shows the mean-field result for the difference in electron density between the two sublattices $\mathrm{\Delta }n$ in the limit that $\beta \to \infty$.

Figure 3

The electron kinetic energies $k_x$ and $k_y$ are shown as functions of $\delta$. Division by the energy scales $t_x$ and $t_y$ isolates the effect of anisotropy on the hopping.

Figure 4

Left panel: Real-space density-density correlations for a moderate strain of $\delta =0.4$ at $T>T_{\mathrm{cdw}}$. Note the enhanced correlations in the $\widehat{y}$ direction relative to the $\widehat{x}$ direction. Right panel: Real-space density-density correlations for $\delta =0.4$ at $T<T_{\mathrm{cdw}}$. Note that the oscillating checkerboard charge-density pattern now persists across the entire lattice.

Figure 5

CDW structure factor versus hopping anisotropy $\delta$. The low-temperature value of the CDW order parameter falls to approximately half of its isotropic value as $\delta \to 0.4$.

Figure 6

(a) $S_{\mathrm{cdw}}$ versus $\beta$ for $\delta =0.3, {\lambda }_D=0.25$ and four different lattice sizes. (b) A finite size scaling where the scaled structure factors $S_{\mathrm{cdw}}{L}^{-\gamma /\nu }$ exhibit a crossing as a function of $\beta$ for different lattice sizes $L$. We infer ${\beta }_c=6.3\pm 0.1$ is slightly increased from the isotropic ${\beta }_c=6.0$. (c) The full data collapse in which the temperature axis is also scaled by $L^{1/\nu }\left(\frac{T-T_{\mathrm{cdw}}}{T_{\mathrm{cdw}}}\right)$. (d) ${\beta }_c$ as a function of $\delta$. The dashed line is a least-squares fit to the data. The value of ${\beta }_c$ at $\delta =0$ (triangle) is from Ref. [22].

Figure 7

Density of states for the isotropic lattice for different inverse temperatures $\beta t$. The phonon frequency ${\omega }_0=t$ and electron-phonon coupling $g=t$. Finite size scaling of $S_{\mathrm{cdw}}$ suggests ${\beta }_{c}t=6.0\pm 0.1$ [22], which is consistent with the $\beta$ value at which a full gap opens in $N\left(\omega \right)$.

Figure 8

Density of states comparing the isotropic lattice with small ($\delta =0.3$) and large ($\delta =0.9$) anisotropy. For $\delta =0.9$, the opening of a gap is delayed until ${\beta }_{c}t\sim 20$.