Localization of long-wavelength acoustic phonons in a new anomalous lattice

数值计算了声子色散关系为抛物型的异常固体中一维弦（长度为L）、准一维条（宽度为W，长度为L）和准一维棒（截面为W×W，长度为L）三种情况下声子的阻尼率γ。在所有情况下，我们都在低频区得到了与阻尼率γ的频率（ω）相关的异常峰结构，证明了长波声子具有很强的局域性。然而，在二维和三维极限情况下，我们无法确定γ（ω,W→∞，L）的数值，因为其峰值在晶格常数逆单位下为10^-5或更小的数量级。这意味着在二维和三维情况下，假设晶格常数为1nm，长波声子在反常无序系统中的定位长度为0.1mm或更多。

We numerically calculated the damping rate γof acoustic phonons in an anomalous solid with parabolic phonon dispersion relation, in the cases of one-dimensional string (of length L), quasi-one-dimensional strip (of width W and length L), and quasi-one-dimensional rod (of cross section W×W and length L). In all cases, we got anomalous peaky structure in the frequency (ω) dependence of the damping rate γin the low frequency region, which proves a strong localization character of the long-wavelength acoustic phonons. However, we could not determine the numerical value of γ(ω,W→∞,L) in the limiting cases of two- and three-dimensions, because the peak values are of the order of 10^-5 or less in the unit of the inverse of the lattice constant. That means the localization length of the long-wavelength acoustic phonons in the anomalous disordered system is of the order of 0.1mm or more, in the cases of two- and three-dimensions, assuming that the lattice constant is 1nm.