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

数值计算了一维弦（长L）、准一维条（宽W，长L）和准一维杆（横截面W×W，长L）情况下，具有抛物型声子色散关系的反常固体中声学声子的阻尼率γ.在所有情况下，在低频区阻尼率γ随频率（ω）的变化都出现了反常的峰结构，证明了长波长声学声子的强局域化特征.然而，我们无法确定二维和三维极限情况下γ（ω，W→∞，L）的数值，因为峰值在晶格常数倒数单位的10^-5量级或更小。这意味着在二维和三维的情况下，在反常无序系统中长波长声学声子的局域长度是0.1mm或更大的数量级，假设晶格常数是1 nm。

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.