The universal form of the density of nonphononic, quasilocalized vibrational modes of frequency $\omega$ in structural glasses, $\mathcal{D}\left(\omega \right)$, was predicted theoretically decades ago, but only recently revealed in numerical simulations. In particular, it has been recently established that, in generic computer glasses, $\mathcal{D}\left(\omega \right)$ increases from zero frequency as ${\omega }^4$, independent of spatial dimension and of microscopic details. However, it has been shown [Lerner and Bouchbinder, Phys. Rev. E 96, 020104(R) (2017)] that the preparation protocol employed to create glassy samples may affect the form of their resulting $\mathcal{D}\left(\omega \right)$: glassy samples rapidly quenched from high-temperature liquid states were shown to feature $\mathcal{D}\left(\omega \right)\sim {\omega }^{\beta }$ with $\beta <4$, presumably limiting the degree of universality of the ${\omega }^4$ law. Here we show that exponents $\beta <4$ are seen only in small glassy samples quenched from high-temperature liquid states—whose sizes are comparable to or smaller than the size of the disordered core of soft quasilocalized vibrations—while larger glassy samples made with the same protocol feature the universal ${\omega }^4$ law. Our results demonstrate that observations of $\beta <4$ in the nonphononic density of states stem from finite-size effects, and we thus conclude that the ${\omega }^4$ law should be featured by any sufficiently large glass quenched from a melt.

中文翻译：

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计算机眼镜中状态的非声密度的有限大小效应

非音速，准局部频率振动模式密度的通用形式 $\omega$ 在结构玻璃中 $\mathcal{d}（\omega ）$理论上是在几十年前预测的，但直到最近才在数值模拟中显示出来。特别地，最近已经确定，在通用计算机眼镜中，$\mathcal{d}（\omega ）$ 从零频率增加 ${\omega }^4$，不受空间尺寸和微观细节的影响。但是，它已经显示[Lerner and Bouchbinder，Phys。启ë 96，020104（R）（2017）]，该制备方案用来产生玻样品可能影响其所得的形式$\mathcal{d}（\omega ）$：从高温液态快速淬灭的玻璃状样品具有以下特征 $\mathcal{d}（\omega ）〜{\omega }^{\beta }$ 与 $\beta <4$，大概限制了 ${\omega }^4$法。这里我们展示指数$\beta <4$仅在从高温液态淬火的小型玻璃状样品中观察到（其尺寸与软准局部振动的无序核的尺寸相当或小于），而使用相同协议制备的较大玻璃状样品具有通用性${\omega }^4$法。我们的结果表明$\beta <4$ 状态的非声子密度中的有限源于有限大小的效应，因此我们得出结论： ${\omega }^4$ 熔化后淬火的任何足够大的玻璃都应具有规律。