Mean-field model of interacting quasilocalized excitations in glasses
Corrado Rainone, Pierfrancesco Urbani, Francesco Zamponi, Edan Lerner, Eran Bouchbinder
Abstract
Structural glasses feature quasilocalized excitations whose frequencies \omega <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ω</mml:mi> </mml:math> follow a universal density of states {D}(\omega)\!\sim\!\omega^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mspace width="-0.167em"/> <mml:mo>∼</mml:mo> <mml:mspace width="-0.167em"/> <mml:msup> <mml:mi>ω</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> . Yet, the underlying physics behind this universality is not fully understood. Here we study a mean-field model of quasilocalized excitations in glasses, viewed as groups of particles embedded inside an elastic medium and described collectively as anharmonic oscillators. The oscillators, whose harmonic stiffness is taken from a rather featureless probability distribution (of upper cutoff \kappa_0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> ) in the absence of interactions, interact among themselves through random couplings (characterized by a strength J <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>J</mml:mi> </mml:math> ) and with the surrounding elastic medium (an interaction characterized by a constant force h <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>h</mml:mi> </mml:math> ). We first show that the model gives rise to a gapless density of states {D}(\omega)\!=\!A_{g}\,\omega^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mspace width="-0.167em"/> <mml:mo>=</mml:mo> <mml:mspace width="-0.167em"/> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mspace width="0.167em"/> <mml:msup> <mml:mi>ω</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> </mml:math> for a broad range of model parameters, expressed in terms of the strength of the oscillators’ stabilizing anharmonicity, which plays a decisive role in the model. Then — using scaling theory and numerical simulations — we provide a complete understanding of the non-universal prefactor A_{g}(h,J,\kappa_0) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , of the oscillators’ interaction-induced mean square displacement and of an emerging characteristic frequency, all in terms of properly identified dimensionless quantities. In particular, we show that A_{g}(h,J,\kappa_0) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> is a non-monotonic function of J <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>J</mml:mi> </mml:math> for a fixed h <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>h</mml:mi> </mml:math> , varying predominantly exponentially with -(\kappa_0 h^{2/3}\!/J^2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mi>h</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>/</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mspace width="-0.167em"/> <mml:mi>/</mml:mi> <mml:msup> <mml:mi>J</mml:mi>