Self-Consistent Current Response Theory of Unjamming and Vibrational Modes in Low-Temperature Amorphous Solids

We study amorphous solids with strong elastic disorder and find an unjamming instability that exists, in a harmonic model built using Euclidean random matrices (ERMs). Employing the Zwanzig-Mori projection operator formalism and Gaussian factorization approximations, we develop a first-principles, self-consistent theory of transverse momentum correlations in athermal disordered materials, extending beyond the standard self-consistent Born approximation. The vibrational anomalies in glass at low temperatures are recovered in the stable solid limit, and floppy modes lacking restoring forces are predicted in unstable states below the jamming transition. Near the unjamming transition, the speed of sound <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mi>v</a:mi><a:mn>0</a:mn><a:mo>⊥</a:mo></a:msubsup></a:math> vanishes with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo>∝</c:mo><c:msqrt><c:mrow><c:mo>−</c:mo><c:mi>ε</c:mi></c:mrow></c:msqrt></c:math>, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>ε</e:mi></e:math> denotes the distance from the critical point. Additionally, the density of states develops a plateau, independent of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>ε</g:mi></g:math> above a frequency <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>ω</i:mi><i:mo>*</i:mo></i:msub></i:math> which vanishes at the transition, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>ω</k:mi><k:mo>*</k:mo></k:msub><k:mo>∝</k:mo><k:mo stretchy="false">|</k:mo><k:mi>ε</k:mi><k:mo stretchy="false">|</k:mo></k:math>. We identify a characteristic length scale in the unjammed phase, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msubsup><o:mi>λ</o:mi><o:mo>−</o:mo><o:mo>⊥</o:mo></o:msubsup><o:mo>∝</o:mo><o:mn>1</o:mn><o:mo>/</o:mo><o:msqrt><o:mi>ε</o:mi></o:msqrt></o:math>, indicating the distance over which injected momentum remains correlated. We confirm the theoretical predictions with numerical solutions of a scalar ERM model, demonstrating overall good qualitative and partly quantitative agreement.