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Liu-Nagel phase diagrams in infinite dimension
by Giulio Biroli, Pierfrancesco Urbani
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We study Harmonic Soft Spheres as a model of thermal structural glasses in the limit of infinite dimensions. We show that cooling, compressing and shearing a glass lead to a Gardner transition and, hence, to a marginally stable amorphous solid as found for Hard Spheres systems. A general outcome of our results is that a reduced stability of the glass favors the appearance of the Gardner transition. Therefore using strong perturbations, e.g. shear and compression, on standard glasses or using weak perturbations on weakly stable glasses, e.g. the ones prepared close to the jamming point, are the generic ways to induce a Gardner transition. The formalism that we discuss allows to study general perturbations, including strain deformations that are important to study soft glassy rheology at the mean field level.
Published as SciPost Phys. 4, 020 (2018)
Author comments upon resubmission
We thank the referee for her/his positive report and for having appreciated our work. We submit an improved presentation in which we took into account all her/his comments and suggestions.
Below we reply to her/his comments.
- There is no Gardner transition in cooling or compression. In fact we strain (infinite dimensional) glasses that are prepared at equilibrium above the Kauzmann transition, thus they are always stable in terms of replica symmetry. We have underlined this in the caption of the figure and in a footnote. We have moreover change the text in the introduction and the caption to better clarify the figure. We tried to draw a surface to join the yielding transition points, as the referee suggested, but we found it was not improving the figure so we preferred to keep the old version.
- We moved the part on FRSB to appendices in order to clarify the presentation as suggested by the referee.
- The reduced temperature and packing fraction are defined in Eq.(17) and Eq.(41). We have underlined this in the caption of Fig.1.
- We fixed the typo. We thank the referee.
- S is defined in Eq.(11) as the number of replicas. The product runs from 1 to s.
- The referee is right. Its definition is in Eq.(8).
- We thank the referee. We corrected the typo.
- Y does not appear because first $\rho(y)$ does not depend on it, and second one can change variable from X+Y->X. The integration over Y produces the $1/\rho$ factor.
- In the limit of infinite dimensions the two last terms of (36) can be evaluated using the central limit theorem since they both correspond to a sum of an infinite number of terms. This is the reasoning we use for eq. 37 and eq. 39. We have added a sentence below eq. 36 to stress this point.
- We did as suggested.
- We fixed the notation.
- We have modified the text skipping the previous equations and we have underlined that the equation (48) can be derived using the same kind of computations in of Eq. (38) and (39).
Submission & Refereeing History
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