Marginal stability of soft anharmonic mean field spin glasses

Abstract

We investigate the properties of the glass phase of a recently introduced spin glass model of soft spins subjected to an anharmonic quartic local potential, which serves as a model of low temperature molecular or soft glasses. We solve the model using mean field theory and show that, at low temperatures, it is described by full replica symmetry breaking (fullRSB). As a consequence, at zero temperature the glass phase is marginally stable. We show that in this case, marginal stability comes from a combination of both soft linear excitations -- appearing in a gapless spectrum of the Hessian of linear excitations -- and pseudogapped non-linear excitations -- corresponding to nearly degenerate two level systems. Therefore, this model is a natural candidate to describe what happens in soft glasses, where quasi localized soft modes in the density of states appear together with non-linear modes triggering avalanches and conjectured to be essential to describe the universal low-temperature anomalies of glasses.

Figures (6)

• Figure 1: The phase diagram of the model at zero temperature for $k\in[0.1,1]$. Below both the dashed green line and the red continuous line, the model is glassy and described by fullRSB. Above both lines the energy landscape is made by a unique minimum. The properties of the zero temperature spin-glass transition depend on the strength of the external field. For small external field, at the critical point, the model has a quadratic gapless density of states $D(\omega)\sim \omega^2$ while for large field one has $D(\omega)\sim\omega^4$. The two colored cuts are the sections investigated at finite temperature in Fig.\ref{['fig:PD']}. Data reprinted from PhysRevB.103.174202.
• Figure 2: The phase diagram of the model at positive temperature for two values of $J$. As in Fig.\ref{['fig:PD_T0']}, we fixed $k_{\min}=0.1$ and $k_{\max}=1$. Above the lines the model is described by a replica symmetric (paramagnetic) solution and right below them the solution is fullRSB describing a critical spin glass phase. For both values of $J$, the point at $T=0,\, H=H_*$ is in the $D(\omega)\sim \omega^4$ universality class.
• Figure 3: We computed numerically the breakpoint $x^*$ and slope of $q(x^*)$ at the transition point as a function of the temperature. Both these quantities show that the transition is of continuous fullRSB type. Remarkably, at zero temperature the breakpoint $x^*$ tends to a finite value. The error bars are due to Monte Carlo evaluation of the integrals that define these quantities. In the inset of the right panel we show a tentative fit with the function $0.5e^{-A T^\nu}$ compatible with $x^*=1/2$ at the transition. A more careful analysis is needed to compute precisely the value of the breakpoint.
• Figure 4: The distribution of local fields computed by the numerical integration of the DMFT equations for $J=0.3$ and $H=0.45$. For $k=0.1001$ (top panels) a depletion is opening around $h_{\textrm{loc}}=0$ when time increases. The panel on the left shows the full distribution, while the panel on the right shows a zoom on the hole which opens as time increases. Conversely for $k=0.9999$ (bottom panel) the distribution is regular and converges to a Gaussian-like shape. We emphasize that, while for large $k$ the local field distribution converges very fast, for small $k$, on the timescales we have access to, the dynamics is far from asymptotic and the hole is still not well formed.
• Figure 5: The distribution of spins. For $k=k_s$ the distribution has a hole opening around $y=0$ and a double peak structure. For large $k=k_l$ the distribution is featureless.