Random field Ising-like effective theory of the glass transition. II. Finite-dimensional models

As in the preceding paper Phys. Rev. B 98, 174205 (2018), hereafter referred to as Paper I we aim at identifying the effective theory that describes the fluctuations of the local overlap with an equilibrium reference configuration close to a putative thermodynamic glass transition. We focus here on the case of finite-dimensional glass-forming systems, in particular supercooled liquids. The main difficulty for going beyond the mean-field treatment comes from the presence of diverging point-to-set spatial correlations. We introduce a variational low-temperature approximation scheme that allows us to account, at least in part, for the effect of these correlations. The outcome is an effective theory for the overlap fluctuations in terms of a random-field + random-bond Ising model with additional, power-law decaying, pair and multibody interactions generated by the point-to-set correlations. This theory is much more tractable than the original problem. We check the robustness of the approximation scheme by applying it to a fully connected model already studied in Paper I. We discuss the physical implications of this mapping for glass-forming liquids and the possibility it offers to determine the presence or not of a finite-temperature thermodynamic glass transition.

Figure 1

2D sketches of the specific pattern of the ${\tau }^i$'s chosen for the implementation of the variational approximation: periodic cluster arrangement described in Sec. 3 (left panel) and slab geometry treated in Appendix pp1 (right panel). ${\tau }^i=1$ in the red regions and 0 in the light blue regions.

Figure 2

Periodic cluster ansatz: correction to the 1-replica action beyond the “zeroth-order” approximation $\mathrm{\Delta }{\mathcal{S}}_1$ as a function of $\ell$ for several values of $\lambda$ and $s_c$ (close to, but below, the mean-field thermodynamic glass transition). The full lines correspond to Eq. (24), and the dashed ones are obtained from the translationally invariant Hamiltonian with effective long-range two-body and multibody interactions and an effective external field, Eq. (26), in three dimensions. The values of the other parameters are $u=0.385,c=0.081,a=4$.

Figure 3

Effective external field $-2H$ as a function of the bare configurational entropy $s_c$. The position of the mean-field critical point is at $s_c=0$ whereas the “renormalized” value for which $H=0$ is at $s_c\approx -0.00786$ [given by Eq. (41) in three dimensions for $u=0.385,c=0.081$, and $a=4]$. The dashed red vertical lines represent the limits of validity of our approach $[s_c<u/3-cd$ to have positive variances and $s_c>cd/(2^d-1)-2u/3$ to ensure that the variational parameter $m_{★}$ stays positive]. For this choice of the parameters, the ratio ${\mathrm{\Delta }}_h/\tilde{J}\approx 11.7$, which is incompatible with the existence of $T_c$.

Figure 4

Sketch of potentially relevant spin configurations of the effective model in Eq. (35) close to the putative thermodynamic transition (i.e., $H\approx 0,J_2$ strong enough and $\overline{\delta h_i^2}$ not too big), showing clusters of $+1$ spins in red (high overlap with the reference configuration) in the background of $-1$ spins in light blue (low overlap with the reference configuration): (a) smooth interfaces, (b) rough interfaces, and (c) one flat interface.

Figure 5

Comparison between the approximate effective theory and the exact result for the fully connected $2^M$-KREM with $M=3$: mean overlap $\langle p\rangle$ with a reference configuration versus $\beta =1/T$. The black curve (circles) represents the values of $p_0$ and $p_1$ given by the exact solution of the model (called $q_0$ and $q_1$ in Appendix A1 of Paper I). The violet curve (crosses) corresponds to the results given by the (quasi)exact effective theory of the model obtained in Paper I: the effective Hamiltonian is exact at and above $T_K$ but is somewhat approximate below $T_K$ because of our use of the annealed approximation for handling the random energies (see Paper I). The red curve (squares) corresponds to the prediction of the approximate effective Hamiltonian, where we neglect the third cumulant of the random field and of the covariance between random field and random bond: the solution is derived in Appendix B of Paper I. The discrepancy between the two curves on the low-$\beta$ side of the transition is due to high-temperature terms that are not included in the low-$T$ approximation scheme. The roughly constant shift actually corresponds to a term that can be calculated in the limit $T\to \infty$: the exact value of the overlap approaches $\langle p\rangle \to 1/2^M$ for $T\to \infty$ (i.e., the probability that two randomly chosen configurations on a given site are in the same state), while the overlap goes to zero when computed by means of the effective random-field + random-bond Ising model. This explains the discrepancy of about $1/2^M\approx 0.11$ on the low-$\beta$ side of the transition. On the high-$\beta$ side, the green curve (diamonds) is the prediction of the approximate effective disordered theory when one accounts for higher-order correlations in the disorder (see Appendix B of Paper I). Finally, the blue curve (triangles) corresponds to the zeroth-order approximation with no account of the glassy correlations [Eq. (B6)].

Figure 6

Estimate for the minimum value of $M$ for the thermodynamic glass transition (RFOT) in the finite-dimensional lattice version of the $2^M$-KREM. The blue curve shows the result of the effective random-field Ising theory: a RFOT exists only for $M$ above this curve. The red circles correspond to the values of $d$ and $M$ for which an ideal-glass fixed point is found within the real-space RG analysis of Ref. [28], and the red dashed line is an exponential fit of these results which yields a lower critical dimension $d_L\approx 4.18$.

Figure 7

Effective theory for the fully connected $2^M$-KREM: transition line in the $\beta =1/T\text{−}M$ diagram for the zeroth-order approximation (when the covariance of the random field and random coupling is moreover neglected). The blue line marks the RFOT, ${\beta }_K^{\left(0\right)}=\sqrt{8({\psi }^{\left(0\right)}\left(2^M\right)+\gamma )/M}$. The green line indicates the limit above which the approximate expression of the variance of the random field becomes negative, ${\beta }_{\mathrm{th}}^{\left(0\right)}=\sqrt{8({\psi }^{\left(1\right)}\left(2^M\right)+{\pi }^2/6)/M}$. Below $M\approx 2.1$ where the two lines cross, the approximate description becomes meaningless. The asymptotic behavior for $M\to \infty$ is exact $({\beta }_K^{\left(0\right)}\to \sqrt{8ln2}$ and ${\beta }_{\mathrm{th}}^{\left(0\right)}\to 0)$.

Figure 8

Correction $\mathrm{\Delta }{\mathcal{S}}_1$ to the first cumulant due to the fluctuations of the $q_{ab}^i$'s within the variational “low-$T$” approximate scheme as a function of the concentration (global overlap) $c$, for $\beta =1.1{\beta }_K^{\left(0\right)}$ and $M=3$. The full black line corresponds to Eq. (48), while the dashed ones correspond to the results obtained after truncating the effective Hamiltonian in Eq. (49) to two- (blue), three- (green), four- (red), and five-body (magenta) interaction terms.