content_image/413841.jpg content_image/413840.jpg content_image/413838.jpg content_image/413843.jpg content_image/413837.jpg
To extract the small g behavior of F$_{0}$ ( g ), we note that at small values of either field, expression (14) yields 

<img src='content_image/413841.jpg'>

implying that the sole effect of the presence of the (small) short-wavelength anharmonic displacement of magnitude g is a renormalization of all acoustic frequencies down- wards by a factor of (1 − g $^{2}$βa$^{3}$/ 6) 1 / 2 ≃ (1 − g $^{2}$βa$^{3}$/ 12). Since the partition function of a classical harmonic os- cillator with frequency ω is ( k$_{B}$T/ ℏ ω ), this reduction in the acoustic frequency results in a free energy shift of − k$_{B}$T ( g $^{2}$βa$^{3}$/ 12) per phonon. As mentioned earlier, the number of the acoustic modes is three times the number of vectors ⃗ g , yielding that at small g , f ( g ) ≃ g $^{2}$/ 2 − 3( k$_{B}$T/a $^{3}$)( g $^{2}$βa$^{3}$/ 12) = g $^{2}$/ 4. Thus at small g , the second derivative of F$_{0}$ ( g ) is positive. 

According to the asymptotic expansion of the Bessel function, I$_{ν}$ ( x ) ∝ e $^{x}$/x 1 / $^{2}$,29 the two leading terms in the large g limit of Eq.(14) are given by 

<img src='content_image/413840.jpg'>

In contrast with Eq.(6), the g -dependent portion of the free energy is now uniform in space and thus no longer has short-wavelength components. As a result, we can always shift the u variable to eliminate the g depen- dence from the integrand in the partition function cor- responding to the free energy (16) so as to make the ex- pression a pure quadratic in the new variable φ $^{′}$: u = u$_{0}$ + u ′ ⇒ ⃗ φ = ⃗ φ$_{0}$ + ⃗ φ ′ where | ⃗ φ$_{0}$ | = g . (This lat- ter reference state physically corresponds to a uniform distribution of stress (not displacement!) at free energy density g $^{2}$/ 2.) The resulting integration is φ ′ becomes asymptotically Gaussian in the large g limit, despite the non-analytical dependence on ⃗ u in Eq.(16). As a result, f$_{g}$ ( g ) g →∞ −−−→ (5 / 2)( k$_{B}$T/a $^{3}$) ln g , i.e. the second deriva- tive of F$_{0}$ ( g ) becomes negative at large g , implying F$_{0}$ ( g ) has an inflection point at an intermediate value of g . 

We thus conclude from the analysis of the asymptotic behavior of the free energy F$_{0}$ ( g ) that with a suitable form of F$_{ex}$ ( g ), such as F$_{ex}$ ( g ) ∝ g $^{2}$, it is possible for the full free energy F ( g ) to have a non-concave portion, also consistent with the mean-field solution in Section III, see Fig.5. We reiterate that the presence of the non-concave portion means the system exhibits a metastable state at a non-zero value of g . The latter quantity therefore can used as an order parameter for a transition between the lowest free energy, fully stable state and the metastable state at non-zero g . 

The free energy of our ansatz, Eq.(14), helps under- stand the origin of the peculiar behavior of F$_{0}$ ( g ). Indeed, at large enough magnitude of anharmonic displacement, g , the equilibrium value of the acoustic displacements be- comes non-zero. According to Eq.(15) this transition oc- curs at a critical value: 

<img src='content_image/413838.jpg'>

<img src='content_image/413843.jpg'>

We show the φ dependence of the free energy from Eq.(14) in Fig.2, for several representative values of g . Note that the critical value of the excess free energy at a single vector, a $^{3}$g2 $_{cr}$/ 2 = 3 k$_{B}$T , is the total energy of a single atom in a harmonic lattice. 

In the metastable phase at g > 0, the thermodynamic potential from Eq.(14) is dominated by aperiodic config- urations of ⃗ g ’s, implying the modulation of the harmonic field is aperiodic, too. This potential becomes particu- larly vivid in the g $^{2}$/k$_{B}$ T → ∞ limit, i.e. ( g − | ⃗ φ | ) $^{2}$/ 2 ≡ [ g − ( u$_{ij}$ Λ$_{ijkl}$ u$_{kl}$ ) 1 / $^{2}$]$^{2}$/ 2. This form means that the ape- riodic phase corresponds to a frozen-in stress pattern at half-wave length a = π/k$_{micro}$ , which we must thus asso- ciate with the bead size of the RFOT theory. Consistent with this identification, the form [ g − ( u$_{ij}$ Λ$_{ijkl}$ u$_{kl}$ ) 1 / $^{2}$]2 is similar to broken-symmetry functionals that arise in sys- tems with self-generated disorder. 30,31 Importantly, the free energy (14) is an equilibrium quantity, implying the metastable aperiodic state corresponds to a melt above the glass transition. 

The aperiodic phase exhibits many distinct states sep- arated by activation barriers. To see this structural de- generacy explicitly, we first integrate out the acoustic de- formation in Eq.(4) at a fixed d$_{ij}$ ( r ) configuration. Anal- ogous calculations have been performed, for instance, by Grannan at el. $^{32,33}$, who have considered defect-phonon couplings in the form of Q$_{ij}$u$_{ij}$ . Using their result or by straightforward path-integration in k -space (see Ap- pendix B), we obtain the following effective Hamiltonian that couples the non-linear oscillators, in the tensorial and Voigt notations respectively: 

<img src='content_image/413837.jpg'>

where the double ( m < n ) sums are over all bead pairs 

