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Student Version of MATLAB The phase diagram of the two-mode PFC model is ob- tained by computing the free-energy density as a func- tion of the mean density ¯ ψ in solid and liquid, denoted by f$_{s}$ ( ¯ ψ ), and f$_{l}$ ( ¯ ψ ), respectively, and then using a stan- dard common tangent construction to obtain equilibrium values of ¯ ψ in solid ( ¯ ψ$_{s}$ ) and liquid ( ¯ ψ$_{l}$ ). 

Since the density is constant in the liquid, f$_{l}$ is ob- tained directly from Eq. (20) 

## B. Phase diagram 

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For small ϵ , the solid free-energy density can be well ap- proximated by only considering the contribution of the 〈 111 〉 and 〈 200 〉 RLVs. Accordingly, the crystal density field is expanded in the form 

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where we have used the fact that all density waves have the same amplitude in the crystal ( | A$_{i}$ | = A$_{s}$ and | B$_{i}$ | = B$_{s}$ ) and the magnitude of the principal RLVs are unity in our dimensionless units so ( q = 1 $_{/}$√$_{3). The parameters}$ A$_{s}$ and B$_{s}$ are solved by substituting Eq. (27) into Eqs. (19) and (20) and by minimizing the resulting free-energy F with respect to A$_{s}$ and B$_{s}$ . This minimization yields the solid free energy density 

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where A$_{s}$ and B$_{s}$ are themselves functions of ¯ ψ . The co- existence densities ¯ ψ$_{s}$ and ¯ ψ$_{l}$ are computed numerically using the standard common tangent construction, which consists of equating the chemical potentials f ′ $_{s}$( ¯ ψ$_{s}$ ) = f ′ $_{l}$( ¯ ψ$_{l}$ ) = µ$_{E}$ and grand potentials f$_{s}$ ( ¯ ψ$_{s}$ ) − µ$_{E}$ ¯ ψ$_{s}$ = f$_{l}$ ( ¯ ψ$_{l}$ ) − µ$_{E}$ ¯ ψ$_{l}$ of the two phases. It is also necessary to compute the solid free-energy curve for bcc since the latter can have a lower free-energy than fcc for some re- gions of the phase diagram. The bcc free-energy density was obtained by expanding the crystal density field us- ing a one-mode approximation, which only involves 〈 110 〉 RL Vs as in Ref. [14], and substituting this expansion into the t wo-mode model defined by Eqs. (19) and (20). 

An example of the phase diagram for R$_{1}$ = 0 . 05 is sho wn in Fig. 1, where we also show for completeness the hexagonal and stripe phases. As desired, we obtain a large ϵ range of fcc-liquid coexistence. For small ϵ , how- ever, bcc becomes favored over fcc. A common tangent construction using fcc and bcc free-energy curves shows that the density range of bcc-fcc coexistence is extremely narrow for small values of ϵ and cannot be resolved on the scale of Fig. 1. As will be explained later in sec- tion III C, the range of ϵ where bcc is favored depends on the value of R$_{1}$ . In the limit R$_{1}$ ≫ 1, the two-mode model reduces to the standard one-mode model after a simple rescaling of parameters, which can be easily seen by comparing Eqs. (20) and (25). Hence, increasing R$_{1}$ reduces the contribution of the second mode. Conversely, reducing R$_{1}$ increases the contribution of this mode and tends to favor the fcc structure, which extends to smaller ϵ for smaller R$_{1}$ . In the extreme case where R$_{1}$ = 0, the region of fcc-liquid coexistence extends all the way to vanishingly small ϵ as shown in Fig. 2. 

