Renormalization-group approach to the metal-insulator transitions in $(\mathrm{DCNQI}{)}_{2}M$ (DCNQI is ${N,N}^{\prime }$-dicyanoquinonediimine and $M=\mathrm{Ag},$ Cu)

Metal-insulator transitions and different ground-state phases in the quasi-one-dimensional materials, ${(R}_1{R}_2-\mathrm{DCNQI}{)}_{2}M$ ${(R}_1{=R}_2={\mathrm{CH}}_3,$ I and $M=\mathrm{Ag},$ Cu), are studied with a renormalization-group method. We use one-dimensional continuum models with backward scatterings, umklapp processes, and couplings with ${2k}_F$ and ${4k}_F$ phonons (not static lattice distortion). We take a quarter-filled band for $M=\mathrm{Ag}$ and a sixth-filled band coupled with a third-filled band for $M=\mathrm{Cu}.$ Depending on electron-electron and electron-phonon coupling strengths, the ground-state phase becomes a Tomonaga-Luttinger liquid or a state with a gap(s). For $M=\mathrm{Ag},$ there appears a spin-gap state with a dominant ${2k}_F$ charge-density-wave correlation, a Mott insulator with a dominant ${4k}_F$ charge-density-wave correlation, or a spin-Peierls state with different magnitudes of spin and charge gaps. Three dimensionality is taken into account by cutting off the logarithmic singularity in either the particle-particle channel or the particle-hole channel. The difference between the ground-state phase of the $R_1{=R}_2={\mathrm{CH}}_3$ salt (spin-Peierls state) and that of the $R_1{=R}_2=\mathrm{I}$ salt (antiferromagnetic state) is qualitatively explained by a difference in the cutoff energy in the particle-particle channel. For $M=\mathrm{Cu},$ there appears a Mott insulator with a charge-density wave of period 3 and a Peierls insulator with a charge-ensity wave of period 6. The conditions for the experimentally observed, Mott insulator phase are strong correlation in the sixth-filled band, moderate electron-phonon couplings, and finite electron-${4k}_F$ phonon coupling. Resistance is calculated as a function of temperature with a memory-function approximation in both cases above. It qualitatively reproduces the differences among the $M=\mathrm{Ag}$ and $M=\mathrm{Cu}$ cases as well as the $R_1{=R}_2={\mathrm{CH}}_3$ and $R_1{=R}_2=\mathrm{I}$ cases.