Multicritical behavior of Abrikosov vortex lattices near the cholesteric–smectic-A–smectic-C*point

The mean-field phase diagram of the cholesteric–smectic-A–smectic-${\mathrm{C}}^{\mathrm{*}}$ (${\mathit{N}}^{\mathrm{*}–}$Sm-A–Sm-${\mathit{C}}^{\mathrm{*}}$) point is derived within the framework of the chiral Chen-Lubensky model. We show that on the Sm-${\mathit{C}}^{\mathrm{*}}$ side of the phase diagram, one or two additional twist-grain-boundary (TGB) (or chiral-smectic) phases will occur. These additional chiral-smectic phases, the ${\mathrm{TGB}}_{\mathit{C}}$ and ${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ phases, are highly dislocated versions of smectic-C and smectic-${\mathit{C}}^{\mathrm{*}}$ phases, respectively. According to the de Gennes superconductor analogy, the grain-boundary phase which occurred on the smectic-A side of the phase diagram (known as the ${\mathrm{TGB}}_{\mathit{A}}$ or chiral Sm-${\mathit{A}}^{\mathrm{*}}$ phase) is the analog of the Abrikosov vortex lattice in type-II superconductors. However, the ${\mathrm{TGB}}_{\mathit{C}}$ and ${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ phases would correspond to Abrikosov vortex lattices in a hypothetical superconductor where the Ginzburg parameter κ is negative and the photon has Bose condensed. These additional phases are predicted to occur near the recently observed ${\mathit{N}}^{\mathrm{*}–}$Sm-${\mathit{A}}^{\mathrm{*}–}$Sm-${\mathit{C}}^{\mathrm{*}}$ multicritical point. We discuss the ${\mathit{N}}^{\mathrm{*}\mathrm{-}}$ ${\mathrm{TGB}}_{\mathit{C}}$, ${\mathrm{TGB}}_{\mathit{A}\mathrm{-}}$ ${\mathrm{TGB}}_{\mathit{C}}$, and ${\mathrm{TGB}}_{\mathit{A}\mathrm{-}}$ ${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ phase transitions. Because of the twist, the XY-like Sm-A–Sm-C is (usually) replaced by the Ising-like ${\mathrm{TGB}}_{\mathit{A}\mathrm{-}}$ ${\mathrm{TGB}}_{\mathit{C}}$. However, if the Sm-${\mathit{C}}^{\mathrm{*}}$ helicoidal pitch length is somewhat smaller than the cholesteric pitch length, then the ${\mathrm{TGB}}_{\mathit{A}\mathrm{-}}$ ${\mathrm{TGB}}_{\mathit{C}}$ transition would be replaced by the ${\mathrm{TGB}}_{\mathit{A}\mathrm{-}}$ ${\mathrm{TGB}}_{\mathit{C}}^{\mathrm{*}}$ transition.