New critical point in smectic liquid crystals

We study a new critical point which terminates a first-order transition line along which two smectic-A (SmA) liquid-crystal phases (or more generally any layered phases with uniaxial symmetry) with different layer spacing coexist. We call this new critical point C the SmA-SmA’ critical point. We develop a model nonlinear elastic Hamiltonian to describe physical properties in the vicinity of C. We study this model using mean-field theory, one-loop-order perturbation theory, and the ε expansion. In mean-field theory, the SmA-SmA’ transition is identical to the mean-field liquid-gas transition. In one-loop perturbation theory, critical corrections to the compressibility become important below an upper critical dimension of 6. In addition, there are important critical corrections to third-order vertex function between 6 and 8 dimensions indicating deviations from mean-field behavior below 8 dimensions. We determine a fixed point in 6-ε dimensions describing C and calculate critical exponents to the first order in ε. This fixed point exhibits anisotropic scaling with different correlation length exponents ${\nu }_{?}$ and ${\nu }_{\perp }$ parallel and perpendicular to the director. Scaling properties and x-ray scattering patterns in the vicinity of the SmA-SmA’ critical point are also considered.