Ferromagnetic transition in a one-dimensional spin-orbit-coupled metal and its mapping to a critical point in smectic liquid crystals

We study the quantum phase transition between a paramagnetic and ferromagnetic metal in the presence of Rashba spin-orbit coupling in one dimension. Using bosonization, we analyze the transition by means of renormalization group, controlled by an $\varepsilon$-expansion around the upper critical dimension of two. We show that the presence of Rashba spin-orbit coupling allows for a new nonlinear term in the bosonized action, which generically leads to a fluctuation driven first-order transition. We further demonstrate that the Euclidean action of this system maps onto a classical smectic-A -- C phase transition in a magnetic field in two dimensions. We show that the smectic transition is second-order and is controlled by a new critical point.