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.

Figure 1

RG flow of the parameters $g_1$ and $g_2$ defined in Eq. (26) describing the FM transition. The flow is given by Eq. (27). Two unstable fixed points are located at the line $g_1=0$, see Table 1. We see that, at large enough RG time $\ell ,g_2$ flows to negative values, thus necessarily resulting in a first-order transition.

Figure 2

Global phase diagram of the RG flow described by Eq. (27) for both the FM and SmA-C transitions. The region $g_1>0$ describes a runaway flow signaling a first-order phase transition. The line $g_1=0$ governs the FM transition in the inversion-symmetric system, which is controlled by the IS fixed point. The region $0<-g_1<g_2$ is controlled by an interacting fixed point (black) and describes the second-order SmA-C transition in a magnetic field. The line $-g_1=g_2$ describes the SmA ‘critical phase,’ with the parameters $g_1,g_2$ flowing to the Sm fixed point (not shown in this figure, see also Fig. 4).

Figure 3

Schematic representation of the smectic-A to smectic-C transition in a magnetic field, $\vec{h}$. The transition occurs via tilting of the layers, and is characterized by a nonzero angle between the molecules' director $\widehat{n}$ and the smectic layer normal, $\widehat{\kappa }$ (see Ref. [20]).

Figure 4

The RG flow for the SmA-C transition in a magnetic field described by Eq. (27) in the region $g_1<0$ $(g_1<0$ part of Fig. 2). A stable fixed point (black) controls the second-order transition. The region $g_2<\left|g_1\right|$ corresponds to the mechanical instability of the system, and, thus, describes a first-order transition. The separatrix $|g_1|=g_2$, which separates two regions (red line), corresponds to the SmA line, and is controlled by the Sm fixed point.