Vestigial nematic order and superconductivity in the doped topological insulator Cu$_{x}$Bi$_{2}$Se$_{3}$

Abstract: If the topological insulator Bi${2}$Se${3}$ is doped with electrons, superconductivity with $T_{{\rm c}}\approx3-4:{\rm K}$ emerges for a low density of carriers ($n\approx10^{20}{\rm cm}^{-3}$) and with a small ratio of the superconducting coherence length and Fermi wave length: $\xi/\lambda_{F}\approx2\cdots4$. These values make fluctuations of the superconducting order parameter increasingly important, to the extend that the $T_{c}$-value is surprisingly large. Strong spin-orbit interaction led to the proposal of an odd-parity pairing state. This begs the question of the nature of the transition in an unconventional superconductor with strong pairing fluctuations. We show that for a multi-component order parameter, these fluctuations give rise to a nematic phase at $T_{{\rm nem}}>T_{c}$. Below $T_{c}$ several experiments demonstrated a rotational symmetry breaking where the Cooper pair wave function is locked to the lattice. Our theory shows that this rotational symmetry breaking, as vestige of the superconducting state, already occurs above $T_{c}$. The nematic phase is characterized by vanishing off-diagonal long range order, yet with anisotropic superconducting fluctuations. It can be identified through direction-dependent para-conductivity, lattice softening, and an enhanced Raman response in the $E_{g}$ symmetry channel. In addition, nematic order partially avoids the usual fluctuation suppression of $T_{c}$.