Title: 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 $T_{c}$.