Spin nematic fluctuations near a spin-density-wave phase

We study an interacting electronic system exhibiting a spin nematic instability. Using a phenomenological form for the spin fluctuation spectrum near the spin-density-wave (SDW) phase, we compute the spin nematic susceptibility in energy and momentum space as a function of temperature and the magnetic correlation length ξ. The spin nematic instability occurs when ξ reaches a critical value {ξ }_{cr}, i.e., its transition temperature {T}_{SN} is always higher than the SDW critical temperature {T}_{SDW}. In particular, {ξ }_{cr} decreases monotonically with increasing {T}_{SN}. Concomitantly, low-energy nematic fluctuations are present in a wider temperature region as {T}_{SN} becomes higher. Approaching the spin nematic instability, the nematic spectral function at zero momentum exhibits a central peak as a function of energy for a finite temperature and a soft mode at zero temperature. These properties originate from the general feature that the imaginary part of the spin-fluctuation bubble has a term linear in energy and its coefficient is proportional to the square of temperature. Furthermore we find that the nematic spectral function exhibits a diffusive peak around zero momentum and zero energy without clear dispersive features. A possible phase diagram for the spin nematic and SDW transitions is also discussed.