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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 xi. The spin nematic instability occurs when xi reaches a critical value xi_{cr}, i.e., its transition temperature T_{SN} is always higher than the SDW critical temperature T_{SDW}. In particular, xi_{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.