Spin nematic phase is a phase of frustrated quantum magnets with a quadrupolar order of electron spins. Since the spin nematic order is usually masked in experimentally accessible quantities, it is important to develop a methodology for detecting the spin nematic order experimentally. In this paper we propose a convenient method for detecting quasi-long-range spin nematic correlations of a quadrupolar Tomonaga-Luttinger liquid state of $S=1/2$ frustrated ferromagnetic spin chain compounds, using electron spin resonance (ESR). We focus on linewidth of a so-called paramagnetic resonance peak in ESR absorption spectrum. We show that a characteristic angular dependence of the linewidth on the direction of magnetic field arises in the spin nematic phase. Measurments of the angular dependence give a signature of the quadrupolar Tomonaga-Luttinger liquid state. In our method we change only the direction of the magnetic field, keeping the magnitude of the magnetic field and the temperature. Therefore, our method is advantageous for investigating the one-dimensional quadrupolar liquid phase that usually occupies only a narrow region of the phase diagram.
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