Photon Echo from Lensing of Fractional Excitations in Tomonaga-Luttinger Spin Liquid

We study theoretically the nonlinear optical response of Tomonaga-Luttinger spin liquid in context terahertz (THz) two-dimensional coherent spectroscopy (2DCS). Using gapless phase XXZ-type chain as an example, we show that its third-order magnetic susceptibilities $\chi^{(3)}_{+--+}$ and $\chi^{(3)}_{-++-}$ exhibit photon echo, where $\pm$ refers to left/right-hand circular polarization with respect $S^z$ axis. The echo arises from a ``lensing'' phenomenon which wave packets fractional excitations move apart then come back toward each other, amounting refocusing excitations' world lines. Renormalization group irrelevant corrections fixed point Hamiltonian result dispersion and/or damping packets, can be sensitively detected by lensing consequently echo. Our results thus unveil strength THz-2DCS probing dynamical properties collective prototypical many-body system.