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Second Harmonic Helicity and Faraday Rotation in Gated Single-Layer 1T$'$-WTe$_2$

Single-layer of 1T$'$ phase of WTe$_2$ provides a rich platform for exotic physical properties such as nonlinear Hall effect and high-temperature quantum spin hall transport. Utilizing a continuum model and diagrammatic method, we calculate the second harmonic conductivity of monolayer 1T$'$-WTe$_2$ modulated by an external vertical electric field and electron doping. We obtain a finite helicity and Faraday rotation for the second harmonic signal in response to linearly polarized incident light in the presence of time-reversal symmetry. The second harmonic signal's helicity is highly controllable by altering the bias potential and serves as an optical indicator of the nonlinear Hall current. Our study motivates future experimental investigation for the helicity spectroscopy of two dimensional materials.