  We investigate the effects of introducing a boost (a Zeeman field parallel to
the spin quantization axis) at the proximitized helical edge of a
two-dimensional (2D) quantum spin Hall insulator. Our self-consistent analysis
finds that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting phase may
emerge at the edge when the boost is larger than a critical value tied to the
induced pairing gap. A non-trivial consequence of retaining the 2D bulk in the
model is that this boundary FFLO state supports a finite magnetization as well
as finite current (flowing along the edge). This has implications for a proper
treatment of the ultra-violet cutoff in analyses employing the effective
one-dimensional (1D) helical edge model. Our results may be contrasted with
previous studies of such 1D models, which found that the FFLO phase either does
not appear for any value of the boost (in non-self-consistent calculations), or
that it self-consistently appears even for infinitesimal boost, but carries no
current and magnetization.
