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We investigate the effects of introducing a boost (a Zeeman field parallel tothe spin quantization axis) at the proximitized helical edge of atwo-dimensional (2D) quantum spin Hall insulator. Our self-consistent analysisfinds that a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconducting phase mayemerge at the edge when the boost is larger than a critical value tied to theinduced pairing gap. A non-trivial consequence of retaining the 2D bulk in themodel is that this boundary FFLO state supports a finite magnetization as wellas finite current (flowing along the edge). This has implications for a propertreatment of the ultra-violet cutoff in analyses employing the effectiveone-dimensional (1D) helical edge model. Our results may be contrasted withprevious studies of such 1D models, which found that the FFLO phase either doesnot appear for any value of the boost (in non-self-consistent calculations), orthat it self-consistently appears even for infinitesimal boost, but carries nocurrent and magnetization.

Quantum spin Hall insulator in proximity with a superconductor: Transition to the Fulde-Ferrell-Larkin-Ovchinnikov state driven by a Zeeman field