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Shot noise in strongly correlated double quantum spin Hall edges

We consider the effects of interactions on the edges of ``double" quantum spin Hall insulators (DQSHIs), motivated by recent experiments on moiré twisted metal dichalcogenides. Without interactions, a DQSHI can be understood as two copies of a conventional quantum spin Hall insulator. If interactions are present and $s^z$-spin is conserved, we show that there are two possible phases for the DQSHI edge. First is a weakly correlated edge which has two pairs of helical modes and is adiabatically equivalent to two conventional quantum spin Hall edges. Second is a strongly correlated edge with only one pair of helical modes. The strongly correlated edge also has a gap to single electrons, but is gapless to pairs of electrons. In a quantum point contact geometry, this single-electron gap leads to a Fano factor of $2$ in shot noise measurements, compared to a Fano factor of $1$ for a weakly correlated edge.