Charge order before superconductivity in the doped kagome Dirac spin liquid

Doping a quantum spin liquid is expected to give a superconductor. On kagome it gives charge order first, because of band structure rather than interaction. In the chargon (Higgs) construction of its U(1) Dirac spin liquid the two gauge components, degenerate on square and triangular lattices, are split by a line-graph effect: one disperses, four minima at $-(1+\sqrt{6})t$; the other has two exactly flat lowest bands at $-2t$. Only the dispersive component condenses, over a bare-mass window of width exactly $(\sqrt{6}-1)t$, in which the doped spin liquid forms charge and loop-current crystals on the $M$, $K$ and $K\mathrm{/}2$ stars, the physical U(1) unbroken: a chargon Higgs phase that does not superconduct. Its pairing channels are those of the ${\mathbb{Z}}_2$ descendants, a pair-density wave, uniform $d+id$ excluded at leading order and at all orders on the $M$ star. The projective symmetry group reduces the quartic competition to one angle, on which projected on-site repulsion sits exactly at the $M$/$K\mathrm{/}2$ boundary, a direction protected to all orders by $O(4)\times U(1)$ under renormalization, though lattice matching gives the physical coupling off-ray components. The competing $M$-star crystals are degenerate through sextic order, split only at eighth. The charge sector carries its own classification, the charged counterpart of the undoped bilinear-and-monopole catalog, which holds no charge-carrying operator; it puts the doped Mott insulator and kagome metals on one $F$-irrep grid, separated by the charge-order irrep ($F_1$ there, $F_3$ here) and the Landau cubic. Predicted: $C_2$-odd ($F_3$) charge order, no cubic invariant; vestigial pair phases of period $h\mathrm{/}4e$ or $h\mathrm{/}6e$; loop-current fields of order ten gauss at herbertsmithite’s ${}^{17}$O and ${}^{35}$Cl; and, if the $\mathrm{\Gamma }$ flux orders with the condensate, charge and current order setting in together below the condensation scale.