We report a new class of algebraic spin liquids, in which the macroscopically degenerate ground state manifold is not Coulombic, like in spin ices, but Ampère-like. The local constraint characterizing an Ampère phase is not a Gauss law, but rather an Ampère law, i.e., a condition on the curl of the magnetization vector field and not on its divergence. As a consequence, the excitations evolving in such a manifold are not magnetically charged scalar quasiparticles, the so-called magnetic monopoles in Coulomb phases, but instead vectorial magnetic loops (or fictional current lines). We demonstrate analytically that in a macroscopically degenerate manifold inheriting the properties of a cooperative paramagnet and subject to a local curl-free constraint, magnetic correlations decay in space with a power law whose exponent is the space dimension 𝑑: the Ampère phase is a 𝑑-algebraic spin liquid. Using Monte Carlo simulations with appropriate cluster dynamics, we confirm this physics numerically in two- and three-dimensional examples and illustrate how the Ampère phase compares to its Coulomb counterpart
