Polarized monopole liquid: A Coulomb phase in a fluid of magnetic charges

The forging of strong correlations on decreasing temperature can take place without the arousal of conventional order. If this happens, as in some geometrically frustrated magnets, disorder can be a phenomenon more interesting than order itself. A Coulomb phase, for example, has critical-like pair-spin correlations, leading to neutron scattering pinch points and emergent electromagnetism. Here we present a peculiar instance of disorder in an Ising pyrochlore lattice: the polarized monopole liquid (PML), a dense monopole fluid with pinch points in the magnetic charge-pair correlations. It is a phase of ``monopole matter'' which, in principle, can be stabilized in real materials using a magnetic field and uniaxial stress along the [100] direction. To explain how the monopole correlations arise, we show that the PML is a Coulomb phase in which spin fluctuations cannot be assigned either to monopoles or to internal magnetic moments, but necessarily comprehend both degrees of freedom. We develop a simple but nontrivial method to Helmholtz decompose the spin field into a divergenceless and a divergenceful part in magnetic charge disordered pyrochlores that shows the appearance of pinch points associated with the divergenceful component in places where Bragg peaks are observed for the ``all-in/all-out'' antiferromagnet.