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 new 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" never considered before 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 to the divergenceful component in places where Bragg peaks are observed for the "all-in/all-out" antiferromagnet.Keywords: spin ice, magnetic monopoles, spin liquid.Introduction. When a fluid of magnetic ions crystallizes in a pyrochlore lattice the nuclear positions get frozen, but their magnetic moments or spins S can fluctuate down to very low temperatures [1,2]. Something analogous happens when the magnetic charge-like quasiparticles or monopoles inhabiting this structure [3][4][5] in turn crystallize in an array of plus and minus monopoles. Quite remarkably, in this state of monopole matter [3, 6-13] both the static charge and the fluctuating moment of the monopoles are derived from a single degree of freedom. It is said that S "fragments" into a conservative static field S m (which describes the crystal of monopoles) and a dipolar-like fluctuating field S d (accounting for the internal magnetic moment of the monopoles), with S m + S d = S [9,14][15]. The divergenceless component S d on this Helmholtz decomposition corresponds then to a "Coulomb phase" [16]. Notably in spin ices, whose low temperature magnetism is described by a divergence free condition, Coulomb phases have been observed through their characteristic signature in neutron scattering patterns: diffuse bowtie shaped scattering known as pinch points [16][17][18][19][20][21][22]. The Helmholtz decomposition of the fragmented Coulomb spin liquid phase (FCSL) [14], however, provided a first example of a Coulomb phase coexisting with magnetic monopole order, a fact that has very recently found experimental counterparts [23][24][25].In this paper we describe a new phase of monopole matter and consider how it could be stabilized in a real material. We call it the Polarized Monopole Liquid (PML), since it can be thought of as the Monopole Liq...

We consider a simple Ising magnetic model in two dimensions with Einstein site phonons and study it using Monte Carlo simulations that take into account both degrees of freedom simultaneously. In non-frustrated systems, like the square lattice with ferromagnetic and antiferromagnetic interactions, we find that the coupling of the magnetic to the elastic degrees of freedom gradually lowers the magnetic ordering transition until it is completely suppressed at a critical value of the coupling constant. Above this the system suffers a simultaneous magnetic and structural transition into a dimerised state with lower crystalline symmetry and ferromagnetic clusters antiferromagnetically aligned. In the case of the Kagomé lattice with antiferromagnetic interactions, which is frustrated, we show that a similar ordered state takes place when the coupling constant is above a critical value.