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.

Figure 1

(a) The pyrochlore lattice, with one “up” and four “down” tetrahedra. The vertical arrow indicates the field direction in all three panels, colored spheres stand for the two types of monopole charge, and big orange arrows mark the “dimer spins” on a PML configuration. Two neighboring tetrahedra inscribed in cubes suggest how the checkerboard lattice can be thought of as a projection of the pyrochlore. The blue links in one cube indicate the relative change in the interaction constant due to stress along [100]. (b) The five possible monopole configurations of two contiguous PML tetrahedra conforming a block, and their degeneracies. The spins in each [100] plane must be multiplied by $\pm 1$ (as indicated by black and red lines) to obtain the divergenceless $\mathit{\Sigma }$ and ${\mathit{\Sigma }}^{\text{dim}}$ fields. (c) Checkerboard lattice with an arbitrary PML configuration. An elongated Gaussian surface marked in green can be used to show that the total topological charge in alternate [100] planes is preserved.

Figure 2

(a) Monopole-monopole correlation in the $\left[lhh\right]$ plane for the PML, showing pinch points in the monopole channel. (b) Diffuse pattern for the lattice spin field $\mathbf{S}$, with Bragg peaks marked with black dots. The diffuse scattering presents pinch points around the same $\mathbf{q}$ values as spin ice, but also where the AIAO antiferromagnetic phase shows Bragg peaks [26]. (c) Diffuse pattern for the approximated divergenceful component ${\mathit{S}}_m$, showing intense pinch points for $\mathbf{q}=\left[311\right]$ and [022]. (d) Diffuse pattern for the approximated divergenceless component ${\mathit{S}}_d$, showing intense pinch points along $\mathbf{q}=\left[200\right]$ and [111] directions. We have allowed for a maximum absolute charge within this channel of ${10}^{-4}$. All colored plots were obtained using $5\times {10}^5$ spins in 330 independent configurations (more details in the Supplemental Material).

Figure 3

Realization of a polarized monopole liquid in an AIAO Ising pyrochlore under the action of a tetragonal distortion and a magnetic field along [100] ($x$ axis). (a) The energy per spin for a single tetrahedron vs magnetic field ($J_{\text{NN}}=0.06\mathrm{K}, \delta =-0.1\mathrm{K}$). On increasing fields we see three different ground states, respectively populated by double monopoles, single monopoles with a positive component of magnetic moment along $x$ (the PML), and neutral tetrahedra with a positive component along $x$. (b) Monte Carlo simulations for Eq. (1), with the previous parameters and $\approx 8000$ spins. The magnetization $M_x$ and monopole density $\rho$ are fully consistent with a PML in the intermediate field interval.