Phase Diagram and Spectroscopic Signatures of a Supersolid in Quantum Ising Magnet K$_2$Co(SeO$_3$)$_2$

Abstract

A supersolid is a quantum-entangled state of matter exhibiting the dual characteristics of superfluidity and solidity. Theory predicts that hard-core bosons with repulsive interactions on a triangular lattice can form supersolid phases at half filling and near complete filling. Leveraging an exact mapping between bosons and spin-$\frac{1}{2}$ degrees of freedom, we investigate these phases in the spin-$\frac{1}{2}$ triangular-lattice antiferromagnet \K212 with exchange constants $J_z = 2.96(2)$~meV and $J_{\perp} = 0.21(3)$~meV. At zero field, neutron diffraction reveals the gradual development for $T<15$~K of quasi-two-dimensional $\sqrt{3}\times\sqrt{3}$ magnetic order with $Z_3$ translational symmetry breaking (solidity) albeit with 44(5)% reduced amplitude at $T=0.3$~K indicating strong quantum fluctuations. These are apparent in equidistant bands of continuum neutron scattering for $\hslash\omega_n\approx n\times J_z$, where $n=0,1,2,3$. The lowest energy ($n=0$) $\bf Q$-dependent continuum has a lower resonant edge and includes a quasi-elastic component at K $(\frac{1}{3}\frac{1}{3})$ consistent with broken $U(1)$ spin rotational symmetry (boson superfluidity). Competing instabilities are apparent in soft albeit finite-energy modes at M $(\frac{1}{2}0)$ and at $\frac{1}{2}$K $(\frac{1}{6}\frac{1}{6})$. For $\bf c$-axis-oriented magnetic fields $17~{\rm T} <\mu_0 H< 21~{\rm T}$ that almost saturate the magnetization, corresponding to nearly filling the lattice with bosons, we find a new phase consistent with a second supersolid. These phases are separated by a pronounced 1/3 magnetization plateau that supports coherent spin waves, from which we determine the spin Hamiltonian.