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 $\hslashω_n\approx n\times J_z$, 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 $17~{\rm T} <μ_0 H< 21~{\rm T}$ 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.

Figures (6)

• Figure 1: Elastic neutron scattering from $\rm K_2Co(SeO_3)_2$ and the inferred magnetic orders. (a-c) Elastic magnetic scattering as a function of momentum, measured at $T=0.1$ K and shown after subtracting the nuclear background scattering acquired at 12 K. (a) shows scattering at 0 T (top right) and 7 T (bottom left). (b) shows 0 T data, while (c) shows 7 T data with the magnetic field applied along the easy $\bf c$-axis. (d) Calculated and measured $l$ dependence of magnetic neutron scattering for ${\bf Q}=(\frac{1}{3}\frac{1}{3}l)$ and $(\frac{2}{3}\frac{2}{3}l)$. Solid lines are fits of the "Y" order with $m_\perp/m_z=0.0(1)$ and $\alpha=0.049(7)$ (Eq. \ref{['Eq: elastic']}). The dashed line shows the expected intensity from a purely transverse component $m_\perp\neq 0$ ($m_z=0$ and $\alpha=0$), calculated using Eq. \ref{['Eq: elastic']}. (e) In-plane correlation length, $\xi$(in units of the lattice constant $a$), and squared staggered magnetization (intensity) as functions of temperature. The data were obtained from fits to elastic neutron scattering data acquired on HYSPEC covering an area of the $(hk0)$ plane surrounding $(\frac{1}{3}\frac{1}{3}l)$ with $|l|<0.4$. Open and closed cycles indicate data taken with $E_i=9.0$ and 3.8 meV, respectively with $|\hslash\omega|<0.5~$meV. Dashed lines are guides to the eye. See SI for details. Error bars in (d, e) indicate the standard deviation. (f-h) Schematic diagrams of the candidate symmetry-breaking supersolid "Y", $\frac{\rm U}{2}\frac{\rm U}{2}$D, and UD0 orders discussed in the text.
• Figure 2: Temperature dependence of magnetization and specific heat capacity, and phase diagram for $\rm K_2Co(SeO_3)_2$. (a) Specific heat capacity as a function of temperature, with curves systematically shifted in proportion to the applied field. The zero-field specific heat capacity is reproduced with permission.zhong2020frustrated (b) Magnetization versus temperature for $\bf c$-axis-oriented DC fields up to 30 T. (c) Interpolated color contour plot of differential susceptibility, $\textup{d}M/\textup{d}T$, versus magnetic field and temperature for ${\bf H}\parallel {\bf c}$ inferred from the data in (b). The labels UUU, UUD, SS, and PM represent the Up-Up-Up (field-polarized), Up-Up-Down, supersolid, and paramagnetic phases, respectively. The inset shows $\textup{d}M/\textup{d}T$ as a function of temperature in a 17.5 T field. The data point on the UUD phase boundary at the lowest temperature was determined from a peak in an isothermal measurement of $C_{\rm p}(H)$. (d) Contour plot of the magnetic entropy change, $\Delta S_m(T,H)$, normalized by the total entropy $R\ln 2$. The map was constructed by first calculating the isothermal change in entropy, $\Delta S_m(T,H)$, from $\textup{d}M(H)/\textup{d}T$ using a Maxwell relation.amaral2010estimating The data were combined with $\Delta S_m(T,H=0)$ inferred from zero field specific heat capacity data to obtain $\Delta S_m(T,H)/R\ln 2$ (see SI).
• Figure 3: Magnetic field dependence of magnetization and specific heat capacity, and phase diagram for $\rm K_2Co(SeO_3)_2$. (a) Magnetization versus $\bf c$-axis-oriented field at various temperatures down to $T=0.5$ K. (b) Differential susceptibility $\textup{d}M/\textup{d}H$ versus field. (c) Specific heat capacity versus field up to $\mu_0 H=14$ T at various temperatures. (d) Low-field specific heat capacity as a function of field for temperatures near $T=4.5$ K. Data are systematically shifted in proportion to temperature to show the onset of a sharp transition. (e) Contour plot of magnetic specific heat capacity $C_{\rm m}$ versus field and temperature. The field-independent lattice contribution to the specific heat capacity was fitted by the Debye model and subtracted from $C_{\rm p}$. The second-order phase transition defined by peak positions in $C_{\rm p}$ versus temperatures (Fig. \ref{['fig:2']}a) and field (Fig. \ref{['fig:3']}c, d) appears to terminate at ($T$, $\mu_0H$) = (4.5 K, 1.1 T). Data in the temperature window from 0.3 K to 1.9 K are reproduced with permission.zhong2020frustrated
• Figure 4: Magnetic neutron scattering from coherent spin waves in the UUD phase of $\rm K_2Co(SeO_3)_2$. (a) The (${\bf Q},\omega$)-dependence of the magnetic neutron scattering cross section along high-symmetry directions in a 7 T magnetic field applied along the $\bf c$-axis at $T=0.1$ K. The path through the Brillouin zone is illustrated in the inset. Data are averaged along the $l$ direction, except for the M$_1-$L$_1$ cut along $l$. The in-plane ${\bf Q}$ integration window is $\pm0.15$ Å$^{-1}$ perpendicular to the trajectory. (b) The neutron scattering cross section for $\rm K_2Co(SeO_3)_2$ in a 7 T field calculated using linear spin-wave theory, as implemented in SpinW.toth2015linear The parameters $J_z = 2.96(2)$ meV and $J_{\perp} = 0.21(3)$ meV in Eqn. \ref{['Eq: Hxxz']} were determined by performing a pixel-to-pixel fit of this model to the measured spectrum in panel (a).
• Figure 5: Zero field magnetic excitations in $\rm K_2Co(SeO_3)_2$ probed by magnetic neutron scattering. (a) The (${\bf Q},\omega$)-dependence of the magnetic neutron scattering cross section along high-symmetry directions in zero field at $T=0.1$ K acquired with $E_i$ = 9.0 meV neutrons. To obtain the zero-field magnetic scattering, two different backgrounds were subtracted: data from 7 T measurement for $\hslash\omega \leq 2.5$ meV, and a constant value for $\hslash\omega > 2.5$ meV, respectively. The in-plane ${\bf Q}$ integration window is $\pm0.15$ Å$^{-1}$ perpendicular to the trajectory. (b, c) Magnetic neutron scattering as a function of momentum in the $(hk0)$ plane for energy transfers $\hslash\omega=3$ meV and 6 meV, respectively. A constant background was subtracted from the data. The energy integration window is $\pm0.5$ meV. (d) The (${\bf Q},\omega$)-dependence of the magnetic neutron scattering cross section along high-symmetry directions at $T=0.29$ K obtained with $E_i=1.0$ meV neutrons. The in-plane ${\bf Q}$ integration window is $\pm0.05$ Å$^{-1}$ perpendicular to the trajectory. (e, f) Low energy magnetic neutron scattering as a function of momentum in the ($hk0$) plane for six values of $\hslash\omega$. The energy integration window is $\pm0.03$ meV. All data shown in (a-f) have been integrated along the $l$ direction, which is justified by the quasi-2D character of the magnetic correlations.