Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$

Inelastic neutron scattering reveals a broad continuum of excitations in ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$, the temperature and magnetic field dependence of which indicate a continuous distribution of quenched transverse fields ($\mathrm{\Delta }$) acting on the non-Kramers ${\mathrm{Pr}}^{3+}$ crystal field ground state doublets. Spin-ice correlations are apparent within 0.2 meV of the Zeeman energy. A random phase approximation provides an excellent account of the data with a transverse field distribution $\rho (\mathrm{\Delta })\propto ({\mathrm{\Delta }}^2+{\mathrm{\Gamma }}^2{)}^{-1}$, where $\mathrm{\Gamma }=0.27(1)\mathrm{meV}$. Established during high temperature synthesis due to an underlying structural instability, it appears disorder in ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ actually induces a quantum spin liquid.

Figure 1

$\mathit{q}$-averaged spectra for $0.5{\AA }^{-1}\le |\mathit{q}|\le 2.2{\AA }^{-1}$ in the corresponding scattering planes. (a) Spectra for $\mathit{B}\Vert [001]$, $T=1.4\mathrm{K}$ and $\mathit{B}\Vert [1\overline{1}0]$, $T=0.2\mathrm{K}$. $|\mathit{B}|=2\mathrm{T}$. Inset shows the relative orientations of the applied fields and the spins. (b) Field dependent spectra for $\mathit{B}\Vert [001]$ at $T=1.4\mathrm{K}$. Inset shows the transverse field distribution $\rho (\mathrm{\Delta })$, and the solid line shows the half Lorentzian fit. Dashed vertical lines show the corresponding energies above which data were used to extract $\rho (\mathrm{\Delta })$, and the solid line shows the elastic scattering contribution at 0 T. (c) $T$-dependent spectra for $\mathit{B}\Vert [001]$, 4 T. The data were divided by $F(T,\omega )$ of Eq. (1). Error bars represent one standard deviation. The energy resolution at the elastic line is 0.11 meV.

Figure 2

Field dependent $\mathit{q}\text{−}\omega$ slices for $\mathit{B}\Vert [001]$, $T=1.4\mathrm{K}$. Data were integrated along $(0K0)$ for 1 (r.l.u.) $\le K\le 3$ (r.l.u.).

Figure 3

$\mathit{q}$ maps in $(HHL)$ plane. Data were folded into the first quadrant for optimal statistics. (a) and (c) show measurements with $E_i=1.7\mathrm{meV}$ at 0.05 K, 0 T with energy integration of [0.3, 0.8] meV and [0.1, 0.3] meV, respectively. (e) shows data with $E_i=3.3\mathrm{meV}$ at 0.2 K, $\mathit{B}=3.5\mathrm{T}\Vert [1\overline{1}0]$ with energy integration of [0.1, 0.5] meV. The energy resolutions at the elastic line are 0.04 meV ($E_i=1.7\mathrm{meV}$) and 0.11 meV ($E_i=3.3\mathrm{meV}$), respectively. (b), (d), and (f) show corresponding RPA calculations.

Figure 4

Elastic longitudinal cuts for the (200) forbidden nuclear peak. Solid lines are fits to the data. Shaded area shows the broad field induced elastic magnetic scattering. Inset shows corresponding cuts through the (220) nuclear peak.