Field-Induced Instability of a Gapless Spin Liquid with a Spinon Fermi Surface

The ground state of the quantum kagome antiferromagnet Zn-brochantite, ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{SO}}_4$, which is one of only a few known spin-liquid (SL) realizations in two or three dimensions, has been described as a gapless SL with a spinon Fermi surface. Employing nuclear magnetic resonance in a broad magnetic-field range down to millikelvin temperatures, we show that in applied magnetic fields this enigmatic state is intrinsically unstable against a SL with a full or a partial gap. A similar instability of the gapless Fermi-surface SL was previously encountered in an organic triangular-lattice antiferromagnet, suggesting a common destabilization mechanism that most likely arises from spinon pairing. A salient property of this instability is that an infinitesimal field suffices to induce it, as predicted theoretically for some other types of gapless SLs.

Figure 1

(a) The temperature dependence of the ${}^{2}D$ NMR spin-lattice relaxation rate $1/T_1$ in the applied field of 4.69 T. The solid arrow indicates the transition temperature $T_c=0.76\mathrm{K}$, while the dashed arrows indicate the temperatures where the ${}^{2}D$ NMR spectra from panel (b) were recorded. These spectra are normalized and shifted vertically for clarity. (c) The temperature dependence of the bulk susceptibility ${\chi }_b$ in 4.69 T measured by a SQUID magnetometer. Note that the decrease of ${\chi }_b$ with temperature is very small. $T_c$ is indicated by the arrow, while the solid line is a guide to the eye.

Figure 2

The temperature dependence of (a) the ${}^{2}D$ NMR spin-lattice relaxation rate $1/T_1$ and (b) the stretching exponent $\beta$ from the magnetization recovery curves [28] in various applied fields. In panel (a) the solid lines demonstrate power-law dependence (numbers correspond to powers), while the dashed lines show the agreement with the gapped model of Eq. (2). We note that $T_c$ roughly corresponds to a common relaxation rate $1/T_1=2.5{\mathrm{s}}^{-1}$ for all except the highest fields. In panel (b) the solid lines are guides to the eye. Arrows in all panels indicate the critical temperatures $T_c$.

Figure 3

The phase diagram of Zn-brochantite with the measured critical temperatures $T_c$ and gaps $\mathrm{\Delta }$ extracted from the low-$T$ $1/T_1$ NMR data. The line shows the $T_c=aB+b{B}^3$ model fit of the phase boundary between two distinct spin liquid states. The inset shows the field evolution of the fraction $f$ of the residual density of states at the Fermi level [Eq. (2)] below $T_c$. The line in the inset is a guide to the eye.