Colloquium: Herbertsmithite and the search for the quantum spin liquid

Quantum spin liquids form a novel class of matter where, despite the existence of strong exchange interactions, spins do not order down to the lowest measured temperature. Typically, these occur in lattices that act to frustrate the appearance of magnetism. In two dimensions, the classic example is the kagome lattice composed of corner sharing triangles. There are a variety of minerals whose transition metal ions form such a lattice. Hence, a number of them have been studied and were then subsequently synthesized in order to obtain more pristine samples. Of particular note was the report in 2005 by Dan Nocera’s group of the synthesis of herbertsmithite, composed of a lattice of copper ions sitting on a kagome lattice, which indeed does not order down to the lowest measured temperature despite the existence of a large exchange interaction of 17 meV. Over the past decade, this material has been extensively studied, yielding a number of intriguing surprises that have in turn motivated a resurgence of interest in the theoretical study of the spin $1/2$ Heisenberg model on a kagome lattice. This Colloquium reviews these developments and then discusses potential future directions, both experimental and theoretical, as well as the challenge of doping these materials with the hope that this could lead to the discovery of novel topological and superconducting phases.

Figure 1

Left: Crystal structure of herbertsmithite ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ ([114]), looking down along the hexagonal $c$ axis. Copper atoms are blue, oxygen (small) red, and zinc atoms are gray if they sit below the copper plane, and brown if they sit above, to better emphasize that the ${\mathrm{CuO}}_4$ units form a buckled pattern (with the octahedral axis tilted 38° relative to the $c$ axis). Chlorine and hydrogen atoms have been suppressed for clarity. Magenta lines (triangular net) emphasize the planar kagome lattice. Right: A side view of the crystal structure using a polyhedral representation, emphasizing the $ABC$ stacking of the kagome layers due to coupling of ${\mathrm{CuO}}_4$ planar units by ${\mathrm{ZnO}}_6$ octahedra (here all zinc atoms are shown as gray). Note, though, that even in nominally stoichiometric compounds, there is a significant percentage of coppers sitting on the zinc sites.

Figure 2

$q=0$ state for the near-neighbor Heisenberg model on the kagome lattice for (a) positive chirality and (b) negative chirality. (c) $q=\sqrt{3}\times \sqrt{3}$ state. In each case, “zero” modes are indicated by the ellipses where the spins can turn with no cost in energy. From [128].

Figure 3

Left: Singlet near-neighbor dimer covering of the kagome lattice. Shown is a 12 site valence bond solid forming a diamond pattern. A quantum superposition of this with all other near-neighbor dimer coverings would be an RVB spin liquid. From [60]. Right: A pair of visons at $a$ and $b$. There is a negative sign in the wave function for this excited state associated with every dimer that the curve $\mathrm{\Omega }$ crosses. From [96].

Figure 4

Phase diagram of the $J_1\text{−}{J}_2\text{−}{J}_{3h}$ Heisenberg model on the kagome lattice (here 1 refers to near neighbors, 2 to next-near neighbors, and $3h$ to coupling across a hexagon, as shown at the top). $q=0$ and $\sqrt{3}\times \sqrt{3}$ are coplanar spin configurations (Fig. 2), $cuboc1$ and $cuboc2$ noncoplanar ones. From [95].

Figure 5

Exact diagonalization energy spectra (27 sites) for the near-neighbor antiferromagnetic (AF) Heisenberg model on the triangular lattice (left) and the kagome lattice (right), as a function of the total spin of the cluster $S$. The crosses denote the “tower of states” that defines the ground state manifold in the infinite lattice limit. From [77].

Figure 6

Left: Crystal structure of clinoatacamite ${\mathrm{Cu}}_4(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ ([43]). The orientation corresponds to that in Fig. 1 to emphasize similarities, and again hydrogen and chlorine atoms have been suppressed. There are three different crystallographic copper sites (blue, cyan, and teal), the last corresponding to the zinc site in herbertsmithite. Note the displacement of these “interlayer” sites relative to the center of the kagome triangles. Right: Proposed magnetic structure of clinoatacamite. Here Cu1 are the cyan sites, Cu2 the blue sites, and Cu3 the teal sites in the left plot. This noncoplanar magnetic structure indicates that one does not have weakly coupled kagome planes, but rather a distorted pyrochlore structure. From [124].

Figure 7

Crystal structures of claringbullite ([13]), ${\mathrm{Cu}}_4(\mathrm{OH}{)}_6\mathrm{ClF}$ (left), and kapellasite ([24]), ${\mathrm{ZnCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ (right). The orientation is as in Fig. 1, and again hydrogen, chlorine, and fluorine atoms have been suppressed. For claringbullite, note the trigonal prismatic coordination of the copper intersites (teal) which are disordered over three crystallographically equivalent locations about the center of the kagome triangles. For kapellasite, note that the zinc atoms (gray) sit in the copper planes at the middle of the hexagonal hole of the kagome lattice.

Figure 8

Left: Specific heat vs temperature for various magnetic fields. The strong variation with the field indicates that the low temperature specific heat is primarily due to spin defects. From [56]. Right: Comparison of the bulk susceptibility (top, green curve) to that inferred from the ${}^{17}\mathrm{O}$ NMR line shift (red squares) for herbertsmithite. At low temperatures, the former is dominated by the copper intersite defect spins, and the latter by the copper spins sitting in the kagome planes. From [92].

Figure 9

Proposed phase diagram for Zn-paratacamite. “Néel” is the long-range canted AF order, “VBS” is a speculated valence bond solid phase, and “VBL” is its melted version. “SG” indicates spin glass behavior, and “RVB” denotes the spin liquid phase. The monoclinic to rhombohedral phase transition occurs near $x=1/3$. From [79].

Figure 10

Left: The imaginary part of the momentum-integrated dynamic spin susceptibility of herbertsmithite for various temperatures, demonstrating quantum critical-like scaling. The solid curve is the scaling function described in the text. From [55]. Right: Momentum structure of the INS data for single crystal samples at 1.6 K for three different energies. Note the evolution from a near-neighbor dimer pattern to a more spotlike pattern as the energy is reduced below 2 meV. From [47].

Figure 11

INS spectra of herbertsmithite at 1.6 K along the $K-\mathrm{\Gamma }-K$ direction as a function of energy. Note that the intensity is almost independent of energy and momentum, in sharp contrast to the magnonlike excitations and zero modes seen in clinoatacamite. This is evidence for a spinon continuum. From [47].

Figure 12

Momentum structure of the INS data at 0.4 and 1.3 meV for single crystal herbertsmithite at 2 K in the ($HK0$) scattering plane (top row) and ($HHL$) scattering plane (bottom row). The plots in the right column are the calculated structure factor for near-neighbor AF correlations between copper defects on the zinc sites, taking into account the copper form factor. These correspond to correlations between the brown and the gray sites of Fig. 1, which sit in successive triangular planes. From [50].

Figure 13

Momentum-integrated INS data ([55]) for herbertsmithite fit to a sum of a low energy damped harmonic oscillator representing the defect spins ([99]) and a gapped continuum representing the kagome spins. The spin gap from the latter is 0.73 meV, close to that inferred from ${}^{17}\mathrm{O}$ NMR data ([39]). From [50].

Figure 14

(a) Synthesis of different copper minerals determined by the relative concentration of various copper tectons in solution. The different polymorphs of ${\mathrm{Cu}}_4(\mathrm{OH}{)}_6{\mathrm{Cl}}_2$ are formed from chains resulting from the condensation of tecton I $\mathrm{Cu}({\mathrm{H}}_2\mathrm{O}{)}_4{\mathrm{Cl}}_2$ and tecton II $\mathrm{Cu}({\mathrm{H}}_2\mathrm{O}{)}_4(\mathrm{OH}{)}_2$ in the ratio of 1 to 3, which then assemble in different ways leading to the formation of (b) atacamite, (c) paratacamite, and (d) clinoatacamite. From [116].