Properties of an algebraic spin liquid on the kagome lattice

In recent work, we argued that a particular algebraic spin liquid (ASL) may be the ground state of the $S=1/2$ kagome lattice Heisenberg antiferromagnet. Furthermore, this state, which lacks a spin gap, is appealing in light of recent experiments on herbertsmithite $\left[{\text{ZnCu}}_3{\left(\text{OH}\right)}_6{\text{Cl}}_2\right]$. Here, we study the properties of this ASL in more detail using both the low-energy effective field theory and Gutzwiller-projected wave functions of fermionic spinons. We identify the competing orders of the ASL, which are observables having slowly decaying power-law correlations—among them we find a set of magnetic orders lying at the $M$ points of the Brillouin zone, the familiar $\mathit{q}=0$ magnetic ordered state, the “Hastings” valence-bond solid (VBS) state, and a pattern of vector spin-chirality ordering corresponding to one of the Dzyaloshinskii–Moriya (DM) interaction terms present in herbertsmithite. Identification of some of these orders requires an understanding of the quantum numbers of magnetic monopole operators in the ASL. We discuss the detection of the magnetic and VBS competing orders in experiments. While we primarily focus on a clean system without DM interaction, we consider the effects of small DM interaction and argue that, surprisingly, it leads to spontaneously broken time-reversal symmetry (for DM interaction that preserves $XY$ spin rotation symmetry, there is also $XY$ magnetic order). Our analysis of the projected wave function provides an estimate of the “Fermi velocity” $v_F$ that characterizes all low-energy excitations of the ASL—this allows us to estimate the specific heat, which compares favorably with experiments. We also study the spin and bond correlations of the projected wave function and compare these results with those of the effective field theory. While the spin correlations in the effective field theory and wave function seem to match rather well (although not completely), the bond correlations are more puzzling. We conclude with a discussion of experiments in herbertsmithite and make several predictions.