  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.
This state, which lacks a spin gap, is appealing in light of recent experiments
on herbertsmithite (ZnCu3(OH)6Cl2). Here, we study the properties of this ASL
in more detail, using both the low-energy effective field theory and
Gutzwiller-projected wavefunctions. We identify the competing orders of the ASL
-- among them we find a set of magnetic orders lying at the M-points of the
Brillouin zone, the 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. We discuss the detection of the magnetic and VBS
competing orders in experiments. While we 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 wavefunction provides an estimate of the
"Fermi velocity" 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
wavefunction and compare these results with those of the effective field
theory. While the spin correlations in the effective field theory and
wavefunction 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.
