the theoretical results, and establish the nature of the $T>0$ phase diagram. We conclude with some brief discussion in Sec. \[sec:conclusions\]. Several appendices present details of calculations underlying some of the results in the main text.

Background {#sec:background}
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Explicit hamiltonian and coordinates {#sec:expl-hamilt-coord}
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For the bulk of this paper, we will adopt a simple orthogonal coordinate system, with $x$ along the crystallographic $b$ direction, parallel to the chains, $y$ along the $c$ axis, perpendicular to the chains within the triangular plane, and $z$ along the $a$ axis, perpendicular to the triangular planes. Moreover, we will adopt a simplified geometry, which respects the topology and interactions between spins, but does not precisely reproduce the actual locations of Cu$^{2+}$ ions. A description of the actual ion locations is given in Appendix \[sec:dmv\]. In our simplified geometry, the spins form a set of regular triangular lattices stacked uniformly along the $z$ axis, with spacing $1$ between spins in the $x$ and $z$ directions, and spacing $\Delta y=1$ between chains in the triangular plane.

In this representation, the lattice Hamiltonian for the standard model is $$\label{eq:117}
  H_{\rm sm} = \sum_{xyz}\Big\{ J {\bm S}_{x,y,z}\cdot {\bm S}_{x+1,y,z}
  - {\bm h}\cdot {\bm S}_{x,y,z} \Big\} + 