  The dynamics of S=1/2 quantum spins on a 2D square lattice lie at the heart
of the mystery of the cuprates
\cite{Hayden2004,Vignolle2007,Li2010,LeTacon2011,Coldea2001,Headings2010,Braicovich2010}.
In bulk cuprates such as \LCO{}, the presence of a weak interlayer coupling
stabilizes 3D N\'{e}el order up to high temperatures. In a truly 2D system
however, thermal spin fluctuations melt long range order at any finite
temperature \cite{Mermin1966}. Further, quantum spin fluctuations transfer
magnetic spectral weight out of a well-defined magnon excitation into a
magnetic continuum, the nature of which remains controversial
\cite{Sandvik2001,Ho2001,Christensen2007,Headings2010}. Here, we measure the
spin response of \emph{isolated one-unit-cell thick layers} of \LCO{}. We show
that coherent magnons persist even in a single layer of \LCO{} despite the loss
of magnetic order, with no evidence for resonating valence bond (RVB)-like spin
correlations \cite{Anderson1987,Hsu1990,Christensen2007}. Thus these
excitations are well described by linear spin wave theory (LSWT). We also
observe a high-energy magnetic continuum in the isotropic magnetic response.
This high-energy continuum is not well described by 2 magnon LSWT, or indeed
any existing theories.
