  Because the three-leg ladder behaves like a renormalized single Heisenberg
chain we argue that a spin-Peierls instability must occur in this system when
it is coupled to three-dimensional phonons. Using the bond-mean-field theory,
we show that this is indeed the case. The dimerized state below the
spin-Peierls transition temperature forms into the columnar dimerized phase not
the staggered one. This contrasts with the argument based on
antiferromagnetism. A physical argument based rather on spin bonding into
singlets explains why the columnar configuration is favored. No quantum
criticality (gaplessness) can occur in the columnar arrangement of the
dimerized chains.
