  We show that condensate-induced transitions between two-dimensional
topological phases provide a general framework to relate one-dimensional spin
models at their critical points. We demonstrate this using two examples. First,
we show that two well-known spin chains, namely the XY chain and the transverse
field Ising chain with only next-nearest-neighbor interactions, differ at their
critical points only by a non-local boundary term and can be related via an
exact mapping. The boundary term constrains the set of possible boundary
conditions of the transverse field Ising chain, reducing the number of primary
fields in the conformal field theory that describes its critical behavior. We
argue that the reduction of the field content is equivalent to the confinement
of a set of primary fields, in precise analogy to the confinement of
quasiparticles resulting from a condensation of a boson in a topological phase.
As the second example we show that when a similar confining boundary term is
applied to the XY chain with only next-nearest-neighbor interactions, the
resulting system can be mapped to a local spin chain with the u(1)_2 x u(1)_2
critical behavior predicted by the condensation framework.
