  Non-compact Conformal Field Theories (CFTs) are central to several aspects of
string theory and condensed matter physics. They are characterised, in
particular, by the appearance of a continuum of conformal dimensions.
Surprisingly, such CFTs have been identified as the continuum limits of lattice
models with a finite number of degrees of freedom per site. However, results
have so far been restricted to the case of periodic boundary conditions,
precluding the exploration via lattice models of aspects of non-compact
boundary CFTs and the corresponding D-brane constructions. The present paper
follows a series of previous works on a $\mathbb{Z}_2$-staggered XXZ spin
chain, whose continuum limit is known to be a non-compact CFT related with the
Euclidian black hole sigma model. By using the relationship of this spin chain
with an integrable $D_2^2$ vertex model, we here identify integrable boundary
conditions that lead to a continuous spectrum of boundary exponents, and thus
correspond to non-compact branes. In the context of the Potts model on a square
lattice, they correspond to wired boundary conditions at the physical
antiferromagnetic critical point. The relations with the boundary parafermion
theories are discussed as well. We are also able to identify a boundary
renormalisation group flow from the non-compact boundary conditions to the
previously studied compact ones.
