  In this paper we study the critical properties of the Heisenberg spin-1/2
model on a comb lattice -- a 1D backbone decorated with finite 1D chains -- the
teeth. We address the problem numerically by a comb tensor network that
duplicates the geometry of a lattice. We observe a fundamental difference
between the states on a comb with even and odd number of sites per tooth, which
resembles an even-odd effect in spin-1/2 ladders. The comb with odd teeth is
always critical, not only along the teeth, but also along the backbone, which
leads to a competition between two critical regimes in orthogonal directions.
In addition, we show that in a weak-backbone limit the excitation energy scales
as $1/(NL)$, and not as $1/N$ or $1/L$ typical for 1D systems. For even teeth
in the weak backbone limit the system corresponds to a collection of decoupled
critical chains of length $L$, while in the strong backbone limit, one spin
from each tooth forms the backbone, so the effective length of a critical tooth
is one site shorter, $L-1$. Surprisingly, these two regimes are connected via a
state where a critical chain spans over two nearest neighbor teeth, with an
effective length $2L$.
