Critical properties of a comb lattice

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.