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In this paper we study the critical properties of the Heisenberg spin-1/2model on a comb lattice --- a 1D backbone decorated with finite 1D chains --the teeth. We address the problem numerically by a comb tensor network thatduplicates the geometry of a lattice. We observe a fundamental difference betweenthe states on a comb with even and odd number of sites per tooth, whichresembles an even-odd effect in spin-1/2 ladders. The comb with odd teeth isalways critical, not only along the teeth, but also along the backbone, whichleads to a competition between two critical regimes in orthogonal directions.In addition, we show that in a weak-backbone limit the excitation energy scales as1/(NL), and not as 1/N or 1/L typical for 1D systems. For even teeth in theweak backbone limit the system corresponds to a collection of decoupledcritical chains of length L, while in the strong backbone limit, one spin from eachtooth forms the backbone, so the effective length of a critical toothis one site shorter, L-1. Surprisingly, these two regimes are connected via astate where a critical chain spans over two nearest neighbor teeth, with an effectivelength 2L. 

In this paper we propose a special type of a tree tensor network that has the geometry of a comb—a one-dimensional (1D) backbone with finite 1D teeth projecting out from it. This tensor network is designed to provide an effective description of higher-dimensional objects with special limited interactions or, alternatively, one-dimensional systems composed of complicated zero-dimensional objects. We provide details on the best numerical procedures for the proposed network, including an algorithm for variational optimization of the wave function as a comb tensor network and the transformation of the comb into a matrix product state. We compare the complexity of using a comb versus alternative matrix product state representations using density matrix renormalization group algorithms. As an application, we study a spin-1 Heisenberg model system which has a comb geometry. In the case where the ends of the teeth are terminated by spin-1/2 spins, we find that Haldane edge states of the teeth along the backbone form a critical spin-1/2 chain, whose properties can be tuned by the coupling constant along the backbone. By adding next-nearest-neighbor interactions along the backbone, the comb can be brought into a gapped phase with a long-range dimerization along the backbone. The critical and dimerized phases are separated by a Kosterlitz-Thouless phase transition, the presence of which we confirm numerically. Finally, we show that when the teeth contain an odd number of spins and are not terminated by spin-1/2's, a special type of comb edge states emerge.