Quantum renormalization of $l_1$-norm and relative entropy of coherence in quantum spin chains

We investigate quantum-phase transitions (QPT) in the Ising transverse field model, the XY-Heisenberg model with staggered Dzyaloshinskii–Moriya (DM) interaction, and the bond alternating Ising model with DM interaction, on a one-dimensional periodic chain using the quantum renormalization group (QRG) method. Based on $l_1$-norm and relative entropy, we employ two quantum coherence measures to identify QPT. Our results show that the QPT can be characterized by the coherence measure calculated for each QRG step. It is found that after large number of QRG iterations, the coherence in the ground state develops two saturated values corresponding to two different phases and changes abruptly at a quantum critical point (QCP). The scaling behavior of the system is investigated by computing the derivative of the coherence measure around the QCP that shows a logarithmic dependence on the length of the chain. The critical exponent of the correlation length is also calculated.