Multipartite entanglement, quantum coherence, and quantum criticality in triangular and Sierpi\'nski fractal lattices

We investigate the quantum phase transitions of the transverse-field quantum Ising model on the triangular lattice and Sierpi'nski fractal lattices by employing multipartite entanglement and quantum coherence along with the quantum renormalization group method. It is shown that the quantum criticalities of these high-dimensional models closely relate to the behaviors of the multipartite entanglement and quantum coherence. As the thermodynamic limit is approached, the first derivatives of multipartite entanglement and quantum coherence exhibit singular behaviors and the consistent finite-size scaling behaviors for each lattice are also obtained from the first derivatives. The multipartite entanglement and quantum coherence are demonstrated to be good indicators for detecting the quantum phase transitions in the triangular lattice and Sierpi'nski fractal lattices. Furthermore, the factors that determine the relations between the critical exponents and the correlation length exponents for these models are diverse. For the triangular lattice, the decisive factor is the spatial dimension, while for the Sierpi'nski fractal lattices, it is the Hausdorff dimension.