  Employing the projective formalism of determinant quantum Monte Carlo (DQMC)
simulations, we meticulously explore the ground-state phase diagram and
critical behavior of the half-filled Hubbard model on a square-hexagon-octagon
(SHO) lattice. This lattice, a two-dimensional (2D) structure comprising
squares, hexagons, and octagons, is representative of the biphenylene network
(BPN). Our findings reveal an intriguing ground-state phase diagram, featuring
an antiferromagnetic (AFM) Mott insulating phase enveloped by three
valence-bond solid-like (VBS-like) insulating phases. Analyzing the
single-particle gap, spin gap, and single-particle spectral function, we
observe that the metallic state in the noninteracting case becomes unstable
under the influence of Hubbard U. This interaction drives the system into a
hexagon insulating phase before transitioning into an AFM Mott insulating
phase. To quantify the critical exponents, we use finite-size scaling
techniques. The critical exponents of quantum critical points between the AFM
Mott insulating phase and two insulating phases, plaquette insulator and
ethylene insulator, closely align with the 3D O(3) universality class. However,
the critical exponents of quantum critical points between the hexagon
insulating phase and the AFM Mott insulating phase deviate from the 3D O(3)
universality class. This deviation is a finite-size effect and can be
attributed to the coupling between the fluctuations of magnetic order parameter
and very low-energy fermionic excitations. Our comprehensive study not only
advances the understanding of correlation effects on the SHO lattice but also
sheds light on the less-explored critical exponents in weakly insulating
quantum critical point.
