We explicitly show that sixth order spatial derivative, Lorentz symmetry violating term in two dimensional space leads to quantum phase transition. We show that around the critical point, the number of zero modes increase dramatically that catalyze the change in the ground state property of the many-body wave function. We extend the analysis to three dimensional cylindrical geometry and show that the entanglement heat-capacity has similar profile to the heat capacity measurements of high temperature superconductors. We explicitly show that the long-range interaction in the two dimensional surface explain key features of high temperature superconductors.