This review reports a series of theoretical and experimental progress on researches of the transverse field Ising chain (TFIC) and the quantum E ₈integrable model. For the TFIC, on one hand, a unique exotic quantum critical behavior of Grüneisen ratio (a ratio from magnetic or thermal expansion coefficient to specific heat) is theoretically established; on the other hand microscopic models can accommodate the TFIC universality class are substantially expanded. These progresses successfully promote a series of experiments collaborations to first-time realize the TFIC universality class in quasi one-dimensional anti-ferromagnetic material BaCo ₂V ₂O ₈and SrCo ₂V ₂O ₈. For the quantum E ₈integrable model, the low temperature local dynamics and the dynamical structure factor with zero transfer momentum of this system are analytically determined, where a cascade of edge singularities with power-law divergences are obtained in the continuum region of the dynamical structure factor. After combining with detailed quantum critical scaling behaviors analysis and large scale iTEBD calculation, it successfully facilitates a series of experiments, including THz spectrum measurements, inelastic neutron scattering and NMR experiments, to realize the quantum E ₈integrable model in BaCo ₂V ₂O ₈for the first time. The experimental realization of the quantum E ₈integrable model substantially extends the frontiers of studying quantum integrable models in real materials. The series of progress and developments on the TFIC and the quantum E ₈integrable model lay down a concrete ground to go beyond quantum integrability, and can inspire studies in condensed matter systems, cold atom systems, statistical field theory and conformal field theory.