This review reports a series of theoretical and experimental progress on researches of transverse field Ising chain and quantum E8 integrable model. For transverse field Ising chain, 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 transverse field Ising chain universality class are substantially expand. These progresses successfully promote a series of experiments collaborations to first-time realize transverse-field Ising chain universality class in quasi one-dimensional anti-ferromagnetic material BaCo2V2O8 and SrCo2V2O8. For the quantum E8 integrable model, the low temperature local dynamics and dynamic 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 dynamic 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 quantum E8 integrable model in BaCo2V2O8 for the first time. The experimental realization of quantum E8 integrable model substantially extends the frontiers of studying quantum integrable models in real materials. The series of progresses and developments on transverse field Ising chain and quantum E8 integrable model not only inspire studies in condensed matter systems, cold atom systems, statistical field theory and conformal field theory, but also lay down a concrete ground to go beyond integrability.
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