Mottness, Phase String, and High-$T_c$ Superconductivity

It is a great discovery in physics of the twentieth century that the elementary particles in Nature are dictated by gauge forces, characterized by a nonintegrable phase factor \cite{Yang1974} that an elementary particle of charge $q$ acquires from A to B points: \begin{equation} P \exp \left( i \frac q {\hbar c}\int_A^B A_{\mu}dx^{\mu}\right) \label{nonit} \end{equation} where $A_{\mu}$ is the gauge potential and $P$ stands for path ordering. In a many-body system of strongly correlated electrons, if the so-called Mott gap is opened up by interaction, the corresponding Hilbert space will be fundamentally changed. A novel nonintegrable phase factor known as phase-string will appear and replace the conventional Fermi statistics to dictate the low-lying physics. Protected by the Mott gap, which is clearly identified in the high-$T_c$ cuprate with a magnitude $> 1.5 \ \mathrm{eV}$, such a singular phase factor can enforce a fractionalization of the electrons, leading to a dual world of exotic elementary particles with a topological gauge structure. A non-Fermi-liquid "parent" state will emerge, in which the gapless Landau quasiparticle is only partially robust around the so-called Fermi arc regions, while the main dynamics are dominated by two types of gapped spinons. Antiferromagnetism, superconductivity, and a Fermi liquid with full Fermi surface can be regarded as the low-temperature instabilities of this new parent state. Both numerics and experiments provide direct evidence for such an emergent physics of the Mottness, which lies in the core of a high-$T_c$ superconducting mechanism.