Hubbard model with Rashba or Dresselhaus spin-orbit coupling and Rotated Anti-ferromagnetic Heisenberg Model

In this work, we investigate Hubbard model subject to Rashba or Dresselhaus spin-orbit coupling (SOC). In the strong coupling limit, it leads to the Rotated Anti-ferromagnetic Heisenberg model (RAFHM) which is a new class of quantum spin model. For a special equivalent class, we identify a new spin-orbital entangled commensurate ground (Y-y) state suffering quantum fluctuations at \(T=0\). We evaluate the quantum fluctuations by the spin wave expansion (SWE) up to order \( 1/S^2 \). It supports a massive relativistic commensurate magnon C-C\(_0\) in one SOC parameter regime and a new massive relativistic elementary excitation: in-commensurate magnon C-IC in the other regime. The C-IC encodes short-range incommensurate orders embedded in a commensurate phase with its gap minimum positions (in momentum space) continuously tuned by the SOC strength. At both \(T=0\) and low temperatures, these relativistic magnons lead to dramatic effects in many physical quantities such as specific heat, magnetization, \((0,\pi) \) and \( (\pi,0) \) susceptibilities, Wilson ratio and spin correlation functions. In the weak coupling limit, any weak repulsive interaction also leads to the Y-y state. The crossover from the weak to the strong coupling is studied. High temperature expansions of the specific heats in both weak and strong coupling are presented. Experimental applications to both condense matter and cold atom systems are discussed.