Critical Phases of the extended isotropic $XY$ chain with four-spin interaction
Using the Jordan-Wigner transformation we calculate exactly the ground state and low-temperature thermodynamic properties of the spin $S=1/2$ isotropic $XX$ chain with four spin interaction. In terms of the equivalent spinless fermion (SF) representation the system is viewed as a lattice fermion gas with nearest-neighbor ($J$) and next-next-next-neighbor ($J^{\ast}/4$) hopping. It is shown that with the increase of four spin coupling, at $J^{\ast}_{c} = 4J/3$ the system experiences the Lifshitz type topological phase transition characterized by the tripling of Fermi points. The quantum phase transition (QPT) point marks transition from a gapless spin-liquid phase of standard $XX$ chain into again a gapless spin-liquid phase with different character of power-low decay of spin correlations. At the transition point the free fermion dispersion relation shows flattering at Fermi points, what determines singular character of density of states $ρ(ω)\sim (ω/J)^{-2/3}$ and as a consequence unconventional temperature dependence of heat capacity of the system $C\sim (T/J^{\ast})^{1/3}$, and singular magnetic susceptibility of the system $χ(H)\sim (H/J^{\ast})^{-2/3}$. In the case of alternating magnetic field the system is characterized by the rich ground state phase diagram which contains fully polarized (ferromagnetic), gapped antiferromagnetic (AFM) and spin liquid phases. At the transition point from the gapped AFM phase into the gapless polarized spin liquid phase the system shows rapid increase of magnetization $m\sim(H-H_c)^{1/6}$ and magnetic susceptibility a singular behavior as $χ(H)\sim (H-H_c)^{-5/6}$.