Pathology Of The Fermionic Representation Of Transverse Ising Model With The Jordan-Wigner Transformation

When studying the quantum Ising chainS(28)2/1in a transverse field,we usually transform the spin operators to fermion operators by applying the Jordan-wigner transformation,i.e we take it for granted that the spin model and corresponding fermionic model are equivalent naively.but deliberate treatment could reveal that the equivalence accompanies some delicate conditions.Besides,we also discussed Ising model in a transverse field.When we introducing afield,the ground states are changed if compared to the previous results.What's more,the system can occur phase transition at the critical pointJh1/(28).We can get the critical point through the divergent behavior of the second derivative of energy spectrum or the image of the energy gap.In addition,the correlation function which can be observed by inelastic neutron scattering experiment also predict the phase transition occurring atJh1/(28),The topologically property also change when the field is introduced The system in one dimension with without field always have topologically nontrivial states.After introducing a field,the topologically nontrivial states or topologically trivial states by adjusting the external field.we will study the Magnetism and the thermodynamics properties of the Three sub-lattice spin chain according to the Jordan-Wigner transformation and Green's function method.Researches have shown that there are three branches for the spin-wave excitation spectrum when in zero temperature.Among them,two are anti-ferromagnetic interaction with nature inspired patterns but the third one is band gap excitation spectrum,with ferromagnetic nature.The heat capacity of the three sub-lattice spin system presents richfull bimodal structures with the changing temperature.Therefore,we can conclude that there is a strong competition between the ferromagnetic and the anti-ferromagnetic interaction,which also makes the different variations of the specific heat bimodal behavior of the system.Finally,we will adopt the transfer matrix to study the magnetism and thermodynamic properties of the triangular ising chain.The results suggest as below: Although the raising in temperature makes the triangular Ising chain system acts unstably,the frustration increasing helps to strengthen thestability of the system.At low temperature,the power of magnetization will present three clear magnetization platforms while the external magnetic changes.When the frustration increases,the magnetization will move to the high magnetic region.That can be said that increasing frustration will not lead the magnetization platform disappear but to make it more difficult to become a magnetization platform.