Study Of Low-dimension Isotropic Heisenberg Spin System

Since some feature of the low-dimension quantum system,such as strongly correlated effects,have no classical counterpart,Heisenberg spin ladders and chains have been thought to be the ideal system to learn the nature of the quantum phases and the phase diagram.For last decades a lot of efforts has been devoted to understanding the properties of the ground state and low-lying energy excitations for low-demension Heisenberg spin system.Theoretical studies such as the O(3) non-linearσ-model and the experiment on real system like Ni(C₂H₈N₂)₂NO₂(ClO₄) have confirm the "Haldane conjecture" that the energy spectrum of an integer spin chain has a finite gap,while that of an half integer spin chain is gapless.What's interesting is that the ladders also have a even-odd scenario analogous to the integer and half integer scenario for Heisenberg spin chains.But Heisenberg spin ladders have more complex properties:with the change of intrachain coupling and rung coupling,system may be of different nature according to the properties of their ground state and low-lying energy excitations.On the other side,it has been found that there stands a chance that even-legged spin ladders become superconductors when doped with charge carriers.So it has a great practical value to study Heisenberg systems.In this paper,we review the pertinent theories of low-dimension isotropic Heisenberg spin system to zero temperature and then select the bond-alternating antiferromagnetic S=1/2 Heisenberg ladder with ferromagnetic diagonal coupling to study.For there is no diagonal coupling,J.Almeida have got a clear phase diagram: There are two gapped phases("Haldane phase" and "dimer phase"),which are separated by the critical line determined by the |J'/J| andγ,where J and J'correspond to intrachain coupling and rung coupling,reletively,andγis the dimerization parameter.We wonder to what extent the influence of a diagonal coupling will change the nature of low-energy properties.We will address the problem by means of density matrix renormalization group,which is very powerful and efficient for a study of low-lying energy properties of low-dimensional quantum lattice models.Besides the energy of ground state and the energy gap,we also calculate other order parameters such as string order parameter and the dimer order parameter to identify the quantum phase.During the study,by taking J' as constant we determined the phase diagram of the system with change of J_c andγ,where J_c is the diagonal coupling.With our results,we can relate the the properties of spin ladder system studied by J.Almeida and Zhu.Our work is helpful for people to understand the quantum phases and the phase transition of the low-dimension isotropic Heisenberg spin system.