Applications Of Density Matrix Renormalization Group

Strongly correlated systems have many interesting physical properties,which can not be well understood by the conventional Landau Fermi liquid theory or band theory.With the development of computer science and technology,numerical calculation plays more and more important role in theoretical research.A variety of numerical methods have been developed.Among them,the density matrix renormalization group method is an important numerical method to study the one-dimensional strongly correlated sys-tems.In chapter 1,we briefly introduce the one-dimensional strongly correlated systems we studied.The t-J model is a theoretical model that can describe high-temperature su-perconducting materials.The spin liquid which is associated with the high-temperature superconducting mechanism may exist in the frustrated spin model,and the spin model shows interesting magnetization behaviors when placed in an external magnetic field.Since almost all the strongly correlated systems can not be solved analytically except for one-dimensional or specific parameter cases,many numerical methods have been developed for theoretical study.We briefly introduce the commonly used numerical methods such as exact diagonalization,quantum Monte Carlo,numerical renormaliza-tion group,density Matrix renormalization group.We also give a brief introduction of the advantages and disadvantages of various methods.In chapter 2,we introduce the exact diagonalization method and density matrix renormalization group method.Exact diagonalization is the basic numerical calculation method.We focus on the density matrix renormalization group method which is used in this thesis.We present its basic ideas and concepts related to density matrix.We also briefly introduced several optimization methods that improve the computational efficiency in actual calculation.In chapter 3,we study the extended one-dimensional t-J model with the nearest-neighbor interaction and density-spin interaction by using density matrix renormaliza-tion group method.This model can describe the copper-oxide high-temperature super-conductor material and it is one of the most important theoretical models in condensed matter physics.We choose three points corresponding to different phases in the ground state phase diagram and calculate some observable quantities,such as the particle num-bers,the spin distribution in real space,the structure factor of density-density correlation function and spin-spin correlation function.The results show that the density-spin in-teraction does not affect the properties of the system.When the density-spin interaction strength is large enough,the system will enter the phase separation.The property of phase separation changes with different parameter.In chapter 4,we investigate the adiabatic magnetization process of the one-dimensional J-Q₂model with XXZ anisotropy g in an external magnetic field h by using density matrix renormalization group（DMRG）method.According to the characteristic of the magnetization curves,we draw a magnetization phase diagram consisting of four phas-es.For a fixed nonzero pair coupling Q,i)when g<-1,the ground state is always ferromagnetic in spite of h;ii)when g>-1 but still small,the whole magnetization curve is continuous and smooth;iii)if further increasing g,there is a macroscopic mag-netization jump from partially-to fully-polarized state;iv)for a sufficiently large g,the magnetization jump is from non-to fully-polarized state.By examining the energy per magnon and the correlation function,we find that the origin of the magnetization jump is the condensation of magnons and the formation of magnetic domains.We also demonstrate that while the experienced states are Heisenberg-like without long-range order,all the jumped-over states have antiferromagnetic or Néel long-range orders,or their mixing.In chapter 5,we investigate the magnetization process of the one-dimensional J₁-J₂model with XXZ anisotropy g in an external magnetic field using density matrix renormalization group method.We consider antiferromagnetic nearest-neighbor J₁>0and ferromagnetic next-nearest-neighbor J₂<0 interactions.In this case,it is found that the sign of g plays an important role in determining the magnetization behaviors of the model due to the fact whether the frustration is available or not depends on the sign of g.For g>0,the frustration is absent.In this case the magnetization process shows one macroscopic magnetization jump beginning from magnetization density m=0 to a finite value.This is quite different to that observed in one-dimensional J-Q₂model without frustration,where the magnetization jump happens usually from an unsaturat-ed to saturation magnetization density.An analysis of the correlation function shows that all jumped-over states have long-range order.For g<0,the frustration is present,which leads to complex and rich magnetization behaviors,showing many magnetiza-tion jumps with fixed?S=2 in each jump.From the limit case of g→-∞,what is the“quasi-particle”consisting of two down spins induces the magnetization jump.Based on the above analysis,we obtain a systematic and comprehensive magnetization phase diagram of this model,which sheds a light in understanding the magnetization behaviors of the related models.In chapter 6,we simply summarize our research and give an outlook about future work.