DMRG method

The DMRG method was introduced in 1992 by White [1] and soon was widely applied to problems involving model Hamiltonians in condensed matter. Early... 

There have also been many theoretical developments that have extended the applicability and functionality of the DMRG method for quantum chemistry. Some have been algorithmic nature, for example, efficient parallel algorithms... 

Among the technical methods proper to the one dimensional geometry, one may cite the Bethe ansatz [19], the bosonization techniques [18], and, more recently, the Density-Matrix Renormalization Group (DMRG) method (20, 21] and a closely related scheme which is directy considered in this note, the Recurrent Variational Approach (RVA) [22, 21], The two first methods are analytical and the third one is numerical the RVA method is in between. 

Recently, the key problems associated with the failure of the old RG method have been identified and a different renormalization procedure based on the eigenvalues of the many-body density matrix of proper subsystems has been developed [64, 65]. This method has come to be known as the density matrix renormalization group (DMRG) method and has found dramatic success in dealing with... 

Finite size algorithms have been used extensively to study the edge states and systems with impurities, where substantial improvement of the accuracy is needed to characterize the various properties of a finite system. The DMRG method has been applied to diverse problems in magnetism study of spin chains with s > 1/2 [72], chains with dimerization and/or frustration [71, 73, 74], coupled spin chains [71, 75, 76], to list a few. The method has also been used to study models with itinerant fermions [77, 78], Kondo systems[79, 80, 81, 82], as well as coupled fermion chains [83, 84], including doping. Formulations for systems with a single impurity [85, 86] as well as randomly distributed impurities [87] and disorder [88] have also been reported. There has also been a study of the disordered bosonic Hubbard model in one dimension [89]. 

To summarize, we have presented a review on the renormalization group theory in the reduced many-body density matrix basis (DMRG method), and we have applied it to conjugated organic systems, with both short range and long range Coulomb interaction potentials. We... 

The density matrix renormalization group (DMRG) method is an efficient and accurate Hilbert space truncation procedure (White 1992 1993) that can be used to solve quantum mechanical models on very large systems. It is particularly suited for one-dimensional quantum lattice models, such as the 7r-electron models discussed in this book. This appendix contains a brief review of the DMRG method relevant for these models. A full discussion of the method and its various applications may be found in (Peschel et al. 1999), (Dukelsky and Pittel 2004), or (Schollwock 2005). 

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