Theoretical Methods for Strongly Correlated Electrons
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The authors also investigate in detail the interesting example of a disordered Mott insulator and argue that intermediate disorder can lead to a novel phase, the Mott glass, intermediate between a Mott and an Anderson insulator. This leads to the next chapter by J. Kroha and P. They review a new systematic many-body method capable of describing both Fermi liquid and non-Fermi liquid behavior of quantum impurity models at low temperatures on the same footing. The method covers the crossover to the high temperature local moment regime as well.
The conserving T -matrix approximation CTMA , discussed in the previous chapter, is used here again, but for the auxiliary particles. The results are compared with the non-crossing approximation NCA and with data obtained by the numerical renormalization group and the Bethe ansatz. Generalizations are discussed as well.
The last chapter by S.
Allen, A. Tremblay, and Y. Vilk presents a formal derivation of a non-diagrammatic approach that was developed a few years ago. The two-particle selfconsistent approach presented in this chapter has produced results that are more accurate, both quantitatively and qualitatively, than other methods when compared with Quantum Monte Carlo calculations, in particular for the so-called pseudogap problem. We are indebted also to Luc Vinet and Yvan Saint-Aubin, who encouraged us to organize this workshop and applied for grants that made it possible.
Yvan Saint-Aubin also gave valuable advice on the philosophy behind these proceedings. Wortis 1 Introduction. Orignac 1 Introduction. Bickers 1 Introduction. Vilk 1 Introduction. Vilk, Valencia Dr. It allows for a very precise calculation of static, dynamical, and thermodynamical properties. White in  and since then DMRG has proved to be a very powerful method for low-dimensional interacting systems. Some calculations have also been performed in 2D quantum systems. We suggest that the interested reader look for further information in the referenced work.
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Our aim here is to give a general overview on the subject. Several methods have been introduced to reduce the size of the Hilbert space to be able to reach larger systems, such as Monte Carlo, renormalization group RG , and DMRG. Each method considers a particular criterion of keeping the relevant information. The DMRG was originally developed to overcome the problems that arise in interacting systems in 1D when standard RG procedures were applied.
These traditional methods consist in putting together two or more blocks e. Although it has been used successfully in certain cases, this procedure, or similar versions of it, has been applied to several interacting systems with poor performance. Other results  were also discouraging. A better performance was obtained  by adding a single site at a time rather than doubling the block size. However, there is one case where a similar version of this method applies very well: the Kondo and Anderson model. Wilson  mapped the one-impurity problem onto a one-dimensional lattice with exponentially descreasing hoppings.
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Based on this consideration, Noack and White  tried including different boundary conditions and boundary strengths. White [1,3] proposed the density matrix as the optimal way of projecting the best states onto part of the system and this will be discussed in the next section. A very easy and pedagogical way of understanding the basic functioning of DMRG is applying it to the calculation of simple quantum problems like one particle in a tight binding chain [10, 11].
The basic idea consists of starting from a small system e. Let us call the collection of N sites the universe and divide it into two parts: the system and the environment see Fig. This is considered as the state of the universe and called the target state. It has components on the system and the environment. We want to obtain the most relevant states of the system, i. To obtain this, the environment is considered as a statistical bath and the density matrix  is used to obtain the desired information on the system. So instead of keeping eigenstates of the Hamiltonian in the block system , we keep eigenstates of the density matrix.
We will be more explicit below. The total Hilbert space of this superblock is the direct product of the individual spaces corresponding to each block and the added sites. In some cases, as the quantum number of the superblock consists of the sum of the quantum numbers of the individual blocks, each block must contain several subspaces several values of S z for example. For closed chains the performance is poorer than for open boundary conditions [3, 13]. When more than one target state is used, i.
The calculation of static properties like correlation functions is easily done by keeping the operators in question at each step and performing the corresponding basis change and reduction, in a similar manner as was done with the Hamiltonian in each block . The energy and measurements are calculated in the superblock. A faster convergence of Lanczos or Davidson algorithm is achieved by choosing a good trial vector [17, 18].
An interesting analysis on DMRG accuracy is done in . Fixed points of the DMRG and their relation to matrix product wave functions were studied in  and an analytic formulation combining the block renormalization group with variational and Fokker—Planck methods in .
There are also 8 Karen Hallberg interesting connections between the density matrix spectra and integrable models  via corner transfer matrices. These articles give a deep insight into the essence of the DMRG method. We would like to mention some applications where this method has proved to be useful.
They obtained a spin correlation length of 6. A related problem, i. For larger integer spins there have also been some studies. Edge excitations for other values of S have been studied in . In  1. For this, very accurate values for the energy and correlation functions were needed. For this model it was found that the data for the correlation function has a very accurate scaling behavior and advantage was taken of this to obtain the logarithmic corrections in the thermodynamic limit.
Other calculations of the spin correlations have been performed for the isotropic [43, 44] and anisotropic cases . In this case a stronger logarithmic correction to the spin correlation function was found. Dimerization and frustration have been considered in [58—66] and alternating spin chains in . The case of several coupled spin chains ladder models have been investigated in [68—72], spin ladders with cyclic four-spin exchanges in [73—76], and Kagome antiferromagnets in .
Zigzag spin chains have been considered in [78—80] and spin chains of coupled triangles in [81—83].
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Magnetization properties and plateaus for quantum spin ladder systems [84—86] have also been studied. An interesting review on the applications to some exact and analytical techniques for quantum magnetism in low dimension, including DMRG, is presented in . The method has been very successful for several band Hubbard models , Hubbard ladders [—] and t-J ladders . Time reversal symmetry-broken fermionic ladders have been studied in  and power laws in spinless-fermion ladders in .
Long-range Coulomb interactions in the 1D electron gas and the formation of a Wigner crystal was studied in . Persistent currents in mesoscopic systems have been considered in . There have been some recent attempts to implement DMRG in two and higher dimensions [—] but the performance is still poorer than in 1D. A recent extension using a two-step DMRG algorithm for highly anisotropic spin systems has shown promising results . Impurity problems have been studied for example in one-  and twoimpurity  Kondo systems, in spin chains  and in Luttinger Liquids .
A momentum representation 1.
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Other recent applications have been in nuclear shell model calculations, where a two-level pairing model has been considered  and in the study of ultrasmall superconducting grains, in this case, using the particle hole states around the Fermi level as the system environment block . A very interesting and successful application is a recent work in highenergy physics .