Density-matrix renormalization group algorithms

The Density Matrix Renormalization Group (DMRG) was developed by White [1, 2] in 1992 to overcome the problems arising in the application of real-space renormalization groups to quantum lattice many-body systems in solid-state physics. Since then the approach has been extended to a great variety of problems in all fields of physics and even in quantum chemistry. The numerous applications of DMRG are summarized in two recent review articles [3, 4]. Additional information about DMRG can be found at http://www.dmrg.info. Originally, DMRG has been considered as an extension of real-space renor-malization group methods. The key idea of DMRG is to renormalize a system using the information provided by a reduced density matrix rather than an effective Hamiltonian (as done in most renormalization groups), hence the name density-matrix renormalization. Recently, the connection between DMRG and matrix-product states has been emphasized (for a recent review, see [5]) and has lead to significant extensions of the DMRG approach. From this point of view, DMRG is an algorithm for optimizing a variational wavefunction with the structure of a matrix-product state. In this chapter I will introduce the basic DMRG algorithms for calculating ground states in quantum lattice many-body systems using the one-dimensional spin-1 2 Heisenberg model as illustration. I will attempt to present these methods in a manner which combines the advantages of both the traditional formulation in terms of renormalized blocks and superblocks and the new description based on matrix-product states. The latter description is physically more intuitive but the former description is more appropriate for writing an actual DMRG program. Pedagogical introductions to DMRG which closely follow the original formulation are available in refs. [6, 7]. The conceptual background of DMRG and matrix-product states is discussed in the previous chapter [8] and should be read before this chapter. Extensions of the basic DMRG algorithms are presented in the following chapters [9, 10, 11]. The outline of this chapter is as follows: First I briefly introduce the DMRG matrix-product state and examine its relation to the traditional DMRG blocks and superblocks in section 1. In the next three sections I present a numerical