# Many-Body Theory

The properties of non-relativistic, many-body quantum systems, as they arise in many branches of physics, are described by the Schrodinger equation. Unfortunately, the Schrodinger equation can be solved exactly in but a few exceptional cases, either involving one-dimensional or symmetry-restricted Hamiltonians or for systems with very few particles. For almost all problems, reliable approximate solutions must be sought. Many methods of approximately solving the Schr¨odinger equation for many-body quantum systems have been developed, often tailored to the specific system or problem under investigation. What has been especially challenging has been to come up with methods that can treat large numbers of particles with great accuracy, particularly in the vicinity of a phase transition. A major breakthrough was the development of the numerical renormalization group (RG) method by Wilson in the mid-1980s, which provided, for the first time, an efficient strategy for solving the Kondo problem. The enormous power of the RG method immediately suggested its application to quantum lattices. It soon became clear, however, that the Wilson RG (WRG) method ran into difficulties when dealing with such systems.

The Density Matrix Renormalization Group (DMRG) was introduced by White in 1992 in an effort to overcome the limitations of Wilson’s RG in describing one-dimensional quantum lattice models. The new method was soon shown to be extremely powerful, producing results for the ground state energy of the S = 1 Heisenberg chain that were accurate to twelve significant figures, well beyond the precision of large-scale diagonalization methods combined with finite size corrections or of Monte Carlo techniques. The DMRG is an approximate variational procedure that is rooted in Wilson’s onion picture. In the context of quantum lattices, the idea is to start with a set of lattice sites and then to iteratively add to it subsequent sites until all have been treated. At each iteration, the dimension of the enlarged space increases as the product of the dimension of the initial subspace and that of the added site. The RG procedure consists of reducing the enlarged space to the same dimension as the initial subspace and then transforming all operators to this new truncated basis (renormalization).

One of the key features that distinguishes the DMRG from Wilson’s RG is the criterion by which the truncation is implemented. While the WRG retains the lowest Hamiltonian eigenstates of the enlarged space, the DMRG uses a very different strategy. Here, the idea is to construct the reduced density matrix for the enlarged space in the presence of a medium that approximates the rest of the Hilbert space, then to diagonalize this density matrix, and finally to maintain only those states with the largest density matrix eigenvalues. This method of truncation is guaranteed to be optimal in the sense that it maximizes the overlap of the approximate (truncated) wave function with the wave function prior to truncation.

Following the ideas just described, one can treat the entire set of ‘onion’ layers. Depending on the manner in which correlations between layers fall off and on the choice of the order in which layers are treated, this can sometimes lead to an accurate representation of the ground state of the system. Usually, it does not, however, since the early layers know nothing of the physics of those treated subsequently. This suggests the use of a ‘sweeping’ algorithm, whereby once all layers have been sampled, we simply reverse direction and update them based on the information of the previous ‘sweep’. Such a sweeping algorithm can be iteratively implemented until acceptable convergence in the results has been achieved.

The original DMRG method of White is based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In a recent Reports on Progress in Physics article, we review recent developments. Following a description of the real-space DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to quantum chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm.