A simple way to avoid metastable configurations in the density-matrix renormalization-group algorithms

The density-matrix renormalization-group [1, 2] (DMRG) is one of the most appropriate techniques to study static properties of the one-dimensional systems at zero temperature (for a review see, for example, Refs. 3 and 4). It is also possible to calculate dynamic properties [3–7] and work at finite-temperature through the DMRG [8–10]. The main advantage of DMRG, compared with the Lanczos exact diagonalization [11], is its capability to obtain the ground-state properties of very large systems in a well controlled way. Note that it is also possible to investigate large systems by Monte Carlo methods. However, the Monte Carlo technique is not appropriated to study frustrated/fermionic systems due to the “sign” problem. Although the DMRG algorithm was developed for onedimensional systems, it has been used to treat twodimensional systems [12–17]. The procedure consists in mapping the low-dimensional model on an one-dimensional model with long range interactions. As first point out by Liang and Pang [12], the energies of two-dimensional systems, converge slower than the ones of one-dimensional systems with short range interactions. The number of states needed to keep a fixed accuracy seems to increase exponentially with the width of the system. A similar effect also appears when we study onedimensional systems with periodic boundary condition (PBC) [1, 2]. Since the DMRG was developed, it was observed that the ground state energy (as a function of the number of states m kept in the truncation process) converge faster for the system with open boundary condition (OBC) than the one with PBC. Although it is not completely understood, it seems that everytime that an operator that acts in the left block is directly connected with an operator that acts in the right block (see Fig. 1), the ground state energy convergence is slower. This has been observed for one-dimensional systems as well for the two-dimensional systems. Another difficulty also appears when the left and right blocks are directly connected. Some times, in the simulations, the system gets stuck in some local minimum of energy (see Fig. 3(a)), even working with large values of m [18, 19]. This is a serious problem. If the energies do not change increasing m, we may think naively that the true ground state energy was reached. But in fact, the energy found is far from the true ground state. Few years ago [19], White proposed a variation of the density-matrix renormalization-group algorithm with a single center site. This new algorithm has the advantage of (i) Right Block Left Block