An efficient density matrix renormalization group (DMRG) algorithm is presented and applied to Y junctions, systems with three arms of
n sites that meet at a central site. The accuracy is comparable to DMRG of chains. As in chains, new sites are always bonded to the most recently added sites and the superblock Hamiltonian contains only new or once renormalized operators. Junctions of up to $N=3n+1\ensuremath{\approx}500$ sites are studied with antiferromagnetic (AF) Heisenberg exchange
J between nearest-neighbor spins
S or electron transfer
t between nearest neighbors in half-filled Hubbard models. Exchange or electron transfer is exclusively between sites in two sublattices with ${N}_{A}\ensuremath{\ne}{N}_{B}$. The ground state (GS) and spin densities ${\ensuremath{\rho}}_{r}=\ensuremath{\langle}{S}_{r}^{z}\ensuremath{\rangle}$ at site
r are quite different for junctions with
S=1/2, 1, 3/2, and 2. The GS has finite total spin
SG=2S(S) for even (odd)
N and for
MG=SG in the
SG spin manifold, ${\ensuremath{\rho}}_{r}>0(<0)$ at sites of the larger (smaller) sublattice.
S=1/2 junctions have delocalized states and decreasing spin densities with increasing
N.
S=1 junctions have four localized
Sz=1/2 states at the end of each arm and centered on the junction, consistent with localized states in
S=1 chains with finite Haldane gap. The GS of
S=3/2 or 2 junctions of up to 500 spins is a spin density wave with increased amplitude at the ends of arms or near the junction. Quantum fluctuations completely suppress AF order in
S=1/2 or 1 junctions, as well as in half-filled Hubbard junctions, but reduce rather than suppress AF order in
S=3/2 or 2 junctions.
The density matrix renormalization group (DMRG) method generates the low-energy states of linear systems of
N sites with a few degrees of freedom at each site by starting with a small system and adding sites step by step while keeping constant the dimension of the truncated Hilbert space. DMRG algorithms are adapted to open chains with inversion symmetry at the central site, to cyclic chains and to weakly coupled chains. Physical properties rather than energy accuracy is the motivation. The algorithms are applied to the edge states of linear Heisenberg antiferromagnets with spin
S≥1 and to the quantum phases of a frustrated spin-1/2 chain with exchange between first and second neighbors. The algorithms are found to be accurate for extended Hubbard and related 1D models with charge and spin degrees of freedom.
The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.
We propose an initialization procedure for the density-matrix renormalization group (DMRG): {\it the recursive sweep method}. In a conventional DMRG calculation, the infinite-algorithm, where two new sites are added to the system at each step, has been used to reach the target system size. We then need to obtain the ground state for a different system size for every site addition, so 1) it is difficult to supply a good initial vector for the numerical diagonalization for the ground state, and 2) when the system reduced to a 1D system consists of an array of nonequivalent sites as in ladders or Hubbard-Holstein model, special care has to be taken. Our procedure, which we call the {\it recursive sweep method}, provides a solution to these problems and in fact provides a faster algorithm for the Hubbard model as well as more complicated ones such as the Hubbard-Holstein model.
We propose an approach to implement the density-matrix renormalization group (DMRG) in two dimensions. With this approach the initial blocks of a $L\ifmmode\times\else\texttimes\fi{}L$ lattice are built up directly from the matrix elements of a $(L\ensuremath{-}1)\ifmmode\times\else\texttimes\fi{}(L\ensuremath{-}1)$ lattice and the topological characteristics of two-dimensional lattices are preserved in the iteration of DMRG. By applying it to the spin-1/2 Heisenberg model on both square and triangular lattices, we find that this approach is significantly more efficient and accurate than other two-dimensional DMRG methods currently in use.
It has proved difficult to extend the density-matrix renormalization-group technique to large two-dimensional systems. I here present an approach where the calculation is done directly in two dimensions. This makes it possible to use an infinite-system method, and the fixed point in two dimensions is studied. By applying several related blocking schemes to the two-dimensional Heisenberg model I find that there exists an algorithm for which the energy per site decreases monotonically as the system size increases, thereby showing the potential feasibility of this method.