DMRG

Density Matrix Renormalization Group (DMRG) is a numerical variational method for obtaining accurate ground states of quantum many-body systems, especially in one dimension. It is built on the matrix product state (MPS) representation of the wavefunction and seeks to minimize the energy by optimizing a set of tensors, with a fixed bond dimension m that governs the amount of entanglement the state can capture.

In practice, DMRG proceeds with iterative sweeps along the lattice. In the common two-site variant, a pair of neighboring tensors is optimized to minimize energy while the rest of the system is represented by an effective environment. The reduced density matrix of a subsystem is diagonalized and truncated by keeping the m largest eigenvalues, discarding the rest. One-site DMRG can follow to refine the state and improve efficiency. The algorithm uses singular value decompositions to absorb indices and maintain the MPS form. The bond dimension m sets both accuracy and computational cost, which scales roughly as O(N m^3) per sweep.

History and extensions: DMRG was introduced by Steven White in 1992 for one-dimensional quantum lattice systems. It has since evolved to time-dependent DMRG (t-DMRG) for real-time evolution and finite-temperature variants. Conceptually, it is a variational method over MPS and is closely related to tensor network approaches such as TEBD and MPO-based representations. Extensions to higher dimensions are active research areas, with tensor network methods like PEPS aiming to address the increased entanglement.

Applications and limitations: DMRG is widely used for spin chains, fermionic lattice models (e.g., Hubbard), and in quantum chemistry for strongly correlated electrons. It delivers highly accurate energies in 1D systems but faces challenges in higher dimensions or highly entangled states, where the required bond dimension grows rapidly and computations become demanding.