The density-matrix renormalization group

Abstract

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 acquired the status of method of choice for numerical studies of one-dimensional quantum systems. Its applications to the calculation of static, dynamic and thermodynamic quantities in such systems are reviewed. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and non-equilibrium statistical physics, and time-dependent phenomena is discussed. This review also 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 DMRG.

Effect of Electron−Electron Interactions on the Charge Carrier Transitions intrans-Polyacetylene

“…DMRG, firstly proposed by White in 1992 [44,45], which uses the eigenvalues of the subsystem's reduced density matrix as the decimation criterion of Hilbert space, has been shown as an extremely accurate technique in solving one-dimensional strongly correlated system with economic computational costs. [42] Both strong electron-electron and electron-phonon integrations are main features of conducting polymers and the important reasons for the conjugated polymers to present novel photoelectronic properties. Therefore, it is obviously not reasonable for the backbone of conjugated polymers to be frozen in dynamic processes with a time of more than several femtoseconds.…”

Dynamical Simulations of Polaron Transport in Conjugated Polymers with the Inclusion of Electron−Electron Interactions

“…Fortunately, the adaptive time-dependent density-matrix renormalization group (TDDMRG) method [41,42,43] can be used instead. In the context of 1D correlated electronic and bosonic systems, the adaptive TDDMRG has been found to be a highly reliable real-time simulation method at economic computational cost, for example in the context of magnetization dynamics [44], of spin-charge separation [45,46], or far-from equilibrium dynamics of ultracold bosonic atoms [47].…”

Relaxation Dynamics in the Spin-1 Heisenberg Antiferromagnetic Chain after a Quantum Quench of the Uniaxial Anisotropy

“…In order to investigate the ground state and its dynamical properties after a sudden change of the Hamiltonian parameters the MPS formalism has been employed [28,29]. The observation that for physical systems only minor part of the Hilbert space is involved [30], resulted in the rapid development of numerical methods based on a variational method within the space of MPS.…”

“…Then the time-evolution algorithm takes a very simple form [28]: one starts from |ψ 0 and repeats the following steps:…”