A detailed description of the time-step-targeting time evolution method within the density-matrix renormalization-group algorithm is presented. The focus of this publication is on the implementation of the algorithm and its generic application. The case of one-site excitations within a Hubbard model is analyzed as a test for the algorithm, using open chains and two-leg ladder geometries. The accuracy of the procedure in the case of the recently discussed holon-doublon photo excitations of Mott insulators is also analyzed. Performance and parallelization issues are discussed. In addition, the full open-source code is provided as supplementary material.
A detailed description of the time-step-targeting time evolution method within the density-matrix renormalization-group algorithm is presented. The focus of this publication is on the implementation of the algorithm and its generic application. The case of one-site excitations within a Hubbard model is analyzed as a test for the algorithm, using open chains and two-leg ladder geometries. The accuracy of the procedure in the case of the recently discussed holon-doublon photo excitations of Mott insulators is also analyzed. Performance and parallelization issues are discussed. In addition, the full open-source code is provided as supplementary material.
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.
In this work we develop a new numerical method, the adaptive time-dependent density-matrix renormalization-group (adaptive t-DMRG), which turns out to be very well suited to investigate time-dependent phenomena in one-dimensional strongly correlated systems. The method is based on the original density-matrix renormalization group method by White and the time evolving block-decimation procedure by Vidal. We further show the applicability of the new method to different physical systems: bosonic, fermionic, and spin chains. We first discuss the consequences of the presence of an external trapping potential on the static properties of trapped cold bosonic atoms in an optical lattice using the Bose-Hubbard model. It is shown that a properly rescaled one-particle density matrix characterizes superfluid versus insulating states just as in the homogeneous system. We note that the superfluid to Mott-insulating transition is seen most directly in the half width of the interference peak. Then we turn to the evolution of density perturbations in these systems. In particular, we discuss the dependence of the velocity and the decay of the amplitude on density, interaction strength, and the extent and height of the perturbation in a numerically exact way, covering a wide range of interaction and perturbation strengths. By comparing our results for the sound velocity to theoretical predictions, we determine the limits of a Gross-Pitaevskii or Bogoliubov type description and the regime where repulsive one-dimensional Bose gases exhibit fermionic behaviour. As a second application of the adaptive t-DMRG, we investigate the phenomenon of spin-charge separation in the Hubbard model and propose an experimental setup for its observation in cold Fermi gases. We show the robustness of this separation beyond the low-energy regime by studying the time evolution of density wave packets of finite strength and at length scales down to a few lattice spacings. A striking signature of spin-charge separation is found in 1D cold Fermi gases in a harmonic trap using the different propagation properties of the liquid and Mott-insulating phases. The third application relates to the area of spintronics. A simplified model for the magnetization transport between two coupled reservoirs with opposite spin polarization is studied. We investigate whether a simple long time limit for the spin transport exists and if so, how it depends on the properties of the system such as integrability and criticality. Time-scales accessible to us are of the order of 100 units of time measured in 1/J while maintaining insignificant error in the observables.
We study the real-time evolution of a charge-excited state in a one-dimensional eg-orbital degenerate Hubbard model, by a time-dependent density-matrix renormalization group method. Considering a chain along the z direction, electrons hop between adjacent 3z2−r2 orbitals, while x2−y2 orbitals are localized. For the charge-excited state, a holon-doublon pair is introduced into the ground state at quarter filling. At initial time, there is no electron in a holon site, while a pair of electrons occupies 3z2−r2 orbital in a doublon site. As the time evolves, the holon motion is governed by the nearest-neighbor hopping, but the electron pair can transfer between 3z2−r2 orbital and x2−y2 orbital through the pair hopping in addition to the nearest-neighbor hopping. Thus holon and doublon propagate at different speed due to the pair hopping that is characteristic of multi-orbital systems.
We study the dynamical density matrix renormalization group (DDMRG) and time-dependent density matrix renormalization group (td-DMRG) algorithms in the ab initio context to compute dynamical correlation functions of correlated systems. We analyze the strengths and weaknesses of the two methods in small model problems and propose two simple improved formulations, DDMRG ++ and td-DMRG ++, that give increased accuracy at the same bond dimension at a nominal increase in cost. We apply DDMRG ++ to obtain the oxygen core-excitation energy in the water molecule in a quadruple-zeta quality basis, which allows us to estimate the remaining correlation error in existing coupled cluster results. Further, we use DDMRG ++ to compute the local density of states and gaps and td-DMRG ++ to compute the complex polarization function, in linear hydrogen chains with up to 50 H atoms, to study metallicity and delocalization as a function of bond length.
We parallelize the density-matrix renormalization group method to directly extend it to two-dimensional ( n -leg) quantum lattice models. The parallelization is performed mainly on the diagonalization for the superblock Hamiltonian since this step requires enormous memory space as the leg number n increases. The superblock Hamiltonian is divided into three parts, and the corresponding superblock vectors are transformed into matrices whose elements are uniformly distributed into processors. Parallel efficiency increases as the number of states kept, m , increases, and the obtained ground-state energy rapidly converges within a few sweeps. Furthermore, the present algorithm applied to doped 4-leg Hubbard ladders reaches their ground states, which satisfy the Lieb–Mattis theorem with m much smaller than those used in different indirect algorithms.
The MPS++ computer code simulates models of strongly correlated quantum lattice systems. The algorithm used is the density-matrix renormalization method formulated in the matrix product states language. These models and methods are used to obtain ground state, dynamic, finite temperature and time evolution properties of the underlying materials that are simulated. These materials are usually one- and quasi-one dimensional strongly correlated electronic systems, such as high temperature superconductors, manganites, and other transition metal oxides. This code is (mostly) an implementation of the paper "The density-matrix renormalization group in the age of matrix product states" by U. Schollwoeck, and references therein, found at http://arxiv.org/abs/1008.3477v2 , and that was also published in Annals of Physics 326,96 (2011).