Sliced-basis DMRG(sb-DMRG) is used to simulate a chain of hydrogen atoms and to construct low-energy effective Hubbard-like models. The downfolding procedure first involves a change of basis to a set of atom-centered Wannier functions constructed from the natural orbitals of the exact DMRG one-particle density matrix. The Wannier function model is then reduced to a fewer-parameter Hubbard-like model, whose parameters are determined by minimizing the expectation value of the Wannier Hamiltonian in the ground state of the Hubbard Hamiltonian. This indirect variational procedure not only yields compact and simple models for the hydrogen chain, but also allows us to explore the importance of constraints in the effective Hamiltonian, such as the restricting the range of the single-particle hopping and two-particle interactions, and to assess the reliability of more conventional downfolding. The entanglement entropy for a model's ground state, cut in the middle, is an important property determining the ability of DMRG and tensor networks to simulate the model, and we study its variation with the range of the interactions. Counterintuitively, we find that shorter ranged interactions often have larger entanglement.
Typical Wannier-function downfolding starts with a mean-field or density functional set of bands to construct the Wannier functions. Here, we carry out a controlled approach, using density matrix renormalization group computed natural orbital bands, to downfold the three-band Hubbard model to an effective single-band model. A sharp drop-off in the natural orbital occupancy at the edge of the first band provides a clear justification for a single-band model. Constructing Wannier functions from the first band, we compute all possible two-particle terms and retain those with significant magnitude. The resulting single-band model includes two-site density-assisted hopping terms with ${t}_{n}\ensuremath{\sim}0.6t$. These terms lead to a reduction of the ratio
U/teff, and are important in capturing the doping-asymmetric carrier mobility, as well as in enhancing the pairing in a single-band model for the hole-doped cuprates.
We investigate the application of the density matrix renormalization group (DMRG) to the Hubbard model in momentum space. We treat the one-dimensional models with dispersion relations corresponding to nearest-neighbor hopping and
1/r hopping and the two-dimensional model with isotropic nearest-neighbor hopping. By comparing with the exact solutions for both one-dimensional models and with exact diagonalization in two dimensions, we first investigate the convergence of the ground-state energy. We find variational convergence of the energy with the number of states kept for all models and parameter sets. In contrast to the real-space algorithm, the accuracy becomes rapidly worse with increasing interaction and is not significantly better at half filling. We compare the results for different dispersion relations at fixed interaction strength over bandwidth and find that extending the range of the hopping in one dimension has little effect, but that changing the dimensionality from one to two leads to lower accuracy at weak-to-moderate interaction strength. In the one-dimensional models at half filling, we also investigate the behavior of the single-particle gap, the dispersion of spinon excitations, and the momentum distribution function. For the single-particle gap, we find that proper extrapolation in the number of states kept is important. For the spinon dispersion, we find that good agreement with the exact forms can be achieved at weak coupling if the large momentum-dependent finite-size effects are taken into account for nearest-neighbor hopping. For the momentum distribution, we compare with various weak-coupling and strong-coupling approximations and discuss the importance of finite-size effects as well as the accuracy of the DMRG.
We introduce a hybrid approach to applying the density matrix renormalization group to continuous systems, combining a grid approximation along one direction with a finite Gaussian basis set for the remaining two directions. This approach is especially useful for chainlike molecules, where the grid is used in the long direction. For hydrogen chain systems, the computational time scales approximately linearly with the number of atoms, as we show with near-exact minimal basis set calculations with up to 1000 atoms. The linear scaling comes from both the localization of the basis and a compression method for the long-ranged two-electron interaction. For shorter hydrogen chains, we show results with up to triple-ζ bases.
We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU(
N) Fermi-Hubbard chain, fully leveraging the SU(
N) symmetry of the problem. This method extends a previously developed approach for the SU(
N) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with
N compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of
U, i.e., the atom-atom interaction amplitude, up to
N=6 for filling
1/N (one particle per site in average), and for one atom (respectively, hole) away from filling
1/N. It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value
Uc (separating the Mott insulator from the metallic phase), which is proved to be finite (i.e.,
Uc>0) and to increase with
N. Central charges
c are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the
SU(N)1 Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and
c=N in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.
The Hubbard model is a simple but profound theoretical tool for understanding the physics underlying strongly correlated systems. This phenomenology includes, but is not limited to, metal-insulating transitions, charge and spin order (charge density/spin density waves), superconducting phases, non-Fermi liquid physics, among others. Examining one dimensional physical models described by the Hubbard Hamiltonian can teach us much about the related physics of higher dimensional systems. Applying advanced numerical computational techniques such as density matrix numerical renormalization group (DMRG) to one-dimensional manypbody model Hamiltonians allows for the calculation of ground state energies and dynamical spectral functions with extremely high precision and accuracy. This thesis explores the application of DMRG numerical techniques to one-dimensional physical systems described by Hamiltonians beyond the conventional Hubbard model, and the nuanced physical effects observed for these systems across various regimes and phases. A Wannier Hubbard chain composed of non-orthogonal overlapping atomic orbitals was transformed to an orthogonal atomic basis using a non-unitary Lowdin transformation. The calculation of energies, dynamical spectral functions and the photoemission spectrum for this Wannier Hubbard chain revealed phases not previously observed for Hubbard chains at half-filling. Additional analysis and calculations were performed for a CuO chain described by the Luther-Emery Hamiltonian that epitomizes charge-transfer physics. The motivation for studying this CuO chain was previous literature on transition metal oxides (TMOs) and the mapping of multiband Hamiltonians to single band models in certain energy regimes. Finally, we performed computational analysis for a spin-3/2 generalized Hubbard chain. By spanning the full phase space of our spin-3/2 Hubbard chain in terms of chemical potential
μ and external magnetic field
h we observed non-typical Cooper pairing states and novel phases for different symmetries of our Hamiltonian.--Author's abstract