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International Journal of the Physical Sciences

Application of density matrix renormalization group to one-dimensional Hubbard model to study strongly correlated electrons system

Olusanmi Ebenezer Odeyemi, Akinola Samson Olayinka, Arthur Imooah Ejere, Samuel Okunzuwa Ikponmwosa, Eddy Enorense Aigbekaen, Okafor John

Published Jun 12, 2019• 0 citations
Publisher

Abstract

In this work, we applied density matrix renormalization group to one-dimensional Hubbard model at five numbers of sweep to solve strongly correlated interacting electrons system, starting from two electrons on two sites up to ten electrons on ten sites at half filling. The results that emerged from the present study is in agreement with that of exact diagonalization, variational and Lanczos solution at the varying values of the Coulomb interaction strength (U/t) at t=1. The total energy, Eg/t, of the ground state increases with the increase in interaction strength for all the numbers of site, N. The spectra intensity increases with increase in the interaction strength but decreases to zero when the interaction strength is made negatively large. This study is extended to more than two electrons on two sites. We equally show effect of interaction strength, U/t, at t = 1 on the energy-dependent entropy, S. Key words: Density matrix renormalization group, Hubbard model, sweep, exact diagonalization, variational, Lanczos, entropy.

DOI
10.5897/ijps2018.4777

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In this work, we applied density matrix renormalization group to one-dimensional Hubbard model at five numbers of sweep to solve strongly correlated interacting electrons system, starting from two electrons on two sites up to ten electrons on ten sites at half filling. The results that emerged from the present study is in agreement with that of exact diagonalization, variational and Lanczos solution at the varying values of the Coulomb interaction strength (U/t) at t=1. The total energy, Eg/t, of the ground state increases with the increase in interaction strength for all the numbers of site, N. The spectra intensity increases with increase in the interaction strength but decreases to zero when the interaction strength is made negatively large. This study is extended to more than two electrons on two sites. We equally show effect of interaction strength, U/t, at t = 1 on the energy-dependent entropy, S. Key words: Density matrix renormalization group, Hubbard model, sweep, exact diagonalization, variational, Lanczos, entropy.
We study a magnetic impurity embedded in a correlated electron system using the density-matrix renormalization group method. The correlated electron system is described by the one-dimensional Hubbard model. At half filling, we confirm that the binding energy of the singlet bound state increases exponentially in the weak-coupling regime and decreases inversely proportional to the correlation in the strong-coupling regime. The spin-spin correlation shows an exponential decay with distance from the impurity site. The correlation length becomes smaller with increasing the correlation strength. We find discontinuous reduction of the binding energy and of spin-spin correlations with hole doping. The binding energy is reduced by hole doping; however, it remains of the same order of magnitude as for the half-filled case.
Phase diagram of the Hubbard-Holstein model in the coexistence of electron-electron and electron-phonon interactions has been theoretically obtained with the density-matrix renormalization group method for one-dimensional systems, where an improved warm-up (the recursive sweep) procedure has enabled us to calculate various correlation functions. We have examined the cases of (i) the systems half-filled by electrons for the full parameter space spanned by the electron-electron and electron-phonon coupling constants and the phonon frequency, (ii) non-half-filled system, and (iii) trestle lattice. For (i), we have detected a region where both the charge and on-site pairing correlations decay with power laws in real space, which suggests a metallic behavior. While pairing correlations are not dominant in (i), we have found that they become dominant as the system is doped in (ii) or as the electronic band structure is modified (with a broken electron-hole symmetry) in (iii) in certain parameter regions.
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In this work, we developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationally strongly correlated interacting elections on a two-dimensional (2D) square lattice. We also showed primarily how to derive possible electronic states available for several 2D N x N square lattices, although, with special emphasis on a 2D 5 x 5 square lattice. The results emerging from our present study was compared with the results of Gutzwiller variational approach (GVA) and correlated variational approach (CVA), at the large limit of the Coulomb interaction strength (U/4t).The approximation to the Hubbard Hamiltonian study is actually necessary because of the strong limitation and difficulty pose by the Hubbard Hamiltonian as we move away from finite - size lattices to larger N - dimensional lattices. Thus this work has provided a means of overcoming the finite - size lattice defects as we pass on to a higher dimension. We have shown in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength. This study reveals that when the Coulomb interaction is zero, that is, for free electron system (non-interacting), the variational parameters which describe the probability distribution of lattice electron system is the same. The spectra intensity increases with increase in the interaction strength and it decreases to zero when the interaction strength is made negatively large. Keywords : unit step Hamiltonian, Hubbard Hamiltonian, 3D cubic lattice, interaction strength, total energy, lattice separation.
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The attractive Hubbard model is investigated in the framework of lattice density-functional theory (LDFT). The ground-state energy E=T+WE=T+WE=T+W is regarded as a functional of the single-particle density matrix ${\ensuremath{\gamma}}_{ij}$ with respect to the lattice sites, where $T[\ensuremath{\gamma}]$ represents the kinetic and crystal-field energies and $W[\ensuremath{\gamma}]$ the interaction energy. Aside from the exactly known functional $T[\ensuremath{\gamma}]$, we propose a simple scaling approximation to $W[\ensuremath{\gamma}]$, which is based on exact analytic results for the attractive Hubbard dimer and on a scaling hypothesis within the domain of representability of $\ensuremath{\gamma}$. As applications, we consider one-, two-, and three-dimensional finite and extended bipartite lattices having homogeneous or alternating onsite energy levels. In addition, the Bethe lattice is investigated as a function of coordination number. Results are given for the kinetic, Coulomb, and total energies, as well as for the density distribution ${\ensuremath{\gamma}}_{ii}$, nearest-neighbor bond order ${\ensuremath{\gamma}}_{ij}$, and pairing energy $\ensuremath{\Delta}{E}_{p}$, as a function of the interaction strength ∣U∣/t|U|/t∣U∣/t, onsite potential $\ensuremath{\varepsilon}/t$, and band filling n=Ne/Nan={N}_{e}/{N}_{a}n=Ne​/Na​. Remarkable even-odd and super-even oscillations of $\ensuremath{\Delta}{E}_{p}$ are observed in finite rings as a function of band filling. Comparison with exact Lanczos diagonalizations and density-matrix renormalization-group calculations shows that LDFT yields a very good quantitative description of the properties of the model in the complete parameter range, thus providing a significant improvement over the mean-field approaches. Goals and limitations of the method are discussed.