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arXiv:1101.0962 · Physical Review B

Ground-state properties of the one-dimensional attractive Hubbard model with confinement: A comparative study

Ji-Hong Hu, Jingjing Wang, Gao Xianlong, Masahiko Okumura, Ryo Igarashi, Susumu Yamada, Masahiko Machida

Published Jul 14, 2010• 20 citations
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Abstract

We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.

DOI
10.1103/physrevb.82.014202

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We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.
Ground-state properties of the attractive Hubbard model in one dimension are studied by means of both the exact Bethe-ansatz formalism and the self-consistent field (SCF) approach with renormalized chemical potential for general band fillings n and a wide range of coupling strength U/t.U/t.U/t. The energy, the concentration of double occupied sites, the kinetic energy and the chemical potential of the ground state are in a good numerical agreement with the exact results over a wide range of parameters U/tU/tU/t and n. The concentration of local pairs or double occupied sites in the Bethe-ansatz solution serves as a suitable parameter measuring the electron pairing correlations. The SCF theory provides a simple analytical relationship between the concentration of double occupied sites, the band filling and the BCS order parameter, valid for arbitrary U/tU/tU/t and n. The calculated energy gap, the BCS order parameter, the phase diagram and the compressibility are also discussed. The SCF theory in one dimension distinguishes the order parameter from the excitation gap and suggests a smooth crossover away from half-filling (at $n\ensuremath{\ne}1)$ from the BCS pairing to the Bose condensation regime under the variation of U/tU/tU/t and n.
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.
The ground-state energy and energy gap to the first excited state are calculated for the attractive Hubbard model in one dimension using both the Bethe ansatz equations and the variational BCS wave function. Comparisons are provided as a function of coupling strength and electron density. While the ground-state energies are always in very good agreement, the BCS energy gap is sometimes incorrect by an order of magnitude, particularly at half-filling. Finite-size effects are also briefly discussed for cases where an exact solution in the thermodynamic limit is not possible. In general, the BCS result for the energy gap is poor compared to the exact result.
We study the ground state of the attractive one-dimensional Bose-Hubbard model, and in particular the nature of the crossover between the weak interaction and strong interaction regimes for finite system sizes. Indicator properties such as the gap between the ground and first excited energy levels, and the incremental ground-state wave function overlaps are used to locate different regimes. Using mean-field theory we predict that there are two distinct crossovers connected to spontaneous symmetry breaking of the ground state. The first crossover arises in an analysis valid for large LLL with finite NNN, where LLL is the number of lattice sites and NNN is the total particle number. An alternative approach valid for large NNN with finite LLL yields a second crossover. For small system sizes we numerically investigate the model and observe that there are signatures of both crossovers. We compare with exact results from Bethe ansatz methods in several limiting cases to explore the validity for these numerical and mean-field schemes. The results indicate that for finite attractive systems there are generically three ground-state phases of the model.
Abstract Context: In the realm of quantum chemistry, the accurate prediction of electronic structure and properties of nanostructures remains a formidable challenge. Density Functional Theory (DFT) and Density Matrix Renormalization Group (DMRG) have emerged as two powerful computational methods for addressing electronic correlation effects in diverse molecular systems. We compare ground-state energies (e0), density profiles (n) and average entanglement entropies (Sˉ\bar SSˉ) in metals, insulators and at the transition from metal to insulator, in homogeneous, superlattices and harmonically confined chains described by the fermionic one-dimensional Hubbard model. While for the homogeneous systems there is a clear hierarchy between the deviations, DD%(\bar S)<D%(e_0)< \bar D%(n)D, and all the deviations decrease with the chain size; for superlattices and harmonical confinement the relation among the deviations is less trivial and strongly dependent on the superlattice structure and the confinement strength considered. For the superlattices, in general increasing the number of impurities in the unit cell represents less precision on the DFT calculations. For the confined chains, DFT performs better for metallic phases, while the highest deviations appear for the Mott and band-insulator phases. This work provides a comprehensive comparative analysis of these methodologies, shedding light on their respective strengths, limitations, and applications. Methods: The DFT calculations were performed using the standard Kohn-Sham scheme within the BALDA approach. It integrated the numerical Bethe-Ansatz (BA) solution of the Hubbard model as the homogeneous density functional within a local-density approximation (LDA) for the exchange-correlation energy. The DMRG algorithms were implemented using the ITensor library, which is based on the Matrix Product States (MPS) ansatz. The calculations were performed until the energy reaches convergence of at least 10-8.
The ground-state properties in one-dimensional Hubbard model with on-site attraction and repulsion of electrons in the presence of magnetic field h are calculated by means of the exact Bethe-ansatz formalism and the generalized self-consistent field (GSCF) approach for general electron concentrations n and arbitrary interaction strength. The ground-state properties, including the energy, the average spin (magnetization) and the kinetic energy are compared over a wide range of parameter space. The GSCF theory is in qualitative and in some cases in good quantitative agreement with the exact results. The GSCF theory at U≤0 (or U≥0) differentiates the spin (or charge) energy gap from the BCS (or antiferromagnetic) order parameter and suggests a smooth crossover from the phase with the itinerant BCS-like behavior to the Bose condensation regime of the local pairs.