The exact diagonalization of the extended Hubbard Model: Six Electrons at Half-Filling (original) (raw)

2000

The application of enhanced quasi-sparse eigenvector methods (EQSE) to the Hubbard model is attempted. The ground state energy for the 4x4 Hubbard model is calculated with a relatively small set of basis vectors. The results agree to high precision with the exact answer. For the 8x8 case, exact answers are not available but a simple first order correction to the quasi-sparse eigenvector (QSE) result is presented.

EQSE diagonalization of the Hubbard model

Nuclear physics, 2000

Organization of the Hilbert space for exact diagonalization of Hubbard model

Computer Physics Communications, 2015

We present an alternative scheme to the widely used method of representing the basis of one-band Hubbard model through the relation I = I ↑ +2 M I ↓ given by H. Q. Lin and J. E. Gubernatis [Comput. Phys. 7, 400 (1993)], where I ↑ , I ↓ and I are the integer equivalents of binary representations of occupation patterns of spin up, spin down and both spin up and spin down electrons respectively, with M being the number of sites. We compute and store only I ↑ or I ↓ at a time to generate the full Hamiltonian matrix. The non-diagonal part of the Hamiltonian matrix given as I ↓ ⊗ H ↑ ⊕ H ↓ ⊗ I ↑ is generated using a bottom-up approach by computing the small matrices H ↑ (spin up hopping Hamiltonian) and H ↓ (spin down hopping Hamiltonian) and then forming the tensor product with respective identity matrices I ↓ and I ↑ , thereby saving significant computation time and memory. We find that the total CPU time to generate the non-diagonal part of the Hamiltonian matrix using the new one spin configuration basis scheme is reduced by about an order of magnitude as compared to the two spin configuration basis scheme. The present scheme is shown to be inherently parallelizable. Its application to translationally invariant systems, computation of Green's functions and in impurity solver part of DMFT procedure is discussed and its extention to other models is also pointed out.

Exact Diagonalization Approach for the infinite D Hubbard Model

1993

We present a powerful method for calculating the thermodynamic properties of the Hubbard model in infinite dimensions, using an exact diagonalization of an Anderson model with a finite number of sites. At finite temperatures, the explicit diagonalization of the Anderson Hamiltonian allows the calculation of Green's functions with a resolution far superior to that of Quantum Monte Carlo calculations. At zero temperature, the Lanczòs method is used and yields the essentially exact zero-temperature solution of the model, except in a region of very small frequencies. Numerical results for the halffilled case in the paramagnetic phase (quasi-particle weight, self-energy, and also real-frequency spectral densities) are presented.

Numerical study of the two-dimensional Hubbard model

Physical review. B, Condensed matter, 1989

We report on a numerical study of the two-dimensional Hubbard model and describe two new al- gorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previ- ously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimerisional Hubbard model with half-and quarter-filled bands. Our re- sults support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occu- pation along with an upper bound to the spin-wave spectrum are given. We find evidence for an at- tractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.

Exact Diagonalization Study of an Extended Hubbard Model for a Cubic Cluster at Quarter Filling

Acta Physica Polonica A, 2017

In the paper the thermodynamics of a cubic cluster with 8 sites at quarter filling is characterized by means of exact diagonalization technique. Particular emphasis is put on the behaviour of such response functions as specific heat and magnetic susceptibility. The system is modelled with extended Hubbard model which includes electron hopping between both first and second nearest neighbours as well as coulombic interactions, both on-site and between nearest-neighbour sites. The importance of hopping between second nearest neighbours and coulombic interactions between nearest neighbours for the temperature dependences of thermodynamic response functions is analysed. In particular, the predictions of Schottky model are compared with the calculations based on the full energy spectrum.

An introduction to the Hubbard model

The Hubbard model is very important for the study of of magnetic phenomena and strongly correlated electron systems. This work serves as an introduction to the Hubbard model and a presentation of the elements necessary to reach it. Here it is applied to a simple case to see how you work with it.

Solutions of the Two-Dimensional Hubbard Model: Benchmarks and Results from a Wide Range of Numerical Algorithms

Physical Review X, 2015

Numerical results for ground state and excited state properties (energies, double occupancies, and Matsubara-axis self energies) of the single-orbital Hubbard model on a two-dimensional square lattice are presented, in order to provide an assessment of our ability to compute accurate results in the thermodynamic limit. Many methods are employed, including auxiliary field quantum Monte Carlo, bare and bold-line diagrammatic Monte Carlo, method of dual fermions, density matrix embedding theory, density matrix renormalization group, dynamical cluster approximation, diffusion Monte Carlo within a fixed node approximation, unrestricted coupled cluster theory, and multireference projected Hartree-Fock. Comparison of results obtained by different methods allows for the identification of uncertainties and systematic errors. The importance of extrapolation to converged thermodynamic limit values is emphasized. Cases where agreement between different methods is obtained establish benchmark results that may be useful in the validation of new approaches and the improvement of existing methods. arXiv:1505.02290v1 [cond-mat.str-el] 9 May 2015

Numerical studies of the Hubbard model

Nuclear Physics B - Proceedings Supplements, 1991

Numerical studies of the two-dimensional Hubbard model have shown that it exhibits the basic phenomena seen in the cuprate materials. At half-filling one finds an antiferromagnetic Mott-Hubbard groundstate. When it is doped, a pseudogap appears and at low temperature d-wave pairing and striped states are seen. In addition, there is a delicate balance between these various phases. Here we review evidence for this and then discuss what numerical studies tell us about the structure of the interaction which is responsible for pairing in this model. stripes, pseudogap behavior, and d x 2 −y 2 pairing. In addition, the numerical studies have shown how delicately balanced these models are between nearly degenerate phases. Doping away from half-filling can tip the balance from antiferromagnetism to a striped state in which half-filled domain walls separate π-phase-shifted antiferromagnetic regions. Altering the next-near-neighbor hopping t ′ or the strength of U can favor d x 2 −y 2 pairing correlations over stripes. This delicate balance is also seen in the different results obtained using different numerical techniques for the same model. For example, density matrix renormalization group (DMRG) calculations for doped 8-leg t-J ladders find evidence for a striped ground state. [12] However, variational and Green's function Monte Carlo calculations for the doped t-J lattice, pioneered by Sorella and co-workers, [23, 24] find groundstates characterized by d x 2 −y 2 superconducting order with only weak signs of stripes. Similarly, DMRG calculations for doped 6-leg Hubbard ladders [14] find stripes when the ratio of U to the near-neighbor hopping t is greater than 3, while various cluster calculations [27, 30-33] find evidence that antiferromagnetism and d x 2 −y 2 superconductivity compete in this same parameter regime. These techniques represent present day state-of-the-art numerical approaches. The fact that they can give different results may reflect the influence of different boundary conditions or different aspect ratios of the lattices that were studied. The n-leg open boundary conditions in the DMRG calculations can favor stripe formation. Alternately, the cluster lattice sizes and boundary conditions can frustrate stripe formation. It is also possible that these differences reflect subtle numerical biases in the different numerical methods. Nevertheless, these results taken together show that both the striped and the d x 2 −y 2 superconducting phases are nearly degenerate low energy states of the doped system. Determinantal quantum Monte Carlo calculations [21] as well as various cluster calculations show that the underdoped Hubbard model also exhibits pseudogap phenomena. [27-32] The remarkable similarity of this behavior to the range of phenomena observed in the cuprates provides strong evidence that the Hubbard and t-J models indeed contain a significant amount of the essential physics of the problem. [34]

Properties of the half-filled Hubbard model investigated by the strong coupling diagram technique

International Journal of Modern Physics B

The equation for the electron Green's function of the fermionic Hubbard model, derived using the strong coupling diagram technique, is solved self-consistently for the near-neighbor form of the kinetic energy and for half-filling. In this case the Mott transition occurs at the Hubbard repulsion U c ≈ 6.96t, where t is the hopping constant. The calculated spectral functions, density of states and momentum distribution are compared with results of Monte Carlo simulations. A satisfactory agreement was found for U > U c and for temperatures, at which magnetic ordering and spin correlations are suppressed. For U < U c and lower temperatures the theory describes qualitatively correctly positions and widths of spectral continua, variations of spectral shapes and occupation numbers with changing wave vector and repulsion. The locations of spectral maxima turn out to be close to the positions of δ-function peaks in the Hubbard-I approximation.