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arXiv:0704.1618 · Physical Review B

Renormalization group approach for highly anisotropic two-dimensional fermion systems: Application to coupled Hubbard chains

S. Moukouri

Published Jun 19, 2008• 0 citations
arXivHTMLPDF

Abstract

I apply a two-step density-matrix renormalization group method to the anisotropic two-dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as $24\ifmmode\times\else\texttimes\fi{}25$. I then apply the method to the interacting case. I find that for strong Hubbard interaction, the ground state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and is not enhanced by frustration. For weak Hubbard couplings, due to interchain interaction, interchain nonlocal singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state. The pairing mechanism is reminiscent of the Kohn-Luttinger mechanism.

DOI
10.1103/physrevb.77.235116

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I apply a two-step density-matrix renormalization group method to the anisotropic two-dimensional Hubbard model. As a prelude to this study, I compare the numerical results to the exact one for the tight-binding model. I find a ground-state energy which agrees with the exact value up to four digits for systems as large as 24\ifmmode×\else\texttimes\fi2524\ifmmode\times\else\texttimes\fi{}2524\ifmmode×\else\texttimes\fi25. I then apply the method to the interacting case. I find that for strong Hubbard interaction, the ground state is dominated by magnetic correlations. These correlations are robust even in the presence of strong frustration. Interchain pair tunneling is negligible in the singlet and triplet channels and is not enhanced by frustration. For weak Hubbard couplings, due to interchain interaction, interchain nonlocal singlet pair tunneling is enhanced and magnetic correlations are strongly reduced. This suggests a possible superconductive ground state. The pairing mechanism is reminiscent of the Kohn-Luttinger mechanism.
The Kato-Bloch perturbation formalism is used to present a density-matrix renormalization-group method for strongly anisotropic two-dimensional systems. This method is used to study Heisenberg chains weakly coupled by the transverse couplings ${J}_{\ensuremath{\perp}}$ and Jd{J}_{d}Jd​ (along the diagonals). An extensive comparison of the renormalization group and quantum Monte Carlo results for parameters where the simulations by the latter method are possible shows a very good agreement between the two methods. It is found, by analyzing ground state energies and spin-spin correlation functions, that there is a transition between two ordered magnetic states. When ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\lesssim}0.5$, the ground state displays a N\'eel order. When ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\gtrsim}0.5$, a collinear magnetic ground state in which interchain spin correlations are ferromagnetic becomes stable. In the vicinity of the transition point, ${J}_{d}∕{J}_{\ensuremath{\perp}}\ensuremath{\approx}0.5$, the ground state is disordered. But, the nature of this disordered ground state is unclear. While the numerical data seem to show that the chains are disconnected, the possibility of a genuine disordered two-dimensional state, hidden by finite size effects, cannot be excluded.
The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.
Abstract The study of ground-state properties of the Fermi-Hubbard model is a long-lasting task in the research of strongly correlated systems, and is crucial for the understanding of notable quantum phenomena including superconductivity and magnetism. Owing to the exponentially growing complexity of the system, a quantitative analysis usually demands high computational cost and is restricted to small samples, especially in two or higher dimensions. Here, we introduce a variational method in the frame of fermionic Gaussian states, and obtain the ground states of one- and two-dimensional attractive Hubbard models via imaginary-time evolution. We calculate the total energy and benchmark the results in a wide range of interaction strength and filling factor with those obtained via exact two-body results, the density matrix renormalization group based on matrix product states (MPS), and projector Quantum Monte Carlo (QMC) method. For both 1D and 2D cases, the Gaussian variational method presents accurate results for total energy with a maximum systematic error ∼ 4 % in the intermediate interaction region. The accuracy of these results has negligible dependence on the system size. We further calculate the double occupancy and find excellent agreement with MPS and QMC, as well as the experimental results of cold quantum gases in optical lattices. The results suggest that the Gaussian pairing state is a good approximation to the ground states of attractive Hubbard model, in particular in the strong and weak coupling limits. Moreover, we generalize the method to the attractive Hubbard model with a finite spin-polarization, which can be mapped to the repulsive interaction case via particle-hole transformation, and obtain accurate results of ground state energy and double occupancy. Our work demonstrates the ability of the Gaussian variational method to extract ground state properties of strongly correlated many-body systems with negligible computational cost, especially of large size and in higher dimensions.
The recent discovery of superconductivity under high pressure in the ladder compound BaFe2S3{\mathrm{BaFe}}_{2}{\mathrm{S}}_{3}BaFe2​S3​ has opened a new field of research in iron-based superconductors with focus on quasi-one-dimensional geometries. In this publication, using the density matrix renormalization group technique, we study a two-orbital Hubbard model defined in one-dimensional chains. Our main result is the presence of hole binding tendencies at intermediate Hubbard UUU repulsion and robust Hund coupling JH/U=0.25{J}_{H}/U=0.25JH​/U=0.25. Binding does not occur either in weak coupling or at very strong coupling. The pair-pair correlations that are dominant near half-filling, or of similar strength as the charge and spin correlation channels, involve hole-pair operators that are spin singlets, use nearest-neighbor sites, and employ different orbitals for each hole. The Hund coupling strength, presence of robust magnetic moments, and antiferromagnetic correlations among them are important for the binding tendencies found here.
We study the anisotropic two-dimensional Hubbard model at and near half filling within a functional renormalization group method, focusing on the structure of momentum-dependent correlations which grow strongly upon approaching a critical temperature from above. We find that a finite nearest-neighbor interchain hopping is not sufficient to introduce a substantial momentum dependence of single-particle properties along the Fermi surface. However, when a sufficiently large second-nearest neighbor inter-chain hopping is introduced, the system is frustrated and we observe the appearance of so-called "hot spots", specific points on the Fermi surface around which scattering becomes particularly strong. We compare our results with other studies on quasi-one-dimensional systems.
The exponent ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ and pair-field-correlation functions are calculated for a two-chain Hubbard model with an exact numerical diagonalization method. Our calculations clearly indicate that a superconducting phase appears for a wide range of Coulomb strength, where ${\mathit{K}}_{\mathrm{\ensuremath{\rho}}}$ and pair-field correlation functions are enhanced compared to the noninteracting system. The interchain pairing appears predominantly and thus our mechanism predicts an anisotropic superconductivity.