Real-space renormalization-group study of the Hubbard model: A modified scheme (original) (raw)
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Real-space renormalization group study of the Hubbard model on a non-bipartite lattice
International Journal of Molecular Sciences, 2002
We present the real-space block renormalization group equations for fermion systems described by a Hubbard Hamiltonian on a triangular lattice with hexagonal blocks. The conditions that keep the equations from proliferation of the couplings are derived. Computational results are presented including the occurrence of a first-order metal-insulator transition at the critical value of U/t ≈ 12.5.
Journal of Physics: Condensed Matter, 1999
We have studied the Hubbard model with bond-charge interaction on a triangular lattice for a half-filled band. At the point of particle-hole symmetry the model could be analyzed in detail in two opposite regimes of the parameter space. Using a real space renormalization group we calculate the ground state energy and the local moment over the whole parameter space. The RG results obey the exact results in the respective limits. In the intermediate region of the parameter space the RG results clearly show the effects of the non-bipartite geometry of the lattice as well as the absence of symmetry in the reversal of the sign of the hopping matrix element.
Solid State Communications, 1994
We present a solution of the infinite dimensional Hubbard model obtained by the numerical renormalization group method. This method gives directly the dynamical response in real frequency at zero temperature, and also gives reliable estimation of static quantities such as the electronic specific heat near absolute zero. We note technical points which need due care in applying this method.
Physical Review B, 2008
We analyze the particle-hole symmetric two-dimensional Hubbard model on a square lattice starting from weak-to-moderate couplings by means of the field-theoretical renormalization group (RG) approach up to two-loop order. This method is essential in order to evaluate the effect of the momentum-resolved anomalous dimension η(p) which arises in the normal phase of this model on the corresponding low-energy single-particle excitations. As a result, we find important indications pointing to the existence of a non-Fermi liquid (NFL) regime at temperature T → 0 displaying a truncated Fermi surface (FS) for a doping range exactly in between the well-known antiferromagnetic insulating and the d x 2 −y 2 -wave singlet superconducting phases. This NFL evolves as a function of doping into a correlated metal with a large FS before the d x 2 −y 2 -wave pairing susceptibility finally produces the dominant instability in the low-energy limit.
Physical Review B, 2007
We analyze the particle-hole symmetric two-dimensional Hubbard model on a square lattice starting from weak-to-moderate couplings by means of the field-theoretical renormalization group (RG) approach up to two-loop order. This method is essential in order to evaluate the effect of the momentum-resolved anomalous dimension eta(textbfp)\eta(\textbf{p})eta(textbfp) which arises in the normal phase of this model on the corresponding low-energy single-particle excitations. As a result, we find important indications pointing to the existence of a non-Fermi liquid (NFL) regime at temperature Tto0T\to 0Tto0 displaying a truncated Fermi surface (FS) for a doping range exactly in between the well-known antiferromagnetic insulating and the dx2−y2d_{x^2-y^2}dx2−y2-wave singlet superconducting phases. This NFL evolves as a function of doping into a correlated metal with a large FS before the dx2−y2d_{x^2-y^2}dx2−y2-wave pairing susceptibility finally produces the dominant instability in the low-energy limit.
Physical Review B, 2003
Models with range-free frustrated Ising spin interaction and additional Hubbard interaction are treated exactly by means of the discrete time slicing method of Grassmann field theory. Critical and tricritical points, spin-and charge correlations, and the fermion propagator, are derived as a function of temperature, chemical potential µ, of the Hubbard coupling U , and of the spin glass energy J. U is allowed to be either repulsive (U > 0) or attractive (U < 0). Cuts through the multi-dimensional phase diagram are obtained. Analytical and numerical evaluations take important replica symmetry breaking (RSB)-effects into account. Results for the ordered phase are given at least in one-step approximation (1RSB), for T = 0 we report the first two-, three-, and four-step calculations (4RSB) for fermionic spin glasses. The use of exact relations and invariances under RSB together with 2RSB-calculations for all fillings and 4RSB-solutions for half filling allow to model exact solutions by interpolation. For T = 0, our numerical results provide strong evidence that the exact spin glass pseudogap obeys ρ(E) = c1|E − EF | for energies close to the Fermi level with c1 ≈ 0.13. Rapid convergence of ρ ′ (EF) under increasing order of RSB is observed and ρ ′′ (E) is evaluated to estimate subleading powers. Over a wide range of the pseudogap and after a small transient regime ρ(E) regains a linear shape with larger slope and a small S-like perturbation. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of two-dimensional localized disordered fermionic systems. Beyond half filling we obtain a ν − 1 ∼ (µ − U) 2 , µ ≥ U, dependence of the fermion filling factor ν. We find a half filling transition between a phase for U > µ, where the Fermi level lies inside the Hubbard gap, into a phase where µ(> U) is located at the center of the upper spin glass pseudogap (SG-gap). For µ > U the Hubbard gap combines with the lower one of two SG-gaps (phase I), while for µ < U it joins the sole SG-gap which exists in this half-filling regime (phase II). Shoulders of the combined gaps are shaped by RSB due to spin glass order. We predict scaling behaviour at the half filling transition which becomes continuous due to ∞RSB. Implications of the half-filling transition between the deeper insulating phase II and phase I for the eventual delocalization by additional hopping processes in itinerant model extensions are discussed. Possible metal-insulator transition scenarios are described. Generalizations to random Hubbard coupling and alloy models as well as frustrated magnetic interactions with ferro-or antiferromagnetic components are also considered separately.
Superconductivity in the attractive Hubbard model: functional renormalization group analysis
New Journal of Physics, 2008
We present a functional renormalization group analysis of superconductivity in the ground state of the attractive Hubbard model on a square lattice. Spontaneous symmetry breaking is treated in a purely fermionic setting via anomalous propagators and anomalous effective interactions. In addition to the anomalous interactions arising already in the reduced BCS model, effective interactions with three incoming legs and one outgoing leg (and vice versa) occur. We accomplish their integration into the usual diagrammatic formalism by introducing a Nambu matrix for the effective interactions. From a randomphase approximation generalized through use of this matrix we conclude that the impact of the 3 + 1 effective interactions is limited, especially considering the effective interactions which are important for the determination of the order parameter. The exact hierarchy of flow equations for one-particle irreducible vertex functions is truncated on the two-particle level, with higher-order selfenergy corrections included in a scheme proposed by Katanin (2004 Phys. Rev. B 70 115109). Using a parametrization of effective interactions by patches in momentum space, the flow equations can be integrated numerically to the lowest scales without encountering divergences. Momentum-shell as well as interaction-flow cutoff functions are used, including a small external field or a 3 Author to whom any correspondence should be addressed.
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]
Physical Review B, 2008
We investigate the Hubbard model on the triangular lattice at half-filling using the dynamical cluster approximation (DCA) and dual fermion (DF) methods in combination with continuous-time quantum Monte carlo (CT QMC) and semiclassical approximation (SCA) methods. We study the one-particle properties and nearest-neighbor spin correlations using the DCA method. We calculate the spectral functions using the CT QMC and SCA methods. The spectral function in the SCA and obtained by analytic continuation of the Pade approximation in CT QMC are in good agreement. We determine the metal-insulator transition (MIT) and the hysteresis associated with a first-order transition in the double occupancy and nearest-neighbor spin correlation functions as a function of temperature. As a further check, we employ the DF method and discuss the advantages and limitation of the dynamical mean field theory (DMFT), DCA and recently developed DF methods by comparing Green's functions. We find an enhancement of antiferromagnetic (AF) correlations and provide evidence for magnetically ordered phases by calculating the spin susceptibility.
The density-matrix renormalization group is employed to investigate a harmonically-trapped imbalanced Fermi condensate based on a one-dimensional attractive Hubbard model. The obtained density profile shows a flattened population difference of spin-up and spin-down components at the center of the trap, and exhibits phase separation between the condensate and unpaired majority atoms for a certain range of the interaction and population imbalance P. The two-particle density matrix reveals that the sign of the order parameter changes periodically, demonstrating the realization of the Fulde-Ferrell-Larkin-Ovchinnikov phase. The minority spin atoms contribute to the quasi-condensate up to at least P ≃ 0.8. Possible experimental situations to test our predictions are discussed.