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arXiv:cond-mat/9704076 · Journal of Physics Condensed Matter

The Hubbard model at intermediate coupling: renormalization of the interaction strength

V. Janiš

Published Apr 6, 1998• 26 citations
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Abstract

We analyse the behaviour of correlated electrons described by Hubbard-like models at intermediate coupling. We argue that with increasing interaction a pole in a generic two-particle Green function is approached. The pole signals condensation of electron-hole pairs and a metal-insulator transition at half-filling. The two-particle singularity calls for a sophisticated renormalization of the interaction strength. A self-consistent diagrammatic technique with renormalized two-particle Green functions is developed. The theory is based on a linked-cluster expansion for the thermodynamic potential with electron-electron interaction as the propagator. The simplest theory with full vertex renormalization, summing self-consistently multiple scatterings from two electron-hole channels, is proposed. We obtain an approximation with a generating functional in closed form enabling us to handle appropriately singularities in two-particle Green functions. The approximation is shown to be asymptotically exact in an external magnetic field close to the fully polarized ferromagnetic state at half-filling and zero temperature.

DOI
10.1088/0953-8984/10/13/010

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Physical review. B, Condensed matter•Jan 15, 1997•3 citations
We analyse the behaviour of correlated electrons described by Hubbard-like models at intermediate coupling. We argue that with increasing interaction a pole in a generic two-particle Green function is approached. The pole signals condensation of electron-hole pairs and a metal-insulator transition at half-filling. The two-particle singularity calls for a sophisticated renormalization of the interaction strength. A self-consistent diagrammatic technique with renormalized two-particle Green functions is developed. The theory is based on a linked-cluster expansion for the thermodynamic potential with electron-electron interaction as the propagator. The simplest theory with full vertex renormalization, summing self-consistently multiple scatterings from two electron-hole channels, is proposed. We obtain an approximation with a generating functional in closed form enabling us to handle appropriately singularities in two-particle Green functions. The approximation is shown to be asymptotically exact in an external magnetic field close to the fully polarized ferromagnetic state at half-filling and zero temperature.
Single-band Hubbard model at criticality of the metal-insulator transition is studied using approximations derived from parquet theory. It is argued that only the electron-hole and interaction two-particle channels in the parquet algebra are relevant. A scheme is proposed how to reduce the parquet equations to a manageable form the complexity of which is comparable with single-channel approximations such as the renormalized RPA. The newly derived approximation, however, contains dynamical vertex corrections, remains self-consistent at the two-particle level and allows only for integrable singularities. A qualitatively new approach for studying two-particle singularities at zero temperature is obtained.
We develop a perturbation expansion in the intersite hopping around the atomic limit of the Hubbard model. It is valid for arbitrary finite temperatures and interaction strengths. Diagrammatic rules that determine the grand-canonical potential and the Green's functions are derived. They reduce the calculation of any finite-order contribution to simple algebra. This opens the way for series extrapolations from computer-aided high-finite-order evaluations. Discrepancies in earlier expansions around the atomic limit are clarified. The present expansion scheme involves only connected diagrams with unrestricted lattice sums. This allows one to perform a vertex renormalization as for the linked-cluster expansion of the Ising model. The renormalized perturbation expansion can be used to construct self-consistent approximations which are automatically exact in the atomic limit. In the limit of high lattice dimensions, only fully two-particle reducible embeddings of diagrams on the lattice contribute. The single-particle properties of the infinite-dimensional Hubbard model reduce to those of independent tight-binding fermions hopping between dressed sites.
We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.
It is shown that the Hubbard chain in a magnetic field of arbitrary strength H can be treated as a liquid of interacting pseudoparticles which at T=0 has only forward scattering. The excitation spectra can be understood in terms of particle-hole processes in the pseudoparticle renormalized bands. Within this framework, we are able to study in detail the low-temperature thermodynamics of the Hubbard model. Explicit expressions for the specific heat cv{\mathit{c}}_{\mathit{v}}cv​ for a nearly half-filled band and for values of the magnetic field close to the ferromagnetic saturated state are derived. The renormalized charge and spin masses which regulate the crossover regime to the exponential behavior of several physical quantities close to the metal-insulator and ferromagnetic transitions, respectively, are calculated.
The equation in the ladder approximation was derived for the two-particle Green function of the itinerant electrons with the Heisenberg ferromagnetic and antiferromagnetic interaction. The general form of its solution was found. The existence of the pole in this solution is related to the condensation of the bound pairs in the corresponding spin state. The condition for the pole determines the critical temperature T c . The effective interaction between the mobile holes due to the mobile hole-localized spin coupling in the two-band Hubbard model was also studied.
An analytic approach is presented of electronic one-particle spectra of the one-band Hubbard model at half filling in the antiferromagnetic phase. Starting from the strong-coupling regime U\ensuremath{\gg}:t, a projection technique is used to set up self-consistent coupled equations for the electron Green function, which are valid down to values U\ensuremath{\approx}t. The self-consistent equation for the hole propagator is a direct generalization of the one found from the t-J model. This gives further support to the ``string'' picture, where propagation of holes creates strings of overturned spins with which the holes interact. In the present work hopping of holes (or electrons) with up spin on the down sublattice is also taken into acount, as well as transitions between the lower and upper Hubbard bands. These are shown to change significantly the incoherent part of the t-J model spectra, by smearing out the shake-off peaks, reminiscent of higher bound string states due to multispin scattering. Coherent (quasiparticle) peaks exist at the band edges, on both sides of the insulating gap. We show that with decreasing U the quasiparticle concept loses its meaning for wave vectors at the center of the magnetic Brillouin zone (MBZ). For large values of U the dispersion of the quasiparticle is found to scale with its band width, which is of order J. Extrema are always found at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The weight of the quasiparticle at this k value decreases logarithmically with increasing U. In the strong-coupling limit the spectrum tends to be symmetric, i.e., to become an even function of the frequency around the chemical potential, for any wave vector. For small values of U the dispersion at the edge of the MBZ flattens away, as expected when approaching the noninteracting limit. The spectral function in this regime, for wave vectors away from the edge of the MBZ, is concentrated mainly on one side of the chemical potential.