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New Journal of Physics

Dynamical mean field theory of the repulsive BCS+Umodel

Kwon Park

Published Jul 15, 2009• 0 citations
Publisher

Abstract

The Gutzwiller-projected BCS Hamiltonian is a useful model for high-temperature superconductivity due to its equivalence to the Heisenberg model at half filling and a close connection to the t – J model at moderate doping. In this work, a dynamical mean field theory (DMFT) is developed for the BCS Hamiltonian with d-wave pairing subject to on-site repulsive interaction, U , which we call the BCS+ U model. The large- U limit corresponds to the Gutzwiller-projected BCS Hamiltonian. It is shown that the equivalence between the Heisenberg and the Gutzwiller-projected BCS model is a manifestation of a broader duality in the BCS+ U model: for any finite U , the local dynamics of the BCS+ U model is dual at half filling with respect to the exchange between the hopping parameter, t , and the pairing amplitude, Δ. It is explicitly demonstrated in our DMFT analysis that the real superconducting gap, determined from the sharp coherence peaks in the local density of states, shows strong renormalization from its bare value as a function of U .

DOI
10.1088/1367-2630/11/7/073027

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The Gutzwiller-projected BCS Hamiltonian is a useful model for high-temperature superconductivity due to its equivalence to the Heisenberg model at half filling and a close connection to the t – J model at moderate doping. In this work, a dynamical mean field theory (DMFT) is developed for the BCS Hamiltonian with d-wave pairing subject to on-site repulsive interaction, U , which we call the BCS+ U model. The large- U limit corresponds to the Gutzwiller-projected BCS Hamiltonian. It is shown that the equivalence between the Heisenberg and the Gutzwiller-projected BCS model is a manifestation of a broader duality in the BCS+ U model: for any finite U , the local dynamics of the BCS+ U model is dual at half filling with respect to the exchange between the hopping parameter, t , and the pairing amplitude, Δ. It is explicitly demonstrated in our DMFT analysis that the real superconducting gap, determined from the sharp coherence peaks in the local density of states, shows strong renormalization from its bare value as a function of U .
One of the most prominent problems in high temperature superconductivity is the nature of the pseudogap phase in underdoped regimes; particularly important is the role of phase fluctuations. The Gutzwiller-projected BCS Hamiltonian is a useful model for high temperature superconductivity due to an exact mapping to the Heisenberg model at half filling and generally a very close connection to the t-J model at moderate doping. We develop the dynamical mean field theory for the d-wave BCS Hamiltonian with on-site repulsive interaction, UUU, physically imposing the partial Gutzwiller projection. For results, two pseudogap energy scales are identified: one associated with the bare pairing gap for the singlet formation and the other with the local phase coherence. The real superconducting gap determined from sharp coherence peaks in the density of states shows strong renormalization from the bare value due to UUU.
A connection between quantum antiferromagnetism and high Tc{T}_{c}Tc​ superconductivity is theoretically investigated by analyzing the $t\text{\ensuremath{-}}J$ model and its relationships to the Gutzwiller-projected BCS Hamiltonian. After numerical corroboration via exact diagonalization, it is analytically shown that the ground state of the $t\text{\ensuremath{-}}J$ model at half filling [i.e., the two-dimensional (2D) antiferromagnetic Heisenberg model] is entirely equivalent to the ground state of the Gutzwiller-projected BCS Hamiltonian with strong pairing. Combined with the high wave function overlap between the ground states of the $t\text{\ensuremath{-}}J$ model and the projected BCS Hamiltonian at moderate doping, this equivalence provides strong support for the existence of superconductivity in the $t\text{\ensuremath{-}}J$ model. The relationship between the ground state of the projected BCS Hamiltonian and Anderson's resonating valence bond state (i.e., the projected BCS ground state) is discussed.
Extended t-t'-t"-J-U models in which the second-nearest-neighbor hopping (t') and third-nearest-neighbor hopping (t") are included are studied using renormalized mean field theory. The models are meant to be low-energy effective models for the Hubbard models, and hence the Heisenberg exchange integral J and Hubbard repulsion U are related by J = 4t(2)/U. The trial wavefunctions for the ground states are partially Gutzwiller projected Hartree-Fock states. The Gutzwiller projection is implemented by means of a Gutzwiller approximation, and the site double occupancy d is taken as a variational parameter. It is found that a large |t'/t| narrows the band filling range that sustains antiferromagnetism (AFM) in the ground state, enhances the d-wave singlet superconductivity (dSC) in hole overdoped systems, but suppresses the dSC in electron overdoped systems. For a system that has large |t'/t| and |t"/t'|, the superconductivity (SC) at the onset of AFM in hole doped band filling is strongly suppressed. On the excitation occurring, when an electron doped system simultaneously contains SC and AFM, the system is found to have a nodeless gap at the Fermi level. Finally, the result of this study is related to experiments on the superconducting cuprates.
We investigate a connection between the $t\text{\ensuremath{-}}J$ model and the strongly correlated Bardeen-Cooper-Schrieffer (BCS) Hamiltonian, with the effect of strong electron correlations accounted by the Gutzwiller projection. We show that in the immediate vicinity of half filling the projected two-dimensional BCS Hamiltonian with strong pairing develops an antiferromagnetically ordered ground state. This result explicitly demonstrates that antiferromagnetism in this model appears as a natural consequence of the strong Coulomb repulsion in a low doped regime. At moderate doping the ground state of the Gutzwiller-projected BCS Hamiltonian is, in finite-system studies, known to become qualitatively similar to Anderson's resonating valence bond state which, in turn, fits nicely with the properties of the $t\text{\ensuremath{-}}J$ model in that regime. Combined together, these two properties indicate that the projected BCS Hamiltonian at least qualitatively captures the essential low-energy physics of the $t\text{\ensuremath{-}}J$ model in the underdoped region.
We investigate analytically a connection between the t-J model and the strongly correlated Bardeen-Cooper-Schrieffer (BCS) Hamiltonian, with the effect of strong electron correlations accounted by the Gutzwiller projection. We show that in the immediate vicinity of half filling the projected 2D BCS Hamiltonian with strong pairing develops an antiferromagnetically (AF) ordered ground state. This result explicitly demonstrates that antiferromagnetism in this model appears as a natural consequence of the strong Coulomb repulsion in a low doped regime. At moderate doping the ground state of the Gutzwiller-projected BCS Hamiltonian becomes qualitatively similar to Anderson's resonating valence bond state which is known to fit nicely the properties of the t-J model in this regime. These two properties taken together indicate that the projected BCS Hamiltonian captures the essential low-energy physics of the t-J model in the whole underdoped region.
A close connection between antiferromagnetism and superconductivity is at the core of high-temperature superconductivity. Here, we put forward the projected BCS theory for the unification of antiferromagnetism and strongly correlated superconductivity. Specifically, it is shown that, with the ddd-wave pairing symmetry, the projected BCS theory can provide excellent trial states at general doping for the exact ground states of the $t\text{\ensuremath{-}}J$ model in the square lattice, generating a unified theory of high-temperature superconductivity as a continuous function of hole concentration. A key to the success of the projected BCS theory is an accurate treatment of the strong correlation between Cooper pairs, which not only causes the breakdown of superconductivity itself at half filling but also defines the precise nature of strongly correlated superconductivity at moderate doping. Also, via the proper incorporation of particle number fluctuations, the projected BCS theory allows direct computation of the superconducting order parameter, shedding important light on the pseudogap phenomenon.