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Physical review. B, Condensed matter

Large-nlimit of the Heisenberg-Hubbard model: Implications for high- Tc superconductors

Ian K. Affleck, John Bradley Marston

Published Mar 1, 1988• 1,308 citations
Publisher

Abstract

The Heisenberg-Hubbard model is solved in the large-n limit to gain insight into its relevance to the coppper-oxide materials. A gapless disordered ground state is found for the Heisenberg model. Electronically induced orthorhombic-tetragonal symmetry breaking is observed as a function of doping, with the greatest tendency towards superconductivity occurring in the orthorhombic phase.

DOI
10.1103/physrevb.37.3774

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The Heisenberg-Hubbard model is solved in the large-n limit to gain insight into its relevance to the coppper-oxide materials. A gapless disordered ground state is found for the Heisenberg model. Electronically induced orthorhombic-tetragonal symmetry breaking is observed as a function of doping, with the greatest tendency towards superconductivity occurring in the orthorhombic phase.
To gain insight into the behavior of the Hubbard model, we define a SU(n)\mathrm{SU}(n)SU(n) invariant generalization of the Hubbard-Heisenberg model and, in the large-nnn limit, solve it in one dimension and in two dimensions on a square lattice. In one dimension the ground state is completely dimerized near half filling. We show that this behavior agrees with a renormalization-group solution of the one-dimensional SU(n)\mathrm{SU}(n)SU(n) Hubbard model. In two spatial dimensions we find several different ground states depending on the size of the hopping term ttt, the doping $\ensuremath{\delta}$, and the biquadratic spin interaction $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. In particular, the undimerized "flux" or "s+ids+ids+id" phase is the ground state at half filling for sufficiently large ttt or $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$. We study the electronic and spin excitations of the various phases and comment on the relevance of the large-nnn problem to the high-Tc{T}_{c}Tc​ superconductors.
We carry out a systematic study of collective spin and charge excitations for the canonical single-band Hubbard, t–J–U, and t–J models of high-temperature copper-oxide superconductors, both on electron- and hole-doped side of the phase diagram. A~recently developed variational wave function approach, combined with the expansion in the inverse number of fermionic flavors, is employed. All three models exhibit a substantial electron–hole asymmetry of magnetic excitations, with a robust paramagnon emerging for hole-doping, in agreement with available resonant inelastic X-ray scattering data for the cuprates. The t–J model yields additional high-energy peak in the magnetic spectrum that is not unambiguously identified in spectroscopy. For all considered Hamiltonians, the dynamical charge susceptibility contains a coherent mode for both hole- and electron doping, with overall bandwidth renormalization controlled by the on-site Coulomb repulsion. Away from the strong-coupling limit, the antiferromagnetic ordering tendency is more pronounced on the electron-doped side of the phase diagram.
Recently developed numerical methods have enabled the explicit construction of the superconducting state of the Hubbard model of strongly correlated electrons in parameter regimes where the model also exhibits a pseudogap and a Mott insulating phase. d(x(2)-y(2)) symmetry superconductivity is found to occur in proximity to the Mott insulator, but separated from it by a pseudogapped nonsuperconducting phase. The superconducting transition temperature and order parameter amplitude are found to be maximal at the onset of the normal-state pseudogap. The emergence of superconductivity from the normal state pseudogap leads to a decrease in the excitation gap. All of these features are consistent with the observed behavior of the copper-oxide superconductors.
A systematic study is made of the t-J model as a working model for copper oxide high-Tc superconductors. The main focus is on the near-half- filling region (low hole concentrations) relevant to these materials. The theory of the magnetic polaron, which is a charge carrier traveling in an antiferromagnetic matrix, and the theory of antiferromagnetic ordering are discussed in a unified framework. The spin liquid state beyond the antiferromagnetic phase is examined. The Hamiltonian parameters – hole concentration phase diagram for the model is described and compared with that for the Hubbard model in the strong correlation limit. Two extensions of the t-J model, the t-t'-J model and the three-center interaction model, are discussed. Mostly analityc strong correlation techniques are employed in this review.
The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt\ensuremath{'} model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard's X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt\ensuremath{'} models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\ensuremath{\alpha}}}^{2}$F in general: The full-symmetric component ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ is in the t model somewhat, in the tt\ensuremath{'} model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\ensuremath{\alpha}}}^{2}$Fi{\mathit{F}}_{\mathit{i}}Fi​ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\ensuremath{\alpha}}}^{2}$F1{\mathit{F}}_{1}F1​ near half-filling.Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt\ensuremath{'} models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compounds. \textcopyright{} 1996 The American Physical Society.
The two-dimensional extended Hubbard model that includes a nearest-neighbor Heisenberg interaction is studied using a mean-field theory where quasiparticles are defined by an U(8) group of canonical transformations permitting both broken gauge, spin, and sublattice symmetry. The theory is further extended to incorporate a possible twist in the spin-quantization axis, so that the competition between superconductivity, charge density waves, and N\'eel and spiral antiferromagnetic order can be monitored within one single theory. Our results for positive Hubbard U and Heisenberg exchange J suggest that antiferromagnetic ordering dominates close to half-filling, while spiral states and d-wave superconductivity compete when doping is introduced. For moderate values of J, we find a phase diagram where a phase transition occurs from an antiferromagnet to a d-wave superconductor as doping is increased. A narrow region of (s+id)-wave superconductor is found for some values of J and U.