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Proceedings of the Estonian Academy of Sciences Physics Mathematics

RENORMALIZATION OF ELEMENTARY EXCITATIONS OF THE t-J MODELWITH DOPING

Alexei Sherman, Michael Schreiber

Published Feb 14, 1995• 0 citations
Publisher

Abstract

Hole and magnon spectral functions of the 2D ¢-J model are self-consistently calculated from a set of Dyson's equations for the hole concentrations z < 0.3.For J < ¢, itis shown that near z = 0.05 the hole spectrum changes drastically, acquiring features of a weakly correlated spectrum.This change is accompanied by the entering of the chemical potential u into a sharp maximum of the density of states.With z increasing the maximum moves together with u upto z =~0.15 and then lags behind.This concentration is also distinctive in that the Fermi surface becomes a simply connected surface and the overdamped magnons which appear near z = 0.02 occupy the widest region in k space.Besides, a limiting frequency of usual magnons passes through a minimum here.Significant deviations from the normal Fermi liquid, which partly resemble the marginal Fermi liquid, are observed for x > 0.05.

DOI
10.3176/phys.math.1995.2/3.21

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Hole and magnon spectral functions of the 2D ¢-J model are self-consistently calculated from a set of Dyson's equations for the hole concentrations z < 0.3.For J < ¢, itis shown that near z = 0.05 the hole spectrum changes drastically, acquiring features of a weakly correlated spectrum.This change is accompanied by the entering of the chemical potential u into a sharp maximum of the density of states.With z increasing the maximum moves together with u upto z =~0.15 and then lags behind.This concentration is also distinctive in that the Fermi surface becomes a simply connected surface and the overdamped magnons which appear near z = 0.02 occupy the widest region in k space.Besides, a limiting frequency of usual magnons passes through a minimum here.Significant deviations from the normal Fermi liquid, which partly resemble the marginal Fermi liquid, are observed for x > 0.05.
The hole dynamics in a quantum antiferromagnetic background for a small doping concentration is studied within the t-J model description. The noncrossing approximation for the hole self-energy is used. The correct quasiparticle pole structure (without artificial broadening) in the spectral density function is employed. Dyson's equation for the hole Green's function is solved numerically by introducing a lattice in momentum space. The evolution of the density of states and the momentum-distribution function for holes is investigated as a function of hole-magnon coupling constant t at fixed J. The momentum distribution function shows a sharp drop at the Fermi surface for several different hole-magnon coupling constants. This indicates the validity of the Fermi liquid description for the t-J model in the low doping concentration. The volume (surface) enclosed by the Fermi surface (curve) is found to be invariant for all the coupling strengths we have studied. The incoherent spectrum below the chemical potential in the density of states becomes more significant for stronger interaction constants. These features numerically verify Luttinger's theorem for a small density of doped holes in the t-J model. The optical conductivity is computed using the lowest-order diagrams. As the hole-magnon coupling constant t increases, absorption becomes stronger at \ensuremath{\omega}\ensuremath{\simeq}2J, the characteristic frequency for spin waves. This feature may be associated with the experimentally observed midinfrared absorption band in underdoped cuprates.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
The hole spectral function of the t-J model on a two-dimensional semi-infinite lattice is calculated using the spin-wave and non-crossing approximations. In the case of small hole concentrations and strong correlations, t >> J, several near-boundary site rows appear to be depleted of holes. The reason is a deformation of the magnon cloud, which surrounds the hole, near the boundary. The hole depletion leads to a more complicated spectral function in the boundary row in comparison with its bulk shape. This is connected with a maximum related to the underlying row, which reveals itself in the boundary spectrum.
The single-particle spectral functions A(k,\ensuremath{\omega}) and self-energies \ensuremath{\Sigma}(k,\ensuremath{\omega}) are calculated within the t-J model using the finite-temperature Lanczos method for small systems. A remarkable asymmetry between the electron and hole part is found. The hole (photoemission) spectra are overdamped, with Im\ensuremath{\Sigma}\ensuremath{\propto}\ensuremath{\omega} in a wide energy range, consistent with the marginal Fermi liquid scenario, and in good agreement with experiments on cuprates. In contrast, the quasiparticles in the electron part of the spectrum show weak damping.
A microscopical theory of electron spectrum and superconductivity within the two-dimensional $t\ensuremath{-}J$ model in paramagnetic state is proposed. By employing the projection technique for the Green functions in terms of the Hubbard operators self-consistent Eliashberg equations are derived. A strong coupling of electrons with spin and charge fluctuations due to exchange and kinematical interactions is obtained. In the normal state, one-electron spectral functions reveal narrow quasiparticle peaks close to the Fermi surface (FS) with an additional broad incoherent band. The FS changes from holelike at low doping to electronlike for hole concentrations $\ensuremath{\delta}>0.3.$ In the superconducting state we observe d-wave pairing with the maximal ${T}_{c}\ensuremath{\simeq}0.04t$ at optimal doping $\ensuremath{\delta}\ensuremath{\simeq}0.3.$
The t-J model in the large-t/J and low-doping limit is approximated by an effective Hamiltonian that treats spin degrees of freedom within the linear-spin-wave approximation and the hole-hopping term with a hole-spin interaction vertex that is linear in the spin-wave operators. The Dyson's equation for the single-hole Green's function is solved numerically on finite- but sufficiently large-size square lattices. Our calculation is based on an approximate equation for the hole motion that has been obtained using a self-consistent perturbation approach where crossing diagrams are neglected. We have studied the contribution of vertex corrections and found that the leading two-loop crossing diagram is exactly zero. Thus, the leading nonzero contribution to the proper self-energy from crossing diagrams is a three-loop correction. In two dimensions, it is possible to solve Dyson's equation numerically including these three-loop corrections. The results demonstrate that the contribution of such vertex corrections is indeed small. Our solution for the hole spectral function, obtained by neglecting such small contributions from vertex corrections, shows that most features describing the hole motion are in close agreement with the results of the exact diagonalization on the 42{4}^{2}42 lattice in the region of J/t\ensuremath{\le}0.2. The results obtained on sufficiently large-size lattices suggest that certain important features of the spectral function remain while others change due to finite-size effects. In particular, we find well-defined peaks above the lowest-energy peak that survive in the thermodynamic limit. We analyze these higher-energy peaks of the spectral functions and find that they can be understood as ``string'' excitations, which are created by the hole moving in a nearly linear potential.