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Physical review. B, Condensed matter

Spectral function of the two-dimensional large-Nt-Jmodel

Marilyn F. Bishop, Zane P. Gibbs, T. McMullen

Published Jun 15, 1999• 2 citations
Publisher

Abstract

Calculations of the one-electron spectral function of the $t\ensuremath{-}J$ model on a two-dimensional square lattice are described. A large-$\mathcal{N}$ generalization of the $t\ensuremath{-}J$ model is used to develop $1/\mathcal{N}$ expansions, and the $t\ensuremath{-}J$ model constraint is enforced by slave bosons. The saddle point about which the $1/\mathcal{N}$ expansion is carried out is chosen to be uniform in both space and time, and self-energies are calculated to next-to-leading order in $1/\mathcal{N}$ as in the work of Z. Wang et al. [Phys. Rev. Lett. 67, 2733 (1991)]. The results of computations of the one-electron electron spectral functions for wave vectors throughout the Brillouin zone are presented for several values of the exchange coupling $J/t$ and doping $\ensuremath{\delta}$. A sharp dispersive quasiparticle peak is found near the Fermi energy. Additional spectral weight occurs at lower frequencies, and this arises from features in the normal self-energy. These features are qualitatively similar to the self-energy obtained by Kampf and Schrieffer [Phys. Rev. B 41, 6399 (1990)] from a parametrized model of an electron coupled to antiferromagnetic spin fluctuations that leads to shadow band structure in the one-electron spectral function.

DOI
10.1103/physrevb.59.14937

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Calculations of the one-electron spectral function of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model on a two-dimensional square lattice are described. A large-N\mathcal{N}N generalization of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model is used to develop 1/N1/\mathcal{N}1/N expansions, and the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model constraint is enforced by slave bosons. The saddle point about which the 1/N1/\mathcal{N}1/N expansion is carried out is chosen to be uniform in both space and time, and self-energies are calculated to next-to-leading order in 1/N1/\mathcal{N}1/N as in the work of Z. Wang et al. [Phys. Rev. Lett. 67, 2733 (1991)]. The results of computations of the one-electron electron spectral functions for wave vectors throughout the Brillouin zone are presented for several values of the exchange coupling J/tJ/tJ/t and doping \ensuremathδ\ensuremath{\delta}\ensuremathδ. A sharp dispersive quasiparticle peak is found near the Fermi energy. Additional spectral weight occurs at lower frequencies, and this arises from features in the normal self-energy. These features are qualitatively similar to the self-energy obtained by Kampf and Schrieffer [Phys. Rev. B 41, 6399 (1990)] from a parametrized model of an electron coupled to antiferromagnetic spin fluctuations that leads to shadow band structure in the one-electron spectral function.
The one-electron spectral function $\ensuremath{\rho}(\mathit{k},\ensuremath{\omega})$ for the large-N\mathcal{N}N $t\ensuremath{-}J$ model typically consists of two distinct peaks: a dispersive quasiparticle and a lower-energy incoherent structure. This is reminiscent of the spectral function of the weak coupling Hubbard model in which the electron is coupled to antiferromagnetic spin fluctuations. In both theories, the two-peaked pattern can be traced to the frequency dependence of the (normal) electron self-energy $\ensuremath{\Sigma}(\mathit{k},\ensuremath{\omega})$. What interaction causes this particular $\ensuremath{\omega}$ dependence of $\ensuremath{\Sigma}$? In the weak coupling case it is clearly the emission and absorption of an antiferromagnetic spin fluctuation by the electron. In the slave boson formulation of the large-N\mathcal{N}N $t\ensuremath{-}J$ model, there are six bosonic excitations: the slave bosons, the constraint field, and the real and imaginary parts of the hopping amplitudes in both the x and y directions. None of these are obviously related to a spin fluctuation propagator. These bosons have the unusual property of transmuting in flight---that is, they are not eigenstates. In order to try to make physical sense of the interaction that is primarily responsible for the frequency dependence of $\ensuremath{\Sigma}$ and the structure of $\ensuremath{\rho}(\mathit{k},\ensuremath{\omega})$, we investigate the transformation to a representation in which the boson propagator matrix is diagonal and extract the contributions of the slave bosons and the constraint field, which appear to be of particular importance.
Using a recently developed perturbative approach, which considers Hubbard operators as fundamental excitations, we have performed electronic self-energy and spectral function calculations for the $t\text{\ensuremath{-}}J$ model on the square lattice. We have found that the spectral functions along the Fermi surface are isotropic, even close to the critical doping where the ddd density wave phase occurs. Fermi liquid behavior with scattering rate $\ensuremath{\sim}{\ensuremath{\omega}}^{2}$ and a finite quasiparticle weight ZZZ was obtained. ZZZ decreases with decreasing doping taking low values for low doping. Results are compared with others obtained by analytical and numerical methods like the slave-boson and Lanczos diagonalization approaches, finding agreement. We discuss our results in the light of recent angle-resolved photoemission spectroscopy experiments in cuprates.
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.
We present an exact diagonalization study of the single-particle spectral function in the one-dimensional (1D) and two-dimensional (2D) ttt-JJJ model. By studying the scaling properties with JJJ and ttt we find a simple building pattern in 1D and show that every spectral feature can be uniquely assigned by a spinon and holon momentum. We find two types of low-energy excitations: A band with energy scale JJJ and high spectral weight disperses upwards in the interior part of the Brillouin zone and reaches EF{E}_{F}EF​ at kF,{k}_{F},kF​, and a band with energy scale ttt and low spectral weight disperses downwards in the outer part of the zone, touching EF{E}_{F}EF​ at 3kF.{3k}_{F}.3kF​. An analogous analysis of the 2D case at half-filling shows that the ttt band exists also in this case, but is diffuse and never reaches the Fermi energy. For the doped case in 2D the picture is more reminiscent of 1D; in particular, the ``main band'' with a dispersion $\ensuremath{\propto}J$ and the ``shadow band'' with energy scale ttt can be identified also in this case. This leads us to propose that the shadow bands discovered by Aebi et al. in Bi2212 are the 2D analog of the 3kF{3k}_{F}3kF​ singularity in 1D systems and unrelated to antiferromagnetic spin fluctuations.
Strong repulsive interactions in a one-dimensional electron system suppress the exchange coupling JJJ of electron spins to a value much smaller than the Fermi energy EF{E}_{F}EF​. The conventional theoretical description of such systems based on the bosonization approach and the concept of Tomonaga-Luttinger liquid is applicable only at energies below JJJ. In this paper, we develop a theoretical approach valid at all energies below the Fermi energy, including a broad range of energies between JJJ and EF{E}_{F}EF​. The method involves bosonization of the charge degrees of freedom, while the spin excitations are treated exactly. We use this technique to calculate the spectral functions of strongly interacting electron systems at energies in the range $J⪡\ensuremath{\epsilon}⪡{E}_{F}$. We show that in addition to the expected features at the wave vector kkk near the Fermi point kF{k}_{F}kF​, the spectral function has a strong peak centered at k=0k=0k=0. Our theory also provides analytical description of the spectral function singularities near 3kF3{k}_{F}3kF​ (the ``shadow band'' features).
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.