The motion of a single hole in a two-dimensional Heisenberg antiferromagnet (hAF) is studied in a representation where holes are described as spinless fermions (holons) and spins as normal bosons. Assuming long-range AF order the spin dynamics is treated in linear spin-wave theory. The formulation highlights the close relation with the conventional polaron problem. The holon Green's function is calculated self-consistently within the Born approximation using finite-cluster geometries for the numerical solution. As a remarkable result we find close agreement with the spectral function A(k,\ensuremath{\omega}) of a hole calculated by exact diagonalization methods. A(k,\ensuremath{\omega}) is characterized by a narrow quasiparticle (QP) peak at the low-energy side of the spectrum, which is well separated from the incoherent part for large enough J values. A complete characterization of our solution is given, including the spectral weight, the dispersion relation, and effective masses of the QP state. A finite-size-scaling study gives a nonvanishing spectral weight of the QP in the thermodynamic limit for values J/t typical for copper oxide superconductors. Our calculations indicate that the self-consistent Born approximation is a valuable scheme for characterizing the dynamics of a hole in the t-J model, even in the strong-coupling regime.
The spectral function of a single hole moving in a two-dimensional antiferromagnetic background is calculated, using the self-consistent Born approximation for the self-energy. For small systems, good agreement is found with the results of exact diagonalization studies. In the thermodynamic limit, the quasiparticle residue, the effective mass, and the incoherent multiple spin-wave background are determined as functions of the ratio of Heisenberg exchange J and hopping amplitude t. The relevance of the single-hole results to the physical problem of a finite hole concentration is briefly discussed.
We revisit the problem of the dispersion of a single hole injected into a quantum antiferromagnet. We applied a spin-density-wave formalism extended to a large number of orbitals and obtained an integral equation for the full quasiparticle Green's function in the self-consistent ``noncrossing'' Born approximation. We found that for $t/J\ensuremath{\gg}1$, the bare fermionic dispersion is completely overshadowed by the self-energy corrections. In this case, the quasiparticle Green's function contains a broad incoherent continuum which extends over a frequency range of $\ensuremath{\sim}6t$. In addition, there exists a narrow region of width
O(JS) below the top of the valence band, where the excitations are mostly coherent, though with a small quasiparticle residue $Z\ensuremath{\sim}J/t$. The top of the valence band is located at $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. We found that the form of the fermionic dispersion, and, in particular, the ratio of the effective masses near $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ strongly depend on the assumptions one makes for the form of the magnon propagator. We argue in this paper that two-magnon Raman scattering as well as neutron-scattering experiments strongly suggest that the zone-boundary magnons are not free particles since a substantial portion of their spectral weight is transferred into an incoherent background. We modeled this effect by introducing a cutoff
qc in the integration over magnon momenta. We found analytically that for small
qc, the strong-coupling solution for the Green's function is universal, and both effective masses are equal to ${(4JS)}^{\ensuremath{-}1}$. We further computed the full fermionic dispersion for
J/t=0.4 relevant for
Sr2CuO2Cl2, and ${t}^{\ensuremath{'}}=\ensuremath{-}0.4J$ and found not only that the masses are both equal to ${(2J)}^{\ensuremath{-}1}$, but also that the energies at
(0,0) and $(0,\ensuremath{\pi})$ are equal, the energy at $(0,\ensuremath{\pi}/2)$ is about half of that at
(0,0), and the bandwidth for the coherent excitations is around
3J. All of these results are in full agreement with the experimental data. Finally, we found that weakly damped excitations only exist in a narrow range around $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$. Away from the vicinity of $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$, the excitations are overdamped, and the spectral function possesses a broad maximum rather than a sharp quasiparticle peak. This last feature was also reported in photoemission experiments.
A finite-size scaling analysis of the spectral function and of the optical conductivity of a single hole moving in an antiferromagnetic background is performed. It is shown that both the low-energy quasiparticle peak and the broad higher-energy structure are robust with increasing cluster size from 4\ifmmode\times\else\texttimes\fi{}4 to \ensuremath{\surd}26 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}26 sites. In the absence of spin fluctuations, for most static or dynamical quantities saturation occurs when the size exceeds a characteristic size
Nc(
Jz). Typically, 16- and 26-site clusters give reliable results for
Jz>0.75 and
Jz>0.3, respectively. The hole optical mass is shown to be very large (>20) in agreement with the small bandwidth. Due to the energy gap to flip a spin in the vicinity of a hole, a small gap \ensuremath{\propto}
Jz separates the low-energy \ensuremath{\delta} function from the rest of the spectrum in the dynamical correlation functions. On the other hand, with ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ this gap seems to disappear with increasing system size as one would expect since the spin waves are gapless in the thermodynamic limit. The large momentum dependence of the quasiparticle weight in the isotropic case is inconsistent with a string picture but agrees well with the self-consistent Born approximation.An accurate estimation of the higher-energy part of the spectral functions of the t-J model can be made for momenta close to (0,0) or (\ensuremath{\pi},\ensuremath{\pi}); it consists of a broad featureless band. For momenta close to the Fermi surface, no evidence for excited string states is found. This may be either because the string scenario breaks down or because exact calculations at momenta near (\ensuremath{\pi}/2,\ensuremath{\pi}/2) are limited by the differing geometries of the finite clusters. In the Ising limit, the spectral function also depends crucially on the momentum, but a feature resembling a string excitation is seen for momenta close to (0,0). Our data on the finite-frequency optical conductivity of a single hole in the t-J model reveal that the total spectral weight at finite frequency has a strong size dependence and decreases with increasing size (for J not too small). This suggests that the optical mass and the optical absorption of the t-J model may have been overestimated in the past.
The motion of a single hole in a two-dimensional triangular antiferromagnet is investigated using the $t\ensuremath{-}J$ model. The one-hole states are described by strings of spin deviations around the hole. Using projection technique the one-hole spectral function is calculated. For large
J/t we find low-lying quasiparticlelike bands which are well separated from an incoherent background by a gap of order J. However, for small
J/t this gap vanishes and the spectrum becomes broad over an energy range of several t. The results are compared with self-consistent Born approximation calculations and numerical data.
Results for the single-particle Green's function for the strong-coupling limit of the single-band Hubbard model ($t\ensuremath{-}J$ model) obtained by numerical and analytical methods are discussed. The numerical calculation of the spectral function is performed for small systems of up to 18 sites using the Lanczos method. A detailed analysis of the spectral functions for the $t\ensuremath{-}J$ model in one and two dimensions is given. We also investigate the effect of the next-nearest-neighbor hopping terms, which appear in order
Ut2 and are omitted in the standard $t\ensuremath{-}J$ model. The role of quantum fluctuations on the shape of the spectra and the fundamental difference between one and two dimensions is elucidated. In 2D we find evidence for a coherent quasiparticle band of width
2J with dispersion $^{\ensuremath{\omega}}\mathrm{k}\ensuremath{\sim}(\frac{J}{2}){(cos{k}_{x}+cos{k}_{y})}^{2}$ and spin
S=21, which emerges at the low-energy edge of the incoherent spectrum. This form of dispersion is due to effective next-nearest-neighbor processes related to spin flips and agrees with our extension of the approach of Brinkman and Rice to finite
J.