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.
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 solved numerically the integral equation for the selfenergy which describes the motion of a single hole in a two-dimensional quantum antiferromagnet (AF) within the selfconsistent Born approximation. This formulation stresses the similarity of the AF-spin polaron with the standard polaron problem. We confine our calculation to finite cluster geometries and compare with results from previous exact diagonalization studies. The spectral function is characterized by a narrow quasiparticle (qp) peak at the low energy side of the spectra, which appears to be well separated from the incoherent band part for large enough J values. For small J we find a reduced width of ~7t for the incoherent band. The bottom of the coherent qp band always occurs at (±π/2, ±π/2). Its bandwidth initially increases with J until J≃t and then decreases as 2t2/J. A complete characterization of our solution is given, including the dispersion relation and effective masses of this quasiparticle. The comparison with exact diagonalization studies for a 4×4 cluster is remarkably good. From our results we see that a lattice of 16×16 sites describes adequately the thermodynamic limit. We conclude that the simple Born approximation is a valuable scheme to characterize the dynamics of one hole in the t-J model. both in the perturbative and the strong coupling regimes.
The spin-wave expansion is used to motivate a simple trial wave function for the N\'eel ground state of a spin-1/2 Heisenberg antiferromagnet on a square lattice; the wave function yields an upper bound on the ground-state energy per bond of (-0.3317\ifmmode\pm\else\textpm\fi{}0.0002)J, where J is the exchange constant. The wave function is easily generalizable to the case where a single hole, with hopping matrix element t, is present. The hole energy-momentum relationship is determined for J/t>0.25; the minimum hole energy is always on the zone boundary at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2). The hole bandwidth, W, has a maximum value of 1.24t at J=0.73t. The small parameter which makes all of these calculations possible is 1/(2Z-2), where Z=4 is the coordination number of the square lattice.
The spectral functions of one hole in the t-J and one-band Hubbard models are calculated using exact diagonalization techniques on small lattices. Results for the t-
Jz model are also presented. For the t-J model we found that there is a quasiparticle at the bottom of the hole spectrum with an energy well approximated by
Eh=-3.17+2.83
J0.73 (for 0.1\ensuremath{\le}J\ensuremath{\le}1.0, t=1) on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The rest of the spectrum is not incoherent: We identified at least two other peaks following a similar power-law behavior with J. We speculate that the J dependence of the results can be explained by a model where the hole is trapped in a confining potential as in the Ising limit. The bandwidth of the hole is linear in J in the region 0.1\ensuremath{\le}J\ensuremath{\le}0.4 although a power-law behavior is not excluded. The spectral weight of the quasiparticle grows like
J0.5 in the same region. We present new analytical results in the large J/t limit to understand the motion of the hole: In perturbation theory it can be shown that the momentum of the hole at large J/t is k=(\ensuremath{\pi},\ensuremath{\pi}) changing to k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) at intermediate J/t in agreement with numerical and spin-waves results. We show analytically and numerically that the bandwidth of the quasiparticle is of order t in the large J/t limit. This result corresponds to a spin-liquid state. The one-hole spectral function of the Hubbard model is obtained for lattices with 8 and 10 sites. A quasiparticle is also observed in this case. The bandwidth and the relation with the t-J model are discussed and a comparison with recent Monte Carlo results is made. We also review and extend previous results for the ground-state properties of the t-J model.
An antiferromagnetic spin model (S=1/2) consisting of a chain of octahedra is considered. In a certain parameter regime of the Hamiltonian the exact ground state is a local resonating-valence-bond (RVB) state. We start with the t-J Hamiltonian for this model describing a single hole in the system and derive exact ground and excited states of the Hamiltonian when the hole is static. The local spin distortion around a hole is termed a spin polaron. The dynamics of the hole is studied through construction of exact eigenstates describing spin polarons in extended states. Coherent states for all momentum wave vectors in the chain direction exist for certain spin polarons. In the Nagaoka limit, the RVB polaronic states have energy greater than that of the ferromagnetic state. For exchange interaction J\ensuremath{\ne}0, the RVB states have lower energy. Other exact states obtained include states in which the hole motion is localized. Structural similarity of the present model to the arrangement of Cu spins in
La2CuO4 is also highlighted.