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.
We have investigated the t-J model and its variants using the string basis (of Shraiman, Siggia, Emery, and Lin) and the hopping basis (of Trugman); both bases are generated by the hopping term in the Hamiltonian applied repeatedly to the N\'eel state with a single hole. For the string basis, states to order
t10 are treated exactly, so that the Green function is obtained to order
t20; the tree approximation (of Brinkman and Rice) is used for the remainder. The hopping basis, which relaxes constraints imposed in the string basis, contains all states generated by up to eight hops. Properties studied include the wave function, the ground-state energy, the effective mass, the bandwidth, the spectral function, the self-energy, and the density of states. For the U=\ensuremath{\infty} model, there are no quasiparticles; the ferromagnetic polaron is missed. For the t-
Jz model, both the string and hopping bases provide excellent results for
Jz>0.1t. The Ising limit of the t-J model is treated well by the string basis, but the Heisenberg limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=
Jz requires the hopping basis, which gives apparently good results for all J>0.1t. The mass is highly anisotropic; for example, at J=0.4t the masses parallel and perpendicular to the magnetic zone face are
m@nN=20.4m and ${\mathit{m}}_{\mathrm{\ensuremath{\perp}}}$=2.1m.
Clusters of the two-dimensional t-J model with two holes and up to 26 sites are diagonalized using a Lanczos algorithm. The behavior of the binding energy with system size suggests the existence of a finite critical value of J above which binding occurs in the bulk. Only the d-wave pair-field operator acting on the Heisenberg ground state has a finite overlap with the two-hole ground state for all the clusters considered. The related spectral function associated with the propagation of a d-wave (singlet) pair of holes in the antiferromagnetic background is calculated. The quasiparticle peak at the bottom of the spectrum as well as some structure appearing above the peak survive with increasing cluster size. Although no simple scaling law was found for the quasiparticle weight the data strongly suggest that this weight is finite in the bulk limit and is roughly proportional to the antiferromagnetic coupling J (for J1).
The energy of one hole in one dimensional Ising antiferromagnet is calculated using exact diagonalization (ED) method on finite systems with number of sites N ranging from 4 to 50. This study gives a power-law description for the hole energy in the weak coupling region (0.001 less than or equal to Jz/t and Jz/t less than or equal to 0.1).This power-law description can otherwise be understood as a weak string-like energy that tends to compromise the motion of the hole by confining it to its birth site. However, for large N, the hole is found to escape from this confinement and hence propagates as a free particle in agreement with theoretical results in the bulk limit and experimental observations from angle resolved photoemission spectroscopy (ARPES) of spin-charge separation. In the strong coupling regime (Jz/t greater than 1), the energy of the hole is found to be independent of the system size. Accordingly, the velocities corresponding to charge and spin degrees of freedom in this regime become comparable. Under this situation, the hole may be treated as a localized particle.
We calculated the binding energy of two holes with antiparallel spin in the t-J model using a variational wave function based on the string or spin-bag picture. The symmetry of the bound state is
dx2-
y2. Although the size of matrices to be diagonalized is only 3\ifmmode\times\else\texttimes\fi{}3 we could reproduce the results of various extensive numerical calculations with reasonable accuracy. We show that a sizeable contribution of the binding energy is due to the quantum fluctuations in the ground state of the Heisenberg antiferromagnet. In contrast to exact diagonalization results on small clusters in the thermodynamical limit the binding energy decreases rapidly with increasing t/J. A detailed discussion of the various finite-size effects in a small cluster is also included.
The coupled-cluster method has recently proved to be efficient in calculating the ground state of strongly interacting spins or fermions on lattices. We apply it to the ground state of one hole in the half-filled Hubbard model on a two-dimensional square lattice. In a wide range for the interaction 10\ensuremath{\lesssim}U/t\ensuremath{\lesssim}80, the ground-state energy of a hole is fitted by a simple function \ensuremath{\epsilon}/t\ensuremath{\approxeq}-3.40+8.9(t/U
)0.70, in good agreement with the results obtained through exact diagonalizations on small clusters. The effective mass of a hole
m∗=8t/W (W represents the bandwidth) is proportional to U/T (
m∗\ensuremath{\approxeq}0.845U/t) for 5\ensuremath{\lesssim}U/t\ensuremath{\lesssim}20 and saturates to a constant value
m∗\ensuremath{\approxeq}84 for large U/t.