We present an exact diagonalization study of the low energy singlet and triplet states for both 1D and 2D t-J model. A scan of the parameter ratio J/t shows that for nearly all low energy states in both 1D and 2D the excitation energy takes the linear form E(t,J) = a t + b J. In 1D this is the natural consequence of the factorization of the low energy wave functions, i.e. spin-charge separation. Examination of the low energy eigenstates in 2D shows that in the singlet sector the lowest states are collective modes, which for larger J correspond to `dynamical hole columns', i.e. a periodic modulation of the hole density. This modulation is well reproduced by treating holes as hard-core Bosons with an attractive interaction.
We present an exact diagonalization study of the dynamical spin and density correlation functions in small clusters of the t-J model, focusing on the regime of intermediate and low electron densities, ${\mathrm{\ensuremath{\rho}}}_{\mathit{e}}$0.5. In two dimensions (2D) both correlation functions agree remarkably well with the convolution of the single-particle spectral function, i.e., the simplest estimate possible within a Fermi-liquid picture. Deviations from the convolution are shown to originate from symmetry-related selection rules, which are unaccounted for in the convolution estimate. For all fillngs under consideration, we show that the low-energy peaks originate from particle-hole excitations between the Fermi momenta, as expected for a Fermi liquid. We contrast this with the behavior in 1D, where spin and density correlation function show the differences characteristic of spin-charge separation and where neither correlation function is approximated well by the convolution.
We investigate the dynamical spin and charge correlation functions, $S(\mathrm{Q}, \ensuremath{\omega})$ and $N(\mathrm{Q}, \ensuremath{\omega})$, for the two-dimensional (2D) $t\ensuremath{-}J$ model by using the exact diagonalization technique. Both quantities are compared with the 1D $t\ensuremath{-}J$ model. At high electron density (
n>0.7), a remarkable feature emerges: $S(\mathrm{Q}, \ensuremath{\omega})$ shows a pronounced peak at low energy of order
J which is consistent with
2kF low-energy excitations, whereas low-energy
2kF-backscattering processes are strongly suppressed in the charge-density response. $N(\mathrm{Q}, \ensuremath{\omega})$ consists of a dense and broad continuum, qualitatively different from the 1D charge response, implying strong interactions between charge and spin degrees of freedom.
The spin and charge excitations of the t-J ladder are studied by exact diagonalization techniques for several electron densities. The various modes are classified according to their spin (singlet or triplet excitations) and their parity under a reflection with respect to the symmetry axis along the chains and a finite-size scaling of the related gaps is performed. At low doping, formation of hole pairs leads to a spin gap for all J/t ratios. This phase is characterized by (at least) one vanishing energy mode only in the charge bonding channel when
Kx\ensuremath{\rightarrow}0 consistent with the existence of superconducting pairing correlations. At larger doping the spin gap disappears. Although the antibonding modes remain gapped, low-energy
Kx\ensuremath{\sim}0 and
Kx\ensuremath{\sim}2
kF spin and charge bonding modes are found consistent with a single-band Luttinger scenario. At sufficient low electron density and above a critical value of J/t we also expect another phase of electron pairs with gapped spin excitations.
We present an exact diagonalization study of the single-particle spectral function in the one-dimensional (1D) and two-dimensional (2D)
t-
J model. By studying the scaling properties with
J and
t 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
J and high spectral weight disperses upwards in the interior part of the Brillouin zone and reaches
EF at
kF, and a band with energy scale
t and low spectral weight disperses downwards in the outer part of the zone, touching
EF at
3kF. An analogous analysis of the 2D case at half-filling shows that the
t 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
t 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 singularity in 1D systems and unrelated to antiferromagnetic spin fluctuations.
Effective models of hard-core hole pair bosons and triplets are derived from the $t\ensuremath{-}J$ model in ladders and on the square lattice by performing a change from site to dimer basis. The Hilbert space is truncated by projecting out single-occupied electron states and only nearest-neighbor interactions are retained. The resulting effective models in one and two dimensions are studied by numerical techniques. In both spatial dimensions, the main result is that each hole pair is surrounded by a singlet cloud expelling triplet excitations from its vicinity. It is suggested an interpretation of this feature as a phase-separated state between a hole-pair-rich singlet phase and an undoped triplet phase with antiferromagnetic correlations. The possible relevance of this result to other theoretical scenarios and experimental results is discussed.