A concise general expression for the dynamical charge susceptibility of the Hubbard model is presented. The results are obtained from the spin-rotation-invariant slave-boson theory, whose paramagnetic saddle point corresponds to Gutzwiller's variational approximation. The charge structure factor and charge susceptibility are calculated for a wide range of interactions and temperatures and are compared thoroughly with the available quantum Monte Carlo and variational Monte Carlo results for the one-dimensional Hubbard model. The comparison shows reasonable agreement. All the calculations are done in a totally self-consistent way, without adjustable parameters.
The charge susceptibility of the two-dimensional repulsive Hubbard model is investigated by using the diagram technique developed for the case of strong correlations. In this technique, a power series in the hopping constant is used. It is shown that once the Fermi level crosses one of the Hubbard subbands, a sharp peak appears in the momentum dependence of the static susceptibility. With further departure from half-filling, the peak transforms to a ridge around the $\ensuremath{\Gamma}$ point. Within the considered range $0\ensuremath{\le}|1\ensuremath{-}\overline{n}|\ensuremath{\lesssim}0.2$ of the electron filling
n, the static susceptibility is finite, which points to the absence of long-range charge ordering. However, for $|1\ensuremath{-}\overline{n}|\ensuremath{\approx}0.12$, the susceptibility maxima are located halfway between the center and the boundaries of the Brillouin zone. In this case, an interaction of the carriers with the tetragonal distortions can stabilize the charge-density wave with the wavelength of four lattice spacings, as experimentally observed in the low-temperature tetragonal phase of lanthanum cuprates. Within the range of parameters inherent in cuprate perovskites, the character of the susceptibility evolution with
n depends only weakly on the ratio of the nearest-neighbor hopping constant to the Hubbard repulsion and on details of the initial band structure. The location of the susceptibility maxima in the Brillouin zone is mainly determined by the value of
n.
We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as
T=0.067 and interactions up to
U=7, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.