We present results for the single-particle spectral weight function for the two-dimensional Hubbard model. Results were obtained using analytic expressions for the first two moments of the spectral weight function, along with numerical calculations using a combination of quantum Monte Carlo and the maximum-entropy method of analytic continuation. The expressions for the first two moments are surprisingly simple, involving only the parameters of the model, the chemical potential, and the filling. At half-filling, the squared width (variance) of the spectral weight is exactly
U2/4. Numerical results are shown for 8\ifmmode\times\else\texttimes\fi{}8 lattices at a variety of fillings and momenta. The insulating antiferromagnetic gap in the spectral weights, readily apparent in the results at half-filling, is absent at a filling of 0.95.
We propose a least-squares-fit method to extract real-frequency correlation functions from imaginary-time Monte Carlo data of a many-particle system. We apply it to the density correlation function in a one-dimensional model of spinless fermions. The spectral functions thus obtained reflect accurately the particle-hole and the soliton-antisoliton excitations of the system in the respective limits of noninteracting and strongly coupled fermions. They are in qualitative agreement with the available analytical results.
We present a method to obtain dynamical correlation functions of quantum many-body systems in the real-frequency domain from Monte Carlo data of the corresponding imaginary-time Green's functions. The method is based on a least-squares-fit procedure to solve the integral equation relating imaginary-time and real-frequency correlation functions. To demonstrate its feasibility, we have applied our method to imaginary-time Monte Carlo data for the density correlation function of a simple model of interacting spinless fermions in one dimension. We compare our results to the analytical results, available in the limits of zero and strong interaction. We find that despite the presence of noise in the input data, the real-frequency spectra obtained with our method are qualitatively correct. They reflect accurately the particle-hole and the soliton-antisoliton excitations, present in the noninteracting and in the strongly coupled fermion system, respectively. We also compare our method to other approaches that have been proposed for obtaining dynamical correlation functions via Monte Carlo simulation.
Implementing an improved method for analytic continuation and working with imaginary-time correlation functions computed using quantum Monte Carlo simulations, we resolve the single-particle dispersion relation and the density of states (DOS) of the two-dimensional Hubbard model at half filling. At intermediate interactions of
U/t=4,6, we find quadratic dispersion around the gap minimum at wave vectors $\mathbf{k}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2)$ (the $\mathrm{\ensuremath{\Sigma}}$ points). We find saddle points at $\mathbf{k}=(\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0),(0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi})$ (the
X points), where the dispersion is approximately quartic, leading to a sharp DOS maximum above the almost flat ledge arising from the states close to $\mathrm{\ensuremath{\Sigma}}$. The fraction of quasiparticle states within the ledge is ${n}_{\mathrm{ledge}}\ensuremath{\approx}0.15$. Upon doping away from half filling, within the rigid-band approximation, these results support Fermi pockets around the $\mathrm{\ensuremath{\Sigma}}$ points, with states around the
X points becoming filled only at doping fractions $x\ensuremath{\ge}{n}_{\mathrm{ledge}}$. The high density of states away from the $\mathrm{\ensuremath{\Sigma}}$ gap edge may be an important clue for a finite minimum doping level for superconductivity and other instabilities of doped Mott insulators.
We present a numerically stable Quantum Monte Carlo algorithm to calculate zero-temperature imaginary-time Green functions $ G(\vec{r}, τ) $ for Hubbard type models. We illustrate the efficiency of the algorithm by calculating the on-site Green function $ G(\vec{r}=0, τ) $ on
4×4 to
12×12 lattices for the two-dimensional half-filled repulsive Hubbard model at
U/t=4. By fitting the tail of $ G(\vec{r}=0, τ) $ at long imaginary time to the form
e−τΔc, we obtain a precise estimate of the charge gap:
Δc=0.67±0.02 in units of the hopping matrix element. We argue that the algorithm provides a powerful tool to study the metal-insulator transition from the insulator side.