Spectral Function of the 2D Hubbard Model Studied with Cluster Perturbation Theory
Master’s Thesis of Nicklas Enenkel at the Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology
Abstract: We present our findings for the single-band Hubbard model in one- and two-dimensional systems on a simple cubic lattice at half-filling. A special interest is taken in the supposed pseudogap regime for the two-dimensional case at intermediate interaction strength (U ≤ 3t) and the metal-insulator transition. We employ Cluster Perturbation Theory (CPT) as our method of choice. It combines the solutions of small individual clusters of an infinite lattice system to give an approximation for the Green’s function in the thermodynamic limit. For this, one needs the full interacting Green’s function of these clusters, which we obtain in form of a Chebyshev expansion. For the calculation of the Chebyshev moments we use a Lanczos-based Exact Diagonalization (ED) code. We first establish the method by considering the 1D Hubbard model in detail and comparing our results to exact ones from Bethe ansatz. Comparing the Mott gap from CPT with the one obtained from the cluster Green’s function, we show that the resolution constrained for the method is set by the finite cluster size. The same considerations also show that CPT gives unreliable results at intermediate U. Following the discussion of the 1D model, we will investigate a pseudogap we found in a CPT calculation of the 2D model at U = 4t. This pseudogap stands as the main motivation for this thesis, as we expect a Mott gap at this interaction strength. We show that following the discussion of the 1D model this pseudogap is in fact just a result of the approximation. In addition, we conclude, that within the current computational limitations, we can not investigate the metal-insulator transition further, as this would require the calculation of much larger clusters. However, we also give an outlook on how to overcome these limitations in future projects, consider additional points that should be taken into account when interpreting CPT results and discuss some more use cases for CPT.