Exact Diagonalization

In the exact diagonalization (ED) method, the host is represented by a finite number of bath sites. If the number of bath sites is small, the resulting Hamiltonian can be solved exactly by diagonalizing the corresponding Hamiltonian matrix. This method makes it possible to avoid the problems of extracting spectral properties from data on the imaginary time axis, e.g., by using a maximum entropy method, as in QMC methods. It also makes it possible to treat $T=0$. The main limitation is the strong restriction on the number of bath sites.

We have derived an exact sum rule, which can be used to check how well the host is described by the finite number of bath sites. We have also developed symmetry considerations for choosing bath sites in an efficient way. The method has been applied to the one- and two-dimensional Hubbard models ( Phys. Rev. B 78, 115102 (2008)).

The results for the density of the one-dimensional Hubbard model greatly improve when the cluster size is increased from one to two sites. This is often cited as a showcase for the power of cluster methods. However, contrary to what was earlier believed, more than two cluster sites are needed to accurately reproduce the exact Bethe Ansatz in the vicinity of the metal-insulator transition. We find that, increasing the cluster size to four and six sites systematically improves the agreement.

We have also studied the two-dimensional Hubbard model using a 2x2=4 site cluster. Since host states of several (two one-dimensional and one two-dimensional representation) different symmetries couple to the cluster, we do not find a fully satisfactory representation of the host in the general case, due to the strong limitations on the number of bath sites.

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For further information contact Olle Gunnarsson (O.Gunnarsson@fkf.mpg.de), Giorgio Sangiovanni (sangiovanni@ifp.tuwien.ac.at) or Erik Koch (E.Koch@fz-juelich.de).