P11: Quantifying electronic correlations in many-body wave-functions
Correlated many-electron wave functions |Ψ> are written as linear combinations of Slater determinants. But not every linear combination of Slater determinants is correlated: To find out whether |Ψ> is correlated, we calculate its density matrix <Ψ|c†α cβ|Ψ>. When it is idempotent, |Ψ> is given by the Slater determinant constructed from the occupied natural orbitals . An uncorrelated wave function that can be written as a single Slater determinant has Slater rank one . Many-electron states are correlated when their Slater rank is larger than one. Since the importance of correlations in different wave-functions with the same Slater rank can vary dramatically, it would be highly desirable to be able to quantify the degree of correlation in a given wave function. There have been many attempts to generalize entanglement measures to many-electron states, but results are mainly limited to two-electron systems .
In this project, we plan to explore a practical method for quantifying the correlation in arbitrary N-electron wave functions, based on an iterative procedure: Given |Ψ>, we start by de-termining the Slater determinant |Φ0> with maximum overlap, max|<Φ0|Ψ>|2, . It will give the best approximation to |Ψ> with Slater rank one. We can improve on this by finding the best approximation to the residual |Ψ>−|Φ0>. Iterating this procedure, we find the expansion |Ψ> = Sm,n |Φn> (S−1)n,m <Φm|Ψ>, in which S is the overlap matrix . The inverse participation ratio of the expansion coefficients then provides a measure of how slowly the expansion converges, i.e., how correlated the state is.
We plan to test this approach by applying it to exact ground-state wave functions for the Hubbard model obtained from Lanczos , in particular across the Mott transition when either the interaction U or the doping is changed. Furthermore, we will compare the correlation measures of the exact wave function to those obtained by variational approaches, in particular matrix-product states, or more general tensor-network states. This will provide us insights into what types of correlation can be captured by the different classes of approximate wave functions. Using our massively parallel Lanczos solver  we plan, in particular, to investigate how the quality of the approximate wave functions differs for one- and higher-dimensional systems.
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