# P11: Quantifying electronic correlations in many-body wave-functions

## Project Description

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 [1]. An uncorrelated wave function that can be written as a single Slater determinant has Slater rank one [2]. 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 [3].

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, [4]. 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 [5]. 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 [6], 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 [6] we plan, in particular, to investigate how the quality of the approximate wave functions differs for one- and higher-dimensional systems.

[1] P.-O. Löwdin, Phys. Rev. **97**, 1474 (1956)

[2] J. Schliemann, D. Loss, and A.H. MacDonald, Phys. Rev. B **63**, 085311 (2011)

[3] L. Amico, R. Fazio, A. Osterloh, and V. Vredal, Rev. Mod. Phys. **80**, 517 (2008)

[4] W. Brenig, Nucl. Phys. **4**, 363 (1957)

[5] E. Koch, “Many-electron states”, in Emergent phenomena in correlated matter, E. Pavarini, E. Koch, and U. Schollwöck (eds.), Modeling and Simulation Vol. **3**, Verlag des Forschungszentrum Jülich (2013)

[6] E. Koch, “The Lanczos method”, in Many-body physics: From Kondo to Hubbard, E. Pavarini, E. Koch, and P. Coleman (eds.), Modeling and Simulation Vol. **5**, Verlag des Forschungszentrum Jülich (2015)