Tutorial for dynamical mean-field theory using exact-diagonalization impurity solver

This is a dynamical mean-field theory (DMFT) tutorial for 2024 Strongly correlated physics – Numerical and Analytical approaches at National Yang-Ming Chiao-Tung University and National Center of Theoretical Science, Taiwan. In this tutorial, you will learn how to implement the DMFT algorithm using exact-diagonalization (ED) as an impurity solver. The repository contains three folders.

• metal: This folder contains a jupyter notebook dmft-ed-metal.ipynb implementing the DMFT loop to investigate the correlated metal behavior. The code shows the behavior of the Green's function and self-energy on the real and Matsubara frequencies. The script dos.py contains a simple semicircular density of states of the Bethe lattice. The script ed.py is a simple but less efficient implementation of the exact-diagonalization algorithm for solving the Anderson impurity model.

• mott: This folder is similar to the metal folder, but it applies the DMFT to the Mott insulating solution and shows the behavior of the Green's function and self-energy on the real and Matsubara frequencies.

• metal_mott_transition: This folder applies the DMFT to investigate the first-order metal-Mott-insulator transition at half-filling. In order to investigate the first-order transition, we run the DMFT calculations from small Coulomb interaction U to large U (the metal2mott folder with dmft-ed-metal2mott.ipynb) as well as from large U to small U (the mott2metal folder with dmft-ed-mott2metal.ipynb). Then, we plot the hysteresis curves of the first order transition (in dmft-ed-metal2mott.ipynb) showing in the double-occupancies vs. U and the quasiparticle weight Z vs. U figures. The density of states as a function of U is also plotted.

• Optional homework: Can you reproduce the finite temperature phase diagram of the DMFT metal-insulator transition using the provided code?

Dependency

The code is implemented in Python. Required packages are jupyter, numpy, scipy, numba.

References

[1] Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, A Georges, G Kotliar, W Krauth, MJ Rozenberg, Reviews of Modern Physics 68 (1), 13 (1996).