Efficient real-frequency solver for dynamical mean-field theory

We here present how a self-consistent solution of the dynamical mean-field theory equations can be obtained using exact diagonalization of an Anderson impurity model with accuracies comparable to those found using renormalization group or quantum Monte Carlo methods. We show how one can solve a correlated quantum impurity coupled to several hundred uncorrelated bath sites, using a restricted active basis set. The number of bath sites determines the resolution of the obtained spectral function, which consists of peaks with an approximate spacing proportional to the bandwidth divided by the number of bath sites. The self-consistency cycle is performed on the real-frequency axis and expressed as numerical stable matrix operations. The same impurity solver has been used on ligand field and finite size cluster calculations and is capable of treating involved Hamiltonians including the full rotational invariant Coulomb interaction, spin-orbit coupling, and low-symmetry crystal fields. The proposed method allows for the calculation of a variety of correlation functions at little extra cost.

Figure 1

Possible bath geometries. The impurity is labeled by $i$ and represented by a square. The bath sites are labeled by $b_i$ [panels (a) and (b)] or by $b,c_i$, and $v_i$. The site occupation is indicated by the filling. For efficient calculations bath sites should either be occupied or empty.

Figure 2

All panels show the DMFT impurity Green's function for different values of $U$ ranging from 0 to 2 in steps of 0.25 in units of the bandwidth of $G_0$. The different columns show the spectral function for 3, 11, 31, 101, or 301 poles in the bath Green's function and thus sites in the Anderson impurity calculation. The impurity Green's function in all calculations consists of 1000 poles. The panels in the top row show the spectral functions broadened with a Lorentzian of full width at half maximum 0.01. The panels in the bottom row show the same spectral functions as the top row, but broadened with a Lorentzian of full width at half maximum 0.1.

Figure 3

Comparison between the NRG results as obtained by Bulla et al. [23, 24, 25, 26, 27] (solid curves with red thin lines) and our calculations (thick blue lines) for $U=0.0$, 0.5, 1.0, and 1.25. Top panel show a Lorentzian broadening of full width at half maximum of 0.01; bottom panels show a Lorentzian broadening of full width at half maximum of 0.2 on both the ED and NRG results (left) or only the ED results (right).

Figure 4

Comparison between the QMC calculations obtained with the CT (triqs package [45, 46, 78]) or HF algorithm and the ED or NRG (Bulla et al. [23, 24, 25, 26, 27]) results for $U=0.0$ to $U=2.0$ in steps of $0.25$. From left to right we show $G\left(\tau \right)$ calculated with the four different methods, the difference between our method and CT-QMC, the difference between our method and HF-QMC, the difference between NRG and CT-QMC, and the difference between NRG and HF-QMC. The difference plots are multiplied by a factor of 100 (20 for $U=1.25)$ compared to the plots in the left panel.

Figure 5

Graphical representation of the evolution of the basis set during the Lanczos cycles which determine the ground-state wave function.