Solving the dynamical mean-field theory at very low temperatures using the Lanczos exact diagonalization

We present an efficient method to solve the impurity Hamiltonians involved in dynamical mean-field theory at low but finite temperature based on the extension of the Lanczos algorithm from ground state properties alone to excited states. We test the approach on the prototypical Hubbard model and find extremely accurate results from $T=0$ up to relatively high temperatures up to the scale of the critical temperature for the Mott transition. The algorithm substantially decreases the computational effort involved in finite temperature calculations.

Figure 1

(Color online) Imaginary part of the local Green’s function on the Matsubara axis for different values of $N_{\mathit{\text{kept}}}$ compared with the full diagonalization of the Hamiltonian matrix for $N_s=6$ and $\beta =50∕D$. The left panel shows $U=2.4D$ and the right one $U=2D$.

Figure 2

(Color online) Imaginary part of the local Green’s function on the Matsubara axis from finite-$T$ ED ($N_s=8$ and $N_{\mathit{\text{kept}}}=40$) and Hirsch-Fye QMC (Ref. 19) for $\beta =60$ and $U=2D$.

Figure 3

(Color online) Phase diagram for the Mott transition in the paramagnetic sector obtained through finite-temperature ED with $N_s=8$ and $N_{\mathit{\text{kept}}}=40$, compared with previous estimates by numerical renormalization group of Ref. 20 and quantum Monte Carlo of Refs. 21, 22. The ED curves are stopped at the highest temperature where the chosen number of states determined a negligible truncation error.

Figure 4

(Color online) Inverse quasiparticle lifetime $1∕\tau$ for $N_s=7$ as a function of square temperature for the displayed values of the correlation strengths. The linear behavior in $T^2$ characteristic of the Fermi liquid is apparent, with a slope that increases with $U∕W$.