Hierarchical equations of motion for an impurity solver in dynamical mean-field theory

A nonperturbative quantum impurity solver is proposed based on a formally exact hierarchical equations of motion (HEOM) formalism for open quantum systems. It leads to quantitatively accurate evaluation of physical properties of strongly correlated electronic systems, in the framework of dynamical mean-field theory (DMFT). The HEOM method is also numerically convenient to achieve the same level of accuracy as that using the state-of-the-art numerical renormalization group impurity solver at finite temperatures. The practicality of the HEOM+DMFT method is demonstrated by its applications to the Hubbard models with Bethe and hypercubic lattice structures. We investigate the metal-insulator transition phenomena, and address the effects of temperature on the properties of strongly correlated lattice systems.

Figure 1

(a) Flow chart of the HEOM based DMFT approach. (b) Schematic diagram of the HEOM impurity solver.

Figure 2

The Lorentzian fit of the hybridization function. The number of Lorentzian functions used is $N=9$ for (a) and $N=16$ for (b).

Figure 3

Spectral function $A(\omega =0)$ as a function of temperature for truncation level $L=2\sim 4.$ The inset shows the minimal $L$ that is necessary to achieve quantitative convergence at different temperatures.

Figure 4

The convergence of the spectral function with respect to the truncation level $L$ for Bethe lattice at (a) $T=0.0125W$ and (b) $T=0.075W$. Here, $U=W$ with $W$ being the effective bandwidth.

Figure 5

The convergence of (a) the self-energy and (b) the spectral function for a hypercubic lattice by iteration. The parameters are $U=W$ and $T=0.0125W$, where $W$ is the effective bandwidth.

Figure 6

Spectral functions of the hypercubic lattice and the Bethe lattice with different values of $U$. The temperature is $T=0.0125W$, where $W$ is the effective bandwidth.

Figure 7

The imaginary part of the self-energy of hypercubic (upper panel) and Bethe (lower panel) lattices at different values of $U$. The temperature is $T=0.0125W$, where $W$ is the effective bandwidth.

Figure 8

Existence of dual solution for the spectral function at $T=0.0125W$ and $U=1.175W$. The two solutions are obtained by increasing and decreasing the value of $U$ gradually to $1.175W$, respectively.

Figure 9

(a) Spectral function $A\left(\omega \right)$, (b) real part of self-energy, and (c) imaginary part of self-energy at different temperatures for a Bethe lattice. Here, $U=W$ with $W$ being the effective bandwidth.

Figure 10

Comparison of $A(\omega =0)$ vs $U$ between DMFT methods using HEOM and NRG impurity solvers. Various temperatures are examined. The inset shows the HEOM obtained $A(\omega =0)$ versus $U$ for temperature $T=0.0168W$ and $T=0.0125W$. The NRG data are extracted from Ref. [13].

Figure 11

Spectral functions obtained by the NRG approach used in Ref. [13] with different values of the controlling parameters $\Lambda$ and $M_s$, at $T=$ (a) $0.0168W$ and (b) $0.0741W$. The expression $M_s=[a,b]$ with numbers $b>a$ means that the actual value of $M_s$ is chosen in the interval $[a,b]$ with the largest energy gap. The hybridization functions are extracted from the HEOM converged results, and are used for both impurity solvers. The corresponding HEOM calculated spectral functions are also shown for comparison.