Projective truncation approximation for equations of motion of two-time Green's functions

In the equation of motion approach to the two-time Green's functions, conventional Tyablikov-type truncation of the chain of equations is rather arbitrary and apt to violate the analytical structure of Green's functions. Here, we propose a practical way to truncate the equations of motion using operator projection. The partial projection approximation is introduced to evaluate the Liouville matrix. It guarantees the causality of Green's functions, fulfills the time translation invariance and the particle-hole symmetry, and is easy to implement in a computer. To benchmark this method, we study the Anderson impurity model using the operator basis at the level of Lacroix approximation. Improvement over conventional Lacroix approximation is observed. The distribution of Kondo screening in the energy space is studied using this method.

Figure 1

(a) $\langle n_{\uparrow }\rangle$ and (b) $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ as functions of impurity energy level ${\epsilon }_d$. Other parameters are $U=2.0, T=0.1, \delta \omega =0.0$, and $\mathrm{\Delta }=0.1$.

Figure 2

(a) $\langle n_{\uparrow }\rangle$ and (b) $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ as functions of magnetic bias $\delta \omega$ of the bath. Other parameters are $U=2.0, {\epsilon }_d=-U/2, T=0.1$, and $\mathrm{\Delta }=0.1$.

Figure 3

(a) $\langle n_{\uparrow }\rangle$ and (b) $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ as functions of $T$. Other parameters are $U=2.0, {\epsilon }_d=-U/2, \delta \omega =-0.2$, and $\mathrm{\Delta }=0.1$.

Figure 4

Comparison of LDOS from pLacroix (dashed lines) and NRG (solid lines). From top to bottom at $\omega =0, U=0.0$, 1.0, 2.0, 3.0, and 4.0, respectively. Other parameters are $T=0.1, {\epsilon }_d=-U/2, \delta \omega =0.0$, and $\mathrm{\Delta }=0.1$.

Figure 5

(a) LDOS obtained from pLacroix at different temperatures. (b) $\rho \left(0\right)$ as functions of temperature. Other parameters are $U=1.0, {\epsilon }_d=-U/2.0, \delta \omega =0.0$, and $\mathrm{\Delta }=0.1$. The numerical parameters are $s=0.2$ and $\eta =0.01$.

Figure 6

Impurity density of states from (a) NRG and (b) pLacroix calculations in the Kondo to mixed valence crossover regime, for the parameters ${\epsilon }_d=-0.7$ (solid black lines), ${\epsilon }_d=-0.3$ (red dashed lines), and ${\epsilon }_d=-0.1$ (green dash-dotted lines). The vertical dashed line marks the Fermi energy. Other parameters are $U=2.0, T=0.001, \delta \omega =0.0$, and $\mathrm{\Delta }=0.1$. The numerical parameters are $s=0.0$ and $\eta =0.02$. Inset: impurity density of states of cLacroix.

Figure 7

Checking the Fermi liquid properties in pLacroix and cLacroix results. (a) Checking $\text{Im}{G}_{\uparrow }^{-1}(0+i\eta )=\pi \mathrm{\Delta }$ for different $U$ values with ${\epsilon }_d=-U/2$. $\eta ={10}^{-3}$ is used. (b) Checking $\text{Re}\left[G_{\uparrow }^{-1}(0+i\eta )\right]=-\pi \mathrm{\Delta }\text{cot}\left(\pi n_{\uparrow }\right)$ in the wide band limit ${\omega }_c=200.0$ for different ${\epsilon }_d$ values at $U=1.0$. Other parameters are $T=0.0, \delta \omega =0.0$, and $\mathrm{\Delta }=0.1$.

Figure 8

Spin-spin correlation function $\langle S_k^{†}{S}_d^{-}\rangle =\langle c_{k\uparrow }^{†}{c}_{k\downarrow }{d}_{\downarrow }^{†}{d}_{\uparrow }\rangle$ of impurity and bath electrons as functions of ${\epsilon }_k$ for different temperatures. The parameters are $U=2.0, {\epsilon }_d=-U/2, \delta \omega =0.0$, and $\mathrm{\Delta }=0.1$. Inset: comparison of $\langle S_k^{†}{S}_d^{-}\rangle$ and $2\langle S_{kz}{S}_{dz}\rangle =\langle n_{k\uparrow }(n_{\uparrow }-n_{\downarrow })\rangle$ at $T=0.001$.