Full density-matrix numerical renormalization group calculation of impurity susceptibility and specific heat of the Anderson impurity model

Recent developments in the numerical renormalization group (NRG) allow the construction of the full density matrix (FDM) of quantum impurity models [see A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99, 076402 (2007)] by using the completeness of the eliminated states introduced by F.  B.  Anders and A.  Schiller [F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 (2005)]. While these developments prove particularly useful in the calculation of transient response and finite-temperature Green's functions of quantum impurity models, they may also be used to calculate thermodynamic properties. In this paper, we assess the FDM approach to thermodynamic properties by applying it to the Anderson impurity model. We compare the results for the susceptibility and specific heat to both the conventional approach within NRG and to exact Bethe ansatz results. We also point out a subtlety in the calculation of the susceptibility (in a uniform field) within the FDM approach. Finally, we show numerically that for the Anderson model, the susceptibilities in response to a local and a uniform magnetic field coincide in the wide-band limit, in accordance with the Clogston-Anderson compensation theorem.

Figure 1

Impurity contribution to the susceptibility from Bethe ansatz (${\chi }_{\mathrm{BA}}$) and FDM (${\chi }_{\mathrm{imp}}$) vs $T/T_{\mathrm{K}}$ for the symmetric Anderson model [$U/{\Delta }_0=12$ and ${\Delta }_0=0.001D$ with $T_{\mathrm{K}}$ defined in Eq. (20)]. Also shown are the impurity contributions $k_{\mathrm{B}}T{\chi }_{\mathrm{S},\mathrm{imp}}\left(T\right)/{\left(g{\mu }_{\mathrm{B}}\right)}^2$ and $k_{\mathrm{B}}T{\chi }_{\mathrm{E},\mathrm{imp}}\left(T\right)/{\left(g{\mu }_{\mathrm{B}}\right)}^2$ as defined at the end of Sec. 3. The calculations are for $\Lambda =10$ with an energy cutoff $e_c\left(\Lambda \right)=15\sqrt{\Lambda }\approx 47$, with $z$ averaging [$n_z=4$, $z=1/8,1/2,3/8,3/4$].

Figure 2

(a) Impurity specific heat, $C_{\mathrm{imp}}\left(T\right)$, and (b), impurity entropy, $S_{\mathrm{imp}}\left(T\right)$, vs reduced temperature $T/T_{\mathrm{K}}$ for the symmetric Anderson model with $U/{\Delta }_0=12,10,8,6,4,2,1$ and ${\Delta }_0=0.001D$. Broken lines: FDM approach. Solid lines: conventional approach. Selected Bethe ansatz results are shown as symbols for $U/{\Delta }_0=12,8,4$ (circles, squares, diamonds, respectively). The Kondo scale is defined in Eq. (20). As a guide to the eye, note that the high-temperature peak in $C_{\mathrm{imp}}$ shifts downwards with decreasing $U$. NRG and $z$-averaging parameters as in Fig. 1.

Figure 3

(a) Impurity specific heat, $C_{\mathrm{imp}}\left(T\right)$, and (b), impurity entropy $S_{\mathrm{imp}}\left(T\right)$, vs reduced temperature $T/T_{\mathrm{K}}$ for the asymmetric Anderson model with $U/{\Delta }_0=12$, ${\Delta }_0=0.001D$ and several values of ${\varepsilon }_d/{\Delta }_0=-5,-3,\cdots ,+5$. Broken lines: FDM approach. Solid lines: conventional approach. Bethe ansatz results are shown as symbols for ${\varepsilon }_d/\Delta$. For simplicity, we used the symmetric $T_{\mathrm{K}}$ of Eq. (20) for all ${\varepsilon }_d$ values. NRG and $z$-averaging parameters as in Fig. 1.

Figure 4

Impurity susceptibility, ${\chi }_{\mathrm{imp}}\left(T\right)$, vs $T/T_{\mathrm{K}}$ for the symmetric Anderson model with $U/{\Delta }_0=12,10,8,6,4,2,1,0.01$ and ${\Delta }_0=0.001D$ with $T_{\mathrm{K}}$ defined in Eq. (20) for $U/{\Delta }_0⩾1$ and $T_K={\Delta }_0$ for the $U/{\Delta }_0=0.01$ case. Broken lines: FDM approach. Solid lines: conventional approach. Symbols: Bethe ansatz (for selected values of $U/{\Delta }_0=12,8,6,4,2$). As a guide to the eye, note that ${\chi }_{\mathrm{imp}}$ is increasingly enhanced with increasing $U$. NRG and $z$-averaging parameters as in Fig. 1.

Figure 5

Impurity susceptibility, ${\chi }_{\mathrm{imp}}\left(T\right)$, vs $T/T_{\mathrm{K}}$ for the asymmetric Anderson model with $U/{\Delta }_0=12$, ${\Delta }_0=0.001D$ and several values of ${\varepsilon }_d/{\Delta }_0$ with $T_{\mathrm{K}}$ defined in Eq. (20). Broken lines: FDM approach. Solid lines: conventional approach. Symbols: Bethe ansatz (for selected values of ${\varepsilon }_d/{\Delta }_0=-5,-3,-1,0,+1,+3$). As a guide to the eye, note that the susceptibility curves shift to higher temperatures with increasing ${\varepsilon }_d$. NRG and $z$-averaging parameters as in Fig. 1.

Figure 6

Comparison of the local, ${\chi }_{\mathrm{loc}}\left(T\right)$, and impurity, ${\chi }_{\mathrm{imp}}$, susceptibilities vs $T/T_{\mathrm{K}}$ for the symmetric Anderson model with $U/{\Delta }_0=12,8,4,2$ and ${\Delta }_0=0.001D$ with $T_{\mathrm{K}}$ defined in Eq. (20). Broken lines: FDM approach. Symbols: Bethe ansatz (for selected values of $U/{\Delta }_0=12,8,4,2$). NRG and $z$-averaging parameters as in Fig. 1.

Figure 7

Comparison of the local, ${\chi }_{\mathrm{loc}}$, and impurity, ${\chi }_{\mathrm{imp}}\left(T\right)$, susceptibilities vs $T/T_{\mathrm{K}}$ for the asymmetric Anderson model with $U/{\Delta }_0=12$, ${\Delta }_0=0.001D$ with $T_{\mathrm{K}}$ defined in Eq. (20) and for several values of the local level position: ${\varepsilon }_d/{\Delta }_0=-5,-4,\cdots ,+5$. Broken lines: ${\chi }_{\mathrm{loc}}$ from the FDM approach. Symbols: ${\chi }_{\mathrm{imp}}$ from Bethe ansatz for selected values of ${\varepsilon }_d$. The NRG calculations are for $\Lambda =4$ with an energy cutoff $e_c(\Lambda =4)=40$ with $z$ averaging ($n_z=2$ with $z=0$ and $0.5$).

Figure 8

(a) Double occupancy $D_{\mathrm{occ}}=\langle n_{d\uparrow }{n}_{d\downarrow }\rangle$ as a function of temperature $T/T_{\mathrm{K}}$ for the symmetric Anderson model and decreasing values of $U/{\Delta }_0=12,10,8,6,4,2,1,0.01$ within FDM (solid lines) and conventional approaches (symbols). (b) Double occupancy $D_{\mathrm{occ}}=\langle n_{d\uparrow }{n}_{d\downarrow }\rangle$ as a function of temperature $T/T_{\mathrm{K}}$ for the asymmetric Anderson model and increasing values of ${\varepsilon }_d/{\Delta }_0=-5,-3,-1,0,+1,+3,+5$ for $U/{\Delta }_0=12$ within FDM (solid lines) and conventional approaches (symbols). $T_{\mathrm{K}}$ is defined in Eq. (20) for $U/{\Delta }_0⩾1$ and is set to ${\Delta }_0$ for the case $U/{\Delta }_0=0.01$. NRG and $z$-averaging parameters as in Fig. 1.