Sum-Rule Conserving Spectral Functions from the Numerical Renormalization Group

We show how spectral functions for quantum impurity models can be calculated very accurately using a complete set of discarded numerical renormalization group eigenstates, recently introduced by Anders and Schiller. The only approximation is to judiciously exploit energy scale separation. Our derivation avoids both the overcounting ambiguities and the single-shell approximation for the equilibrium density matrix prevalent in current methods, ensuring that relevant sum rules hold rigorously and spectral features at energies below the temperature can be described accurately.

Figure 1

Diagram for the kept (or discarded) matrix product state $|s^{\prime }{⟩}_n^K$ (or $|s^{\prime }{⟩}_n^D$): the $n$th box represents the matrix block $A_{KX}^{[{\sigma }_n]}$, its left, bottom, and right legs carry the labels of the states $|s{⟩}_{n-1}^K$, $|{\sigma }_n⟩$, and $|s^{\prime }{⟩}_n^K$ (or $|s^{\prime }{⟩}_n^D$), respectively.

Figure 2

FDM-NRG results for the spectral function ${\mathcal{A}}_T(\omega )$ of the SIAM, with $U=0.12$, $\Gamma =0.01$, ${ϵ}_d=-U/2$ ($T_K=2.185\times {10}^{-4}$), $\Lambda =1.7$, and $M_K=1024$, unless indicated otherwise. Inset of (a): FDM-NRG result for $A_T(\omega )$ with $\omega$ in units of bandwidth. For (a),(b), an unconventionally small smearing parameter was used, ${\omega }_0=0.005T$ [except for thick gray (red) curve in (a)], with ${\omega }_0=0.5T$), leading to spurious low-frequency oscillations. These illustrate the differences (a) between NRG (dashed green curve), DM-NRG [solid thin (blue) curve), and FDM-NRG (black curve) results for the regime $\omega \lesssim T$, and (b) between different choices of $M_K$ and $\Lambda$ for FDM-NRG, which yield different shapes for the weights $w_n$ [shown in inset of (b)]: larger $\Lambda$ reduces the scale ${\delta }_T$ at which oscillations set in, but yields less accurate values for the Kondo peak height in the regime ${\delta }_T\lesssim \omega \lesssim T_K$. (c),(d) Comparison of high-quality FDM-NRG data (dots, solid curves) with exact Fermi-liquid results (black dashed lines) for (c) the conductance $G(T)$ for $T\ll T_K$, and (d) for ${\mathcal{A}}_T^{\mathrm{im}}(\omega )$ for $T,\omega \ll T_K$. In (c), $c_{\mathrm{fit}}$ was found from a data fit to $c_{\mathrm{fit}}(T/T_K{)}^2$ for $T<T_{\mathrm{fit}}$ (arrow). In (d) we plot $\delta {\mathcal{A}}_T(\omega )=[A_T^{\mathrm{im}}(\omega )-A_T^{\mathrm{im}}(0)]/A_0^{\mathrm{im}}(0)$ vs $\omega /T_K$ (curves) and $\delta \mathcal{A}(T)=[A_T^{\mathrm{im}}(0)/A_0^{\mathrm{im}}(0)-1]$ vs $(T/T_K)\pi /\sqrt{3}$ (dots), for a set of 12 temperatures between 0.001 and $0.069T_K$ (with curves and dots having same $T$ in the same color), to illustrate the leading $\omega$ and $T$ behavior of ${\mathcal{A}}_T^{\mathrm{im}}(\omega )$; the dashed black line represents the expected Fermi-liquid behavior in both cases, $-(3c/2{\pi }^2)x^2$ vs $x$.