Interleaved numerical renormalization group as an efficient multiband impurity solver

Quantum impurity problems can be solved using the numerical renormalization group (NRG), which involves discretizing the free conduction electron system and mapping to a “Wilson chain.” It was shown recently that Wilson chains for different electronic species can be interleaved by use of a modified discretization, dramatically increasing the numerical efficiency of the RG scheme [Phys. Rev. B 89, 121105(R) (2014)]. Here we systematically examine the accuracy and efficiency of the “interleaved” NRG (iNRG) method in the context of the single impurity Anderson model, the two-channel Kondo model, and a three-channel Anderson-Hund model. The performance of iNRG is explicitly compared with “standard” NRG (sNRG): when the average number of states kept per iteration is the same in both calculations, the accuracy of iNRG is equivalent to that of sNRG but the computational costs are significantly lower in iNRG when the same symmetries are exploited. Although iNRG weakly breaks SU($N$) channel symmetry (if present), both accuracy and numerical cost are entirely competitive with sNRG exploiting full symmetries. iNRG is therefore shown to be a viable and technically simple alternative to sNRG for high-symmetry models. Moreover, iNRG can be used to solve a range of lower-symmetry multiband problems that are inaccessible to sNRG.

Figure 1

Schematic illustration of standard (left) and interleaved (right) Wilson chains, for a spinful single-channel model ($m=2$). (a) In sNRG, subsites for spin up (red) and spin down (blue) are grouped into supersites (indicated by dashed boxes), which are connected by nearest-neighbor hopping (thin lines). (b) In iNRG, subsites are interleaved in linear fashion and hopping occurs between next-nearest neighbors. $n$ labels sNRG supersites, while $\tilde{n}$ labels iNRG subsites. (c) and (d) Depictions of the MPS-structure used for sNRG and iNRG: boxes represent MPS tensors, vertical thin legs represent local state spaces of dimension $d_{\mathrm{f}}$, and thick diagonal lines represent the state spaces obtained after diagonalizing a Wilson shell and discarding all but the lowest $N_{\mathrm{K}}$ states (the truncation process is indicated by scissors). The matrix size to be diagonalized is reduced from $N_{\mathrm{tot}}^{\mathrm{sNRG}}=N_{\mathrm{K}}\times d_{\mathrm{f}}^m$ in sNRG to $N_{\mathrm{tot}}^{\mathrm{iNRG}}=N_{\mathrm{K}}\times d_{\mathrm{f}}$ in iNRG. The vertical positions of the boxes reflect, on a logarithmic scale, the characteristic energies ${\omega }_n$ (sNRG) and ${\tilde{\omega }}_{\tilde{n}}$ (iNRG) of each shell. The additional energy-scale separation within each supershell justifies the additional truncations in iNRG.

Figure 2

Comparison of sNRG and iNRG for three models: SIAM (left column), 2CKM (middle column), and 3CAHM (right column). (a)–(c) The number of kept multiplets, $N_{\mathrm{K}}^{*}$; (d)–(f) the total number of multiplets generated during an NRG step, $N_{\mathrm{tot}}^{*}$; (g)–(i) the total CPU time for one NRG run; (j)–(l) the relative deviations $\delta A\left(0\right)/A\left(0\right)$ of correlation functions at the Fermi energy $\omega =0$ from their exact values [cf. Eqs. (21)]; and (m)–(o) the discarded weight $\delta {\rho }_{\mathrm{disc}}$ (the horizontal dashed lines indicate the convergence threshold). All quantities (computable with a maximum memory of 128 GB) are plotted versus $E_{\mathrm{trunc}}$ ($\equiv E_{\mathrm{trunc}}^{\mathrm{sNRG}}$), for $\mathrm{\Lambda }=1.7$ (crosses), 2.0 (triangles), and 4.0 (circles). Each symmetry setting is identified by a particular color in iNRG and sNRG. iNRG results have been geometrically averaged over all interleaved flavors. Data for $N_{\mathrm{K}}^{*}$ and $N_{\mathrm{tot}}^{*}$ have been geometrically averaged over even and odd Wilson shells at an energy scale $E_{\mathrm{ref}}=5\times {10}^{-8}D\ll T_K$. We used $z=0$ in all cases except for (j)–(l), where data for $z=0$ and 0.5 have been averaged. An exception is the flavor-iNRG data point at $E_{\mathrm{trunc}}=7$ in (l), which was obtained for $z=0$ without $z$ averaging ($z=0.5$ exceeded memory resources). In (j)–(l), sNRG results with the same $\mathrm{\Lambda }$ but different symmetry settings coincide.

Figure 3

Number of kept multiplets $N_{\mathrm{K}}^{*}$ vs Wilson shell index $n$ for the 2CKM within the $\text{SU}{\left(2\right)}_{\mathrm{spin}}\times {\left[\text{SU}{\left(2\right)}_{\mathrm{charge}}\right]}^2$ symmetry setting. For iNRG (red), the number of multiplets kept after adding channel 1 (dashed) or channel 2 (dash-dotted) are shown separately, as well as their geometric average (solid). sNRG results are shown in blue for comparison. All results are geometrically averaged over even and odd iterations.

Figure 4

Number of kept states $N_{\mathrm{K}}$ at the low-energy fixed point of the $N_{\mathrm{c}}\mathrm{CAHM}$ (black) and $N_{\mathrm{c}}\mathrm{CKM}$ (blue), showing a roughly exponential increase with the number of channels, $N_{\mathrm{c}}$. Results were obtained with sNRG and have been geometrically averaged over even and odd Wilson shells at energy scale $E_{\mathrm{ref}}=5\times {10}^{-8}D\ll T_K$.

Figure 5

Comparison of physical quantities calculated with iNRG and sNRG. (a) Impurity spectral function $\pi \mathrm{\Gamma }A\left(\omega \right)$ at $T=0$ for the SIAM; (b) impurity spectral function $\pi \mathrm{\Gamma }A\left(\omega \right)$ at $T=0$ for the 3CAHM; (c) spectrum of the $t$ matrix $t\left(\omega \right)$ and the correlator $A_{{\mathrm{f}}_0}\left(\omega \right)$ at $T=0$ for the 2CKM; (d) impurity contribution to the entropy, $S_{\mathrm{imp}}\left(T\right)$, for the 2CKM. For the dynamical correlators shown in (a)–(c), the protocol of Ref. [31] was used to broaden discrete data, using a broadening parameter of ${\sigma }_{\mathrm{broad}}=0.8$ or 1 for $\mathrm{\Lambda }=2.0$ or 4.0, respectively. All quantities were $z$-averaged over $z=0$ and $0.5$, except for the flavor-interleaved 3CAHM spectral function in (b), which was only calculated for $z=0$ ($z=0.5$ exceeded memory resources). The observed low-frequency oscillations are therefore an artifact of underbroadening, and would be removed by additional $z$ averaging or use of a larger ${\sigma }_{\mathrm{broad}}$.

Figure 6

Fine tuning in iNRG for the 2CKM (a) Flow of iNRG many-particle energies with Wilson shell index $n$ [solid (dashed) lines for even (odd) $n]$ for the 2CKM. Different colors correspond to states with different quantum numbers. (b) The Fermi liquid crossover scale $T_{\text{FL}}$ can be extracted from the flow of the first excited state (thick black line): we fit its large-$n$ behavior with a power law (dashed red line), take $n\left(T_{\text{FL}}\right)$ to be the iteration number [vertical grey line in (a) and (b)] at which this power law reaches half of the fixed-point value of this state (horizontal grey line), and define the Fermi liquid scale as $T_{\text{FL}}={\omega }_{n\left(T_{\text{FL}}\right)}$. In (c) and (d), the resulting values of $T_{\mathrm{FL}}$ are plotted as function of $J_2-J_2^c$ on a log-log or linear plot, respectively, using red (blue) symbols for $J_2>J_2^c$ ($<J_2^c$). Grey lines give the asymptotic form $T_{\mathrm{FL}}\sim {(J_2-J_2^c)}^2$. By using an extrapolative protocol, the critical coupling $J_2^c$ can be located exponentially rapidly in the number of separate iNRG runs. [Inset to (d)] The difference between the critical coupling $J_2^c$ and $J_1$, plotted as a function of the truncation energy.

Figure 7

Schematic depiction of the sNRG (blue) and iNRG (red) truncation schemes used here, illustrated for a model with $m=3$ flavors. The vertical axis corresponds to absolute energies on a logarithmic scale, while the Wilson (sub)shell index is given on the horizontal axis. The lower part of the sketch depicts the evolution of the characteristic energies ${\omega }_n$ and ${\tilde{\omega }}_{\tilde{n}}$ for sNRG and iNRG. Thin faint blue and red lines depict the excitation eigenenergies (relative to the ground state energy) of sNRG supershells or iNRG subshells. In sNRG, all three subsites comprising the supersite for that iteration are added at once (there is no intermediate truncation), while in iNRG the subsites are added separately, and truncation occurs at each step. The absolute truncation energies ${\text{E}}_{\text{abs-trunc}}^{\mathrm{sNRG}}$ and ${\text{E}}_{\text{abs-trunc}}^{\mathrm{iNRG}}$ therefore form two different staircases, depicted as the thick blue and red lines, respectively (the step width for sNRG is $m$ times longer than that of iNRG). States with higher energies are discarded. The truncation pattern of sNRG, when viewed from the perspective of iNRG, amounts to employing the effective truncation energy ${\text{E}}_{\text{abs-trunc,eff}}^{\mathrm{sNRG}}$, shown as the black dashed line: by using a high truncation threshold (that of the previous iteration, $E_{\mathrm{trunc}}^{\mathrm{sNRG}}\times {\omega }_{n-1}$) for the first $m-1$ subsites, and then dropping to $E_{\mathrm{trunc}}^{\mathrm{sNRG}}\times {\omega }_n$ only for the last subsite, all states are effectively kept until the supersite is complete. Viewed from this iNRG perspective, the truncation energies of iNRG and sNRG are the same on average (green dotted line for supersite $n=1$) and the areas under the red solid, black dashed, and green dotted lines are the same, provided $E_{\mathrm{trunc}}^{\mathrm{sNRG}}$ and $E_{\mathrm{trunc}}^{\mathrm{iNRG}}$ are related via Eq. (15).

Figure 8

Estimating the discarded weight, $\delta {\rho }_{\mathrm{disc}}$, of a single NRG run. Colored dots show the diagonal weights of the reduced density matrices in the energy eigenbasis of NRG. Different colors represent the weights for different NRG iterations. We calculate and plot the cumulative weights (blue open circles) using 16 bins in the energy window $[0,E_{\mathrm{trunc}}]$. The truncation energy $E_{\mathrm{trunc}}=7$ is indicated by the vertical black dashed line. The red dashed line is an exponential fit to the cumulative weights; its slope gives $\kappa$ as defined in Eq. (B1). The black line shows the normalized integrated weight distribution $\rho \left(E\right)=\kappa {\mathrm{e}}^{-\kappa E}$, extrapolated to energies $E>E_{\mathrm{trunc}}$. The shaded grey area under this black line then serves as estimate for the discarded weight: $\delta {\rho }_{\mathrm{disc}}={\mathrm{e}}^{-\kappa E_{\mathrm{trunc}}}$. This example is well-converged, with $\kappa =2.37$ yielding $\delta {\rho }_{\mathrm{disc}}=6.23\times {10}^{-8}$.