Matrix-product-state comparison of the numerical renormalization group and the variational formulation of the density-matrix renormalization group

Wilson’s numerical renormalization group (NRG) method for solving quantum impurity models yields a set of energy eigenstates that have the form of matrix product states (MPS). White’s density-matrix renormalization group (DMRG) for treating quantum lattice problems can likewise be reformulated in terms of MPS. Thus, the latter constitute a common algebraic structure for both approaches. We exploit this fact to compare the NRG approach for the single-impurity Anderson model with a variational matrix product state approach (VMPS), equivalent to single-site DMRG. For the latter, we use an “unfolded” Wilson chain, which brings about a significant reduction in numerical costs compared to those of NRG. We show that all NRG eigenstates (kept and discarded) can be reproduced using VMPS, and compare the difference in truncation criteria, sharp vs smooth in energy space, of the two approaches. Finally, we demonstrate that NRG results can be improved upon systematically by performing a variational optimization in the space of variational matrix product states, using the states produced by NRG as input.

Figure 1

(Color online) (a) The standard spinful or “folded” representation of the Wilson chain of the single-impurity Anderson model, and (b) its spin-resolved or “unfolded” representation. The latter makes explicit that spin-down and -up states are coupled only at the impurity sites and not at any of the bath sites. The dashed boxes indicate the chains described by ${\mathcal{H}}_1$ and ${\mathcal{H}}_n$, respectively.

Figure 2

(Color online) (a) and (b) show the matrix product structure of the state ${|E_{\beta }^N⟩}_f$ of Eq. (7), depicting the site index as a single or composite index, ${\sigma }_n$ or $\left({\sigma }_{n\downarrow },{\sigma }_{n\uparrow }\right)$, respectively. (c) shows the matrix product structure of the state ${|{\Psi }^N⟩}_u$ of Eq. (15). [For the sake of illustrating Eq. (A9) of Appendix A2, the labels ${\left(B_{n\downarrow }\right)}_{\nu {\nu }^{\prime }}$ in the bottom row are purposefully typeset “upside down,” so that they would be right-side up if the chain of boxes were all drawn in one row in the order indicated by Eq. (15). Thus, the latter contains the factors $\dots {\left(B_{n\downarrow }\right)}_{\nu {\nu }^{\prime }}\dots {\left(B_{n\uparrow }\right)}_{{\eta }^{\prime }\eta }\dots$, in that order, compare Eq. (A9).] Each matrix $A$ or $B$ is represented by a box, summed-over indices by links, free indices by terminals, and dummy indices having just a single value, namely 1, by ending in a triangle. The dimensions ($d$, $D$, $d^{\prime }$, $D^{\prime }$, etc.) next to each link or terminal give the number of possible values taken on by the corresponding index, assuming Wilsonian truncation for (a) and (b), and VMPS truncation for (c). Note the similarity in structure between (c) and (b): the dashed boxes in the former, containing $B_{\nu {\nu }^{\prime }}^{\left[{\sigma }_{n\downarrow }\right]}\otimes B_{{\eta }^{\prime }\eta }^{\left[{\sigma }_{n\uparrow }\right]}$, play the role of the $A_{{\alpha }^{\prime }\alpha }^{\left[\left({\sigma }_{n\downarrow },{\sigma }_{n\uparrow }\right)\right]}$ matrices in the latter. Their capacity for entangling neighboring sites is comparable if one chooses $D^{\prime 2}\propto D$ [cf. Eq. (23)], since neighboring dashed boxes in (c) are connected by two links of combined dimension $D^{\prime 2}$, whereas neighboring $A$-matrices in (b) are connected by only a single link of dimension $D$.

Figure 3

(Color online) NRG result for the mutual information $M_n^{\downarrow \uparrow }$ between spin-down and -up degrees of freedom of site $n$ of a folded Wilson chain of lenght $N=50$. The Anderson model parameters are fixed at $U=\frac{1}{2}$, $U/\pi \Gamma =1.013$, ${ϵ}_d=-\frac{1}{2}U$ throughout this paper. Lines connecting data points are guides for the eye. The slight differences in behavior observed for even or odd $n$ are reminiscent of the well-known fact (Ref. 1) that the ground-state degeneracy of a Wilson chain is different for even or odd $N$.

Figure 4

(a) Typical singular value spectrum for site $5\downarrow$ of the unfolded Wilson chain, obtained by singular value decomposition of $B^{\left[{\sigma }_{5\downarrow }\right]}$. It shows, roughly, power-law decrease for large enough $\beta$, modulo steps due to degeneracies in the singular value spectrum. (b) $D^{\prime }$-dependence of the truncation error $\tau \left(D^{\prime }\right)$ [Eq. (18)].

Figure 5

(Color online) Comparison of NRG and VMPS results for (a,b) the ground-state energies and (c) the ground-state overlaps, plotted as functions of $D$ with $D^{\prime }=d^{\prime m}\sqrt{D}$, for three values of $\Lambda$ and, in (a), for two values of $m$. In (a) the reference energies $E_{\text{ref}}^N$ for each $\Lambda$ were obtained by extrapolating the VMPS data points for $m=2$ to $D^{\prime }\to \infty$, which represents the best estimate of the true ground-state energy available within the present set of methods. The power law fits to the numerical data in (b) and (c), shown as dashed lines, were made for the three data points with largest $D$, for which the dimensions are large enough to have reliable NRG data.

Figure 6

(Color online) Contraction patterns used to calculate (a) the overlap ${{}_f⟨E_{\text{G}}^N|E_{\text{G}}^N⟩}_u$ [Eqs. (24)] between folded NRG and unfolded VMPS ground states, and (b) the overlap matrix ${\tilde{S}}_{\alpha \beta }^n={{}_r⟨{\Psi }_{\alpha }^n|E_{\beta }^n⟩}_f$ [Eq. (27a, 27b)] between refolded VMPS basis states and folded NRG eigenstates. The boxes represent $A$ or $B$ matrices in the graphical representation of Figs. 2a, 2c, respectively (including the labeling conventions used there), and links connecting them represent indices that are being summed over.

Figure 7

(Color) Comparison of energy flow diagrams from NRG (dashed red lines) and refolded VMPS data (solid black lines), showing the rescaled energies ${\left({\epsilon }_{\beta }^n\right)}_{f,r}$ [Eq. (9)] versus $n$, calculated for even iteration numbers and four combinations of $m$ ($=0$ or 2) and $\Lambda$ ($=1.5$ or 2.5). The number of NRG states shown (kept and discarded) is $Dd$; the number of refolded VMPS states shown is $D^r=D^{\prime 2}=d^{m}D$, this being the maximal dimension of refolded matrices $B^{\left[{\sigma }_n\right]}$. For $m=2$ and $\Lambda =2.5$, the NRG and DMRG flow diagrams agree very well, see (d).

Figure 8

(Color) Results for the density of states, ${\rho }_n\left(\epsilon \right)$ [Eq. (25)], broadened with a Gaussian broadening function. In each panel, the red vertical dashed and solid lines [which coincide in (a)] indicate the energies of the highest-lying kept and discarded NRG states of that iteration, while the shaded area indicates the range of kept NRG states.

Figure 9

(Color) Plot of the overlap matrix $S_{\alpha \beta }^n={{}_r⟨E_{\alpha }^n|E_{\beta }^n⟩}_f$ [Eq. (26)] between refolded VMPS and NRG energy eigenstates, with a color scale ranging between 0 and 1. In (a), with $n=2$, no truncation occurs at all, and both state labels $\alpha$ and $\beta$ run from 1 to $d^{n+1}=64$. In (b) to (d), truncation does occur: For the folded NRG eigenstates ${|E_{\beta }^n⟩}_f$, the label $\alpha$ runs from 1 to $Dd=1024$, i.e., it includes all kept and discarded NRG states, while for the refolded VMPS eigenstates ${|E_{\beta }^n⟩}_r$, the label $\beta$ runs from 1 to $D_n^r=D^{\prime 2}=1024$ [Eq. (20)].

Figure 10

(Color) For several NRG iteration numbers $n$ and two values of $\Lambda$ (different panels), this figure shows the weights $w_{\beta }^{\left(n\right)}$ [Eq. (28)] with which NRG states ${|E_{\beta }^n⟩}_f$ with rescaled NRG eigenenergies ${\left({\epsilon }_{\beta }^n\right)}_f$ [Eq. (9)] are found to lie in the VMPS-subspace of dimension $D^{\prime }=d^{\prime m}\sqrt{D}$, with $m=0$, 1 or 2 (indicated by $+$, $\times$ or $○$, respectively). In each panel, the red vertical dashed and solid lines indicate the energies of the highest-lying kept and discarded NRG states of that iteration. For $n=3$, both of these lines are missing, since truncation has not yet set in. The choices for $n$ in the left and right panels of each row are related by ${\Lambda }_1^{-n_1/2}={\Lambda }_2^{-n_2/2}$, to ensure that both panels show a comparable energy scale.

Figure 11

(Color online) Integrated weights $W_{\text{X}}^{\left(n\right)}$ [see Eq. (29)] for two different $\Lambda$ and three values of $m$. The dashed lines depict the maximum possible values of the kept and discarded weights, $\frac{1}{4}$ and $\frac{3}{4}$ for $W_{\text{K}}^n$ and $W_{\text{D}}^n$, respectively.

Figure 12

(Color online) The deviation of the overlap $\left|{{}_{\text{c}}⟨\text{G}|\text{G}⟩}_f\right|$ from 1 (red circles) and the cloning truncation error $\tau \left(D^{\prime }\right)$ (blue squares), as functions of the number $D^{\prime }$ of kept states in the cloning procedure. Both approach 0 in power-law fashion, as indicated by the dashed line fits. The inset shows how the overlap deviation from 1 decreases and converges to a small but finite constant in the course of sequential cloning sweeps.

Figure 13

(Color online) Comparison of (a) energies and (b) wave function overlaps for random initialization (squares) vs NRG-cloned initialization (circles), as functions of the number $k_{\text{em}}$ of variational energy minimization sweeps. Results are shown for $\Lambda =1.5$ (green, open symbols, dashed lines) and $\Lambda =2.5$, (blue, filled symbols, solid lines). The energies in (a) and overlaps in (b) are calculated with respect to a reference ground state ${|\text{G}⟩}_{\text{ref}}$ with $D^{\prime }=64$, obtained by performing 50 energy minimization sweeps starting from random initialization. The red horizontal straight lines in (a) (dashed or solid for $\Lambda =1.5$ or 2.5, respectively), show the energy difference $E_{\text{NRG}}-E_{\text{ref}}$, where $E_{\text{NRG}}$ is the energy of the NRG ground state ${|\text{G}⟩}_f$ used as input into cloning. The fact that $E_{\text{NRG}}$ does not completely coincide with the energy $E_{\text{c}}=E_{k_{\text{em}}=0}$ of the cloned state (horizontal straight lines do not meet circles at $k_{\text{em}}=0$) is due to the fact that the deviation of the overlap $\left|{{}_{\text{c}}⟨\text{G}|\text{G}⟩}_f\right|$ from 1 is not strictly equal to 0 (see Fig. 12).

Figure 14

(Color online) Graphical representation of the variational equation used for cloning, Eqs. (32, A14), drawn for the case $\mu =\uparrow$, and assuming all matrix elements to be real, using the labeling conventions of Figs. 2a, 2c. The upper part of the figure represents $\frac{1}{2}\frac{\partial }{\partial B^{\left[k\uparrow \right]}}{{}_{\text{c}}⟨G|G⟩}_{\text{c}}$; it simplifies to $B^{\left[k\uparrow \right]}$ [left hand side of Eq. (A14)] upon realizing that the parts in dashed boxes represent the left-hand sides of Eqs. (A12a, A12b), and hence reduce to unity.