Generalized Spectral Decomposition for Quantum Impurity Problems

TL;DR

This work tackles the challenge of solving multi-impurity quantum impurity problems with a shared bath by introducing a general strategy to disentangle the bath into $N_A$ auxiliary baths via an $\omega$-independent transformation $w_{\mu p}$, recasting the problem as a spectral-decomposition of the matrix $\rho_{\mu\nu}(\omega)$. It establishes exact decompositions for highly symmetric cases: Theorem 1 handles regular $N$-gons with circulant symmetry, and Theorem 2 extends to block circulant matrices using $w^{(n)}\otimes U$, with extensions to $C_n\otimes C_m$; Theorem 3 further shows any configuration whose graph embeds into a larger circulant-symmetric graph can be exactly decomposed (potentially with $N_A>N$). When exact decomposition is impractical, a graph-based embedding approach or an approximate HOSVD-based decomposition (yielding a reduced set of auxiliary baths) enables tractable NRG calculations for realistic multi-impurity systems. Collectively, the paper provides a unifying framework linking quantum impurity problems to graph theory and offers concrete procedures to extend NRG to complex impurity configurations with controlled approximations.

Abstract

Solving quantum impurity problems may advance our understanding of strongly correlated electron physics, but its development in multi-impurity systems has been greatly hindered due to the presence of shared bath. Here, we propose a general operation strategy to disentangle the shared bath into multiple auxiliary baths and relate the problem to a spectral decomposition problem of function matrix for applying the numerical renormalization group (NRG). We prove exactly that such decomposition is possible for models satisfying (block) circulant symmetry, and show how to construct the auxiliary baths for arbitrary impurity configuration by mapping its graph structure to the subgraph of a regular impurity configuration. We further propose an approximate decomposition algorithm to reduce the number of auxiliary baths and save the computational workload. Our work reveals a deep connection between quantum impurity problems and the graph theory, and provides a general scheme to extend the NRG applications for realistic multi-impurity systems.

Figures (4)

• Figure 1: (a) Disentanglement of the shared bath (left) into multiple auxiliary baths (right), each coupled in a similar momentum-independent form to all impurities. (b) Decomposition of the spectral function matrix $\rho_{\mu\nu}(\omega)$ with the $\omega$-independent transformation matrix $w$ as given in Eq. (\ref{['basic']}).
• Figure 2: (a) Examples of regular impurity configurations (regular triangle, rectangle, regular hexagon) on a triangle lattice with the local, nearest, next-nearest, and next-next-nearest-neighbor $\rho_{\mu\nu}(\omega)$ denoted by $\rho_0$, $\rho_1$, $\rho_2$, and $\rho_3$, respectively. (b)(c)(d) Illustrations of the auxiliary-bath models with $N_A = N$ for configurations shown in (a). Also given are their respective transformation parameters $w_{\mu p}$ and densities of states $\tilde{\rho}_p(\omega)$ of the auxiliary baths. For comparison, (b) also shows the local and nearest-neighbor $\rho_{\mu\nu}(\omega)$ of the original bath (the bandwidth is set to 1.5). (e) Extension of a two-impurity configuration into a rectangle, a straight triangular prism, and a cuboid, whose spectral function matrices are block circulant.
• Figure 3: (a) Illustration of the graph method by mapping a general triangle to the subgraph of a rectangle and then doing the decomposition by ignoring the additional impurity. (b)(c) 4-impurity configurations that can be embedded into a regular pentagon or hexagon. (b) shows two quadrilaterals with different graph structures (equivalent edges). (d) Mapping of a general quadrilateral to a tetrahedron and then to the subgraph of a cuboid, giving the decomposition with a reduced $N_A=8$ instead of $3\cdot2^{N-2} = 12$ from regular polygons.
• Figure 4: (a) Illustration of the approximate decomposition procedure using the HOSVD. (b) Comparison of approximate solutions with $N_A=4$ and 5 for a special 4-impurity model where three impurities form a right-angled triangle and the rest one is located at the midpoint of the hypotenuse. Also compared are the errors and their means induced by discarding the off-diagonal elements after the HOSVD.