Tree tensor network solver for real-time quantum impurity dynamics

TL;DR

The paper tackles the challenge of obtaining accurate real-time dynamics and high-resolution real-frequency spectra for quantum impurity models within dynamical mean-field theory. It introduces a tree tensor network impurity solver that maps the bath to a Cayley-tree, distributing entanglement across a hierarchical network. Benchmarking on the single-impurity Anderson model demonstrates faster ground-state convergence, favorable entanglement scaling, and improved low-frequency spectral accuracy compared with matrix-product-state approaches, without requiring analytic continuation. The approach is scalable and adaptable, with clear paths to multi-impurity generalizations and integration into DMFT loops and high-performance computing architectures.

Abstract

We introduce a tree tensor network (TTN) impurity solver that enables highly efficient and accurate real-time simulations of quantum impurity models. By decomposing a noninteracting bath Hamiltonian into a Cayley tree, the method provides a tensor network representation that naturally captures the multiscale entanglement structure intrinsic to impurity-bath systems. This geometry differs from conventional chain-based mappings and yields a substantial reduction of entanglement, allowing accurate ground-state properties and long-time dynamics to be captured at significantly lower bond dimensions. Benchmark calculations for the single-impurity Anderson model demonstrate that the TTN solver achieves markedly enhanced resolution of real-frequency spectral functions, without invoking analytic continuation. This impurity solver provides a balanced, scale-uniform description of impurity physics and offers a versatile approach for real-time dynamical mean-field theory and related applications involving quantum impurity models.

Figures (11)

• Figure 1: Three equivalent representations of the SIAM: (a) a star geometry representation, (b) a one-dimensional chain representation, and (c) a binary-tree representation. The interacting impurity site is shown in orange, while the noninteracting fermionic bath sites are shown in white. Links indicate hopping (hybridization) between sites.
• Figure 2: The SIAM in an MPS (a) and TTN (b) representation with explicit spin separation. The left and right branches correspond to the spin-up and spin-down channels, respectively. Orange nodes denote the impurity spins. The quantity $|R|$ measures the minimal graph distance of a bath site from the impurity along the connecting path, with $R=0$ assigned to the separation between the two impurity spins. Positive (negative) values of $R$ label the coordinate (or layer) of spin-up (down) bath sites.
• Figure 3: Ground-state energy convergence. Ground state energy difference, $\Delta E=E(D)-E_0$, as a function of the bond dimension $D$. The reference energy $E_0$ is obtained by extrapolating the DMRG results to the $D\to\infty$ limit. The inset shows the same data on a log–log scale. Dashed lines denote power-law fits of the form $\Delta E = \alpha D^{-\beta}$, where $(\alpha, \beta) = (11048, 5.39)$ for the MPS results and $(\alpha, \beta) = (31, 4.71)$ for the TTN results.
• Figure 4: Entanglement entropy $S$ as a function of the distance $R$ from the impurity site for MPS with $D=60$ (blue circles) and TTN with $D=30$ (red circles). For the TTN, the data points represent averages over all bonds at the same distance $R$.
• Figure 5: Entanglement entropy $S$ at the bond adjacent to the impurity ($R=1$) as a function of the logarithmic system size $n=\log_2 N$. The top and bottom panels show the convergence of $S$ with bond dimension $D$ for the MPS and TTN calculations, respectively. The inset of the upper panel shows a logarithmic fit to the converged MPS data ($D=480$), yielding $S \approx 0.14 \ln n + 0.58$. The inset of the lower panel compares MPS ($D=480$) and TTN ($D=40$) results and demonstrates quantitative agreement.