Computing excited eigenstates using inexact Lanczos methods and tree tensor network states

TL;DR

This work tackles the challenge of computing excited vibrational eigenstates in quantum many-body systems by introducing a TTNS-based inexact shift-and-invert Lanczos approach. The method adapts approximate linear solves, approximate orthogonalization, block-Lanczos, and state shifting to TTNS representations, enabling targeted extraction of excited states in dense spectral regions. Demonstrations on CH3CN, the Zundel ion, and the Eigen ion show that thousands of states can be obtained with competitive accuracy, using a two-stage refinement to ensure interval completeness and state targeting with high overlaps to reference TTNS-DMRG states. The approach broadens the applicability of TTNS to excited-state spectroscopy and offers a practical, generalizable tool for vibrational dynamics and potentially electronic-structure calculations in high-dimensional, strongly correlated systems.

Abstract

To understand the dynamics of quantum many-body systems, it is essential to study excited eigenstates. While tensor network states have become a standard tool for computing ground states in computational many-body physics, obtaining accurate excited eigenstates remains a significant challenge. In this work, we develop an approach that combines the inexact Lanczos method, which is designed for efficient computations of excited states, with tree tensor network states (TTNSs). We demonstrate our approach by computing excited vibrational states for three challenging problems: (1) 122 states in two different energy intervals of acetonitrile (12-dimensional), (2) Fermi resonance states of the fluxional Zundel ion (15-dimensional), and (3) selected excited states of the fluxional and very correlated Eigen ion (33-dimensional). The proposed TTNS inexact Lanczos method is directly applicable to other quantum many-body systems.

Figures (13)

• Figure 1: Example of a tree tensor network diagram and the corresponding contraction pattern. Compare with \ref{['eq:ttns']}.
• Figure 2: Example for change of root node. The unitary transformation matrix $\mathbf{T}$ orthogonalizes $\mathbf{A}^{[1]}$. The arrows indicate the tensors into which $\mathbf{T}$ and $\mathbf{T}^{-1}=\mathbf{T}^\dagger$ are absorbed. See the text for details.
• Figure 3: Effective eigenvalue problem, \ref{['eq:eigenvalue']}, in tensor dagram notation. The gray box denotes the Hamiltonian, which is not shown in tensor form. The blue node corresponds to the eigenvector to be found. The arrows indicate the first steps of the sweep procedure. The tree corresponds to that used for the CH3CN simulations (see \ref{['sec:results']} below), and the numbers in the orange boxes denote the mode ordering. Adapted from Ref. [ Benchmarking2025larsson]; licensed under a Creative Commons license.
• Figure 4: Convergence of the inexact Lanczos methods for selected states of CH3CN using a random initial guess. Energy convergence as a function of cumulative iteration (a and b), and the absolute value of the energy difference to the reference (c and d). Due to different magnitudes, the convergence is split into two separate panels. In (b), the complete spectrum from the reference computations is shown as dotted gray lines. Note further the different scales in (d), separated by a thick black horizontal line. Iteration 1 contains the initial guess only. The computations are restarted after $N_\text{Krylov}=10$ iterations. For degenerate levels, the energies are shown as straight and dotted lines.
• Figure 5: Same as \ref{['fig:ch3cn_specific_random']} but for a DMRG-based initial guess.