Lanczos Method for QRPA Strength Functions in Atomic Nuclei

Technical Summary: Symmetric Lanczos Method for QRPA Strength Functions

Problem Statement
The calculation of charge-changing nuclear response functions is essential for describing $\beta$ decay, double-$\beta$ decay, and collective excitations in atomic nuclei. For medium-mass and heavy open-shell nuclei, particularly deformed systems, the Quasiparticle Random-Phase Approximation (QRPA) provides a necessary microscopic framework. However, the dimension of the QRPA matrix grows rapidly with the quasiparticle space size, making explicit construction and full eigendecomposition prohibitively expensive.

While the Finite Amplitude Method (FAM) offers a matrix-free alternative by computing responses through induced fields, conventional FAM-based approaches solve the linear-response problem frequency-by-frequency. This requires solving a new linear system for every complex frequency on the chosen contour. Consequently, reconstructing a strength function over a broad energy interval becomes computationally costly. The paper addresses the need for a method that captures spectral information globally over an extended frequency range without the expense of repeated frequency-specific solves.

Methodology
The authors develop a symmetric Lanczos framework to approximate charge-changing QRPA strength functions. The methodology proceeds through the following steps:

1. Spectral Reformulation: Starting from the FAM linear-response equations, the authors derive equivalent spectral representations. In the case where QRPA matrices $A$ and $B$ are real, they formulate a reduced eigenvalue problem involving the matrix products $MK$ and $KM$, where $M = A + B$ and $K = A - B$.
2. Symmetric Lanczos Projection: Unlike the standard QRPA matrix which is non-Hermitian in the Euclidean inner product, the reduced operator $MK$ is self-adjoint with respect to the $K$-inner product ($\langle x, y \rangle_K = x^T K y$). This property allows the application of a symmetric Lanczos process.
3. Single-Run Approximation: The Lanczos algorithm generates a tridiagonal matrix $T_m$ from a single Krylov subspace run. The eigenvalues of $T_m$ approximate the squared excitation energies ($\Omega_i^2$), and the associated Ritz vectors approximate the QRPA transition amplitudes.
4. Strength Function Reconstruction: Using the Ritz pairs, the method reconstructs the Lorentzian-smeared strength function over a broad energy interval. This contrasts with frequency-by-frequency solvers (FAM or GMRES) which must be executed separately for each frequency point.
5. Initialization: The Lanczos recursion is initialized with a vector derived from the external field combination $f_+ = F^{20} + F^{02}$, ensuring the projection respects the symmetry sectors of the axially deformed system.

Key Contributions

• Derivation of Reduced Eigenproblem: The paper establishes the equivalence between the full QRPA linear-response problem and a reduced eigenvalue problem involving $MK$ and $KM$, facilitating the use of symmetric Lanczos projection.
• Matrix-Free Implementation: The method avoids assembling the full QRPA matrix. Matrix-vector products (matvecs) are computed using the induced field formalism, similar to FAM, but applied to the reduced operator structure.
• Efficiency in Broad Spectral Scans: The approach demonstrates that a single Krylov run can capture the dominant spectral information required for a full strength profile, offering a significant alternative to pointwise frequency calculations.

Numerical Results
The method was tested on two nuclei: the medium-mass $^{112}\text{Sn}$ and the heavy rare-earth $^{150}\text{Nd}$, using the Skyrme energy-density functional SkM*.

• Reference Validation: The authors first established Generalized Minimal Residual (GMRES) as an efficient frequency-by-frequency reference. GMRES reproduced the converged iterative FAM (IFAM) strength profiles but required fewer iterations (e.g., 2347 vs. 3781 for $^{112}\text{Sn}$).
• Lanczos Accuracy: Comparing the Lanczos approximation against the GMRES reference:
  • For $^{112}\text{Sn}$, increasing the Krylov dimension from $m=50$ to $m=100$ significantly improved the agreement with the reference, capturing both broad distributions and local peak structures.
  • For $^{150}\text{Nd}$, increasing $m$ from 100 to 200 similarly reduced discrepancies.
  • Kullback-Leibler (KL) divergence metrics confirmed that increasing the Lanczos dimension substantially reduced the distribution-level error (e.g., KL divergence for $^{112}\text{Sn}$ dropped from $2.89 \times 10^{-2}$ at $m=50$ to $2.34 \times 10^{-3}$ at $m=100$).
• Computational Cost: The Lanczos method proved more efficient than frequency-by-frequency solvers for broad spectral scans. A 100-step Lanczos run for $^{112}\text{Sn}$ required approximately 600 matvec-equivalent operations, compared to 2347 for the GMRES reference. Similar savings were observed for $^{150}\text{Nd}$.

Significance and Claims
The paper claims that the symmetric Lanczos projection provides an efficient and accurate approach for QRPA strength-function calculations when spectral information is required over an extended frequency range. The method retains the matrix-free efficiency advantages of FAM-style calculations while exploiting the broader spectral information available through Lanczos projection.

The authors position this work as a practical alternative for heavy and deformed nuclei where explicit QRPA matrices are too large to build and repeated frequency-by-frequency solves become computationally prohibitive. The paper does not claim to replace frequency-specific solvers for cases where only a few discrete frequencies are needed, but rather to optimize the reconstruction of full response profiles. Future work is suggested to extend the method to additional external fields and larger systematic calculations.