Data-Driven Bath Fitting for Hamiltonian-Diagonalization Dynamical Mean-Field Theory

Abstract

We propose a machine-learning-based initialization method to overcome the nonlinear bath-fitting bottleneck in Hamiltonian-diagonalization-based dynamical mean-field theory (HD-DMFT). In HD-DMFT, the continuous hybridization function is approximated by a finite set of bath-site energies and hybridization amplitudes, determined by minimizing a highly non-convex multivariable cost function. As the number of bath sites increases, the optimization becomes more sensitive to the initial guess and more prone to suboptimal local minima, which can slow or destabilize the DMFT self-consistency loop. We reformulate bath fitting as a supervised regression problem and train a kernel ridge regression model to predict near-optimal discrete bath parameters directly from the target hybridization function on the Matsubara axis. To ensure physical relevance and data diversity, we construct the training dataset from tight-binding Hamiltonians of layered-perovskite-like ruthenate models across systematically deformed structures, instead of relying on naive random parameter sampling, and obtain high-quality labels through fully converged conventional bath fitting. Time-reversal symmetry is explicitly incorporated in both feature and target representations to reduce effective dimensionality and enforce physical consistency. Benchmarks in the non-interacting limit show that the learned initialization systematically reduces the initial fitting error, decreases the number of conjugate-gradient iterations, and improves robustness against local minima over a wide range of bath sizes. We further demonstrate transferability to interacting DMFT calculations for $\mathrm{Sr_{2}RuO_{4}}$ solved with an adaptive-truncation impurity solver, where the ML initialization yields consistently faster convergence than a symmetry-preserving heuristic baseline while preserving the final fitted solution.

Figures (10)

• Figure 1: (a) Schematic overview of the DMFT self-consistency cycle, highlighting the two computational challenges: the impurity solver and bath fitting. Bath fitting constitutes a highly non-convex optimization problem whose convergence behavior depends critically on the quality of the initial bath parameters. (b) Cost function $\chi^2$ landscape as a function of two bath onsite energies $(\varepsilon_1, \varepsilon_3)$. Filled circles denote initial parameter values and crosses mark the converged solutions reached by conjugate gradient optimization. Despite starting from nearby positions in parameter space, different trajectories converge to distinct local minima with substantially different cost function values, illustrating the sensitivity of bath fitting to initialization and motivating the need for data-driven strategies to identify favorable starting points.
• Figure 2: Schematic representation of (a) the hybridization-function matrix and (b) the bath-parameter matrix. Only the symmetry-independent (unshaded) components are used as machine-learning inputs and outputs. The remaining (shaded) elements are related by time-reversal (Kramers) symmetry and are reconstructed a posteriori, rather than learned independently.
• Figure 3: Reproduction of the nearest-neighbor DFT band dispersion of Ca$_2$RuO$_4$ using fitted Slater--Koster parameters. The reference hoppings in the local $t^{}_{2g}$ basis $(yz, zx, xy)$ are extracted from Wannier-projected DFT calculations Kim2023. The parameters in Table \ref{['tab:tb_params']} are fitted to match the NN Wannier Hamiltonian, and the resulting band structure is compared with that from the DFT NN hoppings.
• Figure 4: Schematic illustration of the structural sampling procedure. (a) The orientation of each RuO$_6$ octahedron is parameterized by Euler angles $(\alpha, \beta, \gamma)$ specifying the Ru--O bond direction. (b) Starting from a symmetric reference configuration (top, red atoms), oxygen atoms are systematically displaced within their allowed angular ranges (middle), yielding distorted octahedral configurations (bottom, blue atoms) while preserving the glide plane symmetry of the layered perovskite structure.
• Figure 5: Illustration of the heuristic initialization procedure based on cumulative distribution function (CDF) sampling. The spectral function $\mathrm{Tr}[\hat{\Delta}_{\mathrm{latt}}(\omega)]$ from the lattice calculation (grey shaded region) is used to construct the cumulative distribution function $F_{\mathrm{latt}}(\omega)$ (black curve with squares). Bath onsite energies (black squares) are determined by inverting the CDF at uniformly spaced quantiles, automatically concentrating bath sites in regions of high spectral weight. The heuristic initialization result (blue dotted curve) and its corresponding CDF $F_{\mathrm{HR}}(\omega)$ (blue curve with circles) demonstrate that the method approximately reproduces the overall spectral weight distribution while satisfying the trace sum rule by construction.