Power-Law Spectra and Asymptotic $ω/T$ Scaling in the Orbital-Selective Mott Phase of a Three-Orbital Hubbard Model

We thank the referee for the careful reading of our manuscript and for the constructive and detailed feedback. We are pleased that the referee finds the model and scaling analysis interesting and recognizes the relevance of $\omega/T$ scaling in the broader context of strongly correlated materials.

Below we address all points raised by the referee and describe the corresponding changes made in the revised manuscript.

• Stability and fine-tuning of the OSM phase: We have expanded Sec.~4 to clarify under which conditions the destructive hybridization interference is stable. As shown in Ref.[27] and discussed more explicitly now, small deviations in the hybridization matrix elements destroy the OSM phase only below an exponentially small scale. We also added a discussion of asymmetric conduction-band dispersions ($\epsilon_k^1 \neq -\epsilon_k^2$) and explain under which circumstances we expect the interference condition is still met, at least asymptotically near the Fermi surface.

• Magnetic ordering: As is standard for single-site DMFT, our results apply only above any magnetic ordering temperature $T_{N}$. As discussed in the main text, general considerations suggest that both ferromagnetic and antiferromagnetic ground states may occur, depending on the bandwidth of the correlated orbitals. Importantly, close to the corresponding magnetic phase boundaries the ordering temperature $T_{N}$ is expected to be very small, so that the paramagnetic regime is wide and the scaling results presented here remain valid in this regime

• Scaling of the self-energy: We thank the referee for this suggestion. We now present the self-energy and its scaling explicitly in a new appendix D, showing its power-law scaling and $\omega/T$ collapse.

• Dependence on NRG parameters: A detailed scan over NRG hyperparameters has been added to appendix D. We now show the dependence on the number of kept states, the discretization parameter and the broadening parameter. The results confirm that the scaling behavior is robust.

• Motivation and experimental relevance: We have expanded the discussion in Sec.~2 to clarify the scope and motivation of the model. While we do not claim a direct material realization, we highlight that similar interference effects naturally arise in systems with enlarged unit cells and band folding, including depleted Anderson lattices and moiré structures.

• Minor comments: (i) The paragraph referring to NISQ devices has been removed. (ii) Appendix B has been revised to clearly explain why multiprecision numerics is needed: the coefficients $p_0$ and $p_1$ scale as $z^2$, making the cubic polynomial ill-conditioned in double precision for small $z = \omega + i\delta$.

We thank the referee for the positive assessment of our work and for highlighting the quality of the NRG calculations and the interest of the observed $\omega/T$ scaling.

We address the raised points below and have updated the manuscript accordingly.

• Motivation of the model and experiments: We have clarified the physical motivation for the band-structure symmetry in Sec.~2. As explained there and in Appendix~C, the destructive hybridization interference arises naturally in systems with enlarged real-space unit cells and band folding, such as the depleted periodic Anderson model. While we do not claim a direct material realization, related models are commonly used to describe heavy-fermion materials and, more recently, moiré van der Waals systems as mentioned in Ref[28] of the manuscript.

• Wording 'exotic scaling': Following the referee's suggestion, we removed the term “exotic scaling” from the conclusion and now simply refer to “scaling.”

We thank the referee for the careful reading of our manuscript and for the constructive and helpful feedback.
We are pleased that the referee finds the model well defined, the scaling analysis appropriate, and the technical level solid. We fully agree that transport properties are an important and interesting aspect of this model. Indeed, we are currently working on calculating and analyzing the optical conductivity and dc resistivity. However, in the present study we chose to focus on the scaling and especially the origin of $\omega/T$ scaling of single-particle quantities, and we leave a detailed discussion of transport for future work.

Below we address the referee’s comments, all of which have been incorporated into the revised manuscript:

• Relation to Ref[27]: In Ref.[27] we studied a closely related model, but the present work extends this analysis in several important directions. Here we consider a \emph{dispersive} $f$-band, include finite-temperature dynamics, and analyze the dynamical spin susceptibility. Moreover, we originally did not expect to find $\omega/T$ scaling in the OSM phase, because the effective impurity problem entering DMFT is not at a quantum critical point. A key result of the present manuscript is that asymptotic $\omega/T$ scaling nevertheless emerges over a wide range of temperatures and frequencies.
• Algorithmic complexity paragraph: We have removed the discussion related to algorithmic complexity and NISQ computing, as suggested by the referee.
• Missing figure references: We have added the missing references to the corresponding figures in the discussion around Eq.~(6) and Fig.~4.
• Frequency condition: We replaced $\omega>0$ with the physically meaningful condition $\omega > T$ in the corresponding paragraph.
• Multiprecision root finding (Appendix B.1): The coefficients $p_0(z)$ and $p_1(z)$ of the cubic polynomial both scale as $z^{2}$, where $z=\omega+i\delta$ with $\delta = 10^{-15}$. For small $\omega$, this makes the cubic-root problem numerically ill-conditioned in double precision. To reliably resolve the two roots that collapse to zero in this limit, we therefore evaluate Cardano’s formula with high-precision arithmetic. Although alternative approaches may exist, the numerical overhead is negligible, and the use of multiprecision guarantees stability of the DMFT self-consistency loop.