Electronic bounds in magnetic crystals

The authors unify and extend a large body of earlier work on metric–curvature inequalities, sum-rule bounds, and gap-related constraints.

A bit long, the authors could consider to move some part of the algebra of section 5 in an appendix

This manuscript presents a comprehensive and systematic study of inequality relations (“bounds”) between geometric, magnetic, and response properties of electronic states in crystalline solids, with particular emphasis on magnetic systems. Building on the positive semidefiniteness of composite tensors constructed from Bloch states, the authors unify and extend a large body of earlier work on metric–curvature inequalities, sum-rule bounds, and gap-related constraints.

I think the manuscript is suitable for publication in its present form, and I have only some minor comments and remarks:

1) Results in Sec. 5.6 applies to arbitrary manifolds or only to low-lying manifold? 2) Do the authors think that results in sec. 7.4 can be used to materials design with given electron susceptibility? 3) Are the the bounds found in Sec. 6 relate to measurable anomalous Hall responses in 3D metals?

Small typos: 1. Appendix D reference in Sec. 5.7.1: “the saturared orbital-moment bound” should be “saturated” 2. Sec. 6.3.1, layered Haldane model: “the two nodes meet again and anihilate” should be “annihilate” 3. Sec. 5.7.3, discussion around Eq. (70): “and the the second line is the same as Eq. (66).” remove one “the” 4. In Fig.4 the dotted line of right panel is the same of the dashed in the left panel with different parameter \Delta I guess. In this case the authors should use the same line type.

Ask for minor revision

1-The authors systematically derive and generalize a variety of bounds, including previously known inequalities.
2-The currently known bounds are well organized, making the work valuable as a review as well.

This manuscript investigates various electronic bounds in magnetic crystals and contributes to a currently active and timely research topic. While earlier studies have primarily focused on two-dimensional insulating systems, the authors explicitly extend the analysis to three-dimensional and metallic systems. They also systematically derive and generalize bounds, related to physical quantities such as the electric susceptibility and the orbital magnetization, in addition to recovering previously known ones. Furthermore, through several model calculations, the authors clarify multiple conditions required for the saturation of these inequalities.

I believe the manuscript is suitable for publication in its present form; however, I recommend that the authors address the following questions before final acceptance.

1-How does Eq. (9) guarantee gauge invariance? It may be helpful to add a few explanatory sentences in the manuscript.
Related to this point, while T_p(k) is gauge invariant for p ≥ 0, is it also gauge invariant for p<0? In particular, for p=−1, it is related to the electric susceptibility, so I would expect gauge invariance to hold at least when F_k is the ground-state manifold.

2-Regarding the 3D matrix-invariant inequalities, the magnitude of the Chern vector is bounded by the quantum metric in its direction. However, in cases such as the layered Haldane model discussed in Sec. 6.3.1, where the Chern vector takes the form K=(0,0,K_z), the inequality in Eq. (88) is reduced to the conventional 2D-matrix inequality in Eq. (57). What are the nontrivial situations that arise specifically in three dimensions? For example, do such cases correspond to situations where more than one component of the Chern vector is finite, such as K=(K_x, 0, K_z)?

Ask for minor revision