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Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization

Koki Shinada, Naoto Nagaosa

TL;DR

This work establishes a unified framework in which generalized quantum geometric tensors (QGTs) bound diverse linear-response observables across many-body, finite-temperature, and general-parameter settings. By proving the semipositive definiteness of the absorptive part and connecting interband contributions to a generalized QGT, the authors derive universal inequalities that tighten conventional thermodynamic bounds and apply them to concrete cases such as the interband Drude weight and orbital magnetization. They provide detailed model analyses (Landau levels, Haldane model, flat-band systems) and real-material relevance (twisted bilayer graphene, kagome metals), showing that orbital magnetization and related quantities can approach theoretical bounds under band-flatness and Landau-like conditions. The framework also extends to higher-order multipoles (magnetic quadrupole) and clarifies the link between these bounds and quantum-geometry-inspired uncertainty principles, highlighting the essential role of quantum effects. Overall, the paper delivers both fundamental inequalities and practical guidance for diagnosing how closely real materials mimic Landau-level physics through their geometric and response properties.

Abstract

The quantum geometric tensor (QGT) provides nontrivial bounds among physical quantities, as exemplified by the metric-curvature inequality. In this paper, we investigate various bounds for different observables through certain generalizations of the QGT. First, we demonstrate that bounds hold for all linear responses, which are produced by a QGT extended to many-body states, finite temperature, and general parameter space. As an application, we show the thermodynamic inequality originating from the convexity of free energy can be further tightened. Second, we establish a bound between the Drude weight and the orbital magnetization. The equality is exactly satisfied in the Landau level system, and systems with nearly flat bands tend to approach equality as well. We apply the resulting inequality to two orbital ferromagnets and support that the twisted bilayer graphene system is close to the Landau level system. Moreover, we show that an analogous inequality also holds for a higher-order multipole, magnetic quadrupole. Finally, we discuss the analogy between the QGT and the uncertainty principle, emphasizing that the existence of nontrivial bounds necessarily reflects quantum effects.

Quantum geometric bounds for observables: Linear responses, Drude weight, and orbital magnetization

TL;DR

This work establishes a unified framework in which generalized quantum geometric tensors (QGTs) bound diverse linear-response observables across many-body, finite-temperature, and general-parameter settings. By proving the semipositive definiteness of the absorptive part and connecting interband contributions to a generalized QGT, the authors derive universal inequalities that tighten conventional thermodynamic bounds and apply them to concrete cases such as the interband Drude weight and orbital magnetization. They provide detailed model analyses (Landau levels, Haldane model, flat-band systems) and real-material relevance (twisted bilayer graphene, kagome metals), showing that orbital magnetization and related quantities can approach theoretical bounds under band-flatness and Landau-like conditions. The framework also extends to higher-order multipoles (magnetic quadrupole) and clarifies the link between these bounds and quantum-geometry-inspired uncertainty principles, highlighting the essential role of quantum effects. Overall, the paper delivers both fundamental inequalities and practical guidance for diagnosing how closely real materials mimic Landau-level physics through their geometric and response properties.

Abstract

The quantum geometric tensor (QGT) provides nontrivial bounds among physical quantities, as exemplified by the metric-curvature inequality. In this paper, we investigate various bounds for different observables through certain generalizations of the QGT. First, we demonstrate that bounds hold for all linear responses, which are produced by a QGT extended to many-body states, finite temperature, and general parameter space. As an application, we show the thermodynamic inequality originating from the convexity of free energy can be further tightened. Second, we establish a bound between the Drude weight and the orbital magnetization. The equality is exactly satisfied in the Landau level system, and systems with nearly flat bands tend to approach equality as well. We apply the resulting inequality to two orbital ferromagnets and support that the twisted bilayer graphene system is close to the Landau level system. Moreover, we show that an analogous inequality also holds for a higher-order multipole, magnetic quadrupole. Finally, we discuss the analogy between the QGT and the uncertainty principle, emphasizing that the existence of nontrivial bounds necessarily reflects quantum effects.

Paper Structure

This paper contains 24 sections, 66 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. bounds in linear response theory
  3. Proof of inequalities
  4. Relation with quantum geometric tensor
  5. Thermodynamic inequality and tighter relation
  6. Some examples of inequality
  7. Case1
  8. Case3
  9. Noninteracting periodic system
  10. Generalized quantum geometric tensor
  11. A generalized QGT with semipositive definiteness and inequalities
  12. Interband Drude weight and orbital magnetization
  13. Case study: Landau level system
  14. Case study: Haldane model
  15. Case study: flat band model
  16. ...and 9 more sections

Figures (2)

  • Figure 1: (Left) The $M$ dependence of the interband Drude weight and the orbital magnetization in Haldane model. (Right) The $M$ dependence of the Chern number (Ch) and $\sqrt{\det G}$. We set $t_1 = 1.0$, $t_2 = 1/3\sqrt{3}$ for the numerical calculation. We set $M_{\mathrm{orb}}$ in units of $et_1/\hbar$.
  • Figure 2: (Left) The $\Delta$ dependence of the interband Drude weight and the orbital magnetization in the flat band model. (Right) The $\Delta$ dependence of the Chern number (Ch) and $\sqrt{\det G}$. (Inset) The $\Delta$ dependence of the flatness. We set $t_1=1.0,~\phi=\pi/3$, and $t_2 = -\Delta/\sqrt{3}$ for the numerical calculation. We set $M_{\mathrm{orb}}$ in units of $et_1/\hbar$.