Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Average density of Bloch electrons in a homogeneous magnetic field: a second-order response

Benjamin M. Fregoso

TL;DR

The paper develops a gauge-invariant framework to compute the average electronic density in a clean 3D multiband crystal under a weak static magnetic field, yielding a first-order correction consistent with the Streda formula and an additional Fermi-surface orbital-moment term in metals. At second order, a comprehensive expression is derived that includes FS and non-FS processes across multiple bands, with a central role played by the quantum metric tensor which generates a Quantum Geometric Magnetic Moment (QGMM) and drives geometric contributions to the nonlinear density response. In a simple two-band model, the intraband QGMM dominates the nonlinear response and leads to a small but finite fractional density change; the work also connects density changes to magnetovolume and pressure effects, highlighting thermodynamic implications of the magnetic response. The approach is explicitly gauge invariant, treats intra- and interband processes on equal footing, accommodates relaxation, and can be extended to other observables, making it suitable for first-principles calculations and broader applications in nonlinear magnetoelectric phenomena.

Abstract

We compute the average density of a three-dimensional multiband crystal of arbitrary symmetry, metal or insulator, to first and second order in a weak static magnetic field. To linear order and for insulators, the density follows the well-known Streda formula, but for metals there is an extra contribution from the orbital magnetic moments at the Fermi surface. To second order, we find that the average density depends on several microscopic processes. Among these, the quantum metric tensor plays an important role by generating a pseudo-magnetic moment resulting from the rotation of the Bloch wave functions in the complex projective plane. We also discuss the implications of our results for the volume and pressure. The method we develop is explicitly gauge invariant, considers intraband and interband processes on equal footing, accommodates relaxation processes, and can be readily extended to other observables.

Average density of Bloch electrons in a homogeneous magnetic field: a second-order response

TL;DR

The paper develops a gauge-invariant framework to compute the average electronic density in a clean 3D multiband crystal under a weak static magnetic field, yielding a first-order correction consistent with the Streda formula and an additional Fermi-surface orbital-moment term in metals. At second order, a comprehensive expression is derived that includes FS and non-FS processes across multiple bands, with a central role played by the quantum metric tensor which generates a Quantum Geometric Magnetic Moment (QGMM) and drives geometric contributions to the nonlinear density response. In a simple two-band model, the intraband QGMM dominates the nonlinear response and leads to a small but finite fractional density change; the work also connects density changes to magnetovolume and pressure effects, highlighting thermodynamic implications of the magnetic response. The approach is explicitly gauge invariant, treats intra- and interband processes on equal footing, accommodates relaxation, and can be extended to other observables, making it suitable for first-principles calculations and broader applications in nonlinear magnetoelectric phenomena.

Abstract

We compute the average density of a three-dimensional multiband crystal of arbitrary symmetry, metal or insulator, to first and second order in a weak static magnetic field. To linear order and for insulators, the density follows the well-known Streda formula, but for metals there is an extra contribution from the orbital magnetic moments at the Fermi surface. To second order, we find that the average density depends on several microscopic processes. Among these, the quantum metric tensor plays an important role by generating a pseudo-magnetic moment resulting from the rotation of the Bloch wave functions in the complex projective plane. We also discuss the implications of our results for the volume and pressure. The method we develop is explicitly gauge invariant, considers intraband and interband processes on equal footing, accommodates relaxation processes, and can be readily extended to other observables.

Paper Structure

This paper contains 27 sections, 96 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Particle density
  3. Zero order
  4. First order
  5. Second order
  6. Example: intraband contribution
  7. Example: magnetovolume effect
  8. Example: Geometric response
  9. Conclusion
  10. Density matrix
  11. Zero order
  12. First order
  13. Second order
  14. Symmetric Landau gauge
  15. First order $\varsigma^{z}$
  16. ...and 12 more sections

Figures (1)

  • Figure 1: Schematic representation of the mapping from parameter space $\mathbf{k}$ to the complex projective plane of Bloch wave functions. A small shift by $\mathbf{g}$ rotates the vector $|u_{n\mathbf{k}} \rangle$ by an angle $d\theta_{n\mathbf{k}}$ (see (\ref{['eq:ds2']})).