Magnetic Moment Map Overview

• Magnetic moment maps are quantitative representations of magnetic moment distributions, orientations, and magnitudes in physical systems.
• They enable precise analysis of electronic and spin dynamics in Bloch systems, quantum rotors, and high-sensitivity magnetometry applications.
• Construction techniques include DFT-based computations, Bloch wavefunction mapping, and advanced imaging methods to derive critical physical insights.

A magnetic moment map is a representation—typically in real or reciprocal space—of the distribution, orientation, or magnitude of magnetic moments within a physical system. Such maps serve as foundational tools in solid-state theory, quantum magnetism, plasma physics, and high-sensitivity magnetometry, enabling visualization, numerical analysis, and physical interpretation of magnetic phenomena from the microscopic (Bloch electrons, spin rotors) to the macroscopic (geological samples). Magnetic moment maps appear in both theoretical and experimental treatments, and their specific mathematical and methodological construction is shaped by the system of interest and the observable under scrutiny.

1. Momentum-Resolved Magnetic Moment Mapping in Bloch Systems

In multiband metals, the momentum-resolved magnetic moment map $\mathbf{m}_n(\mathbf{k})$ quantifies the intrinsic (orbital plus spin) magnetic moment carried by Bloch electrons at each point $\mathbf{k}$ on the Fermi surface. This object, distinct from the Berry curvature, is central to the theory of the gyrotropic magnetic effect (GME), where a current density $\mathbf{j}^\mathbf{B}$ is induced by a time-dependent, spatially uniform magnetic field. The magnetic moment map is used in the Kubo-formula for the GME tensor $\alpha^{\rm gme}_{ij}$, yielding

$$j_i(\omega) = \alpha^{\rm gme}_{ij}(\omega) B_j(\omega),$$

with

$$\alpha^{\rm gme}_{ij}(\omega) = \frac{i \tau e}{1 - i \omega \tau} \sum_n \int \frac{d^3k}{(2\pi)^3} \frac{\partial f(\varepsilon_{n\mathbf{k}})}{\partial \varepsilon_{n\mathbf{k}}} v_{n,i}(\mathbf{k}) m_{n,j}(\mathbf{k}),$$

where $v_{n,i}(\mathbf{k})$ is the band velocity, $f$ is the Fermi-Dirac function, and $\mathbf{m}_n(\mathbf{k})$ has both orbital and spin components. In the zero-temperature, DC limit, integration is collapsed to the Fermi surface, making $\mathbf{m}_n(\mathbf{k})$ a true Fermi-surface-resolved map. This mapping is essential for quantifying unconventional electromagnetic responses in topological metals, Weyl semimetals, and strongly spin-orbit-coupled systems (Zhong et al., 2015).

2. Construction and Visualization of Momentum-Space Magnetic Moment Maps

Numerically constructing $\mathbf{m}_n(\mathbf{k})$ on the Fermi surface is a multi-step process that leverages advances in electronic structure theory:

• Band structure and wavefunction calculation: Compute $\varepsilon_{n\mathbf{k}}$ and cell-periodic Bloch wavefunctions $\lvert u_{n\mathbf{k}}\rangle$ on a dense grid in the Brillouin zone (via DFT or Wannier interpolation).
• Fermi surface extraction: Identify mesh points in reciprocal space where $\varepsilon_{n\mathbf{k}}$ crosses the chemical potential $\mu$ and build a FS triangulation (e.g., via Marching Cubes).
• Velocity and moment evaluation: Calculate $v_{n,i}(\mathbf{k})$ (finite-difference or Wannier velocity operator) and obtain $\mathbf{m}^{\rm orb}_n(\mathbf{k})$ using either

$$\mathbf{m}^{\rm orb}_n(\mathbf{k}) = \frac{e}{2\hbar} \mathrm{Im} \left\langle \partial_{\mathbf{k}} u_{n\mathbf{k}} \Big| \times (H_{\mathbf{k}} - \varepsilon_{n\mathbf{k}}) \Big| \partial_{\mathbf{k}}u_{n\mathbf{k}} \right\rangle$$

or a sum-over-bands formula for efficiency.

• Spin moment computation: Evaluate

$$\mathbf{m}^{\rm spin}_n(\mathbf{k}) = -\frac{g_s e}{2m_e} \langle u_{n\mathbf{k}} | \mathbf{S} | u_{n\mathbf{k}} \rangle$$

at each point.

• Visualization: Render $\mathbf{m}_n(\mathbf{k})$ as field arrows or via projections ($\mathbf{m}_n \cdot \hat{v}_F$, $|\mathbf{m}_n|$) on the FS mesh using scientific visualization libraries.

This numerical mapping enables both qualitative and quantitative insights into the electronic magnetic structure of materials, and is a crucial step in computing GME and related Fermi-surface responses (Zhong et al., 2015).

3. Magnetic Moment Maps in Rotational Quantum Systems

In the context of quantum rotors, especially systems exhibiting tunneling of the magnetic moment between discrete orientations, a “magnetic-moment map” refers to the phase diagram of the ground-state magnetic moment $\mu(\alpha, \lambda)$ as a function of control parameters:

• $\alpha \equiv 2(\hbar S)^2 / (I_z \Delta)$: magneto-mechanical ratio,
• $\lambda \equiv I_z / I_x$: aspect-ratio of moments of inertia.

For a rigid spin-$S$ rotor, the Hamiltonian is diagonalized for each total angular momentum $J, K$, yielding eigenenergies and analytic expressions for the magnetic moment:

$$\mu_J(\alpha) = -g\mu_B S \frac{\alpha J}{\sqrt{S^2 + (\alpha J)^2}}.$$

Phase boundaries in the $(\alpha, \lambda)$ plane separate domains of well-defined $J$ and thus distinct plateau values of the magnetic moment. The quantum phase diagram exhibits a staircase-like structure, with regime transitions determined by critical curves parameterized by the system’s shape and anisotropy (O'Keeffe et al., 2011).

4. Real-Space Magnetic Moment Mapping in Magnetometry

In magnetometry, particularly the high-sensitivity regime enabled by SQUID magnetic microscopy, a magnetic moment map refers to an empirical reconstruction of a sample’s net (dipole) magnetic moment from spatial measurements of stray magnetic fields above the sample. The mapping is achieved by fitting the 2D field map $\{B_z(x_i, y_i, h)\}$ to the forward model for a point dipole:

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \left[ \frac{3 (\mathbf{m}\cdot(\mathbf{r}-\mathbf{r}'))(\mathbf{r}-\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|^5} - \frac{\mathbf{m}}{|\mathbf{r}-\mathbf{r}'|^3} \right],$$

using nonlinear or separable least-squares minimization algorithms to extract the best-fit dipole position and moment. This protocol enables quantitative mapping of net magnetic moments in geological and paleomagnetic samples down to $\sim 10^{-15}$~A·m$^2$ and supports identification of contamination and multipole errors. Validation on synthetic sources and geological specimens establishes error bounds ($<5\%$ magnitude, $<2^\circ$ direction for moderate SNR) and underpins advanced paleomagnetic analyses (Lima et al., 2016).

5. Magnetic Moment Map in Phase-Space Transformations and Hamiltonian Reduction

For plasma, kinetic, or gyrokinetic theories, a magnetic moment map denotes the canonical transformation from $(\mathbf{q}, \mathbf{v})$ to $(\mathbf{q}, v_\parallel, \mu, \theta)$, where $\mu = |\mathbf{v}_\perp|^2/(2|\mathbf{B}|)$ is the adiabatic invariant associated with cyclotron motion. This transformation isolates the fast (gyro-) angle $\theta$ from the slow guiding-center dynamics. In the Vlasov-Maxwell Hamiltonian structure, replacement of $f(\mathbf{q},\mathbf{v})$ by $f(\mathbf{q},v_\parallel,\mu,\theta)$ modifies the noncanonical Poisson bracket, introduces a Jacobian $|\mathbf{B}|$, and leads (upon averaging over $\theta$ and eliminating fast-scale electric field energy) to a reduced 5D gyrokinetic Hamiltonian. The magnetic moment map is thus a pivotal tool for Hamiltonian reduction and systematic derivation of reduced plasma kinetic models. Consistency of the induced bracket and preservation of the Jacobi identity under these variable changes are explicitly verified, and the physical role of $\mu$ as a collective adiabatic invariant is central to the formalism (Morrison et al., 2012).

6. Applications and Physical Significance

Magnetic moment maps are employed across a diverse set of applications:

• Optical activity and gyrotropic metals: Calculation of GME and its tensorial structure in Weyl and topological semimetals (Zhong et al., 2015).
• Quantum rotors: Mapping ground-state properties and phase transitions in single-molecule magnets, nanoparticles, and rigid spin systems (O'Keeffe et al., 2011).
• Geological and paleomagnetic analysis: Unprecedented sensitivity in net moment measurements of weakly magnetized samples, crucial for resolving cosmochemical and paleomagnetic questions (Lima et al., 2016).
• Plasma theory: Systematic Hamiltonian reduction and derivation of nonlinear gyrokinetics for strongly magnetized plasmas, respecting underlying symplectic structure (Morrison et al., 2012).

Across these fields, the magnetic moment map provides both quantitative and qualitative access to fundamental magnetic degrees of freedom, elucidating transport phenomena, phase boundaries, and collective dynamics. Theoretical and computational developments in mapping procedures, measurement protocols, and bracket transformations underpin ongoing research in quantum materials, magnetometry, and plasma physics.