Electric-Magnetic Cross-Correlations

• Electric-magnetic cross-correlations are defined as nontrivial relationships between electric and magnetic observables that shape physical responses through symmetry-based multipole expansions.
• They are quantified using advanced models such as 6×6 coherence matrices and magnetoelectric tensors, enabling detailed analysis in condensed matter, optics, and nuclear physics.
• Key practical insights include applications in multiferroic materials, non-invasive electromagnetic sensing, and high-energy collision environments where cross-coupling influences chiral and magnetoelectric effects.

Electric-magnetic cross-correlations quantify and mediate the intricate interplay between electric and magnetic field variables at microscopic, mesoscopic, and macroscopic scales. These cross-correlations fundamentally structure electromagnetic response, quantum and classical coherence phenomena, and a wide variety of coupled order-parameter effects in condensed matter, optics, and high-energy nuclear systems. Their rigorous treatment has evolved to rely on advanced frameworks such as multipole expansions, electromagnetic coherence matrices, and explicit symmetry-based order parameter classifications.

1. Fundamental Definitions and Theoretical Formalism

Electric-magnetic cross-correlation refers to any nontrivial statistical or quantum mechanical relationship between electric field observables $\mathbf{E}$ and magnetic field observables $\mathbf{B}$ (or $\mathbf{H}$) within a given physical system. In classical wave theory and coherence optics, this is captured through joint expectation values such as $\langle E_i B_j^*\rangle$, corresponding to off-diagonal elements in the 6×6 electromagnetic polarization (coherency) matrix:

$$W_{\rm EM} = \left\langle \begin{pmatrix} \mathbf{E}(t) \ \mathbf{B}(t) \end{pmatrix} \begin{pmatrix} \mathbf{E}^\dagger(t) & \mathbf{B}^\dagger(t) \end{pmatrix} \right\rangle = \begin{pmatrix} W_E & W_{EB} \ W_{BE} & W_B \end{pmatrix}$$

where $W_{EB}$ encodes the electric-magnetic cross-correlations, and $W_{BE} = W_{EB}^\dagger$ (Gil et al., 2 Dec 2025). In quantum and condensed matter theory, related tensors arise from multipole expansions, symmetry analysis, and response function calculations, manifesting as linear magnetoelectric tensors $\alpha_{ij}$ that link induced polarization $P_i$ to applied $H_j$ and reciprocally $M_j$ to applied $E_i$ (Hayami et al., 14 Mar 2024, Hayami et al., 2017, Ryan et al., 2012).

2. Multipolar Mechanisms and Symmetry Analysis

The cross-correlation between electric and magnetic observables is underpinned by the symmetry structure of multipole moments. Second-quantization and group-theoretical approaches classify all local field interactions into polar (electric) multipoles $Q_{l,m}$, axial (magnetic) multipoles $M_{l,m}$, and two classes of toroidal multipoles: magnetic-toroidal (MT) $T_{l,m}$ and electric-toroidal (ET) $G_{l,m}$ (Hayami et al., 2017, Hayami et al., 14 Mar 2024).

These four families are distinguished by distinct parity (P) and time-reversal (T) properties, summarized as:

Family	P (even/odd)	T (even/odd)
Electric (Q)	$(-1)^l$	even
Magnetic (M)	$(-1)^{l+1}$	odd
Magnetic-Toroidal (T)	$(-1)^l$	odd
Electric-Toroidal (G)	$(-1)^{l+1}$	even

Toroidal multipoles, absent in pure atomic shells but active in orbital-hybridized systems, directly mediate cross-couplings such as magneto-electric (ME) and magneto-elastic effects. For example, an MT dipole ($T_i$) enables a linear $E_iB_j$ coupling in the free energy, while ET multipoles generate similar cross-terms via different symmetry channels (Hayami et al., 2017).

Symmetry constraints imposed by the magnetic point group instruct which components of the magnetoelectric tensor $\alpha_{ij}$ can be nonzero. These constraints are systematically treated through the irreducible representation decomposition of multipolar operators (Hayami et al., 14 Mar 2024).

3. Phenomenology in Materials: Magnetoelectric and Multiferroic Systems

Electric-magnetic cross-correlations are most prominently observed in multiferroic and magnetoelectric materials, where simultaneous or switchable electric and magnetic order parameters coexist.

The linear magnetoelectric tensor $\alpha_{ij}$ is formally defined as:

$$\alpha_{ij} \equiv \left.\frac{\partial P_i}{\partial H_j}\right|_{E=0} = \left.\frac{\partial M_j}{\partial E_i}\right|_{H=0}$$

Electronic structure and symmetry dictate the possible tensor elements:

• Nondissipative (interband) contributions ($\alpha_{ij}^{(E)}$) require both $\mathcal{P}$ and $\mathcal{T}$ breaking.
• Dissipative (intraband) contributions ($\alpha_{ij}^{(J)}$) require only $\mathcal{P}$ breaking (Hayami et al., 14 Mar 2024).

Empirically, cross-coupling exemplified by “giant” magnetoelectric effects is observed in compressively strained EuTiO$_3$, where an external electric field modulates the magnetic ground state by altering superexchange constants via lattice polarization, yielding $\alpha \sim 6.5 \times 10^{-3}\,\mathrm{s/m}$, which vastly exceeds that of classical ME materials (Ryan et al., 2012). Such electronic cross-coupling is realized when both (i) large ionic polarizability, and (ii) high sensitivity of magnetic exchange to polar displacements, are present.

The multipole-based approach enables systematic prediction of ME effects through:

1. Assigning the magnetic point group $G$ of the candidate material.
2. Listing the allowed multipoles ($Q_{l,m}$, $M_{l,m}$, $T_{l,m}$, $G_{l,m}$) in $A_1^+$ of $G$.
3. Identifying which multipoles are electronically active.
4. Expanding the Hamiltonian in the symmetry-adapted basis and computing response (Hayami et al., 14 Mar 2024).

4. Cross-correlation in High-Energy and Nuclear Environments

In high-energy heavy-ion collisions, such as Au+Au at $\sqrt{s_{NN}}=200$ GeV, intense, rapidly fluctuating $\mathbf{E}$ and $\mathbf{B}$ fields are generated at femtometer and sub-femtosecond scales. The event-by-event scalar product $\mathbf{E}\cdot\mathbf{B}$ emerges as a key observable, acting as a source term for chiral transport phenomena (Chiral Magnetic Effect, Chiral Separation Effect, Chiral Magnetic Wave) (Alam et al., 2021).

Within the Monte Carlo Glauber model, event-specific spatial distributions of $$\mathcal{E}(x,y,z;t) \equiv \mathbf{E}\cdot\mathbf{B}$$ define an "EM symmetry-plane" $\psi_{E\cdot B}^{(n)}$, whose harmonic structure ($n=2$--$5$) is compared to participant-plane angles $\psi_P^{(n)}$ derived from nucleon configurations. Strong angular correlation between these electric-magnetic cross-plane observables and participant geometry is demonstrated in central and mid-central collisions, with decoupling and rotation ($\sim \pi/2$) observed in peripheral events and larger impact parameters. The distributional sharpness and systematic shifts are sensitive to system size and charge, as found in comparisons between Au, Ru, and Zr collisions (Alam et al., 2021).

5. Cross-correlation Formulation in Electromagnetic Coherence and Sensing

The complete characterization of random or structured electromagnetic fields necessitates a joint electric-magnetic coherence formalism beyond traditional electric-only descriptions (Gil et al., 2 Dec 2025). The 6×6 polarization matrix encodes all second-order moments, with the off-diagonal $W_{EB}$ systematically decomposed into:

• Active (radiative) flux (antisymmetric real part), directly linked to the real Poynting vector $$\operatorname{Re}\langle \mathbf{E}\times\mathbf{B}^*\rangle$$.
• Reactive flux (antisymmetric imaginary part), corresponding to $$\operatorname{Im}\langle \mathbf{E}\times\mathbf{B}^*\rangle$$.
• In-phase (symmetric real) and quadrature (symmetric imaginary) parallel-component alignment matrices.

Dimensionless global indices $p$, $q$, $p_u$, $p_i$ are constructed from these submatrices to quantify the strength and type of electric-magnetic coupling, subject to a generalized Pythagorean constraint $c^2 = p^2 + q^2 + p_u^2 + p_i^2$, where $c$ measures the total cross-correlation (Gil et al., 2 Dec 2025).

Concrete field configurations, such as superpositions of orthogonally propagating plane waves with common polarization, can exhibit nontrivial off-diagonal cross blocks despite zero mutual energy flow in any individual beam. Only the full 6×6 description reveals and quantifies these correlations.

6. Cross-correlation Retrieval and Measurement in Complex Media

In experimentally complex or chaotic environments, such as reverberation chambers or random media, electric-magnetic cross-correlation manifests in the measured passive field cross-correlation functions. The scattering-matrix formalism, when applied to equipartitioned or randomly stirred electromagnetic fields, demonstrates that the cross-correlation between two receiving probes reproduces the difference between the causal and anti-causal Green's functions of the system (Davy et al., 2021).

Although these experiments typically treat each antenna port as a generic probe (sensitive to an unspecified E/H mix), the formalism extends to cases where distinct electric and magnetic sensitivities are engineered at the receiving channels. In such scenarios, the cross-correlation framework directly captures the nontrivial electric-magnetic Green's function elements and, upon ensemble averaging, enables the non-invasive recovery of the full response characteristics between arbitrary pairs of E- or H-sensitive probes.

This approach underpins passive mutual coupling calibration, non-destructive antenna characterization, and imaging in complex electromagnetic environments, with convergence properties and sensitivity determined by stirring (ensemble size), absorption, and medium complexity (Davy et al., 2021).

7. Material Realizations, Applications, and Outlook

Material realization of pronounced electric-magnetic cross-correlations is achieved in systems exhibiting strong orbital hybridization, toroidal multipole order, or pronounced structural and symmetry-tunable transitions, including:

• Strongly hybridized $f$-electron compounds, U-based cage systems, and pyrochlores.
• Non-centrosymmetric semiconductors and chiral phases (e.g., Te hosting ET monopole order).
• Thin films under epitaxial strain (e.g., EuTiO$_3$) tuned through competing exchange and polar distortions to achieve field-switchable magnetic states (Ryan et al., 2012).
• Complex “designer” photonic or microwave environments enabling direct electromagnetic Green's function retrieval via passive cross-correlation measurement (Davy et al., 2021).

The unification of multipolar symmetry analysis, electromagnetic coherence theory, and advanced computational approaches underpins the prediction and manipulation of electric-magnetic cross-correlations in both quantum and classical domains. Contemporary research continues to expand the range of known coupled phases, optimize device-level control of cross-correlated phenomena, and refine the measurement and analysis of such effects through both direct experimental and theoretically guided protocols (Hayami et al., 14 Mar 2024, Gil et al., 2 Dec 2025, Hayami et al., 2017).