Spin-Induced Multipole Moments (SIQM)

• Spin-induced multipole moments (SIQM) are multipolar structures generated by spin degrees of freedom, manifesting in magnetic, electromagnetic, and gravitational contexts.
• They are described using higher-rank operators that capture spin-dependent electric and magnetic multipoles, influencing waveforms and polarization responses.
• Applications of SIQM include characterizing symmetry-breaking in noncentrosymmetric crystals and modeling post-Newtonian corrections in compact binary dynamics.

Spin-induced multipole moments (SIQM) are multipolar structures whose leading nontrivial contributions are generated by spin degrees of freedom. The term spans several research traditions: in noncentrosymmetric magnets it denotes spin-dependent electric dipoles and higher electric multipoles induced by single-spin or two-spin operators; in relativistic electromagnetic theory it denotes higher electric and magnetic moments generated by spin-magnetization currents; in gravitational theory it denotes spin-generated mass and current multipoles of compact objects and binaries that enter post-Newtonian dynamics and gravitational-wave emission. Across these settings, SIQM are controlled by symmetry, by the representation of the underlying degrees of freedom, and by the response of the system to external electromagnetic or gravitational probes (Matsumoto et al., 2017, Delgado-Acosta et al., 2012, Porto et al., 2010).

1. General theoretical structure

A common structural feature of SIQM is that spin enters not only as a dipolar quantity but also through higher-rank operators. In the relativistic electromagnetic formulation, the current takes the Gordon-like form

$$j^\mu = e\,\bar u(p')\Big[(p'+p)^\mu + i g\,M^{\mu\nu}(p'-p)_\nu\Big]u(p),$$

and the spin-magnetization term $i g\,M^{\mu\nu}(p'-p)_\nu$ is the source of the higher multipoles beyond the charge monopole and magnetic dipole (Delgado-Acosta et al., 2012).

In post-Newtonian gravity, the waveform is organized in mass-type multipoles $I^L$ and current-type multipoles $J^L$. Spin-induced contributions modify both classes of source moments, either linearly in spin through spin-orbit couplings or quadratically through spin-spin and self-spin terms, and these corrections propagate into the gravitational-wave amplitude and phase (Porto et al., 2012, Porto et al., 2010).

In periodic crystals, an analogous bulk formulation is obtained from a nonlocal spin density $\chi^i(\mathbf q)$. The arbitrary-order spin magnetic multipole moments are defined as long-wavelength coefficients,

$$\mathcal{M}^i_{j_1\ldots j_n} = -\lim_{\mathbf q\to 0} \left[ (-i\partial_{q_{j_1}})\cdots(-i\partial_{q_{j_n}})\chi^i(\mathbf q) \right],$$

so that the multipole hierarchy is extracted directly from the $\mathbf q\to0$ expansion rather than from ill-defined position moments in an infinite crystal (Chen et al., 4 Apr 2026).

2. Local electric SIQM: single-spin and two-spin mechanisms

A central condensed-matter realization of SIQM is the spin-dependent electric dipole in a crystal or molecule without inversion symmetry. For a single spin, the most general local polarization up to quadratic order is

$$p_{\rm S}^\alpha = K^\alpha_{\beta\gamma} S^\beta S^\gamma,$$

with $$K^\alpha_{\beta\gamma}=K^\alpha_{\gamma\beta}\in\mathbb R$$. Because $S^\beta S^\gamma$ is even under inversion, it transforms as an electric quadrupole rather than as an odd-parity electric dipole. Consequently, a centrosymmetric environment forbids a single-spin electric dipole, while a noncentrosymmetric environment allows even- and odd-parity mixing, so that the dipole can be expressed through quadrupole operators such as $i g\,M^{\mu\nu}(p'-p)_\nu$0 and $i g\,M^{\mu\nu}(p'-p)_\nu$1 (Matsumoto et al., 2017).

This symmetry principle is fully classified for all 32 crystallographic point groups. In noncentrosymmetric groups, the local dipole components $i g\,M^{\mu\nu}(p'-p)_\nu$2 are linear combinations of quadrupoles in the same irreducible representation; in centrosymmetric groups all single-spin components vanish up to quadratic order. The same framework explains why electromagnon excitation, static magnetoelectric polarization, and directional dichroism appear naturally in materials such as Ba$i g\,M^{\mu\nu}(p'-p)_\nu$3CoGe$i g\,M^{\mu\nu}(p'-p)_\nu$4O$i g\,M^{\mu\nu}(p'-p)_\nu$5, where the local site symmetry allows a direct identification of the active quadrupole channels (Matsumoto et al., 2017).

For two spins $i g\,M^{\mu\nu}(p'-p)_\nu$6, the bilinear polarization separates into a symmetric part and an antisymmetric part. The antisymmetric contribution is controlled by the vector spin chirality

$i g\,M^{\mu\nu}(p'-p)_\nu$7

and in the presence of an inversion center between the two spins only the antisymmetric piece survives, reducing at higher symmetry to the Katsura–Nagaosa–Balatsky form

$i g\,M^{\mu\nu}(p'-p)_\nu$8

If the inversion center is absent, symmetric bond bilinears $i g\,M^{\mu\nu}(p'-p)_\nu$9 are also allowed and behave as bond electric quadrupoles. In this sense, local SIQM and bond SIQM are unified by the same point-group logic: odd-parity dipoles are generated from even-parity quadrupolar spin operators once inversion is broken (Matsumoto et al., 2017).

3. SIQM in solids, Bloch bands, and multipolar order

In correlated solids, SIQM are often discussed as odd-parity magnetic multipoles. The standard electric and magnetic multipole operators $I^L$0 and $I^L$1 become crystal-order parameters after decomposition into point-group irreducible representations. Odd-parity magnetic multipoles are time-reversal odd and inversion odd, preserve $I^L$2, and include the magnetic monopole \