Orbital Magnetic Quadrupole Moment

• Orbital magnetic quadrupole moment is a rank-2 magnetic multipole defined by the second spatial moment of orbital currents, crucial for understanding magnetic field gradient responses.
• Its formulations range from gauge-covariant bulk theories and Berry-phase methods to boundary manifestations in higher-order topological phases, highlighting diverse computational and experimental approaches.
• Material estimates and response theories reveal that the MQM underpins key effects such as magnetoelectric and gravito-magnetoelectric responses, thus playing a critical role in modern condensed matter physics.

Orbital magnetic quadrupole moment (MQM) is the rank-2 magnetic multipole associated with the second spatial moment of orbital current distributions and, in periodic crystals, with the response of the local free-energy density to a magnetic-field gradient. In the contemporary literature it appears under the notations $M_{ij}$, $\mathcal Q_{ij}$, and $Q^{ij}$, and more than one normalization convention is used. The concept has acquired several distinct but connected formulations: a gauge-invariant bulk theory for Bloch bands, a boundary-based characterization in higher-order topological phases, and response-theoretic roles in magnetoelectric, gravito-magnetoelectric, and nonlinear anomalous thermoelectric phenomena (Shitade et al., 2018, Gao et al., 2018, Gliozzi et al., 2022, Shinada et al., 2023).

1. Classical tensor and thermodynamic definition

For a finite current distribution $J(\mathbf r)$, the magnetic quadrupole tensor is written in the cited literature in two closely related forms. One convention uses

$$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$

while another uses

$$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$

The coexistence of these expressions indicates that notation and normalization vary across subliteratures (Gao et al., 2018, Gliozzi et al., 2022).

As with the electric quadrupole, origin independence requires the net dipole to vanish. In the higher-order-topology formulation this appears as the condition that the net magnetic dipole

$M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$

vanish, while in the crystalline-response formulation the same issue motivates abandoning direct use of the position operator in favor of a thermodynamic definition (Gliozzi et al., 2022).

In periodic crystals the position operator is ill-defined in the Bloch basis, so the orbital MQM is defined as a response of the local free-energy density to a weak, spatially inhomogeneous magnetic field. One formulation introduces

$$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$

and another writes the local gradient expansion

$dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$

In this sense the MQM is the coefficient conjugate to the first spatial derivative of the magnetic field rather than to the field itself (Gao et al., 2018, Shinada et al., 2023).

A complementary wave-packet interpretation is also available. In the Berry-phase framework,

$$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$

where $\mathcal Q_{ij}$0 is the packet center and $\mathcal Q_{ij}$1 is the orbital angular-momentum operator. The fully gauge-invariant evaluation of this second moment reproduces the Brillouin-zone expression for $\mathcal Q_{ij}$2 (Shinada et al., 2023).

2. Microscopic formulations in periodic Bloch bands

A fully gauge-invariant quantum-mechanical formula for the orbital MQM in crystalline solids was derived by gauge-covariant gradient expansion. In that construction one starts from the two-point Keldysh Green’s function with a Wilson line, rewrites the Dyson equation with the gauge-covariant Moyal product, expands in gradients of the electromagnetic field, and identifies the coefficient of $\mathcal Q_{ij}$3 in the thermodynamic potential. The resulting tensor is

$\mathcal Q_{ij}$4

with the kernels $\mathcal Q_{ij}$5, $\mathcal Q_{ij}$6, and $\mathcal Q_{ij}$7 defined by interband-projector matrix elements of Bloch derivatives and the Hamiltonian. The first term is a Fermi-sea itinerant contribution, while the second and third are local contributions. The formula is stated to hold for insulators and metals, and at zero and finite temperature (Shitade et al., 2018).

A complementary Bloch-band expression emphasizes explicitly geometric objects. Summing over occupied bands $\mathcal Q_{ij}$8, one obtains a gauge-invariant formula in which $\mathcal Q_{ij}$9 contains three types of terms: an interband term proportional to $Q^{ij}$0, a Hessian term involving $Q^{ij}$1, and a metric term involving $Q^{ij}$2. Here $Q^{ij}$3 is the Berry connection, $Q^{ij}$4 the interband orbital moment, $Q^{ij}$5 the band-energy Hessian, and $Q^{ij}$6 the quantum metric. The same work also gives an equivalent compact “interband-Berry-velocity” form, with the metric and Hessian pieces required for full gauge invariance under $Q^{ij}$7 (Gao et al., 2018).

The Berry-phase formulation of the uniform tensor $Q^{ij}$8 in metals and insulators likewise yields a Brillouin-zone integral built from the Berry connection $Q^{ij}$9, the quantum metric $J(\mathbf r)$0, the interband orbital object $J(\mathbf r)$1, the Fermi function $J(\mathbf r)$2, and the grand-potential density $J(\mathbf r)$3. In this form, the MQM is not a purely Fermi-surface quantity: it contains both occupation and grand-potential pieces, and the latter are essential for thermodynamic consistency (Shinada et al., 2023).

These formulations collectively establish the orbital MQM as a bona fide bulk quantity of band theory, albeit one whose implementation depends strongly on whether the physical problem is a generic periodic crystal or a higher-order topological phase.

3. Boundary manifestation in higher-order topological phases

In three-dimensional higher-order topological phases, the diagonal components of the MQM manifest as surface-localized magnetization and hinge currents. For an orthorhombic crystal with vanishing bulk magnetization $J(\mathbf r)$4, a nonzero bulk $J(\mathbf r)$5 produces a surface magnetization density on a face $J(\mathbf r)$6 with outward normal $J(\mathbf r)$7,

$J(\mathbf r)$8

and a hinge current on the intersection of faces $J(\mathbf r)$9,

$$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$0

This boundary description is the direct magnetic analogue of the way an electric quadrupole produces boundary charge structure (Gliozzi et al., 2022).

A central point is that the hinge current is generally not equal to the difference of surface magnetizations that intersect at the hinge. The mismatch is precisely quantified by the bulk MQM. In practice, one may glue arbitrary surface layers, such as 2D Chern insulators, which contribute extra surface magnetizations $$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$1 and hinge currents $$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$2 satisfying the Ampère law

$$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$3

Appropriate combinations of $$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$4 and $$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$5 then cancel the decoration-dependent pieces and isolate the bulk quadrupole contribution (Gliozzi et al., 2022).

The resulting bulk-sensitive quantities are the three independent quadrupole differences

$$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$6

$$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$7

$$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$8

Each $$Q_{ij}=\frac{1}{3V}\int d^3r\, r_i [r\times J(r)]_j,$$9 is invariant under any surface decoration with zero net $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$0, and therefore functions as a bulk characterization of the higher-order phase (Gliozzi et al., 2022).

A microscopic computation of the required surface magnetizations uses slab geometry. For a slab infinite in $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$1 but finite in $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$2, the layer-resolved magnetization is

$$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$3

with

$$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$4

and $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$5 the projector onto sites in layer $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$6. The surface magnetization on the $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$7 face is then

$$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$8

Analogous formulas apply to $$M_{ij}=\frac{2}{3V}\int_V d^3r\, r_i [r\times J(r)]_j.$$9- and $M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$0-slabs (Gliozzi et al., 2022).

Within this framework, the MQM can distinguish phases in some intrinsic and boundary-obstructed higher-order topological insulators. The formalism also shows that a fully bulk formula valid for higher-order phases remains an open question, so the boundary-based construction is not merely calculational convenience but part of the present conceptual structure of the subject (Gliozzi et al., 2022).

4. Relation to magnetoelectric and gravito-magnetoelectric response

One of the main outcomes of the modern theory is that the orbital MQM is directly tied to magnetoelectric (ME) response. In a zero-temperature insulator, the gauge-covariant theory gives the Maxwell relation

$M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$1

and the band-theory approach gives the equivalent statement

$M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$2

Thus the MQM is identified as the microscopic origin of the linear ME effect in insulating crystals (Shitade et al., 2018, Gao et al., 2018).

The same connection acquires a thermal analogue in the orbital gravito-magnetoelectric effect (OGME), where a temperature gradient induces orbital magnetization according to

$M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$3

A naive Kubo treatment based only on the current–energy–density correlator leads to

$M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$4

but this expression diverges like $M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$5 at low temperature. The correction is supplied by the equilibrium magnetization current arising from $M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$6 in the perturbed Hamiltonian, which injects precisely the magnetic-quadrupole term $M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$7. The full intrinsic response is

$M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$8

Once this correction is included, the $M_k=\tfrac12\epsilon_{k\ell m}\int r_\ell J_m$9 divergences cancel, and the intrinsic OGME acquires a finite entropy-density representation (Shinada et al., 2023).

The same analysis proves the Mott relation. With the zero-temperature intrinsic orbital ME tensor written as

$$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$0

the intrinsic OGME satisfies

$$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$1

and therefore, at low temperature,

$$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$2

An analogous Mott relation holds for the extrinsic OGME and extrinsic OME (Shinada et al., 2023).

This response-theoretic role corrects a common misconception that the Kubo formula alone captures the intrinsic thermal magnetoelectric response. In the cited analysis, the MQM is the missing bulk correction required both to eliminate an unphysical zero-temperature divergence and to recover the correct thermodynamic relation (Shinada et al., 2023).

5. Symmetry structure, quantization, and representative models

The orbital MQM transforms as a rank-2 pseudotensor. In the Bloch-band formulation, under time reversal $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$3 one has $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$4, $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$5, and $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$6, implying $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$7; under inversion $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$8, $$\mathcal Q_{ij}(r)=-\lim_{B\to0}\frac{\partial F(r)}{\partial(\partial_i B_j)}\Big|_{B(r)},$$9 flips $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$0 while leaving $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$1 axial, again giving $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$2. Hence $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$3 is even under combined $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$4, and only states that break $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$5 and $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$6 separately but preserve $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$7 can host a nonzero quadrupole. For any group containing $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$8 or $dF=M_{0i}\,dB_i+Q^{ij}\,d(\partial_i B_j)+\cdots.$9 separately, $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$0 (Gao et al., 2018).

Magnetic-point-group constraints for the OGME sharpen this classification. In any pure $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$1-symmetric group the extrinsic response vanishes because $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$2, while the intrinsic response can remain nonzero. In any group with full $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$3, only the extrinsic response survives and the intrinsic one is forbidden. A subset of groups, including $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$4 and $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$5, allows a monopole (Chern–Simons) term, so that $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$6; in all others $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$7 (Shinada et al., 2023).

In higher-order topology, derivatives of the bulk quadrupole differences define quantized invariants

$$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$8

Along a given hinge, $$Q^{ij}=-e\,\langle W|(r^i-r_c^i)\,(L^j)|W\rangle_c+\cdots,$$9 counts net chiral hinge modes. On a gapped surface, $\mathcal Q_{ij}$00 is the half-integer magneto-electric polarizability. Consequently, $\mathcal Q_{ij}$01 whenever surfaces and hinges are gapped and the bulk $\mathcal Q_{ij}$02 can be fractional. For a $\mathcal Q_{ij}$03-protected intrinsic chiral HOTI, the individual slopes satisfy

$\mathcal Q_{ij}$04

so that $\mathcal Q_{ij}$05; numerically, $\mathcal Q_{ij}$06 reproduces the usual $\mathcal Q_{ij}$07 invariant. For a boundary-obstructed HOTI with all three surfaces gapped by two-fold rotations, numerics give $\mathcal Q_{ij}$08 (Gliozzi et al., 2022).

Several model systems illustrate these symmetry principles. In a minimal two-band tilted Dirac-cone model,

$\mathcal Q_{ij}$09

only $\mathcal Q_{ij}$10 survives because mirror-$\mathcal Q_{ij}$11 symmetry forces $\mathcal Q_{ij}$12 while the $\mathcal Q_{ij}$13 tilt breaks mirror-$\mathcal Q_{ij}$14. At $\mathcal Q_{ij}$15 and $\mathcal Q_{ij}$16,

$\mathcal Q_{ij}$17

so $\mathcal Q_{ij}$18 jumps as $\mathcal Q_{ij}$19 crosses the Dirac point, and a finite gap smooths the jump (Gao et al., 2018).

In the $\mathcal Q_{ij}$20-symmetric three-orbital CuO$\mathcal Q_{ij}$21 loop-current model,

$\mathcal Q_{ij}$22

the magnetic point group is $\mathcal Q_{ij}$23, so only $\mathcal Q_{ij}$24 and $\mathcal Q_{ij}$25 survive. Numerically, $\mathcal Q_{ij}$26 at low $\mathcal Q_{ij}$27 shows sharp peaks at each Dirac-point energy $\mathcal Q_{ij}$28, and $\mathcal Q_{ij}$29 at fixed $\mathcal Q_{ij}$30 is strictly proportional to $\mathcal Q_{ij}$31 for $\mathcal Q_{ij}$32, in accord with the Mott law (Shinada et al., 2023).

A related multipolar-ordering literature treats the magnetic quadrupole as a rank-2, time-reversal-odd, axial tensor with five independent components and shows that spontaneous odd-parity magnetic quadrupole order in a three-orbital system produces antisymmetric spin-orbital polarization in momentum space, or spin-orbital momentum locking. Under $\mathcal Q_{ij}$33 these components transform as $\mathcal Q_{ij}$34, $\mathcal Q_{ij}$35, $\mathcal Q_{ij}$36, and $\mathcal Q_{ij}$37. This establishes a symmetry bridge between quadrupolar order, momentum-space textures, and cross-correlated responses such as the magnetoelectric effect and current-induced distortion (Hayami et al., 2021).

6. Computation, material estimates, and open issues

For higher-order topological phases, the practical computation of the MQM proceeds under explicit assumptions. Zero bulk magnetization is enforced by requiring $\mathcal Q_{ij}$38 so that $\mathcal Q_{ij}$39. Surfaces must be gapped, or gapped by benign decorations, so that $\mathcal Q_{ij}$40 is well-defined. Hinges must carry well-localized currents, and conservation together with $\mathcal Q_{ij}$41 symmetry ensures that parallel hinges have equal current. The Bloch gauge must be smooth in the two-dimensional Brillouin zone of the slab. One then computes layer-resolved $\mathcal Q_{ij}$42 in a large slab, integrates half the slab to obtain $\mathcal Q_{ij}$43, computes hinge currents $\mathcal Q_{ij}$44 in a geometry with two open directions, combines them through the quadrupole differences $\mathcal Q_{ij}$45, and numerically differentiates with respect to $\mathcal Q_{ij}$46 to extract $\mathcal Q_{ij}$47 (Gliozzi et al., 2022).

In crystalline antiferromagnets, the gauge-covariant theory yields concrete material estimates for the orbital part of the magnetoelectric susceptibility. For BaMn$\mathcal Q_{ij}$48As$\mathcal Q_{ij}$49, focusing on the nonzero diagonal component in the AFM-ordered state with $\mathcal Q_{ij}$50, one finds

$\mathcal Q_{ij}$51

while the spin contribution is

$\mathcal Q_{ij}$52

For CeMn$\mathcal Q_{ij}$53Ge$\mathcal Q_{ij}$54Si$\mathcal Q_{ij}$55 with $\mathcal Q_{ij}$56, the estimates include

$\mathcal Q_{ij}$57

and

$\mathcal Q_{ij}$58

These values support the statement that the orbital contribution can be comparable with, or even dominant over, the spin contribution (Shitade et al., 2018).

For the thermal response problem, possible detection routes have been proposed rather than established. In the loop-current context, experimental detection might proceed by torque-magnetometry in a well-controlled $\mathcal Q_{ij}$59 field, or by ultrafast pump-probe schemes that effectively create transient $\mathcal Q_{ij}$60’s and then read off $\mathcal Q_{ij}$61. Since $\mathcal Q_{ij}$62 couples to $\mathcal Q_{ij}$63, one may also envisage applying small inhomogeneous magnetic fields and measuring the ensuing orbital polarization. These suggestions are consistent with the response definition but remain proposals in the cited discussion (Shinada et al., 2023).

Two conceptual boundaries of the present theory are explicit. First, in higher-order topological phases a fully bulk formula valid for higher-order phases remains an open question, so boundary signatures continue to play a defining role. Second, the various formulations emphasize different physical regimes: Bloch-band thermodynamics for periodic crystals, boundary reconstruction for higher-order topology, and correlation-function corrections for thermal responses. A plausible implication is that the orbital MQM is best viewed not as a single computational object but as a unifying multipolar response concept whose precise representation depends on which aspect of orbital magnetism is physically accessible.