---
title: Orbital Magnetic Quadrupole Moment
url: https://www.emergentmind.com/topics/orbital-magnetic-quadrupole-moment
type: topic
---

# Orbital Magnetic Quadrupole Moment

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 [1803.00217] [1803.06726] [2211.08438] [2302.06047].

## 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 [1803.06726] [2211.08438].

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 [2211.08438].

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 [1803.06726] [2302.06047].

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 $r_c$ is the packet center and $L^j$ is the orbital angular-momentum operator. The fully gauge-invariant evaluation of this second moment reproduces the Brillouin-zone expression for $Q^{ij}$ [2302.06047].

## 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 $\partial_{X^l}B^k$ in the thermodynamic potential. The resulting tensor is
$$
M^{l}{}_{k}
=\frac{q}{\hbar}\,\frac12\,\epsilon_{ijk}
\sum_{n}\int\frac{d^{d}p}{(2\pi\hbar)^{d}}
\left[
-\,A_{n}^{lij}\int_{\epsilon_{n}-\mu}^{\infty}dz\,f(z)
+ m_{n}^{lij}f_{n}
+ \gamma_{n}^{lij}f'_{n}
\right],
$$
with the kernels $A_n^{lij}$, $m_n^{lij}$, and $\gamma_n^{lij}$ 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 [1803.00217].

A complementary Bloch-band expression emphasizes explicitly geometric objects. Summing over occupied bands $n$, one obtains a gauge-invariant formula in which $\mathcal Q_{ij}$ contains three types of terms: an interband term proportional to $\mathrm{Re}(\mathcal A_{nm,i} M_{mn,j})$, a Hessian term involving $\partial_k \Gamma^n_{i\ell}$, and a metric term involving $\partial_k g^n_{i\ell}$. Here $\mathcal A$ is the Berry connection, $M_{nm,j}$ the interband orbital moment, $\Gamma$ the band-energy Hessian, and $g$ 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 $|u_n\rangle\to e^{i\phi_n(\mathbf k)}|u_n\rangle$ [1803.06726].

The Berry-phase formulation of the uniform tensor $Q^{ij}$ in metals and insulators likewise yields a Brillouin-zone integral built from the Berry connection $A^i_{nm}$, the quantum metric $g_n^{ij}$, the interband orbital object $M^j_{mn}$, the Fermi function $f(\epsilon)$, and the grand-potential density $g(\epsilon)$. 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 [2302.06047].

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 $M=0$, a nonzero bulk $M_{ij}$ produces a surface magnetization density on a face $\alpha$ with outward normal $n^\alpha$,
$$
K_j^\alpha=n_k^\alpha M_{jk},
$$
and a hinge current on the intersection of faces $\alpha,\beta$,
$$
J_i^{\alpha\beta}
=\tfrac12\epsilon_{ikl}(n^\alpha_j n^\beta_l+n^\beta_j n^\alpha_l)\,M_{jk}.
$$
This boundary description is the direct magnetic analogue of the way an electric quadrupole produces boundary charge structure [2211.08438].

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 $\tilde K^\alpha$ and hinge currents $\tilde J^{\alpha\beta}$ satisfying the Ampère law
$$
\tilde J^{\alpha\beta}=\tilde K^\alpha\times n^\beta+\tilde K^\beta\times n^\alpha.
$$
Appropriate combinations of $J$ and $K$ then cancel the decoration-dependent pieces and isolate the bulk quadrupole contribution [2211.08438].

The resulting bulk-sensitive quantities are the three independent quadrupole differences
$$
\mathcal M_{12}\equiv \tfrac12(M_{11}-M_{22})
=K_1^{+x}-K_2^{+y}-J_3^{+x,+y},
$$
$$
\mathcal M_{23}\equiv \tfrac12(M_{22}-M_{33})
=K_2^{+y}-K_3^{+z}-J_1^{+y,+z},
$$
$$
\mathcal M_{31}\equiv \tfrac12(M_{33}-M_{11})
=K_3^{+z}-K_1^{+x}-J_2^{+z,+x}.
$$
Each $\mathcal M_{ij}$ is invariant under any surface decoration with zero net $M$, and therefore functions as a bulk characterization of the higher-order phase [2211.08438].

A microscopic computation of the required surface magnetizations uses slab geometry. For a slab infinite in $x,y$ but finite in $z$, the layer-resolved magnetization is
$$
M_3(z)=\frac12\sum_{k_x,k_y}\mathrm{Im\,Tr}\Bigl\{
\tilde P_z[g_{k,xy}+h_{k,xy}]-(x\leftrightarrow y)
\Bigr\},
$$
with
$$
g_{k,jl}=\partial_jQ_k\,Q_k(H_k-\mu)Q_k\,\partial_lQ_k,\qquad
h_{k,jl}=\partial_lP_k\,P_k(H_k-\mu)P_k\,\partial_jP_k,
$$
and $\tilde P_z$ the projector onto sites in layer $z$. The surface magnetization on the $+i$ face is then
$$
K_i=\int_0^{L_i/2}dr_i\,M_i(r_i).
$$
Analogous formulas apply to $x$- and $y$-slabs [2211.08438].

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 [2211.08438].

## 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
$$
\alpha^{l}{}_{k}\equiv \frac{\partial P^l}{\partial B^k}
=-\,q\,\frac{\partial M^{l}{}_{k}}{\partial\mu},
$$
and the band-theory approach gives the equivalent statement
$$
e\,\frac{\partial \mathcal Q_{ij}}{\partial\mu}=-\alpha_{ij}.
$$
Thus the MQM is identified as the microscopic origin of the linear ME effect in insulating crystals [1803.00217] [1803.06726].

The same connection acquires a thermal analogue in the orbital gravito-magnetoelectric effect (OGME), where a temperature gradient induces orbital magnetization according to
$$
M^i=\chi^{i\mathrm{OGME}}_{ij}(-\partial_j T).
$$
A naive Kubo treatment based only on the current–energy–density correlator leads to
$$
\chi^{i\mathrm{OGME},(\mathrm{naive})}_{ij}=-(2i/T)\,\beta^{JH}_{ij},
$$
but this expression diverges like $1/T$ at low temperature. The correction is supplied by the equilibrium magnetization current arising from $(1+\psi)J_0(x)$ in the perturbed Hamiltonian, which injects precisely the magnetic-quadrupole term $Q_{ij}$. The full intrinsic response is
$$
\chi^{i\mathrm{OGME}}_{ij}=-(2i\,\beta^{JH}_{ij}+Q_{ij})/T.
$$
Once this correction is included, the $O(1/T)$ divergences cancel, and the intrinsic OGME acquires a finite entropy-density representation [2302.06047].

The same analysis proves the Mott relation. With the zero-temperature intrinsic orbital ME tensor written as
$$
\chi^{i\mathrm{OME}}_{ij}(\mu,T=0)
=-e^2\int_{BZ}\frac{d^3k}{(2\pi)^3}\sum_n f(\epsilon_{nk}-\mu)\,W^n_{ij},
$$
the intrinsic OGME satisfies
$$
\chi^{i\mathrm{OGME}}_{ij}(\mu,T)
=\int d\epsilon\,\frac{\epsilon-\mu}{eT}\,
[-\partial_\epsilon f(\epsilon-\mu)]\,
\chi^{i\mathrm{OME}}_{ij}(\epsilon,T=0),
$$
and therefore, at low temperature,
$$
\chi^{i\mathrm{OGME}}_{ij}\simeq
\left[-\frac{\pi^2T}{3e}\right]\partial_\mu \chi^{i\mathrm{OME}}_{ij}(\mu,0).
$$
An analogous Mott relation holds for the extrinsic OGME and extrinsic OME [2302.06047].

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 [2302.06047].

## 5. Symmetry structure, quantization, and representative models

The orbital MQM transforms as a rank-2 pseudotensor. In the Bloch-band formulation, under time reversal $\mathcal T$ one has $\mathcal A\to-\mathcal A^*$, $v\to -v$, and $M\to -M$, implying $\mathcal Q_{ij}\to -\mathcal Q_{ij}$; under inversion $\mathcal I$, $\mathbf k\to -\mathbf k$ flips $\mathcal A$ while leaving $M$ axial, again giving $\mathcal Q\to -\mathcal Q$. Hence $\mathcal Q$ is even under combined $\mathcal T\mathcal I$, and only states that break $\mathcal T$ and $\mathcal I$ separately but preserve $\mathcal T\mathcal I$ can host a nonzero quadrupole. For any group containing $\mathcal T$ or $\mathcal I$ separately, $\mathcal Q=0$ [1803.06726].

Magnetic-point-group constraints for the OGME sharpen this classification. In any pure $\mathcal P\mathcal T$-symmetric group the extrinsic response vanishes because $m_{n\mathbf k}=0$, while the intrinsic response can remain nonzero. In any group with full $\mathcal T$, only the extrinsic response survives and the intrinsic one is forbidden. A subset of groups, including $23$ and $432$, allows a monopole (Chern–Simons) term, so that $\mathrm{Tr}\,\chi^{i\mathrm{OGME}}\neq 0$; in all others $\mathrm{Tr}\,\chi=0$ [2302.06047].

In higher-order topology, derivatives of the bulk quadrupole differences define quantized invariants
$$
n_{ij}\equiv 2\pi\,\partial \mathcal M_{ij}/\partial\mu.
$$
Along a given hinge, $\partial J/\partial\mu$ counts net chiral hinge modes. On a gapped surface, $\partial K/\partial\mu=\theta/2\pi \ (\mathrm{mod}\ 1)$ is the half-integer magneto-electric polarizability. Consequently, $n_{ij}\in\mathbb Z$ whenever surfaces and hinges are gapped and the bulk $\theta$ can be fractional. For a $C_4T$-protected intrinsic chiral HOTI, the individual slopes satisfy
$$
2\pi\,\partial (K_1-K_2)/\partial\mu\in \mathbb Z+\tfrac12,\qquad
2\pi\,\partial J_3/\partial\mu\in \mathbb Z+\tfrac12,
$$
so that $n_{12}\in 2\mathbb Z$; numerically, $n_{12}/2\ \mathrm{mod}\ 2$ reproduces the usual $\mathbb Z_2$ invariant. For a boundary-obstructed HOTI with all three surfaces gapped by two-fold rotations, numerics give $(n_{12},n_{23},n_{31})=(-2,+2,0)$ [2211.08438].

Several model systems illustrate these symmetry principles. In a minimal two-band tilted Dirac-cone model,
$$
\hat H(\mathbf k)=v'k_x+v_xk_x\sigma_x+v_yk_y\sigma_y+\Delta\sigma_z,
$$
only $\mathcal Q_{yz}$ survives because mirror-$y$ symmetry forces $\mathcal Q_{xz}=0$ while the $k_x$ tilt breaks mirror-$x$. At $T=0$ and $\Delta=0$,
$$
\mathcal Q_{yz}=+\frac{ev'}{16\pi}\left| \frac{v_y}{v_x}\right| \quad (\mu<0),\qquad
\mathcal Q_{yz}=-\frac{ev'}{16\pi}\left| \frac{v_y}{v_x}\right| \quad (\mu>0),
$$
so $\mathcal Q_{yz}$ jumps as $\mu$ crosses the Dirac point, and a finite gap smooths the jump [1803.06726].

In the $\mathcal{PT}$-symmetric three-orbital CuO$_2$ loop-current model,
$$
H_k=
\begin{pmatrix}
0 & i t s_x+i r c_x & i t s_y+i r c_y\\
-i t s_x-i r c_x & 0 & t' s_x s_y\\
-i t s_y-i r c_y & t' s_x s_y & 0
\end{pmatrix},
$$
the magnetic point group is $m'mm$, so only $\chi^{i\mathrm{OGME}}_{y'z}$ and $\chi^{i\mathrm{OGME}}_{z y'}$ survive. Numerically, $\chi^{i\mathrm{OGME}}_{y'z}(\mu)$ at low $T$ shows sharp peaks at each Dirac-point energy $E_D$, and $\chi^{i\mathrm{OGME}}_{y'z}(T)$ at fixed $\mu$ is strictly proportional to $T$ for $T\ll |\mu-E_D|$, in accord with the Mott law [2302.06047].

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 $D_{4h}$ these components transform as $A_{1u}^-: M_u$, $B_{1u}^-: M_v$, $B_{2u}^-: M_{xy}$, and $E_u^-:\{M_{yz},M_{zx}\}$. This establishes a symmetry bridge between quadrupolar order, momentum-space textures, and cross-correlated responses such as the magnetoelectric effect and current-induced distortion [2105.09444].

## 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 $C_{2x,y,z}$ so that $\int r\times J=0$. Surfaces must be gapped, or gapped by benign decorations, so that $K$ is well-defined. Hinges must carry well-localized currents, and conservation together with $C_2$ 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 $M_i(r_i)$ in a large slab, integrates half the slab to obtain $K_i$, computes hinge currents $J_i$ in a geometry with two open directions, combines them through the quadrupole differences $\mathcal M_{ij}$, and numerically differentiates with respect to $\mu$ to extract $n_{ij}$ [2211.08438].

In crystalline antiferromagnets, the gauge-covariant theory yields concrete material estimates for the orbital part of the magnetoelectric susceptibility. For BaMn$_2$As$_2$, focusing on the nonzero diagonal component in the AFM-ordered state with ${\bf h}\parallel z$, one finds
$$
\alpha^{1}{}_{1}=-\alpha^{2}{}_{2}\approx 4.2\times10^{-7}\,\frac{q^2}{\hbar}
=1.3\times10^{-4}\,\mathrm{ps/m},
$$
while the spin contribution is
$$
\alpha^{1}{}_{1}|_{\rm spin}\approx -1.5\times10^{-3}\,\mathrm{ps/m}.
$$
For CeMn$_2$Ge$_{2-x}$Si$_x$ with ${\bf h}\parallel x$, the estimates include
$$
\alpha^{3}{}_{1}({\rm orbital})\approx -3.3\times10^{-5}\,\mathrm{ps/m},\qquad
\alpha^{3}{}_{1}({\rm spin})\approx 1.9\times10^{-6}\,\mathrm{ps/m},
$$
and
$$
\alpha^{1}{}_{3}({\rm orbital})\sim 2.9\times10^{-4}\,\mathrm{ps/m}
\quad\text{versus}\quad
\alpha^{1}{}_{3}({\rm spin})\sim 1.5\times10^{-3}\,\mathrm{ps/m}.
$$
These values support the statement that the orbital contribution can be comparable with, or even dominant over, the spin contribution [1803.00217].

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 $\nabla T$ field, or by ultrafast pump-probe schemes that effectively create transient $\nabla T$’s and then read off $M(t)$. Since $Q^{ij}$ couples to $\partial_i B_j$, 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 [2302.06047].

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.

Source: https://www.emergentmind.com/topics/orbital-magnetic-quadrupole-moment