Interband Quadrupole: Mechanisms & Manifestations

• Interband quadrupole is a rank-2 degree of freedom that couples distinct spectral bands, enabling diagnostic and mediating roles in superconductors, atomic systems, and topological phases.
• In superconductivity, interband quadrupole effects are observed via nuclear quadrupole resonance and spin-lattice relaxation, revealing the influence of sign-reversing interband processes on coherence phenomena.
• In atomic and topological materials, quadrupole phenomena manifest through inter-level mixing, resonant optical transitions, and quantized electronic geometry, providing insights for quantum gating and material characterization.

Interband quadrupole denotes a family of phenomena in which a quadrupole degree of freedom mediates, weights, or diagnoses processes connecting distinct bands, manifolds, or Fermi-surface sectors. In the cited literature, the expression is context-dependent: it can refer to a quadrupole-selected nucleus that senses interband superconducting coherence through spin-lattice relaxation, to electric-quadrupole operators that couple different atomic or Rydberg manifolds, to quadrupole-like optical tensors whose dominant contributions are interband, to electric-quadrupole $E2$ transitions between nuclear collective bands, or to higher-order topological phases whose quadrupole moment is encoded in the interband geometry of occupied states (Mukuda et al., 2010, Higgins et al., 2020, Sato et al., 2020, Cheng et al., 2016, Matsuzaki et al., 2016, Peng et al., 2019).

1. Conceptual scope and operator structure

Across these settings, the quadrupole degree of freedom is always rank-2, but the physical object carrying it differs. In $^{75}$As NQR the relevant operator is the nuclear quadrupole Hamiltonian

$$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$

with NQR frequency

$$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$

In trapped-ion spectroscopy and Rydberg physics the central operator is $er^2Y_2^m$, entering quadrupole polarizability and quadrupole–quadrupole interactions. In collective nuclear models the electric quadrupole transition operator is

$$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$

In electronic solids, by contrast, “quadrupole” may refer to cluster magnetic quadrupole order, to electric quadrupole-like optical tensors $S_Q^{dabc}$, or to a bulk quadrupole moment extracted from nested Wilson loops rather than from a local multipole operator (Mukuda et al., 2010, Higgins et al., 2020, Han et al., 2022, Inci et al., 2011, Sato et al., 2020, Cheng et al., 2016, Peng et al., 2019).

Setting	Quadrupole object	Interband manifestation
Iron pnictide NQR	$^{75}$As nuclear quadrupole	Interband quasiparticle coherence in $1/T_1$
Trapped Rydberg ion	Quadrupole polarizability $\alpha_{22}$	Inter-level mixing and Floquet sidebands
Ultracold Rydberg gas	$^{75}$0	Quadrupole blockade and manifold mixing
Spin–orbit metal	Cluster magnetic quadrupole order	Selected low-energy interband Hall transitions
Graphene	Electric quadrupole-like tensor $^{75}$1	Interband SHG, photon drag, DFG resonances
Nuclear collective bands	$^{75}$2 operator	Interband $^{75}$3 systematics
Quadrupole insulator	Nested Wilson loops	Interband geometry of occupied states

A plausible implication is that “interband quadrupole” is best understood as a structural motif rather than as a single subfield: a rank-2 degree of freedom is coupled to transitions between distinct sectors of the spectrum, and the observable of interest is controlled by that coupling.

2. Multiband superconductivity and quadrupole-selected NQR

In heavily electron-overdoped LaFeAsO$^{75}$4F$^{75}$5, $^{75}$6As NQR provides a local quadrupole probe of the electric field gradient while the spin-lattice relaxation rate $^{75}$7 measures low-energy magnetic excitations across all Fermi-surface sheets. For a spin-$^{75}$8 nucleus in zero field there is a single NQR transition between the $^{75}$9 and $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$0 manifolds. The $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$1As line in the heavily overdoped sample occurs at $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$2 MHz, and previous work had shown that $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$3 increases monotonically with electron doping, so the large $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$4 was taken as evidence of strong electron overdoping and a homogeneous electronic environment. The same experiment found that the Hebel–Slichter peak, absent in optimally doped La1111 with $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$5 K, partially recovers in the heavily overdoped regime with $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$6 K (Mukuda et al., 2010).

The multiband analysis uses

$$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$7

so that

$$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$8

contains intraband terms and an interband cross term $$\mathcal{H}_Q = \frac{e^2 q Q}{4I(2I-1)} \left[3I_z^2 - I(I+1) + \frac{\eta}{2}(I_+^2 + I_-^2) \right],$$9. In an $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$0 state, with $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$1 and $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$2, this interband term is negative,

$$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$3

and it cancels the positive intraband coherence contribution when interband scattering is strong. The paper introduces a phenomenological weighting parameter $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$4: $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$5 corresponds to intraband-dominated coherence, $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$6 to nearly complete suppression of the coherence term by sign-reversing interband processes, and intermediate $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$7 to partial cancellation. Fits give $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$8 for optimally doped La1111 and $$\nu_Q = \frac{3 e^2 q Q}{2h I(2I-1)} \sqrt{1 + \frac{\eta^2}{3}}.$$9 for the heavily overdoped sample.

The interpretation is tied to Fermi-surface evolution. At optimal doping, hole and electron Fermi surfaces are well nested by $er^2Y_2^m$0 or $er^2Y_2^m$1, so interband $er^2Y_2^m$2 processes dominate and the Hebel–Slichter peak is essentially absent. In the heavily overdoped sample, nominal $er^2Y_2^m$3, the hole Fermi surface is nearly gone, nesting is badly degraded, interband scattering is strongly suppressed, and some conventional coherence effect reappears. The normal-state comparison

$er^2Y_2^m$4

shows a substantially reduced DOS at $er^2Y_2^m$5, while the low-temperature relaxation gives $er^2Y_2^m$6, with $er^2Y_2^m$7 and $er^2Y_2^m$8, indicating very weak coupling in the overdoped compound. A frequent misconception in this literature is that absence of a Hebel–Slichter peak by itself implies nodes; here the same fully gapped two-band structure yields either no peak or a small finite peak depending on the strength of interband sign-cancelling processes. Within this NQR framework, interband scattering is not merely a relaxation channel but part of the mechanism that stabilizes the sign-reversing $er^2Y_2^m$9 state and enhances $$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$0.

3. Atomic and Rydberg realizations: quadrupole polarizability, mixing, and blockade

In a single trapped $$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$1 ion excited to Rydberg states, the electric potential of the linear Paul trap has a dominant quadrupole component,

$$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$2

and the interaction with the ion’s charge distribution is

$$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$3

The second-order response is governed by the quadrupole polarizability

$$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$4

which is explicitly an inter-level quantity built from virtual quadrupole transitions. For the $$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$5 Rydberg states studied, there is no permanent quadrupole moment, but there is a second-order quadrupole response described by a scalar $$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$6. The time-dependent shift takes the form

$$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$7

leading to sidebands at

$$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$8

For $$\hat{T}_\mu(E2) = t\,\beta\Big[ \mathcal{D}^{(2)}_{\mu,0}(\Omega)\cos\gamma +\frac{1}{\sqrt{2}}\big(\mathcal{D}^{(2)}_{\mu,2}(\Omega)+\mathcal{D}^{(2)}_{\mu,-2}(\Omega)\big)\sin\gamma \Big].$$9 the experiment extracts $S_Q^{dabc}$0, in good agreement with the calculated $S_Q^{dabc}$1; for $S_Q^{dabc}$2 it reports $S_Q^{dabc}$3 a.u. For $S_Q^{dabc}$4, a “forest” of sidebands is observed with extracted $S_Q^{dabc}$5 rad, and a full diagonalization including $S_Q^{dabc}$6, $S_Q^{dabc}$7, and nearby $S_Q^{dabc}$8 manifolds is required, showing that the perturbative single-parameter polarizability picture breaks down in the strong-mixing regime. The observed scaling $S_Q^{dabc}$9 reflects stronger quadrupole mixing as principal quantum number increases (Higgins et al., 2020).

In ultracold $^{75}$0Rb Rydberg gases the corresponding two-body interaction contains an explicit first-order quadrupole–quadrupole term,

$^{75}$1

with $^{75}$2, $^{75}$3. The essential distinction from resonant dipole–dipole physics is explicit in the paper: on-resonance dipole–dipole interactions cannot exist in the same state, whereas on-resonance quadrupole–quadrupole interactions can exist in the same state. For $^{75}$4 Rydberg atoms, the nearest dipole-coupled pair state lies $^{75}$5 GHz above, so dipole coupling contributes only an attractive van der Waals tail, while quadrupole–quadrupole interactions remain on resonance within the $^{75}$6 manifold. Experimentally, increasing density from $^{75}$7 to $^{75}$8 broadens the spectral line in both directions, with somewhat larger high-frequency broadening, and suppresses excitation per atom. The authors interpret this as evidence for partial quadrupole blockade. A one-dimensional few-body calculation with nearest-neighbor spacing $^{75}$9 finds that the maximal shift grows approximately linearly with atom number $1/T_1$0, indicating that many-body quadrupole effects enhance the blockade. This suggests a short-range, anisotropic route to compact quantum-gate architectures that differs structurally from Förster-type dipole blockade (Han et al., 2022).

4. Interband optical response in correlated and Dirac materials

For spin–orbit-coupled metals with cluster magnetic multipoles, the optical Hall conductivity is purely interband: $1/T_1$1 with $1/T_1$2 terms absent. In the quadrupole-ordered state of the square four-site-cluster model,

$1/T_1$3

the cluster magnetic moments are parallel or antiparallel to the ASOC effective field $1/T_1$4. This yields a sharp interband selection rule in the low-energy manifold: $1/T_1$5 so the dominant Hall response comes from the interband pairs $1/T_1$6 and $1/T_1$7. The corresponding resonance scale is

$1/T_1$8

set by kinetic energy rather than the exchange splitting $1/T_1$9. Numerically, with $\alpha_{22}$0, $\alpha_{22}$1, $\alpha_{22}$2, $\alpha_{22}$3, $\alpha_{22}$4, $\alpha_{22}$5, and $\alpha_{22}$6, the quadrupole case is dominated by low-energy optical Hall weight below $\alpha_{22}$7, while the monopole case is dominated by higher-energy interband transitions across the exchange gap and the toroidal case is weak and dispersed. A common misconception is to treat all magnetic multipole orders as optically similar once time-reversal symmetry is broken; the interband decomposition shows instead that ASOC and multipole texture impose distinct selection rules (Sato et al., 2020).

In graphene the second-order response vanishes in the dipole approximation because the crystal is inversion symmetric, but it survives at finite in-plane wavevector through electric quadrupole-like and magnetic dipole-like terms. The small-$\alpha_{22}$8 expansion is written in terms of

$\alpha_{22}$9

The electric quadrupole-like sector is extracted from the scalar-potential calculation and controls terms proportional to $^{75}$00. Within the linear-dispersion approximation, the analytic second-order conductivities contain denominators $^{75}$01 and $^{75}$02, which identify one-photon and sum-frequency interband resonances at $^{75}$03 and $^{75}$04. The paper therefore attributes the strong $^{75}$05-dependence of SHG, photon drag, and DFG to dominant interband optical transitions. For SHG, the effective susceptibility can reach $^{75}$06 pm/V at $^{75}$07, $^{75}$08 meV, $^{75}$09 eV, and $^{75}$10 eV, while room-temperature or larger-$^{75}$11 values are of order $^{75}$12 pm/V. In DFG, the independent-particle quadrupole/magnetic model reproduces the resonant structure but underestimates some experimentally inferred values by several orders of magnitude. Here “interband quadrupole” is not a local multipole moment but a symmetry-allowed, finite-$^{75}$13 electric-quadrupole-like channel in the nonlinear conductivity tensor (Cheng et al., 2016).

5. Nuclear interband $^{75}$14 transitions and collective quadrupole structure

In deformed Gd isotopes, interband electric quadrupole strengths between the $^{75}$15 state and the ground-state band are analyzed microscopically with the Nilsson+BCS mean field plus RPA. For the lowest $^{75}$16 phonon,

$^{75}$17

the basic intrinsic interband matrix element is

$^{75}$18

and, neglecting rotational corrections,

$^{75}$19

The generalized intensity relation adds a rotational term proportional to

$^{75}$20

With this correction, the calculated $^{75}$21 values reproduce an isotopic variation of nearly two orders of magnitude. In $^{75}$22Gd and $^{75}$23Gd the interband strength is large and supports the interpretation of $^{75}$24 as a collective $^{75}$25 vibration on a deformed ground state rather than a shape-coexisting configuration; in $^{75}$26Gd, by contrast, the observed strength is fragmented over many $^{75}$27 states and the simple P+Q·Q RPA fails to reproduce the detailed pattern (Matsuzaki et al., 2016).

A different collective treatment, the Bohr Hamiltonian with a Morse potential in $^{75}$28, obtains analytic wavefunctions via the Asymptotic Iteration Method and computes $^{75}$29 values for both $^{75}$30-unstable and rotational nuclei. Its systematic conclusion is notable: intraband transitions within the ground band are described reasonably well, but some interband transitions are systematically underpredicted in $^{75}$31-unstable nuclei and overpredicted in rotational nuclei. The issue persists despite the Morse potential’s improved description of $^{75}$32-band spacings, suggesting that the mismatch is not cured simply by replacing Davidson or harmonic forms with an asymmetric finite-depth $^{75}$33 potential. This suggests that interband quadrupole strengths are more sensitive than energies to missing $^{75}$34–$^{75}$35 coupling, deformation-dependent masses, or more complicated transition operators (Inci et al., 2011).

An important nuclear controversy concerns whether the disappearance of observed $^{75}$36 lines in negative-parity bands is direct evidence for tetrahedral symmetry. In a nine-dimensional quadrupole–octupole collective Hamiltonian for $^{75}$37Dy, the calculated intraband $^{75}$38 values in one-phonon negative-parity bands remain large: for example, $^{75}$39, $^{75}$40, $^{75}$41, and $^{75}$42 in the $^{75}$43 band are $^{75}$44, $^{75}$45, $^{75}$46, and $^{75}$47 W.u., while the corresponding $^{75}$48 “tetrahedral” values are $^{75}$49, $^{75}$50, $^{75}$51, and $^{75}$52 W.u. The model therefore concludes that the apparent disappearance of low-spin $^{75}$53 transitions in the experimental band cannot be attributed to tetrahedral symmetry; rather, the realistic quadrupole-deformed minimum sustains strong quadrupole collectivity, and non-observation is interpreted through branching competition and sensitivity thresholds (Dobrowolski et al., 2017).

6. Interacting quadrupole insulators and Green’s-function interband geometry

The spinful BBH quadrupole-insulator model on a square lattice with four orbitals per unit cell and on-site Hubbard interaction

$^{75}$54

provides a higher-order-topological realization in which quadrupole topology is encoded in the interband geometry of the occupied subspace rather than in a local transition operator. At half filling, two of the four bands are occupied, and the Wilson loop

$^{75}$55

acts within this occupied subspace. Its eigenvalues $^{75}$56 define Wannier bands, and nested Wilson loops within a Wannier sector define the Wannier-sector polarizations $^{75}$57 and $^{75}$58. The quadrupole phase is characterized by

$^{75}$59

with quantization enforced by mirror symmetries and inversion (Peng et al., 2019).

For the interacting case, the paper replaces the noninteracting Bloch Hamiltonian by the topological Hamiltonian

$^{75}$60

and then recomputes the Wilson loops and nested Wilson loops from its eigenvectors. This Green’s-function formalism successfully characterizes the interacting quadrupole topology: up to $^{75}$61, the Wannier-band gap closings that separate quadrupole, dipole, and trivial phases occur at the same $^{75}$62 values as at $^{75}$63. Along cuts through the phase diagram, $^{75}$64 closes at $^{75}$65 for fixed $^{75}$66, and $^{75}$67 closes at $^{75}$68 for fixed $^{75}$69, with the corresponding components of $^{75}$70 jumping from $^{75}$71 to $^{75}$72.

At stronger coupling, however, the quadrupole insulator is destroyed by antiferromagnetic order. Projector QMC yields a continuous transition at

$^{75}$73

with critical exponents

$^{75}$74

distinct from those of the known AFM transitions and interpreted as evidence for a new universality class. Under a tiny staggered field $^{75}$75, which explicitly breaks mirror symmetry, the Wannier-sector polarization $^{75}$76 decreases smoothly from $^{75}$77 to $^{75}$78 as $^{75}$79 increases, while the AFM structure factor extrapolates to a finite thermodynamic value. A plausible implication is that, in this setting, “interband quadrupole” is the interacting, Green’s-function-renormalized non-Abelian geometry of the occupied states: it is stable against weak local correlations, but it loses quantization once the symmetry structure required by nested Wilson loops is broken.