Interband LOC in Quantum Materials

• Interband LOC is the finite-frequency optical response arising from transitions between occupied and unoccupied states, revealing band topology and selection rules.
• It is defined via the full microscopic dielectric matrix in the q→0 limit, where anisotropy, tilt, and dimensionality dictate distinct spectral signatures.
• First-principles and experimental studies show that interband LOC probes quantum metrics, pseudogap effects, and even Floquet-driven nonequilibrium states.

Interband longitudinal optical conductivity (LOC) is the finite-frequency optical response associated with transitions between occupied and unoccupied electronic states in the optical long-wavelength limit, conventionally resolved into tensor components such as $\sigma_{xx}(\omega)$, $\sigma_{yy}(\omega)$, or $\sigma_{zz}(\omega)$ for electric fields polarized along specified directions. In the macroscopic regime relevant to reflectivity, ellipsometry, and related probes, it is defined through $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$ with $\mathbf{q}\to 0$, and is linked to the macroscopic longitudinal dielectric function by $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$ in Gaussian units (Schindlmayr, 2011). Across metals, Dirac and Weyl systems, nodal-line semimetals, semi-Dirac bands, flat-band lattices, and Floquet-engineered states, interband LOC serves as a compact probe of band topology, matrix-element selection rules, Pauli blocking, lifetime broadening, and lattice-scale structure.

1. Optical-limit definition and formal structure

The longitudinal optical conductivity is a tensor quantity in general, reducing to a scalar only in sufficiently symmetric cases such as cubic metals (Schindlmayr, 2011). Its experimentally relevant component is the diagonal response to a longitudinal probe field in the $\mathbf q\to0$ limit, rather than a fully microscopic conductivity retaining unit-cell-scale spatial variation. In that limit, the absorptive optical spectrum is carried by the real part of $\sigma(\omega)$, while $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$ when the dielectric response is written in Gaussian units (Schindlmayr, 2011).

A recurrent formal distinction is between intraband and interband sectors. In band-based Kubo formulations, the intraband terms come from $s'=s$, whereas the interband terms come from $\sigma_{yy}(\omega)$0 or, more generally, from transitions between distinct bands weighted by current matrix elements and occupation differences (Barati et al., 2017). In first-principles dielectric formulations, this split need not be written as a separate formal equation, because both sectors are already embedded in the full susceptibility built from occupied and unoccupied Kohn–Sham states; the interband contribution then emerges directly from the occupied-to-unoccupied transition structure of $\sigma_{yy}(\omega)$1 (Schindlmayr, 2011).

Several works emphasize that interband LOC is not determined by the joint density of states alone. The conductivity always includes matrix-element factors, which can suppress, enhance, or even nullify transitions otherwise allowed by phase space. This is explicit in models where the current operator becomes diagonal in the band basis, producing vanishing interband response despite an apparent two-band structure, and in systems where spinor texture or pseudospin geometry enforces exact cancellations (Barati et al., 2017).

Another important technical point is that longitudinal optical response is properly macroscopic only after constructing the full microscopic dielectric matrix, inverting it, and then taking the $\sigma_{yy}(\omega)$2 projection in the long-wavelength limit. Replacing this by a single matrix element $\sigma_{yy}(\omega)$3 neglects local-field effects (Schindlmayr, 2011). This distinction is usually modest in the metallic examples treated in first-principles calculations, but it is formal rather than optional.

2. Dimensionality, tilt, and universal Dirac–Weyl structures

A unified analytical theory for tilted Weyl and Dirac fermions shows that, for spatial dimension $\sigma_{yy}(\omega)$4, the interband LOC factorizes into a dimensional prefactor and a dimensionless tilt- and doping-dependent function, with intrinsic scaling $\sigma_{yy}(\omega)$5 (Hou et al., 2022). This yields a frequency-independent intrinsic scale in $\sigma_{yy}(\omega)$6, a linear-in-$\sigma_{yy}(\omega)$7 scale in $\sigma_{yy}(\omega)$8, and a vanishing interband LOC in $\sigma_{yy}(\omega)$9 within this framework; the latter is also derived explicitly in the $\sigma_{zz}(\omega)$0 treatment of tilted Dirac bands, where the vanishing follows from the absence of a transverse current-matrix-element structure in one dimension (Hou et al., 2022).

Tilt reshapes Pauli blocking and therefore the frequency window of allowed interband absorption. In doped type-I tilted Dirac bands, a single untilted threshold at $\sigma_{zz}(\omega)$1 broadens into a finite interval bounded by $\sigma_{zz}(\omega)$2 and $\sigma_{zz}(\omega)$3; in type-II bands, the high-frequency background remains reduced because part of the resonant contour stays blocked even asymptotically (Hou et al., 2022). A particularly robust result is the fixed point

$\sigma_{zz}(\omega)$4

which survives dimensionality, polarization sector, and tilt strength up to $\sigma_{zz}(\omega)$5 after normalization by the intrinsic scale (Hou et al., 2022). The same article recasts the interband LOC in terms of the joint density of states in dimensions $\sigma_{zz}(\omega)$6 and $\sigma_{zz}(\omega)$7, making the phase-space origin of the result explicit (Hou et al., 2022).

In two-dimensional tilted Dirac bands, the Lifshitz transition at $\sigma_{zz}(\omega)$8 reorganizes the interband LOC into tilt-dependent low-frequency windows bounded by two characteristic frequencies, $\sigma_{zz}(\omega)$9 and $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$0, with distinct behavior on the type-I and type-II sides (Tan et al., 2021). In the undoped case the normalized interband backgrounds are constant for $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$1 but become tilt dependent for $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$2, while at finite doping the $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$3- and $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$4-polarized conductivities differ because tilt redistributes the allowed angular phase space differently for the two current operators (Tan et al., 2021). The separation $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$5 increases with tilt in the type-I phase and decreases in the type-II phase, providing an optical signature of the Lifshitz transition (Tan et al., 2021).

A tight-binding re-examination of two-dimensional tilted Dirac bands extends this picture beyond the linearized $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$6 regime. In addition to conventional threshold frequencies, the lattice model exhibits partner frequencies, a robust sharp-peak frequency $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$7, and a cutoff frequency $$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$8, all absent in the corresponding linearized theory (Tan et al., 7 Apr 2026). The sharp peak is tied to high-symmetry-point van Hove structure, whereas the cutoff follows from the finite Brillouin-zone boundary and Pauli blocking (Tan et al., 7 Apr 2026). This suggests that interband LOC can distinguish local Dirac-cone physics from full-band lattice effects.

3. Nodal-line and quasi-one-dimensional semimetals

In three-dimensional topological nodal-line semimetals, interband LOC is intrinsically anisotropic. For the continuum nodal-ring model with Hamiltonian

$$\mathbf{j}(\mathbf{q},\omega)=\sigma(\mathbf{q},\omega)\mathbf{E}(\mathbf{q},\omega)$$9

the independent longitudinal components are $\mathbf{q}\to 0$0 and $\mathbf{q}\to 0$1, and both are Pauli blocked for $\mathbf{q}\to 0$2 at $\mathbf{q}\to 0$3 (Barati et al., 2017). For low doping, there is a plateau region $\mathbf{q}\to 0$4 in which both components are frequency independent, but the high-frequency asymptotics differ qualitatively: $\mathbf{q}\to 0$5 saturates to a constant, whereas $\mathbf{q}\to 0$6 (Barati et al., 2017). The anisotropy originates in distinct current operators and pseudospin textures around the nodal ring, not merely in a scalar density-of-states effect.

The same work shows that a two-dimensional nodal-line model can have identically vanishing interband LOC. For

$\mathbf{q}\to 0$7

the current operator is diagonal in the band basis and $\mathbf{q}\to 0$8, so $\mathbf{q}\to 0$9 exactly (Barati et al., 2017). This is a useful correction to the common assumption that every nominal two-band crossing automatically supports interband optical absorption.

A distinct nodal-line realization appears in the layered $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$0 family, modeled by a Dirac SSH Hamiltonian for coupled $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$1 chains (Ahn, 21 Apr 2026). Here the longitudinal direction is explicitly the nodal-line or chain direction, $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$2, and the low-energy interband longitudinal conductivity at charge neutrality obeys

$$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$3

valid for $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$4 in the clean limit (Ahn, 21 Apr 2026). Thus $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$5 with no optical gap at neutrality, while finite doping imposes a sharp threshold at $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$6 through Pauli blocking (Ahn, 21 Apr 2026). The transverse in-plane response is also linear in $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$7, though with a different slope, so the interband sector is less anisotropic than the Drude sector in this quasi-one-dimensional nonsymmorphic nodal-line semimetal (Ahn, 21 Apr 2026).

Finite temperature enters this quasi-one-dimensional case through a multiplicative factor $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$8, so the zero-temperature result remains valid for $$\varepsilon_{\mathrm M}(\omega)=1+\frac{4\pi i}{\omega}\lim_{\mathbf q\to0}\sigma(\mathbf q,\omega)$$9, whereas at neutrality and $\mathbf q\to0$0 the longitudinal interband LOC crosses over from $\mathbf q\to0$1 to $\mathbf q\to0$2 (Ahn, 21 Apr 2026). A plausible implication is that polarization-resolved low-frequency spectroscopy can separate intrinsic nodal-line scaling from thermal filling effects without relying on a fitted Drude background.

4. Semi-Dirac dispersions, pseudospin-1 models, and flat-band channels

Semi-Dirac systems furnish one of the clearest examples of qualitatively distinct longitudinal interband channels. In the clean two-dimensional semi-Dirac model with quadratic dispersion along one axis and linear dispersion along the other, the absorptive interband conductivities obey

$\mathbf q\to0$3

and band tilt modifies only the dimensionless envelope, not these exponents (Yan et al., 2023). The same study finds a robust fixed point at $\mathbf q\to0$4 for $\mathbf q\to0$5, together with tilt-controlled thresholds $\mathbf q\to0$6, kinks, and asymptotic backgrounds (Yan et al., 2023). This distinguishes tilted semi-Dirac bands from tilted Dirac bands, where the intrinsic interband scale is not split into opposite $\mathbf q\to0$7 and $\mathbf q\to0$8 laws.

The earlier semi-Dirac optical analysis in the untilted case gives separate clean-limit formulas for both longitudinal directions,

$\mathbf q\to0$9

and shows that the geometric mean $\sigma(\omega)$0 is frequency independent and universal within the continuum model (Carbotte et al., 2019). At finite doping, a sum rule holds in the relativistic direction $\sigma(\omega)$1, where the Drude gain equals the blocked interband loss, but no such sum rule applies in the quadratic direction $\sigma(\omega)$2 (Carbotte et al., 2019). With a gap $\sigma(\omega)$3, both channels acquire a threshold at $\sigma(\omega)$4, but the edge exponents remain anisotropic: $\sigma(\omega)$5 and $\sigma(\omega)$6 (Carbotte et al., 2019).

A complementary formulation based on velocity correlators, interpreted as zitterbewegung, arrives at exact longitudinal optical conductivities for semi-Dirac and pseudospin-1 models (Oriekhov et al., 2022). For the semi-Dirac merging model, it reproduces the strong anisotropy and identifies a logarithmic interband van Hove singularity in $\sigma(\omega)$7 at $\sigma(\omega)$8 in the two-cone phase (Oriekhov et al., 2022). For the gapped dice lattice, the interband LOC arises only from transitions to and from the flat band, with threshold $\sigma(\omega)$9, while direct lower-to-upper dispersive-band transitions are absent (Oriekhov et al., 2022). In the gapped Lieb lattice, by contrast, all three channels contribute—lower to upper dispersive, lower to flat, and flat to upper—with a $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$0 threshold for dispersive-to-dispersive absorption (Oriekhov et al., 2022).

Stacking in bilayer dice lattices further restructures interband LOC. In the aligned $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$1 bilayer, the monolayer dice selection rule survives, so only transitions involving the intermediate flat-derived band contribute and the effective-model conductivity has the simple step form

$$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$2

with onset at $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$3 (Sukhachov et al., 2023). In the hub-aligned and mixed stackings, the intermediate band becomes optically inert in the low-energy effective theory and the monolayer selection rule is lifted; the relevant interband LOC is then governed by a semi-Dirac or tilted-Dirac reduced two-band model, respectively (Sukhachov et al., 2023). In the cyclic stacking, all three low-energy bands participate and the interband LOC develops an onset–plateau–offset structure not present in the other stackings (Sukhachov et al., 2023).

5. Symmetry selection rules, finite-$$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$4 coherence, and pseudogap precursors

Interband LOC can be exactly suppressed by symmetry. For a $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$5-symmetric topological-insulator surface with Rashba- and Dresselhaus-like spin-orbit terms,

$$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$6

the interband matrix element along $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$7 becomes proportional to $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$8, and the resulting longitudinal interband conductivity vanishes exactly at the persistent-spin-helix point $$\operatorname{Re}\sigma(\omega)=\frac{\omega}{4\pi}\operatorname{Im}\varepsilon_{\mathrm M}(\omega)$$9 (Sengupta, 2015). The paper interprets this as zero interband absorption produced by a selection-rule cancellation in the spin texture rather than by a trivial density-of-states effect (Sengupta, 2015). The same framework yields unequal $s'=s$0 and $s'=s$1 for the anisotropic $s'=s$2-point surface Dirac cone of SmB$s'=s$3 (Sengupta, 2015).

A different interband correction appears in the general momentum-block-diagonal two-band model with constant band-independent scattering rate $s'=s$4. There the longitudinal conductivity decomposes into intraband quasiparticle transport and a symmetric interband contribution

$s'=s$5

which is controlled by the quantum metric $s'=s$6 (Mitscherling, 2020). This term is a finite-$s'=s$7 coherence correction: it vanishes in the clean DC limit, scales as $s'=s$8 for small $s'=s$9, and decays as $\sigma_{yy}(\omega)$00 in the dirty limit (Mitscherling, 2020). In gapped situations it can become the main longitudinal contribution because intraband transport is suppressed (Mitscherling, 2020). A plausible implication is that interband LOC in broadened multiband systems need not be reducible to a clean quasiparticle transition picture.

In thermally disordered two-dimensional spin-density-wave metals, the longitudinal optical conductivity develops a broad finite-frequency peak around twice the pseudogap scale when the spin correlation length is large (Lin et al., 2010). The paper identifies this as the natural analogue of the ordered-state SDW interband feature: in the ordered limit it corresponds to transitions between the reconstructed bands $\sigma_{yy}(\omega)$01, whereas in the fluctuation regime it survives as a broadened precursor generated by transitions between pseudogap-split spectral branches (Lin et al., 2010). As the correlation length decreases, spectral weight moves from this finite-frequency interband-like structure back into the low-frequency region (Lin et al., 2010). This suggests that interband LOC can remain meaningful even when long-range order is absent, provided the single-particle spectrum retains a momentum-selective reconstruction.

6. First-principles metals, spectral separation, and Floquet nonequilibrium

First-principles calculations place interband LOC on the same footing as dielectric response. In a full-potential linearized augmented-plane-wave implementation, the macroscopic dielectric function is obtained from the full dielectric matrix within the random-phase approximation and then converted into conductivity without empirical Drude parameters (Schindlmayr, 2011). This scheme includes interband transitions directly from the Kohn–Sham spectrum, retains local-field effects by matrix inversion, and yields quantitative spectra for Al, Cu, Ag, Fe, and Ni over a wide frequency range (Schindlmayr, 2011). In these metals, the interband sector already dominates in the visible range; in Al, resonances near $\sigma_{yy}(\omega)$02 and $\sigma_{yy}(\omega)$03 eV are attributed to interband transitions, while in noble metals the visible/UV structure is controlled by occupied $\sigma_{yy}(\omega)$04-band transitions, and in Fe and Ni low-energy spin-dependent $\sigma_{yy}(\omega)$05-$\sigma_{yy}(\omega)$06 transitions become essential (Schindlmayr, 2011).

An experimental counterpart appears in PdCoO$\sigma_{yy}(\omega)$07, where the in-plane free-carrier response lies below about $\sigma_{yy}(\omega)$08 and interband transitions begin only above about $\sigma_{yy}(\omega)$09, producing what the authors call a “perfect separation of intraband and interband excitations” (Homes et al., 2018). This unusually clean spectral separation allows a nearly model-independent extraction of the plasma frequency from the sum rule, while the interband conductivity is represented by a set of broad electronic oscillators whose lowest component near $\sigma_{yy}(\omega)$10 already affects the screened plasma edge (Homes et al., 2018). The same spectra also contain strong features attributed not to ordinary interband transitions but to coupling of in-plane carriers to out-of-plane longitudinal optic modes, illustrating that longitudinal optical spectroscopy can mix electronic interband structure with LO-mode electrodynamics (Homes et al., 2018).

Periodic driving generalizes interband LOC into a probe of quasienergy bands rather than equilibrium bands. In driven graphene, the longitudinal optical conductivity was proposed as the key observable for detecting light-induced Floquet band gaps, with resonances at probe frequencies matching quasienergy splittings within a Floquet zone and across neighboring Floquet zones (Broers et al., 2021). The analysis identifies resonant structures at $\sigma_{yy}(\omega)$11 and $\sigma_{yy}(\omega)$12, together with population inversion near some Floquet gaps, which can suppress or even reverse the sign of the conductivity (Broers et al., 2021). In a Floquet graphene antidot lattice, the homodyne longitudinal conductivity similarly reorganizes across phases: the quasi-equilibrium regime shows one broad absorption feature, the Floquet Dirac regime shows two narrower peaks, and the Floquet semi-Dirac regime shows one narrow peak, with additional intervals of negative $\sigma_{yy}(\omega)$13 interpreted as gain (Cupo et al., 2023).

These nonequilibrium results clarify a common misconception: negative $\sigma_{yy}(\omega)$14 in a driven system does not by itself signal an ill-defined conductivity. In the Floquet analyses, it is tied to sideband-assisted interband processes and to nonequilibrium occupations that allow the material to amplify, rather than absorb, the probe field (Broers et al., 2021). More broadly, they show that interband LOC is not restricted to equilibrium band structures; it remains well defined in quasienergy space, provided the response is formulated around a driven steady state.