Lindhard Continuum in Fermi Liquids

• The Lindhard continuum is the kinematic envelope for particle–hole excitations in Fermi liquids, defined via the non-interacting dynamical charge susceptibility.
• It captures how band-structure and dimensionality shape the excitation boundaries and redistribute spectral weight across intra- and interband transitions.
• RIXS experiments in MgB₂ have directly observed its dispersive features, clarifying its central role in screening, transport, and damping in metals.

The Lindhard continuum is the kinematic envelope of elementary particle–hole excitations in a Fermi liquid: the set of energy–momentum points $(\mathbf q,\omega)$ at which moving an electron across the Fermi surface is allowed. It is encoded in the non-interacting dynamical charge susceptibility, or Lindhard function, and appears in the spectral density of charge fluctuations measured by momentum-resolved probes. In the simplest single-band parabolic case it is bounded at small $q$ by $\omega \approx v_F q$, whereas in multiband and anisotropic metals band-structure effects distort the boundaries, redistribute spectral weight through matrix elements, and split the continuum into intra- and interband contributions. Although the continuum is central to screening, dissipation, transport, and the damping of collective charge modes, direct experimental observation has historically been difficult; ultra-soft resonant inelastic x-ray scattering in MgB$_2$ has now been reported to resolve it directly (Lomeli et al., 12 Sep 2025).

1. Formal definition and response-function framework

For a multi-band metal, the Lindhard continuum is encoded in the non-interacting dynamical charge susceptibility

$$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$

where $n,m$ are band indices, $f$ is the Fermi function, and $M_{nm}$ encodes the orbital and matrix-element content of the probed density operator (Lomeli et al., 12 Sep 2025). In the single-band non-interacting limit, the corresponding expression reduces to the familiar Lindhard function

$$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$

so that the continuum is precisely the region where $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$ (Ma et al., 2024).

The dynamic structure factor,

$q$0

is the spectral density of charge fluctuations for probes that couple to the relevant charge operator (Lomeli et al., 12 Sep 2025). In inelastic x-ray scattering in periodic crystals, the same object is expressed in reciprocal-lattice-vector space through the diagonal susceptibility elements $q$1, with the extended-zone momentum transfer selecting the appropriate reciprocal-lattice sector (Wang et al., 2012). This formulation makes clear that the continuum is not an auxiliary construction: it is the directly measurable support of the two-particle charge response.

Within this framework, the continuum is a statement about kinematics rather than collectivity. It identifies where elementary particle–hole excitations are allowed. Collective modes such as plasmons emerge only after Coulomb dressing, typically within RPA, through zeros of the dielectric function. The continuum therefore forms the background against which collective charge excitations propagate and, when overlap occurs, are Landau damped (Mihaila, 2011).

2. Kinematics, dimensionality, and band-structure dependence

For a parabolic band, the continuum boundaries follow from extremizing the energy transfer at fixed momentum. In both two and three dimensions, the allowed window is

$q$2

with $q$3 vanishing outside this interval (Ma et al., 2024). In the small-$q$4 limit this reduces to the familiar onset near $q$5. In one, two, and three dimensions, exact closed forms for the Lindhard function can be obtained, and the support of $q$6 defines the continuum in each case; the same algebraic $q$7 structure appears, but the detailed line shape is dimension-dependent (Mihaila, 2011).

Dimensionality changes the analytic structure substantially. In three dimensions the continuum is a wedge bounded by $q$8, with the lower edge touching $q$9 for $\omega \approx v_F q$0. In two dimensions the same kinematic window applies, but the response exhibits stronger nonanalytic behavior at the boundaries. In one dimension the continuum collapses to a single interval at fixed $\omega \approx v_F q$1, with constant $\omega \approx v_F q$2 inside and logarithmic edge singularities in the real part (Mihaila, 2011). These differences matter because they determine threshold behavior, damping rates, and the visibility of collective modes.

In crystals, especially anisotropic or multiband ones, the simple parabolic picture is only a baseline. In MgB$\omega \approx v_F q$3, for example, the $\omega \approx v_F q$4 bands derived from B $\omega \approx v_F q$5 orbitals form two quasi-2D cylinders along $\omega \approx v_F q$6 at $\omega \approx v_F q$7, while the $\omega \approx v_F q$8 bands from B $\omega \approx v_F q$9 form a 3D network near the Brillouin-zone boundary; the observed continuum therefore reflects multiband, strongly anisotropic fermiology rather than a single isotropic $_2$0 (Lomeli et al., 12 Sep 2025). Likewise, in Nd$_2$1Ce$_2$2CuO$_2$3, the continuum in extended momentum space shows strong anisotropy, a zero-intensity region near $_2$4 at $_2$5 eV, and pronounced weight near $_2$6 associated with a van Hove–like singularity and enhanced joint density of states (Wang et al., 2012). This suggests that, in realistic materials, the continuum is best regarded as a band- and matrix-element-weighted kinematic manifold rather than a universal wedge.

3. Screening, plasmons, and why the continuum is often hard to see

The continuum is frequently conflated with the plasmon, but the two are distinct. The continuum is the support of elementary particle–hole excitations; the plasmon is a collective mode arising from Coulomb interactions. In RPA one writes

$_2$7

or equivalently

$_2$8

and plasmons occur where the dielectric function vanishes while damping remains small (Lomeli et al., 12 Sep 2025). In layered Lindhard metals, the low-$_2$9 response can be dominated by long-lived standing-wave plasmon modes arising from interlayer Coulomb coupling, while at larger $$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$0 the response increasingly reflects the microscopic properties of individual layers (Vries et al., 23 Jul 2025).

The historical elusiveness of the Lindhard continuum in metals follows from screening. EELS and non-resonant IXS measure the longitudinal screened charge response, commonly written through the loss function

$$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$1

and are subject to the $$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$2-sum and screening sum rules

$$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$3

At small $$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$4, these concentrate nearly all low-energy spectral weight into the plasmon and leave the intraband continuum vanishingly weak (Lomeli et al., 12 Sep 2025). In that sense, the absence of low-$$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$5 continuum weight in total-charge probes does not imply the absence of the underlying particle–hole phase space.

By contrast, RIXS can couple to an orbitally projected, non-conserved density and therefore is not constrained by the same screening sum-rule collapse of low-$$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$6 spectral weight into the plasmon (Lomeli et al., 12 Sep 2025). In cuprates, this distinction has been central to the interpretation of apparently broad, weakly dispersing continua. One line of analysis argues that many material-specific results from transmission EELS, reflection EELS, RIXS, and optical spectroscopy can be understood within an extended Lindhard model in which the continuum is shaped by electron–hole excitations inside a lifetime-broadened conduction band (Fink, 2021). A later layered comparison found agreement with IR and R-EELS at low $$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$7 but not with published EELS spectra at large $$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$8, emphasizing unresolved discrepancies rather than a settled consensus (Vries et al., 23 Jul 2025).

4. Direct observation in MgB$$\chi_0(\mathbf q,\omega)=\frac{2}{N}\sum_{n,m,\mathbf k} \frac{f\!\left(\varepsilon_{n\mathbf k}\right)-f\!\left(\varepsilon_{m,\mathbf k+\mathbf q}\right)} {\hbar\omega+\varepsilon_{n\mathbf k}-\varepsilon_{m,\mathbf k+\mathbf q}+i0^+} \,\big|M_{nm}(\mathbf k,\mathbf q)\big|^2,$$9

The direct observation reported for MgB$n,m$0 used ultra-soft RIXS at the B $n,m$1-edge, with the incident photon tuned to a sharp pre-edge resonance at approximately $n,m$2 eV, combined energy resolution of approximately $n,m$3 meV, sample temperature $n,m$4 K, and in-plane momentum transfers up to approximately $n,m$5 \AA$n,m$6 along $n,m$7–M and $n,m$8–K (Lomeli et al., 12 Sep 2025). High-quality MgB$n,m$9 films were grown by hybrid physical–chemical vapor deposition. Within the measured window, the central observation was a linearly dispersing low-energy excitation visible within a few hundred meV of the elastic line and shifting to higher energies with increasing $f$0.

Voigt fits of the peak position yielded a slope $f$1 eV \AA, of the same order as—and roughly half—the $f$2-band Fermi velocity $f$3 eV \AA. The excitation was observed from the smallest measured $f$4 \AA$f$5 up to approximately $f$6 \AA$f$7, with characteristic energies of a few $f$8 eV, consistent with $f$9 and with the expected boundaries of the intraband particle–hole continuum. The linewidth and linear dispersion identified the feature as particle–hole kinematics rather than a collective mode, while the MgB$M_{nm}$0 plasmon sits near $M_{nm}$1 eV and is nearly nondispersive over the measured small-$M_{nm}$2 window.

The identification rested on several mutually consistent calculations. An eight-orbital tight-binding Hamiltonian retaining B $M_{nm}$3, $M_{nm}$4, $M_{nm}$5, and $M_{nm}$6 orbitals was used to compute $M_{nm}$7 on a dense $M_{nm}$8-grid with a small broadening $M_{nm}$9 eV, and the resulting susceptibility, broadened to the experimental $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$0 meV resolution, matched the low-energy dispersive feature in both location and slope. BSE-based RIXS simulations on top of DFT reproduced the XAS pre-edge and the momentum- and polarization-dependent RIXS spectra, including the suppression of intensity at grazing incidence due to polarization rotating out of the $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$1-plane. Projecting the intermediate-state exciton density onto the band structure showed that the resonance selectively enhances $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$2 and, to a lesser extent, $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$3 states near $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$4 at $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$5 and M, respectively.

A band-resolved decomposition of $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$6 cleanly separated intraband and interband channels. At representative $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$7 \AA$$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$8, the spectral weight below $$\chi_0(\mathbf q,\omega)=2\sum_{\mathbf k} \frac{f(\epsilon_{\mathbf k})-f(\epsilon_{\mathbf k+\mathbf q})} {\omega+i0^+ + \epsilon_{\mathbf k}-\epsilon_{\mathbf k+\mathbf q}},$$9 eV was overwhelmingly intraband, with significant contributions from both $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$0 and $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$1 sectors, while interband weight was negligible. This pinned the observed feature near $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$2 eV to intraband particle–hole excitations and therefore to the Lindhard continuum. The broader non-dispersive features above $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$3 eV were instead attributed to interband $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$4 transitions.

The MgB$\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$5 result also sharpened the methodological point. Ultra-soft RIXS, through element and orbital selectivity together with resonance enhancement of low-lying B $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$6 states, provided a momentum-resolved probe of fermiology in a weakly correlated metal. A plausible implication is that continuum spectroscopy in metals need not be restricted to indirect inference from plasmons or transport, but can instead proceed through direct comparison of measured spectra with realistic $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$7.

5. Model extensions and non-Fermi-liquid departures

In models related to Mott physics, the Lindhard continuum can be modified rather than merely broadened. In the band Hatsugai–Kohmoto and orbital Hatsugai–Kohmoto models, the charge susceptibility takes the form of a modified Lindhard function with lower and upper Hubbard bands, leading to a multi-pole structure in which the continuum contains both intraband sectors and strong interband sectors (Ma et al., 2024). Because the Hubbard bands are non-rigid and their spectral weights depend on occupancy, interband terms remain unsuppressed as $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$8, unlike the non-interacting multiband case where orthogonality suppresses them. The resulting $\mathrm{Im}\,\chi_0(\mathbf q,\omega)\neq 0$9 displays hot spots near $q$00, and the RPA plasmon dispersion becomes inversely dependent on momentum, $q$01 at small $q$02.

The same analysis also yields an additional contribution to the conventional $q$03-sum rule, traced to a long-range diamagnetic contribution to the current, together with a non-commutativity of the long-wavelength and thermodynamic limits (Ma et al., 2024). These are model-specific statements, but they demonstrate that the phrase “Lindhard continuum” can designate a family resemblance rather than a unique analytic form once Mottness, spectral-weight transfer, and nonlocal currents enter.

A different departure appears in bad and slow metals. In a bad metal with short mean free path, the low-energy Lindhard continuum is destroyed; low-energy charge dynamics are then collective and diffusive over essentially the full Brillouin zone (Glorioso et al., 2022). In a slow metal, defined by $q$04, particle–hole pairs are kinematically unable to emit acoustic phonons, because the phonon dispersion lies outside the Lindhard continuum. The energy-relaxation problem must then be reformulated in terms of the density spectral weight $q$05 evaluated on the phonon dispersion, or in terms of the optical conductivity in the $q$06 limit (Glorioso et al., 2022). These cases mark genuine breakdowns of the standard continuum-controlled picture of metallic energy relaxation.

6. Computation, finite temperature, and spectroscopic applications

For homogeneous Fermi gases, the Lindhard function is available in closed form in one, two, and three dimensions, providing the standard baseline for dielectric response in noninteracting fermion systems (Mihaila, 2011). At finite temperature in the three-dimensional homogeneous electron gas, explicit expressions for the real and imaginary parts of the Lindhard dielectric function can be written in terms of Dawson integrals and logarithmic closed forms, respectively, enabling direct calculation of thermal smearing, screening, and Raman response in systems such as $q$07-type GaAs (Andrade-Neto, 2014). At $q$08, the continuum boundaries remain sharp; at finite $q$09, the Fermi–Dirac factors smear the edges on the scale of $q$10.

In periodic crystals, efficient evaluation of the continuum over extended momentum space can be achieved by rewriting the Lindhard susceptibility as a convolution of spectral functions and then using FFTs on products of real-space functions (Wang et al., 2012). For Nd$q$11Ce$q$12CuO$q$13, this scheme produced $q$14 across several Brillouin zones and connected the anisotropy of the continuum to Bloch-state coefficients, orbital character, and the underlying Cu–O band structure. The dynamic structure factor then follows from the fluctuation–dissipation relation and serves as the direct baseline for interpreting IXS or related spectra.

The same computational logic extends to more elaborate settings. Realistic tight-binding or DFT-derived band structures generate material-specific $q$15; BSE or Kramers–Heisenberg treatments incorporate resonance and geometry in RIXS; RPA dressing reorganizes spectral weight into collective modes; and spectral-function formulations allow lifetime broadening and self-energy effects to feed back on the continuum (Lomeli et al., 12 Sep 2025). This suggests that the Lindhard continuum is best understood not as a single formula, but as a hierarchy: exact free-gas benchmark, band-structure realization, interaction-dressed charge background, and, in some systems, a quantity whose very suppression or destruction becomes physically diagnostic.

The modern significance of the concept therefore lies in its dual role. It is simultaneously a formal object—the support of $q$16—and an empirical organizing principle for screening, dissipation, transport, phonon emission, and Landau damping. Its recent direct observation in MgB$q$17 has not closed the subject. Rather, it has clarified that continuum features can be isolated experimentally when the probe couples to projected densities and the band structure is sufficiently well controlled, while theoretical generalizations continue to track how this baseline is deformed in layered, strongly scattered, or Mottness-dominated systems (Lomeli et al., 12 Sep 2025).