Lattice Vibrations and Collective Modes

A peak in a spectrum is not automatically a quasiparticle, and the word collective does not identify which degrees of freedom are moving. A material may support lattice vibrations, charge-density oscillations, spin waves, bound electron–hole pairs, or a carrier dressed by its environment. Each object comes from a different kernel, has different quantum labels, and reaches an experiment through a different forward model.

This gateway owns the routing problem. Given a declared material state, candidate excitation, regime, and observable, it selects the narrowest live owner, identifies the missing calculation or probe model, and states when a mode or quasiparticle interpretation must stop. It does not repeat normal-mode quantization, response theory, band theory, or material-specific spectroscopy.

Helpful background. Use Lattices, Reciprocal Space, and Bloch Electrons for cells, reciprocal vectors, and Brillouin-zone language. Use Band Theory and Electronic Structure for declared electron and hole states, fillings, and band-energy conventions. Use Quasiparticles and Collective Modes for the general distinctions among a normal mode, a pole, a continuum, a lifetime, and a breakdown of particle-like language. None is a universal hard prerequisite: a reader may enter directly with the capabilities needed for one branch.

Enter This Chapter

Phonons is the chapter’s current substantive materials page. It owns force constants, dynamical matrices, real-crystal dispersions and polarizations, phonon density of states, vibrational thermodynamics, probe-dependent spectra, computational checks, and stability claims. Phonons as Many-Body Excitations owns the reusable quantization and operator construction.

Six narrower material treatments remain planned: Debye–Einstein model reductions, anharmonic lattice dynamics, electron–phonon coupling, polarons in materials, plasmons in materials, and excitons in materials. Until each is substantive, this gateway routes to live canonical owners and states the coverage boundary. Sidebar order is a catalog, not a prerequisite chain.

Readiness Check

Before choosing a page, answer five questions.

Reference state. What structure, phase, filling, temperature, pressure, strain, dimensionality, dielectric environment, and boundary conditions define the background about which the excitation is constructed?

Moving object. Is the candidate variable an ionic displacement, charge or spin density, particle–hole amplitude, electromagnetic field, order parameter, or a mobile carrier plus a dressing cloud?

Generating problem. Does the calculation diagonalize a dynamical matrix, find a vanishing eigenvalue of a dielectric matrix—equivalently a zero of its determinant at the appropriate generally complex frequency—locate a pole of a response or inverse dielectric function, solve a Maxwell boundary determinant, bind a two-particle state, or find a pole of a dressed one-particle propagator?

Regime. Are momentum, frequency, temperature, field, disorder, damping, system size, and experimental resolution inside the range where that description is controlled?

Observable. Is the requested output a dispersion, density of states, free-energy contribution, linewidth, conductivity, loss peak, optical intensity, scattering cross section, or phase-level claim?

Use How Quantum Matter Is Measured when the route also needs the probe-independent chain from a calibrated record through a forward model to a bounded material inference.

If one answer is missing, repair only that capability. Do not infer an excitation’s identity from its plotted color, historical name, or one bright experimental feature.

Write the Excitation-Claim Ledger

Record ten fields before interpreting a mode.

1. Material background. State the equilibrium structure, electronic or magnetic reference state, dimension, boundaries, strain, fields, temperature, and ensemble or preparation.
2. Microscopic channel. Name the coordinate or operator that moves: displacement, density, current, spin, electron–hole amplitude, or a mobile carrier coupled to a host.
3. Candidate object. Say whether the claim concerns a harmonic normal mode, renormalized quasiparticle pole, collective response mode, bound composite, relaxational feature, continuum, or probe-weighted peak.
4. Labels and conventions. Fix frequency or energy, primitive cell, branch, polarization, degeneracy, volume or cell normalization, and the distinction between reduced momentum $\mathbf q$ and total transfer $\mathbf Q=\mathbf q+\mathbf G$.
5. Kernel or Hamiltonian. Declare the force constants, anharmonic vertices, electron–phonon matrix elements, screened interaction, dielectric matrix, or electron–hole kernel retained in the calculation.
6. Approximation and control window. Name the Born–Oppenheimer, harmonic, quasiharmonic, weak-anharmonic, effective-mass, RPA, Migdal, or other reduction, together with its small parameter or empirical validation and its momentum, energy, temperature, field, damping, and size regime.
7. Analytic evidence. Identify the stable or bound-state eigenvalue, complex pole, response-matrix eigenchannel, dielectric zero, threshold, branch cut, or finite-system line that supports the proposed object.
8. Weight and width. Record eigenvector character, oscillator strength, residue or structure-factor weight, intrinsic linewidth and lifetime convention, nearby multiparticle continua, and instrumental resolution.
9. Requested output. Name the dispersion, heat capacity, expansion, thermal conductivity, scattering rate, pairing input, dressing parameter, loss spectrum, optical threshold, or measured intensity being requested.
10. Audit. Record units, parameter provenance, finite-size and order-of-limits checks, numerical and model uncertainty, credible alternatives, and a falsification or escalation test.

A bounded claim has the form: “for this background, kernel, regime, and probe, the observed or computed feature is consistent with this excitation to the stated accuracy.” That is stronger than naming a peak and weaker than claiming that the same object controls every response.

Read the Dependency Graph

The chapter branches rather than forming one long sequence.

1. For lattice motion, combine crystal, reciprocal-lattice, and Brillouin-zone preparation, then use the material Phonons page. The universal many-body phonon construction is helpful when the claim needs its operator or propagator language; it is not a hard gate to this branch.
2. Debye–Einstein reductions and anharmonic lattice dynamics begin after the phonon spectrum and its normalization conventions are declared.
3. Electron–phonon coupling combines phonon eigenvectors with an electronic band or quasiparticle description. A material-polaron route adds the generic dressing theory only after the vertex and target observable are specified.
4. Charge collective modes begin with the general Plasmons Preview; a later material bridge will specialize the dielectric environment, dimensionality, surfaces, and experimental inference.
5. Excitons combine a declared electron–hole sector with screening and an optical or transport question. They do not require first reading the phonon branch.
6. Spin waves leave this chapter’s local sequence: use the live material and many-body magnon owners directly.

Choose the Shortest Live Route

Compute or interpret a lattice vibration

Use Phonons for the force-constant eigenproblem, acoustic and optical branches, longitudinal– transverse splitting, density of states, thermodynamics, scattering weights, and computational stability checks. Use Structure Factors when the question is about the normalized correlation function measured in scattering, and use Raman and Optical Spectroscopy when selection rules, matrix elements, polarization geometry, or resolution control the intensity.

Stop calling a broad feature a well-defined phonon quasiparticle with one frequency and lifetime when no reasonably isolated renormalized pole remains. An overdamped lattice resonance or response with identifiable phonon ancestry may still remain. Do not turn an imaginary harmonic frequency into a particle with negative energy.

Reduce vibrational thermodynamics

The live Phonons page owns the low-frequency density of states, harmonic free energy, the Debye law, and the Dulong–Petit limit. Heat Capacity and Thermodynamics owns calorimetry, addenda and background subtraction, entropy accounting, and fit-to-data claims. The planned Debye–Einstein comparison will own the two approximate spectral measures, cutoff and state-count normalization, complete temperature curves, mixed reductions, and identifiability limits.

Until then, do not treat a fitted Debye or Einstein temperature as a unique microscopic frequency or a temperature-independent material constant outside the fitted model and range.

Interpret linewidths, expansion, or lattice thermal transport

Begin with the anharmonic and quasiharmonic limits already stated in Phonons. Use Phonons as Many-Body Excitations for number-changing vertices and propagator language. Add the relevant Raman, neutron, x-ray, or thermal-transport forward model before comparing a computed self-energy with a measured width or conductivity.

A linewidth needs a convention and can contain anharmonic, electronic, magnetic, defect, boundary, and instrumental contributions. Thermal expansion and thermal conductivity are not determined by a phonon density of states alone. A unified material treatment of these controls remains planned.

Route an electron–phonon claim

Use Transport, Response, and Optics to decide whether a transport or electrodynamic claim needs a bulk coefficient, kinetic closure, correlation kernel, coherent terminal, or optical response.

Combine material Phonons with Band Theory and Electronic Structure and declare the electronic state, phonon branch, momentum transfer, and coupling vertex. Use Boltzmann Transport for a kinetic collision problem, Lifetime and Spectral Weight for one-particle renormalization and decay language, or BCS Theory when a retarded attractive channel is reduced to a pairing model. Once the question moves from that coupling input to pairing, stiffness, electrodynamics, or phase evidence, enter Superfluidity and Superconductivity; this chapter retains the excitation and coupling route.

These are different outputs. One deformation potential does not by itself determine mobility, a superconducting transition temperature, a phonon Kohn anomaly, and an electronic mass enhancement.

Route a polaron

Use Polarons Preview for dressing, conserved total momentum, energy, residue, effective mass, emission thresholds, and the distinctions among electron–phonon, Fermi, and Bose polarons. Add material Phonons and the electronic band owner only when the claim needs a real lattice vertex, dielectric environment, optical line, or transport signature. The future material bridge will not repeat the generic Lee–Low–Pines or impurity derivations.

Route a plasmon or screening question

Use Plasmons Preview for the dielectric zero, density-response pole, dimensional scaling, damping, sum rules, surface modes, and experimental identification criteria. Use Susceptibilities for response conventions and static-versus-dynamic limit order. Use Random Phase Approximation when the requested result depends on that closure, and Drude Theory only in its declared local long-wavelength regime.

Static screening and a dynamic plasmon are different limits. A peak in a loss function is evidence filtered by damping, background response, and probe kinematics; it is not automatically a new boson. A future material page will specialize band structure, dielectric surroundings, surfaces, and nanostructure geometry. Bulk longitudinal plasmons are diagnosed through a vanishing dielectric-matrix eigenvalue, equivalently a zero determinant at the appropriate generally complex frequency, together with a compatible pole of the inverse dielectric or density response. Surface and retarded plasmon–polaritons instead require the relevant Maxwell boundary determinant and need not be purely longitudinal.

Route an exciton

Start with Band Theory and Electronic Structure and Particle–Hole Excitations. Then declare whether the target is a bound neutral eigenstate, a resonance in a two-particle response, or an optical peak. Spectral Functions owns the one-particle addition and removal spectrum. Raman and Optical Spectroscopy owns the measured optical intensity, while Angle-Resolved Photoemission Spectroscopy owns the momentum-resolved single-particle comparison. Quantum Wells and Two-Dimensional Materials provide live platform context.

Binding energy, optical transition energy, quasiparticle gap, and peak separation are not interchangeable. The cross-platform material construction, screening audit, and evidence ladder remain a local coverage gap until the planned exciton page is substantive.

Route spin or generic quasiparticle language

Use Spin Waves and Magnons for the material Hamiltonian, linear spin-wave calculation, scattering signatures, and validation. Use Magnons for the universal bosonic construction. Use Quasiparticles Overview for the taxonomy of residues, lifetimes, continua, and breakdown. These treatments supply the complementary material and general-theory perspectives.

Material Specializations and Canonical Owners

The retained planned pages have nonduplicative jobs: compare Debye and Einstein model reductions; audit anharmonic material lattice dynamics; connect a material electron–phonon vertex to specific observables; specialize polaron and plasmon theory to material inputs and probes; and construct and test excitons across material platforms. These entries are marked Planned; the substantive treatments above provide the current explanations.

Harmonic-lattice dynamics and acoustic and optical branches are developed in Phonons and its many-body foundation. Debye and Einstein approximations form one comparative model treatment. General screening belongs to response theory and the transport chapter. Magnons connect the material magnetism and general many-body treatments; the broader quasiparticle taxonomy belongs in Many-Body.

Worked Routing Audit: Coupled LO Phonon and Plasmon Features

Suppose an $n$-doped polar semiconductor shows two longitudinal loss features that approach and repel as carrier density is tuned through the frequency of a longitudinal optical phonon. Calling one branch “the phonon” and the other “the plasmon” at every density discards the possibility that both are hybrid modes.

Declare the background and bare scales. Fix the crystal phase, carrier density and distribution, temperature, dielectric environment, sample geometry, and boundary conditions. Record the uncoupled lattice eigenvector and LO frequency, the electronic particle–hole continuum, and the density scaling expected for the uncoupled charge mode. Keep reduced momentum $\mathbf q$ distinct from the total transfer $\mathbf Q$ used by the probe.

Use the coupled generator. In a bulk, nonretarded calculation, inspect the full complex longitudinal dielectric matrix containing both ionic and electronic polarization. A candidate bulk mode needs a vanishing dielectric-matrix eigenvalue, equivalently a zero determinant at the appropriate generally complex frequency, together with a pole of the inverse dielectric or density response. A zero of only $\operatorname{Re}\epsilon$ is insufficient. For a film, interface, or retarded geometry, replace that bulk test by the appropriate Maxwell boundary determinant and coupled material response.

Track character as well as energy. Follow the ionic and charge components of each eigenvector, pole residue or oscillator strength, spectral-weight transfer, and complex linewidth through the avoided crossing. Test how the modes depend on carrier density and enter the particle–hole continuum. Energy repulsion alone can also result from branch sorting or an incomplete model; a hybrid claim requires the expected exchange of character and weight.

Apply the probe forward model. An electron-loss spectrum weights an inverse dielectric response and includes projectile kinematics, multiple scattering, surface sensitivity, and resolution. Infrared or Raman data use different operators and selection rules. A defensible conclusion states which coupled kernel accounts for the dispersions, weights, widths, density evolution, and boundary dependence, or else reports the surviving alternatives.

Worked Claim Audit: A Soft Branch at Finite Temperature

Suppose a zero-temperature harmonic calculation gives $\omega^2(\mathbf q_*)<0$ for one branch, while finite-temperature scattering shows a broad low-energy feature near $\mathbf q_*$ and a zone-center Raman line remains finite. “Anharmonicity stabilizes the soft phonon that drives the transition” combines several claims that require different checks.

Audit the harmonic result. First declare and hold fixed the reference structure, stress, electronic or magnetic state, Hamiltonian, and constraints. Then converge the force constants, supercell or wave-vector mesh, and acoustic and rotational sum rules. If the negative eigenvalue disappears under that fixed-problem convergence, it was a numerical or constraint artifact. If it survives, it identifies a negative-curvature direction of the declared reference structure, not a negative-energy phonon. Changing the structure, stress, state, Hamiltonian, or constraints defines a different physical problem rather than repairing the original calculation. Map the energy along the unstable eigenvector and test candidate lower-symmetry structures.

Test finite-temperature stabilization. A controlled anharmonic calculation must evaluate the temperature-dependent free-energy curvature in the relevant coupled coordinate space. Positive curvature at the same $\mathbf q_*$ and symmetry establishes stability only of that channel. Local structural stability additionally requires a stationary free energy and a nonnegative full coupled Hessian; global stability requires comparison with competing-phase free energies. A reasonably isolated complex pole in the dynamical response is separate evidence for renormalized-phonon language, not an alternative proof of thermodynamic stability. A broad central or relaxational feature need not have one frequency and lifetime, even when it descends from the same harmonic coordinate.

Audit probe visibility. Neutron or x-ray scattering can access $\mathbf q_*$ through a total transfer $\mathbf Q=\mathbf q_*+\mathbf G$. First-order Raman scattering is normally near the zone center, so its line is not automatically the same branch; folding, disorder, or multiphonon processes would need separate evidence. Match temperature, pressure, polarization, matrix elements, backgrounds, and resolution before comparing widths.

The bounded result states whether the harmonic instability is converged, whether one channel, the full local structure, or the globally preferred phase is stabilized at finite temperature, whether a renormalized pole remains identifiable, and which probe actually sees its symmetry channel. None of those conclusions alone proves a transition mechanism.

Canonical Boundaries

• This gateway owns material-excitation routing, readiness, current coverage, and stopping rules. It owns no detailed derivation.
• Phonons owns real-crystal lattice dynamics, thermodynamics, probe weights, and computational stability. The Many-Body phonon page owns universal quantization and operator structure.
• The Many-Body quasiparticle chapter owns generic poles, residues, lifetimes, damping, and collective-mode theory. Its polaron and plasmon pages remain the current canonical detailed owners.
• Band Theory owns electronic dispersions and state filling. Response and correlation pages own susceptibilities, dielectric functions, spectral functions, structure factors, and sum rules.
• Magnetism owns material spin waves and magnons. Probe pages own count rates, matrix elements, backgrounds, geometry, and resolution.
• Future local pages may specialize material parameters, computational workflows, and evidence audits, but they must link rather than repeat these canonical foundations.

Exit Checkpoint

Before leaving the gateway, you should be able to:

• name the background state and actual dynamical variable;
• distinguish a normal mode, pole, resonance, continuum, bound state, and probe-weighted peak;
• state the generating kernel and its approximation level;
• give momentum, symmetry, polarization, damping, and normalization data;
• select the shortest substantive theory and probe owners;
• distinguish static screening from a dynamic mode and a computed spectrum from a measured intensity;
• state a sum rule, limiting check, convergence test, or alternative mechanism;
• identify where the local chapter remains planned; and
• write a bounded claim with a failure or escalation condition.

Common Routing Errors

• A normal mode is not one atom, electron, or spin moving independently.
• “Collective” does not mean macroscopic occupation, long lifetime, or experimental brightness.
• An optical phonon is not automatically optically active.
• An imaginary harmonic frequency indicates a negative-curvature instability direction or a convergence or sum-rule artifact, not a negative excitation energy.
• A fitted Debye or Einstein temperature is model-dependent reduction data, not a unique microscopic thermometer.
• Static screening, a dielectric zero, a loss peak, and a density pole are related but not identical objects.
• A sharp spectral peak may be a resonance without being a stable eigenstate; a broad continuum can still carry most of a sum rule.
• An exciton is neutral overall but can couple strongly to light; an electron and a hole visible separately do not by themselves prove a bound exciton.
• A polaron effective mass is not a transport mobility.
• A linewidth is not a lifetime until the spectral convention and all broadening contributions are declared.
• A high-symmetry path does not prove full-zone stability.
• Agreement with one probe does not establish a phase, mechanism, or canonical quasiparticle across every scale.

Routing Exercises

1. Four signals, four owners

A Raman-active lattice peak, an electron-energy-loss peak, a neutron magnetic branch, and an optical bound-state threshold are reported for one material. Route each feature to its theory and probe owners, and name one missing check that prevents a peak label from becoming an excitation claim.

Solution

• Route the lattice eigenvector and dispersion to Phonons, and the measured intensity to Raman and Optical Spectroscopy. The Raman tensor, symmetry, near-zone-center momentum transfer, polarization geometry, and resolution must support the assignment.
• Route the loss feature to Plasmons Preview and Susceptibilities. Test the full complex dielectric or density response, particle–hole continuum, projectile kinematics, boundary geometry, multiple scattering, and resolution; a bright loss peak alone is not a plasmon.
• Route the material magnetic branch to Spin Waves and Magnons, its reusable bosonic construction to Magnons, and its counts to Neutron Scattering. The magnetic structure factor, polarization factor, and competing continuum must be checked.
• Route the charged quasiparticle inputs to Band Theory and Electronic Structure, the neutral electron–hole continuum and its generic kinematics to Particle–Hole Excitations, and the measured threshold to Raman and Optical Spectroscopy. A two-particle kernel, oscillator strength, and matched quasiparticle continuum are needed before calling it a bound exciton.

2. Four meanings of a soft result

Distinguish a harmonic imaginary frequency, a renormalized soft phonon, a numerical sum-rule artifact, and finite-temperature anharmonic stabilization. What evidence identifies each case?

Solution

1. A converged negative eigenvalue of the harmonic dynamical matrix is a negative-curvature direction of the declared reference structure. It is not a negative-energy phonon.
2. A renormalized soft phonon is supported by a low-frequency, reasonably isolated complex pole with trackable eigenvector character, weight, and linewidth after interactions are included. A response peak may be its probe-weighted manifestation, but a peak alone is insufficient.
3. At fixed reference structure, stress, electronic or magnetic state, Hamiltonian, and constraints, a numerical artifact disappears or moves materially when the supercell, wave-vector mesh, force-constant range, thresholds, or acoustic and rotational sum rules are converged.
4. Finite-temperature stabilization of the named channel requires positive controlled anharmonic free-energy curvature at the same $\mathbf q_*$ and symmetry. Full local stability also requires stationarity and a nonnegative full coupled Hessian; global stability requires competing-phase free energies. A stable response pole is separate dynamical evidence, and a fitted positive peak alone proves neither kind of thermodynamic stability.

3. A heat-capacity fit

A low-temperature crystal heat capacity is fitted by one Debye term and two Einstein terms. What can the fitted parameters support, and what can they not identify?

Solution

The fit can provide a compact approximation to the vibrational spectral weight over its declared temperature range if mode counts, units, electronic and other backgrounds, and uncertainties are controlled. For $r$ atoms per primitive cell in three dimensions, the per-cell measure must satisfy $\int_0^\infty g(\omega)\,d\omega=3r$. The Debye cutoff and coefficient must obey their assigned acoustic state count; the weights of all Einstein delta peaks and Debye continua must add to the declared total.

At low temperature, the Debye coefficient tests the low-frequency density of states, equivalently the branch-weighted inverse-cubed sound-velocity combination. Einstein terms can summarize concentrated higher-frequency weight. Their fitted temperatures do not by themselves identify branches or atoms.

The fit does not uniquely reconstruct a phonon dispersion, polarization, atomic eigenvector, or microscopic force constant. Different spectral measures can produce similar finite-range heat capacities, and fitted characteristic temperatures can covary. Compare with the actual phonon density of states or a scattering probe before assigning a term to one branch or atom.

4. Five electron–phonon outputs

The same calculated electron–phonon matrix elements are proposed as sufficient to predict an ARPES kink, a transport scattering rate, a phonon linewidth, a pairing kernel, and a polaron mass. Explain why these are not one transferable coupling number.

Solution

• An ARPES kink uses the momentum- and energy-dependent electronic self-energy, spectral matrix elements, occupations, backgrounds, and resolution along the measured cut.
• A transport rate weights scattering by current relaxation, often including a factor such as $1-\cos\theta$, as well as occupations, phase space, and a justified kinetic or response closure.
• A phonon linewidth comes from the phonon self-energy and electron–hole phase space for the specified branch and wave vector, with a declared linewidth convention.
• A pairing kernel uses a retarded Fermi-surface projection together with Coulomb competition, gap symmetry, and the controlled reduction behind the chosen superconducting model.
• A polaron mass comes from the curvature of a dressed one-particle pole and an approximation valid in the relevant coupling regime.

The material vertex is shared input, not a shared answer. Route the claims to Angle-Resolved Photoemission Spectroscopy, Boltzmann Transport, Phonons and Lifetime and Spectral Weight, BCS Theory, or Polarons Preview according to the observable.

5. An LO phonon–plasmon avoided crossing

In a doped polar semiconductor, two loss branches repel as carrier density is increased through the bare LO phonon scale. What must be tracked before this is called an LO phonon–plasmon avoided crossing?

Solution

Solve the coupled ionic and electronic response, not two uncoupled dispersions. For a bulk longitudinal mode, track a vanishing eigenvalue of the full complex dielectric matrix, equivalently its zero determinant at the appropriate generally complex frequency, together with the compatible pole of the inverse dielectric or density response. For a film, interface, or retarded mode, use the appropriate Maxwell boundary determinant.

Across the crossing, follow the ionic-displacement and charge-density parts of the eigenvectors, residues or oscillator strengths, exchange of spectral weight, complex linewidths, carrier-density dependence, and entry into the particle–hole continuum. Apply the actual electron-loss, infrared, or Raman forward model and its resolution. Level repulsion alone does not establish hybridization.

6. Four semiconductor gaps

An optical peak lies below a Kohn–Sham band gap. Construct a ledger for the fundamental charge gap, quasiparticle gap, optical threshold, and exciton binding energy without assuming a hydrogenic model.

Solution

The exact fundamental charge gap is $E_0^{N+1}+E_0^{N-1}-2E_0^N$, with charged-state boundary and finite-size conventions stated. The separation between the exact one-particle addition and removal thresholds encodes the same charged excitation scale; an approximation-labelled quasiparticle gap is a computational estimate of it. A Kohn–Sham eigenvalue gap is an auxiliary-system quantity and is not silently substituted for either.

The optical threshold is a neutral, matrix-element- and resolution-weighted transition; the lowest neutral state may be dark. An ordinary exciton binding energy is defined only for a bound two-particle state below the matching free electron–hole quasiparticle continuum, with consistent momentum, spin, symmetry, screening, and environment. An excitonic resonance embedded in a continuum instead requires a resonance energy and width; it does not share that binding-energy definition. The bound-state difference does not require a hydrogenic wavefunction, and an optical peak below a Kohn–Sham gap does not determine it.

7. Finite lines or intrinsic decay?

A finite-supercell calculation gives discrete sharp lines, interpolation gives a broad feature, and experiment reports a still broader peak. Diagnose finite-size lines, interpolation artifacts, instrumental broadening, and true intrinsic decay. When should particle-like language stop?

Solution

First vary supercell size, wave-vector mesh, and accessible momenta. Continuum-derived finite-system levels generally shift and become denser toward a declared thermodynamic limit, whereas a genuine bound state or stable quasiparticle pole may converge to an isolated delta contribution with finite residue. Track energies and weights to distinguish those cases. Then vary the interpolation mesh, range, algorithm, and broadening while checking symmetries and sum rules; a feature that depends on those choices is not an intrinsic linewidth. Convolve a candidate intrinsic response with the documented instrumental resolution and compare measurements at more than one resolution when possible.

True decay requires a size- and interpolation-stable complex pole or spectral width tied to an allowed microscopic continuum or scattering channel, with a declared linewidth-to-lifetime convention. When the pole is no longer isolated or its width is comparable to its frequency, report the broad response, continuum thresholds, and redistributed weight. Resonance language may remain useful, but one frequency, residue, and lifetime no longer define a robust quasiparticle.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976).
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• M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon Press, 1954).
• J. T. Devreese and A. S. Alexandrov, “Fröhlich polaron and bipolaron: recent developments,” Reports on Progress in Physics 72, 066501 (2009), doi:10.1088/0034-4885/72/6/066501.
• G. F. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid (Cambridge University Press, 2005).
• F. Giustino, “Electron–phonon interactions from first principles,” Reviews of Modern Physics 89, 015003 (2017), doi:10.1103/RevModPhys.89.015003.
• X. Gonze and C. Lee, “Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory,” Physical Review B 55, 10355–10368 (1997), doi:10.1103/PhysRevB.55.10355.
• G. Grimvall, The Electron–Phonon Interaction in Metals (North-Holland, 1981).
• C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, 2005).
• G. D. Mahan, Many-Particle Physics, 3rd ed. (Springer, 2000).
• G. Onida, L. Reining, and A. Rubio, “Electronic excitations: density-functional versus many-body Green’s-function approaches,” Reviews of Modern Physics 74, 601–659 (2002), doi:10.1103/RevModPhys.74.601.
• D. Pines and P. Nozières, The Theory of Quantum Liquids, Volume I: Normal Fermi Liquids (W. A. Benjamin, 1966).
• J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, 1960).