Collective Modes

A collective mode is a coherent pattern of motion of many microscopic degrees of freedom that behaves as one dynamical coordinate or response eigenchannel.

The defining feature is organization, not merely particle number. Instead of tracking every microscopic fluctuation $\delta x_j(t)$ separately, one identifies a pattern

$$Q_{\lambda\mathbf q}(t) ={} \sum_j \ell_{\lambda j}^*(\mathbf q) \times \delta x_j(t),$$

whose amplitude evolves approximately independently over a stated range of wavelength, frequency, temperature, and perturbation strength. The label $\lambda$ identifies a branch or polarization, and $\mathbf q$ identifies momentum when translation symmetry is available. The coefficients $\ell_{\lambda j}^*$ project microscopic motion onto the mode amplitude; they need not equal the displacement pattern itself in a nonorthogonal or damped problem.

Examples include:

• compression waves in a fluid;
• lattice vibrations in a solid;
• coherent spin precession in an ordered magnet;
• plasma oscillations of charge density;
• phase and amplitude oscillations of an order parameter;
• in-phase and out-of-phase motion in multicomponent systems;
• diffusive relaxation of a conserved density.

Some collective modes are sharp propagating excitations whose quanta are quasiparticles. Others are overdamped or purely relaxational. A collective mode therefore need not be gapless, ballistic, long lived, or even oscillatory.

A mode can be described by a coordinate, a solution of linearized equations, or a pole of a response matrix. Connecting these descriptions requires the same variables, background, and boundary conditions.

Helpful background. Retarded response supplies the pole and Fourier conventions; susceptibilities explains how a specified source and detector select a mode.

Collective Coordinates and Polarizations

Microscopic variables

A microscopic description may involve:

$$\left\{ x_j, p_j, n_j, \mathbf S_j, \psi_j, \ldots \right\},$$

for a large number of sites, particles, orbitals, or field components. Exact dynamics couples these quantities through the many-body Hamiltonian.

Even when the underlying equations are local, an arbitrary microscopic disturbance generally excites many frequencies and dephases into a complicated pattern.

Collective variables

A collective description selects combinations

$$X_a(\mathbf q,t) = \sum_j f_{aj}(\mathbf q) O_j(t)$$

that capture the slow, coherent, or strongly responding sector. Typical $X_a$ include:

• number density;
• momentum density;
• energy density;
• magnetization;
• lattice displacement;
• condensate phase;
• order-parameter amplitude;
• relative density or phase between components.

The index $a$ labels coupled variables before their dynamics is diagonalized. A mode is an eigencombination

$$Q_\lambda = \sum_a \ell_{\lambda a}^*X_a.$$

The reconstruction $X_a=\sum_\lambda e_{\lambda a}Q_\lambda$ defines the right polarization vectors $\mathbf e_\lambda$. For an orthonormal basis, $\boldsymbol\ell_\lambda=\mathbf e_\lambda$; for generalized or non-Hermitian dynamics, projection uses the appropriate mass metric or left vectors.

Resolution dependence

Collectivity is scale dependent. A sound wave is a coherent density and velocity field at wavelengths long compared with microscopic collision or lattice scales. At atomic resolution, the same disturbance is a complicated superposition of particle motion.

The chosen variables, background state, wavelength and frequency range, dynamical approximation, and probe determine whether a collective description is useful.

Distinguishing Nearby Concepts

Collective mode versus normal mode

A normal mode is an eigenpattern of a linearized dynamical problem. A collective mode emphasizes that the pattern organizes many microscopic degrees of freedom or macroscopic fields.

Every harmonic phonon is a normal mode and a collective mode. A two-level atom has a transition frequency, but calling that isolated internal transition a collective mode would usually add no useful information.

The terms overlap strongly, but they answer different questions:

Term	Main question
normal mode	does the linearized problem diagonalize into this pattern?
collective mode	which many-body coordinate moves coherently?
quasiparticle	can the excitation be treated as a particle-like object?
resonance	does the response have a finite-lifetime pole or enhancement?
exact eigenstate	is the state stationary under the exact Hamiltonian?

Collective mode versus quasiparticle

A phonon, magnon, or underdamped plasmon can be both:

$$\begin{gathered} \text{collective origin} \\ + \\ \text{particle-like quantum}. \end{gathered}$$

A diffusive density mode is collective but generally not a ballistic quasiparticle. A Landau fermionic quasiparticle is particle-like but is not primarily described as a coherent oscillation of a macroscopic coordinate.

The categories are overlapping, not mutually exclusive.

Collective mode versus coherent state

A mode is a dynamical degree of freedom or branch. A coherent state is one possible quantum state of an approximately harmonic bosonic mode.

One quantum of a phonon mode is already collective because its displacement pattern extends over many sites. A classical elastic wave corresponds instead to a state with a large coherent occupation of that mode. Large occupation is not required for collectivity.

Collective mode versus exact eigenstate

In a finite closed system, exact eigenstates have real discrete energies. A linearized collective mode may be:

• an exact eigenexcitation of a quadratic Hamiltonian;
• an approximate eigenstate concentrated in a narrow spectral window;
• a resonance that decays into other exact states;
• a hydrodynamic pole defined only after a thermodynamic and long-wavelength limit.

Calling a mode “exact” requires a stronger statement than identifying a branch in an approximation.

Collective mode versus many-body entanglement

Collective motion does not by itself specify an entanglement pattern. A classical normal mode of coupled oscillators is collective. A quantum one-mode excitation can be entangled in a site partition, but the amount depends on the state and partition.

Conversely, a highly entangled state need not possess a sharp collective mode. Collectivity is a dynamical organization, not an entanglement measure.

Linearized Dynamics

Expansion around a background

Choose a stationary or slowly varying background $X_a^{(0)}$ and write

$$X_a = X_a^{(0)} + \delta X_a.$$

The background may be:

• an equilibrium fluid;
• a crystal;
• an ordered magnet;
• a condensate;
• a mean-field saddle;
• a thermal state characterized by conserved densities.

Linearization keeps terms first order in $\delta X_a$. This step is controlled only when the perturbation remains small in the variables used.

Conservative second-order form

For real coordinates with inertia, a common equation is

$$\sum_b \left[ M_{ab}\partial_t^2 + K_{ab}(\mathbf q) \right] \delta X_b = 0.$$

With

$$\delta X_b(t) = e_b e^{-i\omega t},$$

one obtains the generalized eigenproblem

$$\sum_b K_{ab}(\mathbf q)e_b = \omega^2 \sum_b M_{ab}e_b.$$

For Hermitian matrices with $M$ positive definite and $K$ positive semidefinite, $M^{-1/2}KM^{-1/2}$ is Hermitian and nonnegative. Thus $\omega^2\ge0$ and the frequencies can be chosen real. A zero eigenvalue is a neutral direction at linear order; it is not a positive-frequency harmonic oscillator.

The mode condition is

$$\det \left[ K(\mathbf q) -\omega^2M \right] = 0.$$

First-order form

Not every mode has Newton-like second-order dynamics. Canonically conjugate fields, spin precession, kinetic equations, and quantum amplitudes often obey

$$i\partial_t \delta\mathbf X = \mathcal L(\mathbf q) \delta\mathbf X.$$

Then

$$\mathcal L(\mathbf q)\mathbf e_\lambda = \omega_\lambda(\mathbf q) \mathbf e_\lambda.$$

The operator $\mathcal L$ need not be Hermitian in the ordinary Euclidean metric. Bosonic Bogoliubov problems use an indefinite particle–hole metric, while dissipative problems are genuinely non-Hermitian.

Damped matrix form

A useful schematic interpolation is

$$\sum_b \left[ -\omega^2M_{ab} -i\omega\Gamma_{ab} +K_{ab} \right] e_b = 0.$$

Here $\Gamma$ represents damping only within the stated effective description. It may arise from collisions, decay into omitted modes, disorder, viscosity, or a bath.

With the Fourier convention

$$e^{i(\mathbf q\cdot\mathbf r-\omega t)},$$

stable damped poles satisfy

$$\operatorname{Im}\omega < 0.$$

Stability is part of the solution

A linearized eigenfrequency can reveal that the chosen background is unstable:

• $\omega^2<0$ in a conservative second-order problem;
• $\operatorname{Im}\omega>0$ in the retarded convention;
• a negative stiffness or compressibility;
• nonlinear behavior along a zero mode, which requires further analysis rather than proving instability by itself.

An exponentially growing solution is a collective instability, not a stable excitation branch.

Coupled-Oscillator Benchmark

Two equal masses already show the essential change of variables. Take $m>0$, $k_0>0$, and $k_c\ge0$, and consider

$$L ={} \frac{m}{2} \left( \dot x_1^2+\dot x_2^2 \right) - \frac{k_0}{2} \left( x_1^2+x_2^2 \right) - \frac{k_c}{2} \left( x_1-x_2 \right)^2.$$

Define

$$Q_+ = \frac{x_1+x_2}{\sqrt2}, \qquad Q_- = \frac{x_1-x_2}{\sqrt2}.$$

The Lagrangian becomes

$$L ={} \frac{m}{2} \left( \dot Q_+^2+\dot Q_-^2 \right) - \frac{k_0}{2}Q_+^2 - \frac{k_0+2k_c}{2}Q_-^2.$$

Therefore

$$\omega_+ = \sqrt{\frac{k_0}{m}}, \qquad \omega_- = \sqrt{ \frac{k_0+2k_c}{m} }.$$

The $+$ mode moves in phase and does not stretch the coupling spring. The $-$ mode moves out of phase and pays the additional stiffness $2k_c$.

This example teaches four durable lessons:

1. a mode is a pattern, not one coordinate;
2. eigenvectors and frequencies are separate pieces of information;
3. the source can couple differently to the two patterns;
4. in-phase and relative modes recur in multiband, multisublattice, and multicomponent quantum systems.

Quantizing a Collective Mode

Harmonic mode coordinate

After mass normalization, a stable mode with $\omega_\lambda>0$ may have

$$H_\lambda = \frac12 \left( P_\lambda^2 + \omega_\lambda^2Q_\lambda^2 \right).$$

Introduce

$$\begin{aligned} Q_\lambda ={}& \sqrt{ \frac{\hbar}{2\omega_\lambda} } \left( b_\lambda+b_\lambda^\dagger \right), \\ P_\lambda ={}& -i \sqrt{ \frac{\hbar\omega_\lambda}{2} } \left( b_\lambda-b_\lambda^\dagger \right). \end{aligned}$$

Then

$$H_\lambda = \hbar\omega_\lambda \left( b_\lambda^\dagger b_\lambda + \frac12 \right).$$

The operator $b_\lambda^\dagger$ creates one quantum of the entire pattern.

Approximate bosonic character

The harmonic algebra is exact only for an exactly quadratic canonical mode. In an interacting many-body system:

• mode operators can contain nonlinear corrections;
• different modes interact;
• mode number may not be conserved;
• high occupation can invalidate the linearization;
• a branch can decay into other branches or continua.

Bosonic quasiparticle operators are therefore often leading effective variables rather than exact microscopic creation operators.

One quantum can move many constituents

For a normalized extended mode, the local vacuum-to-one-quantum displacement matrix element scales as $N^{-1/2}$ at fixed frequency. The corresponding extra local mean-square fluctuation is of order $1/N$, while the total excitation energy remains of order one. In a phonon number state, the mean displacement itself is zero; a coherent superposition is needed for an oscillating expectation value.

This is the useful heuristic:

In a collective quantum, many degrees of freedom move a little in a correlated pattern.

It is a heuristic, not a universal definition. Localized collective modes, boundary modes, and strongly inhomogeneous systems require more careful language.

Response-Matrix Viewpoint

Source and detector

For Hermitian coordinates and real time-domain sources, let

$$\delta H(t) = -\sum_b f_b(t)X_b.$$

Use the physical susceptibility $\chi_{ab}^{\mathrm R}(t)=(i/\hbar)\theta(t)\langle[X_a(t),X_b]\rangle$, appropriate to this minus-sign coupling. Linear response gives

$$\delta\langle X_a\rangle = \sum_b \chi_{ab}^{\mathrm R}f_b.$$

For translation-invariant systems, $\chi^{\mathrm R}$ depends on $(\mathbf q,\omega)$.

The source vector chooses how the system is driven. The detector vector chooses which response is read out. A mode can exist while remaining dark in a particular source–detector channel.

Inverse response kernel

For Gaussian fluctuations or a linearized effective theory, write

$$\sum_b \mathcal K_{ab}^{\mathrm R} \delta X_b = f_a.$$

When no contact or vertex convention changes the relation,

$$\chi^{\mathrm R} = \left( \mathcal K^{\mathrm R} \right)^{-1}.$$

Collective modes solve the homogeneous equation

$$\mathcal K^{\mathrm R} \mathbf e_\lambda = 0,$$

so their complex frequencies satisfy

$$\det \mathcal K^{\mathrm R}(\mathbf q,\omega) = 0.$$

This determinant condition is necessary but not sufficient for visibility. The numerator, source projection, detector projection, and possible cancellations must also be checked.

Isolated pole

Near a simple positive-frequency pole,

$$\chi_{ab}^{\mathrm R} (\mathbf q,\omega) \simeq{} \frac{ R_{ab}^{(\lambda)}(\mathbf q) }{ \omega-\Omega_\lambda(\mathbf q) +i\kappa_\lambda(\mathbf q) } + \text{negative-frequency pole} + \chi_{ab,\mathrm{reg}}^{\mathrm R}.$$

For a well-isolated mode, the residue matrix often factorizes:

$$R_{ab}^{(\lambda)} \propto u_{\lambda a} v_{\lambda b}^*,$$

where right and left mode vectors can differ in a damped non-Hermitian problem.

The residue matrix encodes polarization and visibility. The pole position encodes frequency and damping. Neither alone is a complete mode identification.

Real-coordinate pole pair

For a Hermitian coordinate in a conservative system, positive and negative frequencies occur together. A schematic oscillator response is

$$\chi^{\mathrm R}(\omega) = \frac{1}{ \Omega_0^2 - (\omega+i0^+)^2 }.$$

Its two poles reflect one real oscillatory coordinate, not two independent positive-energy species.

Finite-volume caution

Every exact transition in a finite isolated system contributes a real-axis delta line. That fact does not make every transition a collective mode.

A finite-size collective identification should show some combination of:

• a coherent branch across momentum or another control parameter;
• concentration of weight in a physically motivated collective operator;
• stable polarization or symmetry quantum numbers;
• scaling toward a thermodynamic pole or continuum feature;
• agreement with a linearized effective description.

Polarization, Weight, and Participation

Mode polarization

If several variables mix, the eigenvector

$$\mathbf e_\lambda = \left( e_{\lambda1}, e_{\lambda2}, \ldots \right)^{\mathsf T}$$

states which variables oscillate and with what relative amplitude and phase.

Examples include:

• longitudinal versus transverse displacement;
• density versus spin density;
• in-phase versus out-of-phase component motion;
• order-parameter amplitude versus phase;
• charge-like versus neutral combinations.

The polarization can change continuously along a branch and exchange character near an avoided crossing.

Operator-dependent weight

For a detector

$$O_{\mathbf d} = \sum_a d_a^*X_a,$$

the observed pole weight depends on

$$\mathbf d^\dagger R^{(\lambda)} \mathbf f,$$

where $\mathbf f$ is the source vector. A mode with nonzero residue matrix can be invisible when symmetry or destructive interference makes this projection vanish.

“No peak in this channel” is not equivalent to “no mode exists.”

Participation ratio

For a normalized discrete pattern

$$\sum_j \lvert e_{\lambda j}\rvert^2 = 1,$$

one possible participation measure is

$$P_\lambda = \frac{1}{ \sum_j \lvert e_{\lambda j}\rvert^4 }.$$

It gives $P_\lambda=1$ for weight on one coordinate and $P_\lambda=N$ for equal magnitude on $N$ coordinates.

This number is useful but not invariant:

• it depends on the basis and normalization;
• internal components and nonorthogonal coordinates require a metric;
• a long-wavelength mode can be spatially extended but dark;
• an edge or defect mode can be collective within a localized region.

Participation measures spatial organization in the chosen basis; it is not a universal definition of collectivity.

Extensive versus normalized operators

For

$$O_{\mathbf q} = \sum_{j=1}^{N} e^{-i\mathbf q\cdot\mathbf r_j} O_j,$$

spectral weight can scale with $N$. If instead one uses $O_{\mathbf q}/\sqrt N$, the same coherent branch can carry order-one weight.

Claims about “macroscopic oscillator strength” must therefore state the operator normalization.

Why Collective Modes Arise

Coupling and restoring forces

Interactions or mechanical couplings make motion of one degree of freedom alter the forces on others. The resulting feedback can synchronize a pattern and supply a restoring force.

Examples:

• elastic bonds restore lattice displacement;
• compressibility restores density compression;
• exchange restores spin twists;
• Coulomb fields restore charge separation;
• condensation energy restores order-parameter amplitude.

Conservation laws

Conserved densities cannot relax locally without transport. If

$$\partial_t n + \nabla\cdot\mathbf j = 0,$$

then, with local constitutive relations and finite long-wavelength coefficients, density fluctuations become slow as gradients become small. Conservation can therefore produce hydrodynamic modes without broken symmetry. Unscreened Coulomb interactions are an important exception to the inference of a gapless density mode: their singular long-range restoring field can produce a plasmon.

Spontaneous symmetry breaking

When an exact continuous global symmetry is spontaneously broken, slow variations along the degenerate order-parameter manifold can produce Goldstone modes.

The full assumptions and counting rule belong to Goldstone Modes in Many-Body Systems. Not every collective mode is a Goldstone mode:

• plasmons can be gapped;
• amplitude modes are generally gapped;
• diffusion follows conservation rather than broken symmetry;
• optical phonons need not soften at zero momentum.

Long-range fields

A fluctuation can generate a field that acts back on the complete system. Long-range Coulomb interaction is the canonical example:

$$\delta n \longrightarrow \delta\phi \longrightarrow \text{collective restoring force}.$$

The feedback can move a neutral acoustic branch to a nonzero plasma frequency.

Geometry and confinement

Boundaries, traps, finite samples, and interfaces create global shape, sloshing, breathing, surface, and edge modes. Their frequencies depend on geometry and boundary conditions as well as bulk constitutive data.

A center-of-mass oscillation can be highly collective while remaining protected from internal interactions by a separation theorem. “Collective” does not mean “strongly renormalized.”

What Can Oscillate?

Collective variable	Typical mode	Long-wavelength character
mass or number density	sound	acoustic and propagating
conserved density without inertia	diffusion	relaxational
charge density	plasmon	often gapped in three dimensions
lattice displacement	phonon	acoustic or optical
spin orientation	magnon or spin wave	linear or quadratic, depending on dynamics
condensate or order-parameter phase	phase mode	often Goldstone-like when neutral
order-parameter magnitude	amplitude mode	generally gapped and decay sensitive
relative phase of components	Leggett-type mode	usually gapped
relative density of components	spin-density or counterflow mode	acoustic, gapped, or diffusive
shape of a confined cloud	breathing or quadrupole mode	geometry dependent

This table classifies by the primary collective coordinate. Real modes can mix several entries.

Conserved-Density Modes

Diffusion

Combine the continuity equation

$$\partial_t n + \nabla\cdot\mathbf j = 0$$

with Fick’s constitutive law

$$\mathbf j = -D\nabla n.$$

Linearizing around a uniform state gives

$$\partial_t\delta n = D\nabla^2\delta n.$$

For

$$\delta n \propto e^{i(\mathbf q\cdot\mathbf r-\omega t)},$$

the mode is

$$\omega(\mathbf q) = -iDq^2.$$

It is collective because it governs coherent long-wavelength relaxation of a conserved density. It is not an oscillatory wave and generally does not define a ballistic quasiparticle.

Why diffusion becomes slow

As $q\to0$,

$$\lvert\omega\rvert \sim Dq^2 \to 0.$$

The slow rate is enforced by conservation: a nearly uniform excess can relax only by transporting density over a distance of order $q^{-1}$.

The diffusion constant is not fixed by conservation alone. It depends on transport coefficients and thermodynamic susceptibilities.

Multiple conserved densities

In a purely diffusive sector, conserved scalar densities such as charge, energy, and component numbers can mix. After excluding or separately treating propagating momentum–density sectors, linearized hydrodynamics produces a matrix

$$-i\omega\, \delta\mathbf n = -q^2 \mathsf D\, \delta\mathbf n.$$

Its eigenvalues determine diffusive combinations. Off-diagonal thermoelectric or spin–charge couplings rotate the hydrodynamic eigenvectors away from the original microscopic labels.

Sound as a Collective Mode

Continuity and inertia

For a neutral fluid of particles with mass $m$, linearized number conservation gives

$$\partial_t\delta n + n_0\nabla\cdot\mathbf v = 0.$$

Neglecting viscosity, the Euler equation is

$$mn_0\partial_t\mathbf v = -\nabla\delta P.$$

For an adiabatic disturbance, hold the entropy per particle $\bar s=S/N$ fixed, not the entropy density:

$$\delta P = \left( \frac{\partial P}{\partial n} \right)_{\!\bar s} \delta n.$$

Eliminating $\mathbf v$ gives

$$\partial_t^2\delta n = c_s^2\nabla^2\delta n,$$

with

$$c_s^2 = \frac{1}{m} \left( \frac{\partial P}{\partial n} \right)_{\!\bar s}.$$

The two propagating roots are

$$\omega_\pm(\mathbf q) = \pm c_sq.$$

The density and longitudinal velocity are not separate modes here. They are components of one sound eigenvector.

Dissipative correction

Viscosity and thermal transport broaden the poles:

$$\omega_\pm(\mathbf q) = \pm c_sq -i\Gamma_sq^2 + \cdots.$$

At sufficiently small $q$,

$$\frac{ \lvert\operatorname{Im}\omega\rvert }{ \lvert\operatorname{Re}\omega\rvert } \sim \frac{\Gamma_s}{c_s}q \to 0,$$

so hydrodynamic sound can become asymptotically sharp even in an interacting finite-temperature fluid.

First sound and zero sound

The same density channel can support different collective regimes.

For collision time $\tau_{\mathrm{coll}}$:

$$\begin{array}{c|c} \text{regime} & \omega\tau_{\mathrm{coll}} \\ \hline \text{hydrodynamic} & \ll1 \\ \text{collisionless} & \gg1 \end{array}$$

The hydrodynamic regime also requires $q\ell_{\mathrm{coll}}\ll1$. First sound is controlled by local thermodynamics and transport. Zero sound is a collisionless collective oscillation sustained by self-consistent quasiparticle interactions.

Fermi Liquid Theory derives the Landau kinetic equation and zero-sound condition. The important lesson here is that a mode name is incomplete without its dynamical regime.

More than one sound mode

Multicomponent and superfluid systems can support:

• in-phase density sound;
• counterflow or spin-density sound;
• first and second sound;
• entrainment-mixed branches;
• additional diffusive thermal or concentration modes.

Counting conserved quantities alone is not enough. One must include constitutive relations, inertia, broken symmetries, and couplings among the variables.

Superfluidity discusses first sound, second sound, and roton branches in helium and other neutral superfluids.

Spin Collective Modes

Precession

For magnetization $\mathbf M$, a schematic nondissipative equation is

$$\partial_t\mathbf M = -\gamma \mathbf M \times \mathbf H_{\mathrm{eff}}.$$

Linearization around an ordered state couples transverse components. The resulting spin wave describes coherent precession of many local moments or spin densities.

Different dynamical structures

Ferromagnetic and antiferromagnetic order can break similar spin symmetries while producing different low-energy dynamics:

$$\omega_{\mathrm{FM}} \propto q^2, \qquad \omega_{\mathrm{AFM}} \propto q.$$

The difference reflects canonical structure and background charge densities, not merely the number of broken generators.

Goldstone Modes explains type-A/type-B counting; Heisenberg Model gives exact one-magnon examples; Magnons develops spin-wave quantization and its limits.

Itinerant magnets

In an itinerant system, a spin collective mode can coexist with a particle–hole continuum. Its fate depends on:

• spin-rotation symmetry;
• exchange enhancement;
• momentum;
• available decay phase space;
• disorder and spin–orbit coupling.

A sharp transverse branch outside a continuum can become Landau damped after entering it.

Phase and Amplitude Modes

Polar coordinates of an order parameter

For a complex order parameter,

$$\Psi = \left( \Psi_0+h \right) e^{i\theta},$$

the fluctuation $h$ changes the magnitude and $\theta$ changes the phase.

Geometrically:

• $h$ is radial motion;
• $\theta$ is tangential motion along the order-parameter manifold.

Minimal neutral effective theory

A schematic real-time quadratic Lagrangian density, with positive inertial and gradient coefficients and amplitude stiffness $\Delta_h^2>0$, is

$$\mathcal L ={} \frac{\chi_\theta}{2} \left( \partial_t\theta \right)^2 - \frac{\rho_s}{2} \left( \nabla\theta \right)^2 + \frac{\chi_h}{2} \left( \partial_th \right)^2 - \frac{\rho_h}{2} \left( \nabla h \right)^2 - \frac{\Delta_h^2}{2}h^2 + \lambda h\partial_t\theta.$$

When $\lambda=0$, phase and amplitude separate at quadratic order. The neutral phase branch has

$$\omega_\theta^2 = \frac{\rho_s}{\chi_\theta} q^2,$$

while the amplitude branch has

$$\omega_h^2 = \frac{ \Delta_h^2+\rho_hq^2 }{ \chi_h }.$$

Here $\Delta_h$ is a parameter of the stiffness, not by itself an excitation energy: at zero momentum the decoupled amplitude energy is $\hbar\Delta_h/\sqrt{\chi_h}$.

Mixing

Particle–hole asymmetry, multiple order parameters, explicit symmetry breaking, and nonequilibrium driving can mix amplitude and phase.

Then “phase mode” and “amplitude mode” describe dominant polarization, not exact pure coordinates. The correct object is the eigenvector of the complete fluctuation kernel.

Amplitude-mode caution

A radial order-parameter fluctuation is often called a Higgs mode by analogy. The analogy is useful only when:

• an amplitude-like eigenchannel is well defined;
• the mode is sufficiently isolated from decay continua;
• its symmetry and selection rules are stated;
• the measured feature cannot be explained by a threshold or pair-breaking edge alone.

Not every broad bump near twice a gap is a sharp amplitude collective mode.

Charged phase fluctuations

In a neutral superfluid, phase stiffness can produce an acoustic Goldstone mode. In a charged superconductor, long-range electromagnetic coupling reorganizes the longitudinal phase response and moves spectral weight toward a plasma scale.

This is not ordinary explicit breaking of the global symmetry by a small source. It is dynamical coupling to gauge and charge-density fields. BCS Mean-Field Theory and Goldstone Modes in Many-Body Systems explain the neutral and charged cases. In an ungated two-dimensional charged layer, the long-wavelength plasmon can remain gapless with $\omega\propto\sqrt q$; a nonzero plasma gap is not universal in dimension.

Charge-Density Modes

Neutral compression versus charged oscillation

For a short-range neutral fluid, long-wavelength density motion is usually acoustic:

$$\omega \sim cq.$$

For a three-dimensional electron fluid with unscreened Coulomb interaction, a uniform displacement of electrons relative to the positive background creates a macroscopic electric restoring field.

The long-wavelength scale is

$$\omega_p^2 = \frac{ n_0e^2 }{ m\epsilon_0 }$$

in the simplest continuum SI convention.

Thus

$$\lim_{q\to0} \omega_{\mathrm{pl}}(q) = \omega_p \ne 0.$$

The precise dispersion changes with dimension, screening, lattice band mass, dielectric environment, and geometry. Plasmons develops those charge-mode cases.

Collective and individual sectors

The Bohm–Pines construction made the distinction vivid: long-range Coulomb motion can be reorganized into collective plasma coordinates, while residual short-range screened interactions act among particle-like degrees of freedom.

Modern response language expresses the same idea through zeros of a dielectric function or poles of a screened density response.

Landau damping

A charge mode can decay into particle–hole excitations when its frequency and momentum overlap their continuum and the matrix element is nonzero.

The mode then acquires a complex pole:

$$\omega_{\mathrm{pl}} \longrightarrow \Omega_{\mathrm{pl}} -i\kappa_{\mathrm{pl}}.$$

Entering a continuum does not always erase the mode immediately, but it removes any automatic guarantee of a narrow Lorentzian.

Matrix Modes and Hybridization

Coupled channels

Near two isolated positive-frequency resonances, a rotating-wave model for complex mode amplitudes can have inverse kernel

$$\mathcal K(\omega) = \begin{pmatrix} \omega-\omega_1 & -g \\ -g & \omega-\omega_2 \end{pmatrix}.$$

The collective frequencies solve

$$\det\mathcal K = 0,$$

giving

$$\omega_\pm ={} \frac{\omega_1+\omega_2}{2} \pm \sqrt{ \left( \frac{\omega_1-\omega_2}{2} \right)^2 +g^2 }.$$

At the uncoupled crossing $\omega_1=\omega_2$,

$$\omega_+-\omega_- = 2\lvert g\rvert.$$

This is an avoided crossing for real nonzero $g$. This first-order amplitude model is not the full real-coordinate oscillator response, which also contains the negative-frequency partners; its positive-energy interpretation requires the lower branch to remain positive.

Character exchange

Far to one side of the crossing, the upper branch may resemble coordinate 1 and the lower branch coordinate 2. Far to the other side, their characters exchange.

A reliable mode label should therefore track:

• frequency;
• eigenvector overlap;
• symmetry;
• source and detector weight;
• continuity across parameters.

Sorting only by energy can mislabel branches near hybridization.

Dark and bright combinations

For two identical components, a uniform source may couple only to

$$Q_+ = \frac{X_1+X_2}{\sqrt2},$$

while

$$Q_- = \frac{X_1-X_2}{\sqrt2}$$

is dark in that channel.

The relative mode still exists. A component-selective or antisymmetric source is required to reveal it.

Coupling to a continuum

If one coordinate couples to many excitations, its inverse response acquires a self-energy:

$$\mathcal K^{\mathrm R} = \mathcal K_0^{\mathrm R} - \Sigma^{\mathrm R}(\omega).$$

Rapid frequency dependence or an imaginary part of $\Sigma^{\mathrm R}$ can:

• shift the mode;
• broaden it;
• distort the line shape;
• transfer weight to a continuum;
• split a feature into several resonances;
• remove an isolated pole.

An avoided crossing describes two identifiable modes. A threshold cusp or branch-cut feature is a different analytic structure.

Dispersion and Softness

Acoustic and gapped branches

A gapless branch satisfies

$$\omega(\mathbf q) \to 0$$

as $\mathbf q$ approaches a specified soft momentum. “Acoustic” more specifically describes a sound-like branch, usually linear near zero momentum; a quadratic Goldstone mode is gapless without having a nonzero sound speed.

A gapped branch satisfies

$$\omega(\mathbf q) \to \Delta/\hbar > 0.$$

“Optical” often means that sublattices or components move against one another and that the zone-center frequency is nonzero. It does not mean the mode is necessarily excited by visible light.

Linear and quadratic dispersion

Common long-wavelength forms include

$$\omega = cq +\cdots$$

and

$$\omega = D_sq^2 +\cdots.$$

The exponent reflects symmetry, conservation, canonical structure, dimensionality, and interactions. It should be derived rather than guessed from the word “collective.”

Soft modes

A mode softens when

$$\Omega_\lambda(\mathbf Q) \to 0$$

as a control parameter approaches an instability or phase transition.

Softening can indicate:

• loss of a restoring stiffness;
• onset of spatial order at $\mathbf Q$;
• restoration or breaking of a symmetry;
• structural instability;
• critical slowing down.

The mode may also become strongly damped. At a critical point, a relaxational pole or broad scaling continuum can replace a sharp oscillator.

Roton-like minima

A branch can remain gapped but develop a local minimum at finite momentum. Such a minimum signals favorable correlations or an incipient instability, but its interpretation is model dependent.

The name “roton-like” is useful only when the system, dispersion, and excitation channel are specified.

Damping and Mode Quality

Damped oscillator

Consider

$$\ddot Q + 2\kappa\dot Q + \Omega_0^2Q = F(t).$$

The retarded response is proportional to

$$\chi^{\mathrm R}(\omega) = \frac{1}{ \Omega_0^2 - \omega^2 - 2i\kappa\omega }.$$

The poles are

$$\omega_\pm = -i\kappa \pm \sqrt{ \Omega_0^2-\kappa^2 }.$$

Underdamped regime

For

$$\kappa < \Omega_0,$$

the response oscillates while decaying. A useful mode frequency is

$$\Omega_d = \sqrt{ \Omega_0^2-\kappa^2 }.$$

The quality factor in the weak-damping convention is

$$Q_{\mathrm{qual}} \simeq \frac{\Omega_0}{2\kappa}.$$

Overdamped regime

For

$$\kappa > \Omega_0,$$

both poles lie on the negative imaginary axis. The response relaxes without completing an oscillation.

The collective coordinate remains meaningful, but particle-like language becomes poor.

Critical damping

At

$$\kappa = \Omega_0,$$

the two poles coincide in the elementary model. Real many-body systems can have additional memory kernels and frequency-dependent damping, so the simple threshold need not remain exact.

Sources of damping

Collective modes can decay or broaden through:

• anharmonic mode–mode scattering;
• particle–hole creation;
• quasiparticle collisions;
• disorder and boundary scattering;
• viscosity and thermal conduction;
• radiation into electromagnetic or substrate modes;
• additional decay channels opened by symmetry breaking;
• coupling to an external environment;
• inhomogeneous broadening.

Different mechanisms have different momentum, temperature, and frequency dependence.

Landau damping versus collisions

Landau damping is collisionless transfer of collective motion into resonant microscopic excitations. Collisional damping comes from scattering processes that redistribute occupations.

Both broaden response, but they are physically and parametrically distinct.

Width convention

If a positive-frequency pole is written

$$\omega_\star = \Omega-i\kappa,$$

then the amplitude decays as $e^{-\kappa t}$. A Lorentzian in angular frequency has HWHM $\kappa$ and FWHM $2\kappa$ under the simplest isolated-pole convention.

Lifetime and Spectral Weight explains the conversion to energy width and why population decay additionally requires a model for dephasing.

The upper patterns distinguish the two coupled-oscillator modes and their source selection rules. The lower curves solve $\ddot Q+2\kappa\dot Q+Q=0$ with the same initial displacement and zero velocity. The underdamped, critically damped, and overdamped cases all describe one collective coordinate; only the first oscillates. The critical curve is $(1+t)e^{-t}$, not a pure exponential.

Collective Modes in Exact Spectra

Lehmann representation

For a collective operator $O_{\mathbf q}$ at zero temperature,

$$S_O(\mathbf q,\omega) \propto{} \sum_n \left| \langle n| O_{\mathbf q} |0\rangle \right|^2 \times \delta \left( \omega-\omega_{n0} \right).$$

A collective mode appears when substantial operator weight organizes into a coherent branch or controlled pole.

The exact eigenstates remain the foundation. “Collective mode” is an interpretation of their energies, matrix elements, scaling, and dynamical organization.

Single-mode dominance

If one state exhausts most of the relevant spectral weight,

$$S_O(\mathbf q,\omega) \simeq Z_O(\mathbf q) \delta \left( \omega-\Omega(\mathbf q) \right)$$

in an ideal stable limit.

The single-mode approximation can be powerful, but its quality must be checked against:

• omitted weight;
• multiparticle continua;
• sum rules;
• finite-size level spacing;
• operator dependence.

Feynman-type variational state

A density collective trial state has the schematic form

$$|\mathbf q\rangle \propto \rho_{\mathbf q}|0\rangle.$$

Its variational energy connects static structure to an excitation estimate. This does not prove exact single-mode saturation.

Sum Rules derives the exact moments and Feynman bound. Subtract any elastic ground-state component before using the density-created state as an excitation trial state.

Continuum without a mode

A collective operator can couple mainly to a continuum:

$$S_O = S_{\mathrm{cont}}$$

with no isolated pole. The operator is collective, but the spectrum need not contain a sharp collective mode.

This distinction matters near quantum critical points, in one-dimensional fractionalized systems, and in strongly damped metals.

Collective Modes and Particle–Hole Continua

Coherent superposition

In a fermionic reference state, a density-like collective excitation can have the leading form

$$|\lambda,\mathbf q\rangle \sim{} \sum_{\mathbf k} \phi_{\lambda\mathbf q}(\mathbf k) \times c_{\mathbf k+\mathbf q}^\dagger c_{\mathbf k} |\Psi_0\rangle + \cdots.$$

The coherent superposition across many $\mathbf k$ values distinguishes a collective eigenvector from one selected particle–hole configuration.

Continuum kinematics

The set of individual particle–hole energies

$$\omega = \frac{ \epsilon_{\mathbf k+\mathbf q} - \epsilon_{\mathbf k} }{ \hbar }$$

fills a continuum as $\mathbf k$ varies.

Interactions can pull or organize a pole outside that continuum. Inside it, resonant decay often broadens the mode.

No universal percentage

There is no universal threshold such as “more than half the particles participate.” A mode can be collective because:

• its eigenvector is a coherent superposition;
• its frequency is set by a self-consistent field;
• its residue scales coherently;
• it is governed by a hydrodynamic variable;
• it is protected by symmetry or conservation.

The relevant criterion depends on the problem.

Methods for Finding Collective Modes

Exact response

In principle, collective modes are read from exact retarded correlators or structure factors. In practice, exact spectra are available only in special models or finite numerical systems.

Time-dependent mean field

Linearizing a time-dependent mean-field equation around a stationary saddle produces small-oscillation modes. Examples include:

• time-dependent Hartree or Hartree–Fock;
• Gross–Pitaevskii and Bogoliubov–de Gennes equations;
• time-dependent density-functional approximations;
• linearized Landau kinetic theory.

The mode solution is only as controlled as the background and closure.

Random phase approximation

For the physical density susceptibility defined by $H'=-f\,n$, a repulsive interaction $V$ generates an opposing Hartree source. Linear self-consistency gives

$$\delta n=\chi_0(f-V\delta n),\qquad \chi_{\mathrm{RPA}}=\frac{\chi_0}{1+V\chi_0}.$$

The collective condition is $1+V(\mathbf q)\chi_0(\mathbf q,\omega)=0$. If instead one defines the polarization $\Pi_0=-\chi_0$ as response to an added potential energy $+Un$, the same condition reads $1-V\Pi_0=0$.

For example, take $\chi_0=C/[\Omega_0^2-(\omega+i0^+)^2]$ with $C>0$. Self-consistency gives

$$\chi_{\mathrm{RPA}}= \frac{C}{\Omega_0^2+VC-(\omega+i0^+)^2}.$$

A positive restoring interaction raises the squared frequency by $VC$ and decreases the static susceptibility. This checks both the sign and its physical meaning. The numerator and continuum absorption still control visibility and damping. Random Phase Approximation develops the density-response derivation and its range of validity.

Effective action

For fields $\Phi_a$, expand an effective action about a saddle:

$$S_{\mathrm{eff}} = S_0 + \frac12 \sum_{ab} \delta\Phi_a \mathcal K_{ab} \delta\Phi_b + \cdots.$$

After the correct real-time continuation, zeros of $\det\mathcal K^{\mathrm R}$ give collective poles.

An imaginary-time Hessian alone does not determine retarded damping without analytic continuation.

Hydrodynamics

At long wavelength and low frequency, conservation laws and constitutive relations can determine pole structure without solving microscopic dynamics.

Hydrodynamics predicts the functional form of slow modes. Microscopic theory supplies thermodynamic derivatives and transport coefficients.

Equation-of-motion and moment methods

Commutators generate exact spectral moments:

$$\langle [O^\dagger,[H,O]] \rangle.$$

Moment constraints can bound or estimate collective energies, but a finite number of moments cannot by itself distinguish a narrow pole from every possible continuum.

Numerical spectra

Exact diagonalization, tensor networks, quantum Monte Carlo with analytic continuation, and real-time simulation can reveal candidate modes.

Necessary controls include:

• system-size scaling;
• frequency broadening or time-window scaling;
• momentum resolution;
• operator normalization;
• boundary conditions;
• analytic-continuation uncertainty;
• sum-rule closure;
• comparison across several channels.

Observing Collective Modes

Scattering probes

Neutron, x-ray, electron-energy-loss, and Bragg scattering access momentum- and frequency-resolved correlations.

For an equilibrium scattering channel at $\omega>0$, the intensity is related to the unsymmetrized correlation spectrum. With the physical susceptibility used here, a schematic form is

$$I(\mathbf q,\omega)\ \propto\ \left[ \frac{\mathbf f^\dagger\chi''(\mathbf q,\omega)\mathbf f} {1-e^{-\beta\hbar\omega}} \right]*\mathcal R+\text{background}, \qquad \chi''=\frac{\chi^{\mathrm R}-\chi^{\mathrm R\dagger}}{2i}.$$

The omitted proportionality includes the chosen Fourier normalization, $\hbar$, and cross-section factors. $\mathbf f$ contains probe vertices and $\mathcal R$ is the resolution kernel. The absorptive matrix is Hermitian; for a passive equilibrium system its positive-frequency quadratic form is nonnegative. Using $-\operatorname{Im}\chi^{\mathrm R}$ here would reverse the sign for this source convention. Fluctuation–Dissipation Theorem derives the thermal factor and treats zero-frequency elastic weight separately.

Optical and Raman probes

Photons carry little crystal momentum in ordinary optical geometries, so these probes emphasize near-zone-center modes and symmetry-selected channels.

A mode can be Raman active, infrared active, both, or neither depending on symmetry and effective charge.

Pump–probe dynamics

A short perturbation can launch coherent oscillations:

$$\delta O(t) \sim A e^{-\kappa t} \cos \left( \Omega t+\varphi \right).$$

The observed frequency and damping can identify a mode, but nonlinear driving, heating, inhomogeneity, and multiple beating frequencies must be controlled.

Transport and thermodynamics

Collective modes affect:

• thermal conductivity;
• sound attenuation;
• optical conductivity;
• heat capacity;
• drag and viscosity;
• critical relaxation.

These integrated observables constrain a mode description but usually do not determine its complete dispersion and polarization.

To identify a mode, follow its frequency and polarization across momentum or a control parameter, calculate its projection onto the probe, and distinguish intrinsic damping from resolution and finite-size effects. A threshold can imitate a peak. Comparing another operator channel or a time-domain response helps distinguish the alternatives.

Exercises

Exercise 1: In-phase and out-of-phase oscillations

For the coupled-oscillator Lagrangian

$$L ={} \frac{m}{2} \left( \dot x_1^2+\dot x_2^2 \right) - \frac{k_0}{2} \left( x_1^2+x_2^2 \right) - \frac{k_c}{2} \left( x_1-x_2 \right)^2,$$

take $m,k_0>0$ and $k_c\ge0$. Derive the normal coordinates and frequencies. Which mode couples to the source term $+F(x_1+x_2)$ in the Lagrangian?

Solution

Use

$$Q_\pm = \frac{x_1\pm x_2}{\sqrt2}.$$

The inverse transformation is

$$x_1 = \frac{Q_++Q_-}{\sqrt2}, \qquad x_2 = \frac{Q_+-Q_-}{\sqrt2}.$$

Then

$$x_1^2+x_2^2 = Q_+^2+Q_-^2$$

and

$$\left( x_1-x_2 \right)^2 = 2Q_-^2.$$

Therefore

$$L ={} \frac{m}{2} \left( \dot Q_+^2+\dot Q_-^2 \right) - \frac{k_0}{2}Q_+^2 - \frac{k_0+2k_c}{2}Q_-^2.$$

The frequencies are

$$\omega_+ = \sqrt{\frac{k_0}{m}}, \qquad \omega_- = \sqrt{ \frac{k_0+2k_c}{m} }.$$

The uniform force couples through

$$F(x_1+x_2) = \sqrt2FQ_+.$$

It excites only the in-phase mode. The out-of-phase mode exists but is dark to a perfectly uniform symmetric source.

Exercise 2: A denominator zero can be dark

Suppose the response in the $Q_\pm$ basis is

$$\chi_\pm^{\mathrm R}(\omega) = \frac{1}{ \Omega_\pm^2 - (\omega+i0^+)^2 }.$$

A source and detector both couple only to $O=X_1+X_2$. Show which pole appears in $\chi_{OO}^{\mathrm R}$.

Solution

Since

$$O = X_1+X_2 = \sqrt2Q_+,$$

the source and detector have no projection onto $Q_-$. Therefore

$$\chi_{OO}^{\mathrm R} = 2\chi_+^{\mathrm R}.$$

The poles at $\omega=\pm\Omega_+$ appear, while the poles at $\pm\Omega_-$ have zero residue in this channel.

The antisymmetric mode is not absent from the system. It requires an antisymmetric source or detector such as $X_1-X_2$.

Exercise 3: Underdamped or overdamped?

For

$$\ddot Q + 2\kappa\dot Q + \Omega_0^2Q = 0,$$

classify the motion for:

1. $\kappa=\Omega_0/10$;
2. $\kappa=\Omega_0$;
3. $\kappa=2\Omega_0$.

Find the poles in each case.

Solution

The poles are

$$\omega_\pm = -i\kappa \pm \sqrt{ \Omega_0^2-\kappa^2 }.$$

For $\kappa=\Omega_0/10$,

$$\omega_\pm = -\frac{i\Omega_0}{10} \pm \frac{\sqrt{99}}{10} \Omega_0.$$

The poles have nonzero real parts and the mode is underdamped.

For $\kappa=\Omega_0$,

$$\omega_+ = \omega_- = -i\Omega_0.$$

This is critical damping in the elementary model. The general homogeneous solution is $(A+Bt)e^{-\Omega_0t}$, so coincident roots do not give two independent pure exponentials.

For $\kappa=2\Omega_0$,

$$\omega_\pm = -2i\Omega_0 \pm i\sqrt3\,\Omega_0.$$

Both poles lie on the negative imaginary axis:

$$\omega_\pm = -i \left( 2\mp\sqrt3 \right) \Omega_0.$$

The response is overdamped and relaxes as a sum of two exponentials.

Exercise 4: Diffusion from conservation

Starting from

$$\partial_t n + \nabla\cdot\mathbf j = 0$$

and

$$\mathbf j = -D\nabla n,$$

derive the dispersion and explain why the mode is collective but not a ballistic quasiparticle.

Solution

Substitution gives

$$\partial_t n = D\nabla^2n.$$

For

$$\delta n = \delta n_0 e^{i(\mathbf q\cdot\mathbf r-\omega t)},$$

one finds

$$-i\omega = -Dq^2,$$

so

$$\omega = -iDq^2.$$

The mode is collective because it describes coherent long-wavelength relaxation of the conserved density field. Its pole has no real oscillation frequency, so there is no ballistic wave packet with a group velocity. Particle-like quasiparticle language is therefore generally inappropriate.

Exercise 5: Derive neutral sound

Use

$$\partial_t\delta n + n_0\nabla\cdot\mathbf v = 0$$

and

$$mn_0\partial_t\mathbf v = -\nabla \left[ \left( \frac{\partial P}{\partial n} \right)_{\!\bar s} \delta n \right]$$

to derive the sound speed.

Solution

Differentiate continuity with respect to time:

$$\partial_t^2\delta n + n_0\nabla\cdot \partial_t\mathbf v = 0.$$

From the Euler equation,

$$\partial_t\mathbf v = - \frac{1}{mn_0} \left( \frac{\partial P}{\partial n} \right)_{\!\bar s} \nabla\delta n.$$

Substitution gives

$$\partial_t^2\delta n - \frac{1}{m} \left( \frac{\partial P}{\partial n} \right)_{\!\bar s} \nabla^2\delta n = 0.$$

Thus

$$\partial_t^2\delta n = c_s^2\nabla^2\delta n$$

with

$$c_s^2 = \frac{1}{m} \left( \frac{\partial P}{\partial n} \right)_{\!\bar s}.$$

Plane waves have

$$\omega_\pm = \pm c_sq.$$

Exercise 6: Participation is basis dependent

In a site basis, let

$$e_j = \frac{1}{\sqrt N}$$

for every $j$. Compute the participation ratio. Then transform to a basis whose first vector is exactly $\mathbf e$. What participation ratio does the same mode have there?

Solution

In the site basis,

$$\sum_j \lvert e_j\rvert^4 = N \left( \frac{1}{N^2} \right) = \frac1N.$$

Therefore

$$P = \frac{1}{1/N} = N.$$

Now choose an orthonormal basis with first basis vector $\mathbf u_1=\mathbf e$. The same mode has components

$$\left( 1,0,\ldots,0 \right).$$

In that basis,

$$P = 1.$$

The physics did not change. The example proves that participation ratio is basis dependent and must be interpreted in a physically motivated local or component basis.

Exercise 7: Avoided crossing and character exchange

For

$$\mathcal K = \begin{pmatrix} \omega-\omega_1 & -g \\ -g & \omega-\omega_2 \end{pmatrix},$$

take real $g$ and derive the two frequencies. At $\omega_1=\omega_2=\omega_0>|g|$, find normalized eigenvectors.

Solution

The determinant condition is

$$\left( \omega-\omega_1 \right) \left( \omega-\omega_2 \right) - g^2 = 0.$$

Solving the quadratic gives

$$\omega_\pm ={} \frac{\omega_1+\omega_2}{2} \pm \sqrt{ \left( \frac{\omega_1-\omega_2}{2} \right)^2 +g^2 }.$$

At degeneracy,

$$\omega_\pm = \omega_0 \pm \lvert g\rvert.$$

For $g>0$, the normalized eigenvectors can be chosen as

$$\mathbf e_+ = \frac{1}{\sqrt2} \begin{pmatrix} 1\\1 \end{pmatrix}, \qquad \mathbf e_- = \frac{1}{\sqrt2} \begin{pmatrix} 1\\-1 \end{pmatrix}.$$

The splitting is $2g$. Away from the crossing, the eigenvectors rotate continuously between the original coordinate characters.

Exercise 8: Test a proposed collective mode

A numerical calculation on one finite lattice shows a broad maximum in a density structure factor at one momentum. The plot uses an artificial Lorentzian kernel with HWHM $\eta=0.15J$. The fitted FWHM is $0.18J$, and the feature is labeled a long-lived collective mode.

List the missing checks.

Solution

The kernel FWHM is $2\eta=0.30J$, larger than the reported $0.18J$. For an isolated positive line, this is inconsistent with the stated convolution model. The width convention and fitting procedure need correction before any intrinsic lifetime can be inferred. A broad maximum at one momentum can be:

• one broadened discrete line;
• several unresolved levels;
• a continuum threshold;
• a density-of-states enhancement;
• the finite-size precursor of a resonance;
• a genuine collective branch.

A responsible analysis should:

1. repeat the calculation for several $\eta$ values;
2. inspect raw finite-size energies and matrix elements;
3. scale system size and boundary conditions;
4. trace the feature across momentum;
5. compare with the particle–hole or multiparticle continuum;
6. identify its polarization or dominant operator eigenchannel;
7. check spectral moments and total weight;
8. compare density response with another appropriate channel;
9. test a linearized effective or hydrodynamic prediction;
10. show that an intrinsic width remains after resolution and finite-size effects are controlled.

Until then, “candidate density-response feature” is more accurate than “long-lived collective mode.”

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