Phonons as Many-Body Excitations

A phonon is one quantum of a normal mode of a many-body vibrational system. In a crystal, the microscopic coordinates are atomic displacements, but the harmonic eigencoordinates are extended patterns labeled by crystal momentum, branch, and polarization. Quantization promotes each pattern to an oscillator with creation and annihilation operators.

The conceptual chain is

$$\begin{gathered} \text{coupled atomic displacements} \\ \downarrow \\ \text{normal modes}, \\ \downarrow \\ \text{quantized oscillators}, \\ \downarrow \\ \text{one phonon}. \end{gathered}$$

This definition separates three ideas that are often blurred:

• a normal mode is an available harmonic degree of freedom;
• a phonon is a quantum occupying that mode;
• a classical lattice wave is usually represented by a coherent state containing an indefinite, often large, number of phonons.

Phonons are bosonic quasiparticles even when the atoms or ions are not bosons. Their Bose algebra follows from quantizing harmonic normal coordinates, not from exchanging the microscopic constituents. In the ideal harmonic Hamiltonian they are exact, infinitely long-lived excitations. Anharmonicity, disorder, boundaries, electrons, spins, and other modes can shift, scatter, or destroy that idealization.

Scope and Canonical Ownership

This page owns the universal many-body construction:

• the harmonic expansion around a stable equilibrium;
• mass weighting and the dynamical-matrix eigenproblem;
• branch counting, polarization, and acoustic versus optical motion;
• canonical quantization of lattice normal modes;
• phonon creation, annihilation, number states, and coherent waves;
• zero-point and thermal displacement correlations;
• crystal momentum and the distinction from mechanical momentum;
• the one-phonon content of scattering spectra;
• the first controlled effects of anharmonicity;
• the relation between lattice phonons and oscillator modes in quantum field theory.

Neighboring pages retain narrower ownership:

• Harmonic Chain Model Dossier owns the convention-complete periodic monatomic baseline, center-of-mass audit, finite-chain examples, and matrix benchmark.
• Coupled Oscillators: First Encounter owns the elementary two-oscillator change of coordinates.
• Bosonic Operators in Many-Body Models owns abstract bosonic Fock-space algebra.
• Collective Modes owns the cross-system language of collective coordinates, response eigenchannels, and hybridization.
• Quasiparticles Overview owns residue, lifetime, propagation, and quasiparticle breakdown.
• Goldstone Modes in Many-Body Systems owns symmetry-enforced acoustic modes and spacetime-symmetry counting.
• Structure Factors owns scattering normalization, detailed balance, and exact spectral representations.
• Spectral Functions owns interacting peak, continuum, linewidth, and sum-rule conventions.
• Bogoliubov Theory owns paraunitary diagonalization and neutral-superfluid phonons.
• Polarons Preview owns the distinct problem of a mobile carrier or impurity dressed by phonons or other host excitations.
• Harmonic Oscillator to Fields owns the broader free-field mode expansion.

Detailed crystallography, empirical and first-principles force constants, polar-crystal nonanalyticities, material dispersions, vibrational thermodynamics, and probe-specific spectra belong in Phonons in Quantum Matter. Here the crystal is a clean setting in which the oscillator quantization and many-body excitation picture can be derived once.

Geometric and Fourier Conventions

Cells, basis atoms, and displacements

Let $\mathbf R_l$ denote the Bravais-lattice vector of cell $l$, and let $\boldsymbol\tau_\kappa$ locate basis atom $\kappa$ inside the cell. Its equilibrium position is

$$\mathbf r^{\,0}_{l\kappa} := \mathbf R_l+\boldsymbol\tau_\kappa.$$

Write the instantaneous position as

$$\hat{\mathbf r}_{l\kappa} := \mathbf r^{\,0}_{l\kappa} + \hat{\mathbf u}_{l\kappa},$$

where $\hat{\mathbf u}_{l\kappa}$ is the displacement operator and $M_\kappa$ is the mass. Cartesian components are labeled by $\alpha,\beta=1,\ldots,d$.

The system contains:

$$\begin{gathered} N_{\mathrm c} := \text{number of cells}, \\ r := \text{basis atoms per cell}, \\ d r N_{\mathrm c} := \text{displacement coordinates}. \end{gathered}$$

Periodic boundary conditions give $N_{\mathrm c}$ allowed crystal momenta $\mathbf q$ in the first Brillouin zone.

A cell-phase convention

This page uses the Fourier transform

$$u_{l\kappa\alpha} := \frac{1}{\sqrt{N_{\mathrm c}M_\kappa}} \sum_{\mathbf q} e^{i\mathbf q\cdot\mathbf R_l} U_{\kappa\alpha}(\mathbf q).$$

The basis position $\boldsymbol\tau_\kappa$ is not included in the phase. Including it is an equally valid convention, but it moves phase factors between polarization vectors and scattering form factors.

Because a classical displacement is real,

$$U_{\kappa\alpha}(-\mathbf q) := U_{\kappa\alpha}(\mathbf q)^*.$$

The quantum operator will be Hermitian for the analogous reason: its positive- and negative-frequency pieces occur together.

Why mass weighting is useful

The factor $M_\kappa^{-1/2}$ makes the kinetic energy Euclidean in the Fourier amplitudes. It also turns the normal-mode equation into an ordinary Hermitian eigenproblem. Without mass weighting one instead solves a generalized eigenproblem with the mass matrix on the right-hand side.

Both approaches give the same frequencies and physical displacements. Mixing their normalization formulas does not.

Harmonic Expansion About Equilibrium

Force constants

Expand the Born–Oppenheimer potential energy in displacements:

$$\begin{aligned} V :={}& V_0 + \sum_{l\kappa\alpha} F_{l\kappa\alpha} u_{l\kappa\alpha} \\ & + \frac12 \sum_{\substack{l\kappa\alpha\\l'\kappa'\beta}} u_{l\kappa\alpha} \Phi_{l\kappa\alpha,l'\kappa'\beta} u_{l'\kappa'\beta} + V_{\mathrm{anh}}. \end{aligned}$$

At a stationary equilibrium,

$$F_{l\kappa\alpha}=0.$$

The harmonic force-constant matrix is the Hessian

$$\Phi_{l\kappa\alpha,l'\kappa'\beta} := \left. \frac{\partial^2 V} {\partial u_{l\kappa\alpha} \partial u_{l'\kappa'\beta}} \right|_{\mathbf u=0}.$$

Translation symmetry implies dependence only on the cell separation:

$$\Phi_{l\kappa\alpha,l'\kappa'\beta} := \Phi_{\kappa\alpha,\kappa'\beta} (\mathbf R_{l'}-\mathbf R_l).$$

Harmonic and anharmonic pieces

The vibrational Hamiltonian is organized as

$$\hat H_{\mathrm{vib}} := \hat H_2+\hat H_3+\hat H_4+\cdots,$$

where

$$\begin{aligned} \hat H_2 :={}& \sum_{l\kappa\alpha} \frac{\hat p_{l\kappa\alpha}^2}{2M_\kappa} \\ & + \frac12 \sum_{\substack{l\kappa\alpha\\l'\kappa'\beta}} \hat u_{l\kappa\alpha} \Phi_{l\kappa\alpha,l'\kappa'\beta} \hat u_{l'\kappa'\beta}. \end{aligned}$$

The cubic and quartic terms contain third- and fourth-order force constants. They are interactions among harmonic phonons, not corrections to the canonical commutator.

Stability

At a stable equilibrium, the quadratic potential must be nonnegative:

$$\sum_{ij} x_i\Phi_{ij}x_j \ge 0.$$

Exact translations of an isolated crystal cost no energy and generate zero eigenvalues. All other harmonic eigenvalues should be positive for a strict local minimum.

An imaginary reported frequency means

$$\omega_{\mathbf q s}^2<0.$$

It signals that the chosen reference structure is unstable within the harmonic expansion. It is not a phonon with negative energy and not an oscillator to quantize with the usual Fock construction.

The Acoustic Sum Rule

Uniform translation

If every atom is shifted by the same constant vector, the potential energy of an isolated translation-invariant crystal is unchanged. Therefore

$$\sum_{l'\kappa'} \Phi_{l\kappa\alpha,l'\kappa'\beta} =0$$

for every $l,\kappa,\alpha,\beta$.

This is the acoustic sum rule.

Consequence at zero momentum

For a uniform displacement in direction $\beta$, the mass-weighted polarization is proportional to

$$e_{\kappa\alpha}^{(\beta)}(\mathbf0) \propto \sqrt{M_\kappa}\, \delta_{\alpha\beta}.$$

The acoustic sum rule makes it a zero-frequency eigenvector:

$$\omega_{\mathbf0,\mathrm{ac}}=0.$$

Numerical force constants often violate the sum rule slightly because of finite convergence tolerances. Enforcing it can remove an unphysical small gap, but it must not be used to hide a genuine pinning potential or an unstable structure.

The Dynamical Matrix

Definition

In the cell-phase convention, define

$$\begin{aligned} D_{\kappa\alpha,\kappa'\beta}(\mathbf q) :={}& \frac{1}{\sqrt{M_\kappa M_{\kappa'}}} \\ & \times \sum_{\mathbf R} \Phi_{\kappa\alpha,\kappa'\beta}(\mathbf R) e^{i\mathbf q\cdot\mathbf R}. \end{aligned}$$

For real conservative force constants,

$$\begin{aligned} D(\mathbf q)^\dagger &=D(\mathbf q), \\ D(-\mathbf q) &=D(\mathbf q)^*. \end{aligned}$$

Thus $D(\mathbf q)$ has real eigenvalues and an orthonormal eigenbasis whenever ordinary diagonalization applies.

Eigenproblem

The normal-mode equation is

$$\sum_{\kappa'\beta} D_{\kappa\alpha,\kappa'\beta}(\mathbf q) e_{\kappa'\beta}^{\,s}(\mathbf q) := \omega_{\mathbf q s}^2 e_{\kappa\alpha}^{\,s}(\mathbf q).$$

The branch label $s=1,\ldots,dr$ enumerates eigenvectors at fixed $\mathbf q$. We choose

$$\sum_{\kappa\alpha} e_{\kappa\alpha}^{\,s}(\mathbf q)^* e_{\kappa\alpha}^{\,s'}(\mathbf q) := \delta_{ss'}.$$

Completeness gives

$$\sum_s e_{\kappa\alpha}^{\,s}(\mathbf q) e_{\kappa'\beta}^{\,s}(\mathbf q)^* := \delta_{\kappa\kappa'} \delta_{\alpha\beta}.$$

For a time-reversal-invariant nonmagnetic lattice one may choose, away from gauge obstructions and degeneracies,

$$\mathbf e^{\,s}(-\mathbf q) := \mathbf e^{\,s}(\mathbf q)^*.$$

Polarization gauge

Each nondegenerate eigenvector can be rephased:

$$\mathbf e^{\,s}(\mathbf q) \longrightarrow e^{i\theta_s(\mathbf q)} \mathbf e^{\,s}(\mathbf q).$$

Inside a degenerate subspace, any unitary rotation is allowed. Individual components and phases are therefore not observables by themselves. Frequencies, projectors onto isolated mode subspaces, and consistently transformed matrix elements are gauge invariant.

This becomes important near crossings. A branch should be tracked by symmetry, continuity, and polarization character, not merely by sorting frequencies at each momentum.

From Normal Coordinates to Independent Oscillators

Mode amplitudes

Expand the mass-weighted displacement in polarization vectors:

$$U_{\kappa\alpha}(\mathbf q) := \sum_s e_{\kappa\alpha}^{\,s}(\mathbf q) Q_{\mathbf q s}.$$

Define conjugate normal momenta $P_{\mathbf q s}$ with the corresponding unitary transformation. The reality constraints are

$$\begin{aligned} Q_{-\mathbf q s} &=Q_{\mathbf q s}^*, \\ P_{-\mathbf q s} &=P_{\mathbf q s}^*. \end{aligned}$$

Diagonal harmonic Hamiltonian

The quadratic Hamiltonian becomes

$$H_2 := \frac12 \sum_{\mathbf q s} \left[ P_{\mathbf q s} P_{-\mathbf q s} + \omega_{\mathbf q s}^2 Q_{\mathbf q s} Q_{-\mathbf q s} \right].$$

This form is a collection of independent real oscillators even though traveling-wave coordinates are complex. One can instead use real standing-wave coordinates built from each pair $\{\mathbf q,-\mathbf q\}$.

No extra particles have been inserted

Diagonalization changes coordinates; it does not add degrees of freedom. The count remains

$$N_{\mathrm{modes}} := d r N_{\mathrm c}.$$

At each allowed $\mathbf q$ there are $dr$ branches. Quantization then gives arbitrarily many occupation states for each oscillator, but it does not create extra oscillator species.

Coupled microscopic displacements are reorganized into momentum- and branch-labeled normal modes. Each mode is then quantized as an oscillator; the raising operator adds one phonon without adding an atom.

Canonical Quantization

Microscopic canonical algebra

Promote displacements and momenta to operators satisfying

$$\begin{aligned} [ \hat u_{l\kappa\alpha}, \hat p_{l'\kappa'\beta} ] &= i\hbar \delta_{ll'} \delta_{\kappa\kappa'} \delta_{\alpha\beta}, \\ [ \hat u_{l\kappa\alpha}, \hat u_{l'\kappa'\beta} ] &=0, \\ [ \hat p_{l\kappa\alpha}, \hat p_{l'\kappa'\beta} ] &=0. \end{aligned}$$

Normal-mode quantization is a canonical change of basis preserving these relations.

Phonon operators

Introduce

$$[ \hat b_{\mathbf q s}, \hat b_{\mathbf q' s'}^\dagger ] := \delta_{\mathbf q\mathbf q'} \delta_{ss'},$$

with all annihilator–annihilator and creator–creator commutators zero.

The displacement operator is

$$\begin{aligned} \hat u_{l\kappa\alpha} :={}& \frac{1}{\sqrt{N_{\mathrm c}}} \sum_{\mathbf q s} \sqrt{ \frac{\hbar} {2M_\kappa\omega_{\mathbf q s}} } \\ & \times \left[ e_{\kappa\alpha}^{\,s}(\mathbf q) e^{i\mathbf q\cdot\mathbf R_l} \hat b_{\mathbf q s} \right. \\ & \left. \qquad + e_{\kappa\alpha}^{\,s}(\mathbf q)^* e^{-i\mathbf q\cdot\mathbf R_l} \hat b_{\mathbf q s}^\dagger \right]. \end{aligned}$$

The conjugate momentum is

$$\begin{aligned} \hat p_{l\kappa\alpha} :={}& \frac{-i}{\sqrt{N_{\mathrm c}}} \sum_{\mathbf q s} \sqrt{ \frac{\hbar M_\kappa\omega_{\mathbf q s}}{2} } \\ & \times \left[ e_{\kappa\alpha}^{\,s}(\mathbf q) e^{i\mathbf q\cdot\mathbf R_l} \hat b_{\mathbf q s} \right. \\ & \left. \qquad - e_{\kappa\alpha}^{\,s}(\mathbf q)^* e^{-i\mathbf q\cdot\mathbf R_l} \hat b_{\mathbf q s}^\dagger \right]. \end{aligned}$$

The two terms in each bracket make the microscopic operators Hermitian.

The zero-mode caveat

The formulas contain $1/\sqrt{\omega_{\mathbf q s}}$ and therefore do not apply directly to an exact translation mode with $\omega=0$. A finite isolated crystal’s center of mass is a free coordinate, not a harmonic oscillator.

Common controlled choices are:

• fix the center of mass;
• add an infinitesimal pinning potential and remove it after computing relative observables;
• work at nonzero momentum before taking the thermodynamic limit.

Simply dividing by $\sqrt{\omega}$ at the exact zero mode is not a valid quantization.

Diagonal Hamiltonian

For every positive-frequency mode,

$$\hat H_2 := E_{\mathrm{eq}} + \sum_{\mathbf q s} \hbar\omega_{\mathbf q s} \left( \hat b_{\mathbf q s}^\dagger \hat b_{\mathbf q s} + \frac12 \right).$$

The harmonic vacuum obeys

$$\hat b_{\mathbf q s}|0\rangle=0.$$

A normalized occupation state is

$$|\{n_{\mathbf q s}\}\rangle := \prod_{\mathbf q s} \frac{ (\hat b_{\mathbf q s}^\dagger)^{n_{\mathbf q s}} } {\sqrt{n_{\mathbf q s}!}} |0\rangle.$$

Its vibrational energy is

$$E_{\mathrm{vib}} := \sum_{\mathbf q s} \hbar\omega_{\mathbf q s} \left( n_{\mathbf q s}+\frac12 \right).$$

What One Phonon Means

One quantum in one mode

The one-phonon state is

$$|1_{\mathbf q s}\rangle := \hat b_{\mathbf q s}^\dagger|0\rangle.$$

Relative to the harmonic vacuum, it carries energy

$$\Delta E=\hbar\omega_{\mathbf q s}$$

and crystal momentum

$$\Delta\mathbf K = \hbar\mathbf q \quad \text{modulo } \hbar\mathbf G,$$

where $\mathbf G$ is a reciprocal-lattice vector.

It does not identify one atom as excited. The mode polarization distributes the displacement matrix element over every cell:

$$\begin{aligned} \langle0| \hat u_{l\kappa\alpha} |1_{\mathbf q s}\rangle :={}& \sqrt{ \frac{\hbar} {2N_{\mathrm c}M_\kappa\omega_{\mathbf q s}} } \\ & \times e_{\kappa\alpha}^{\,s}(\mathbf q) e^{i\mathbf q\cdot\mathbf R_l}. \end{aligned}$$

This extended matrix element is the precise sense in which a harmonic phonon is collective.

Mean displacement is not the diagnostic

In a number state,

$$\langle n_{\mathbf q s}| \hat u_{l\kappa\alpha} |n_{\mathbf q s}\rangle =0.$$

The absence of a mean displacement does not mean the lattice is motionless. Quadratic fluctuations and transition matrix elements are nonzero.

For one oscillator coordinate

$$\hat Q := \sqrt{\frac{\hbar}{2\omega}} \left( \hat b+\hat b^\dagger \right),$$

the variance is

$$\langle n|\hat Q^2|n\rangle := \frac{\hbar}{\omega} \left( n+\frac12 \right).$$

A phonon is not a tiny classical wave

A number state has definite occupation and undefined oscillator phase. A classical-looking wave requires phase coherence between number sectors. That is supplied by a coherent state, not by a single phonon.

Coherent Lattice Waves

Coherent state

For a chosen mode,

$$\hat b_{\mathbf q s} |\beta_{\mathbf q s}\rangle := \beta_{\mathbf q s} |\beta_{\mathbf q s}\rangle.$$

Its mean occupation and number variance are

$$\begin{aligned} \langle \hat n_{\mathbf q s}\rangle &= |\beta_{\mathbf q s}|^2, \\ (\Delta n_{\mathbf q s})^2 &= |\beta_{\mathbf q s}|^2. \end{aligned}$$

The expected displacement contains

$$\begin{aligned} \langle\hat u_{l\kappa\alpha}(t)\rangle \propto{}& e_{\kappa\alpha}^{\,s}(\mathbf q) \beta_{\mathbf q s} \\ & \times e^{i( \mathbf q\cdot\mathbf R_l - \omega_{\mathbf q s}t )} + \text{c.c.} \end{aligned}$$

This is the corresponding classical normal-mode wave.

Standing versus traveling waves

A real standing wave can be formed from coherent amplitudes at $\mathbf q$ and $-\mathbf q$. A traveling wave can be represented by a phase-correlated choice whose mean displacement moves with phase velocity.

The distinction is state preparation, not a difference in the underlying normal-mode spectrum.

Coherent phonons in experiments

Ultrafast optical excitation can prepare a macroscopic phase-coherent lattice oscillation. Calling it a “coherent phonon” refers to the state of a vibrational mode. It does not mean that each constituent phonon is a different quasiparticle species.

A Monatomic Chain Benchmark

Hamiltonian

Consider $N$ equal masses $M$ on a periodic one-dimensional chain with spacing $a$:

$$H := \sum_{n=0}^{N-1} \left[ \frac{p_n^2}{2M} + \frac{K}{2} (u_{n+1}-u_n)^2 \right].$$

The allowed momenta are

$$q_m := \frac{2\pi m}{Na},$$

chosen modulo $2\pi/a$.

Classical mode equation

For

$$u_n(t) := u_q e^{i(qna-\omega t)},$$

the equation of motion gives

$$-M\omega^2u_q := K \left( e^{iqa}+e^{-iqa}-2 \right) u_q.$$

Therefore

$$\omega(q) := 2\sqrt{\frac{K}{M}} \left| \sin\frac{qa}{2} \right|.$$

The frequency is periodic in reciprocal space:

$$\omega\left(q+\frac{2\pi}{a}\right) := \omega(q).$$

Long-wavelength limit

For $|q|a\ll1$,

$$\omega(q) \approx c|q|,$$

with sound speed

$$c:=a\sqrt{\frac{K}{M}}.$$

The $q=0$ mode is uniform translation. The zone-boundary mode has neighboring atoms moving in opposite phase and frequency

$$\omega\left(\frac{\pi}{a}\right) := 2\sqrt{\frac{K}{M}}.$$

Quantized chain

For nonzero-frequency modes,

$$\hat H_2 := \sum_q \hbar\omega(q) \left( \hat b_q^\dagger\hat b_q+\frac12 \right).$$

One application of $\hat b_q^\dagger$ adds $\hbar\omega(q)$ of vibrational energy and crystal momentum $\hbar q$ modulo $2\pi\hbar/a$.

This simple chain is a benchmark, not a complete theory of a material. Real crystals require multiple polarizations, basis atoms, longer-range and tensor force constants, electronic screening, and three-dimensional symmetry.

Acoustic and Optical Branches

Branch counting

With $r$ atoms per primitive cell in $d$ dimensions, there are

$$d r$$

branches at each $\mathbf q$.

For an ordinary unpinned crystal near $\mathbf q=0$, these are commonly organized as:

$$\begin{aligned} d &\text{ acoustic branches}, \\ d(r-1) &\text{ optical branches}. \end{aligned}$$

The labels describe long-wavelength motion, not the measurement method.

Acoustic motion

At small momentum, basis atoms in an acoustic mode move approximately together. In a stable three-dimensional bulk crystal,

$$\omega_{\mathbf q s} \approx c_s(\widehat{\mathbf q})|\mathbf q|.$$

The sound speed depends on propagation direction and polarization. In an isotropic medium there is one longitudinal and two transverse acoustic branches.

Optical motion

In an optical mode, basis atoms move relative to one another inside the unit cell. Such a branch generally has

$$\omega_{\mathbf0 s}>0.$$

The word “optical” arose because some zone-center relative motions couple to electromagnetic radiation. An optical phonon need not be optically active; symmetry and effective charges decide that.

A diatomic-chain illustration

For alternating masses $M_1$ and $M_2$ connected by equal nearest-neighbor springs $K$, with primitive-cell length $a$, the two branches satisfy

$$\begin{aligned} A &:= \frac{1}{M_1}+\frac{1}{M_2}, \\ B(q) &:= \frac{4}{M_1M_2} \sin^2\frac{qa}{2} , \\ \omega_\pm^2(q) &:= K \left[ A \pm \sqrt{ A^2-B(q) } \right]. \end{aligned}$$

At $q=0$,

$$\begin{aligned} \omega_-(0)&=0, \\ \omega_+^2(0) &= 2K \left( \frac{1}{M_1}+\frac{1}{M_2} \right). \end{aligned}$$

The lower branch is acoustic and the upper branch is optical. This model illustrates branch splitting; detailed lattice dynamics and material interpretation require the full crystal symmetry and interaction range.

Longitudinal and transverse are conditional labels

A polarization is longitudinal when

$$\mathbf e_{\mathbf q s} \parallel \mathbf q,$$

and transverse when

$$\mathbf e_{\mathbf q s} \perp \mathbf q.$$

In anisotropic crystals and along generic directions, modes can be mixed rather than purely longitudinal or transverse. At degeneracies, the basis within the degenerate polarization subspace is not unique.

Goldstone Interpretation and Its Limits

Crystal acoustic phonons

In an ideal self-organized continuum crystal, continuous translations are spontaneously broken. The displacement field is the corresponding long-wavelength coordinate, and acoustic phonons are spacetime Goldstone modes.

Broken rotations do not normally add independent acoustic branches because a slowly varying local rotation is encoded in spatial derivatives of the displacement field.

A lattice model is not the same symmetry problem

If particles live on an externally imposed lattice, continuous translations are already absent from the Hamiltonian. A gapless mode must then be justified by the actual symmetry, conservation laws, or tuning of that model.

Likewise, a substrate or pinning potential can give a translational mode a nonzero gap.

Optical modes are not required Goldstone modes

Relative motion inside a unit cell is not a uniform translation of the entire solid. Optical branches can soften at structural transitions, but their generic finite zone-center frequency does not contradict Goldstone reasoning.

The symmetry theorem and its spacetime caveats belong to Goldstone Modes in Many-Body Systems.

Crystal Momentum Is Not Mechanical Momentum

Translation eigenvalue

Under translation by a Bravais vector $\mathbf R$,

$$\hat b_{\mathbf q s}^\dagger \longrightarrow e^{i\mathbf q\cdot\mathbf R} \hat b_{\mathbf q s}^\dagger.$$

Thus $\mathbf q$ labels the representation of discrete translation symmetry. It is defined modulo a reciprocal vector:

$$\mathbf q\sim\mathbf q+\mathbf G.$$

Why crystal momentum is not generally center-of-mass momentum

The crystal background can exchange reciprocal-lattice momentum with an excitation. A phonon’s quasimomentum is conserved modulo $\mathbf G$, while the mechanical momentum of atoms, supports, and external fields must be tracked for a full momentum balance.

It is therefore unsafe to assign each phonon a literal atom-like momentum $\hbar\mathbf q$ in every mechanical calculation.

Normal and umklapp kinematics

A three-phonon process can obey

$$\mathbf q_1+\mathbf q_2 := \mathbf q_3+\mathbf G.$$

If $\mathbf G=\mathbf0$, it is called a normal process. If $\mathbf G\ne\mathbf0$, it is called an umklapp process after all momenta are represented in the first Brillouin zone.

Energy conservation separately requires, for a coalescence process,

$$\omega_1+\omega_2=\omega_3.$$

Both constraints and the interaction matrix element determine whether a process actually occurs.

Zero-Point Motion

Harmonic vacuum fluctuations

Even in the phonon vacuum,

$$\langle0| \hat u_{l\kappa\alpha}^2 |0\rangle \ne0.$$

For a diagonal Cartesian component,

$$\begin{aligned} \langle0| \hat u_{l\kappa\alpha}^2 |0\rangle :={}& \frac{\hbar} {2N_{\mathrm c}M_\kappa} \\ & \times \sum_{\mathbf q s} \frac{ |e_{\kappa\alpha}^{\,s}(\mathbf q)|^2 } {\omega_{\mathbf q s}}, \end{aligned}$$

with exact zero modes treated separately.

Low-frequency modes contribute strongly because of the factor $1/\omega$. Whether the thermodynamic-limit integral converges depends on dimension, observable, branch dispersion, and pinning.

Ground-state energy

The harmonic zero-point contribution is

$$E_{\mathrm{zp}} := \frac12 \sum_{\mathbf q s} \hbar\omega_{\mathbf q s}.$$

Absolute values depend on the effective model and reference energy, but differences in zero-point energy can affect isotope trends, phase competition, and equilibrium structure.

The harmonic approximation alone does not produce thermal expansion because its frequencies are fixed at the reference geometry. Geometry-dependent frequencies in the quasiharmonic approximation and explicit anharmonicity generate such effects.

Thermal Occupation and Displacement Correlations

Bose occupation

For independent harmonic phonons in equilibrium,

$$\overline n_{\mathbf q s} := \frac{1} {e^{\beta\hbar\omega_{\mathbf q s}}-1}.$$

The phonon chemical potential is ordinarily

$$\mu_{\mathrm{ph}}=0$$

because anharmonic and environmental processes can create and destroy phonons while conserving the microscopic quantities of the full system.

An approximately conserved driven phonon population can sometimes be described by a transient effective chemical potential. That is a nonequilibrium statement, not the equilibrium default.

Thermal covariance

The equal-time displacement covariance is

$$\begin{aligned} & \langle \hat u_{l\kappa\alpha} \hat u_{l'\kappa'\beta} \rangle_\beta \\ &\quad:= \frac{\hbar} {2N_{\mathrm c}\sqrt{M_\kappa M_{\kappa'}}} \sum_{\mathbf q s} \frac{ e_{\kappa\alpha}^{\,s}(\mathbf q) e_{\kappa'\beta}^{\,s}(\mathbf q)^* } {\omega_{\mathbf q s}} \\ &\qquad\quad\times \coth \left( \frac{\beta\hbar\omega_{\mathbf q s}}{2} \right) e^{i\mathbf q\cdot( \mathbf R_l-\mathbf R_{l'} )}. \end{aligned}$$

The identity

$$2\overline n_{\mathbf q s}+1 := \coth \left( \frac{\beta\hbar\omega_{\mathbf q s}}{2} \right)$$

unifies zero-point and thermal fluctuations.

Debye–Waller suppression

Scattering from a fluctuating atom contains a factor schematically of the form

$$e^{-W_\kappa(\mathbf Q)},$$

where, in a Gaussian harmonic state,

$$W_\kappa(\mathbf Q) := \frac12 \left\langle \left( \mathbf Q\cdot \hat{\mathbf u}_{l\kappa} \right)^2 \right\rangle.$$

The same displacement covariance that defines phonon fluctuations therefore suppresses elastic Bragg intensity and weights inelastic channels.

Density of States and Heat Capacity

Vibrational density of states

Define

$$g(\omega) := \sum_{\mathbf q s} \delta( \omega-\omega_{\mathbf q s} ).$$

Its integrated mode count is

$$\int_0^\infty g(\omega)\,d\omega := d r N_{\mathrm c},$$

up to separately treated exact zero coordinates.

Thermal energy

The harmonic vibrational energy is

$$U_{\mathrm{vib}}(T) := \sum_{\mathbf q s} \hbar\omega_{\mathbf q s} \left( \overline n_{\mathbf q s} + \frac12 \right).$$

The constant-volume heat capacity is

$$\begin{aligned} C_V :={}& k_{\mathrm B} \sum_{\mathbf q s} \left( \beta\hbar\omega_{\mathbf q s} \right)^2 \\ & \times \frac{ e^{\beta\hbar\omega_{\mathbf q s}} } { \left( e^{\beta\hbar\omega_{\mathbf q s}}-1 \right)^2 }. \end{aligned}$$

Debye scaling

For linearly dispersing acoustic modes in $d$ dimensions,

$$g(\omega)\propto\omega^{d-1}$$

at low frequency. Consequently,

$$C_V\propto T^d.$$

In three dimensions this gives the Debye law

$$C_V\propto T^3.$$

The Debye cutoff is a mode-counting approximation, not a physical abrupt end shared by every real branch. Full material thermodynamics requires the actual spectrum and anharmonic corrections.

How Phonons Appear in Scattering

Momentum transfer

Let a probe transfer momentum $\mathbf Q$. In a crystal, one-phonon scattering is organized by

$$\mathbf Q := \mathbf G+\mathbf q,$$

where $\mathbf q$ lies in the first Brillouin zone.

The reciprocal vector identifies the Brillouin zone in which the measurement is made, while $\mathbf q$ identifies the reduced phonon momentum.

One-phonon form factor

For a density-like probe, a typical one-phonon amplitude has the structure

$$\begin{aligned} F_s(\mathbf Q) :={}& \sum_\kappa \frac{ f_\kappa(\mathbf Q) e^{-W_\kappa(\mathbf Q)} } {\sqrt{M_\kappa}} \\ & \times e^{i\mathbf Q\cdot\boldsymbol\tau_\kappa} \, \mathbf Q\cdot \mathbf e_\kappa^{\,s}(\mathbf q). \end{aligned}$$

Here $f_\kappa$ represents the probe-dependent atomic scattering amplitude. The dot product supplies a polarization selection rule, and interference among basis atoms can make a real phonon dark at a particular $\mathbf Q$.

Creation and annihilation sides

Suppressing instrument and normalization factors, the harmonic one-phonon contribution is

$$\begin{aligned} S^{(1)}(\mathbf Q,\omega) \propto{}& \sum_s \frac{|F_s(\mathbf Q)|^2} {2\omega_{\mathbf q s}} \\ & \times \left[ (\overline n_{\mathbf q s}+1) \delta( \omega-\omega_{\mathbf q s} ) \right. \\ & \left. \qquad + \overline n_{\mathbf q s} \delta( \omega+\omega_{\mathbf q s} ) \right]. \end{aligned}$$

For the convention in which positive $\omega$ means energy deposited into the sample:

• $\omega=+\omega_{\mathbf q s}$ creates a phonon and carries $\overline n+1$;
• $\omega=-\omega_{\mathbf q s}$ annihilates a thermally occupied phonon and carries $\overline n$.

Their ratio is

$$\frac{ S(\mathbf Q,-\omega) } { S(-\mathbf Q,\omega) } := e^{-\beta\hbar\omega},$$

subject to the precise momentum and operator convention stated by the experiment.

Visibility is not existence

A missing peak at one momentum transfer does not prove that the mode is absent. The polarization factor, basis interference, probe coupling, resolution, and multiphonon background all matter.

Neutron scattering, inelastic x-ray scattering, Raman spectroscopy, infrared absorption, Brillouin scattering, and ultrafast probes sample different momentum, symmetry, and frequency windows.

Propagators and Spectral Peaks

Harmonic retarded propagator

For a dimensionless normal coordinate proportional to

$$\hat X_{\mathbf q s} := \hat b_{\mathbf q s} + \hat b_{-\mathbf q s}^\dagger,$$

one common retarded convention gives

$$D_{0,\mathbf q s}^{R}(\omega) := \frac{ 2\omega_{\mathbf q s} } { (\omega+i0^+)^2 - \omega_{\mathbf q s}^2 }.$$

It has poles at positive and negative frequency. A different normalization of $X$ changes the numerator but not the pole locations.

Interacting propagator

Anharmonicity and coupling to other sectors generate a self-energy:

$$\begin{aligned} D_{\mathbf q s}^{R}(\omega) &:= \frac{ 2\omega_{\mathbf q s} } { \mathcal D_{\mathbf q s}(\omega) }, \\ \mathcal D_{\mathbf q s}(\omega) &:= (\omega+i0^+)^2 - \omega_{\mathbf q s}^2 \\ & \quad - 2\omega_{\mathbf q s} \Pi_{\mathbf q s}^{R}(\omega). \end{aligned}$$

Near an isolated positive-frequency pole, write

$$\omega_\star := \Omega_{\mathbf q s} - i\gamma_{\mathbf q s}.$$

Then:

• $\Omega_{\mathbf q s}$ is the renormalized mode frequency;
• $\gamma_{\mathbf q s}>0$ is the amplitude decay rate in this convention;
• the narrow Lorentzian has half width at half maximum approximately $\gamma_{\mathbf q s}$;
• the amplitude decays as $e^{-\gamma_{\mathbf q s}t}$.

Population and energy decay rates can differ by a factor of two in a simple damped oscillator. Any quoted “lifetime” must state which convention is being used.

Quasiparticle criterion

A sharp phonon quasiparticle requires at least

$$\gamma_{\mathbf q s} \ll \Omega_{\mathbf q s}.$$

Propagation also depends on group velocity

$$\mathbf v_{\mathbf q s} := \nabla_{\mathbf q} \omega_{\mathbf q s}$$

and on whether the corresponding mean free path is long compared with the wavelength and microscopic scales.

Lifetime and Spectral Weight owns the systematic width, decay-rate, residue, and propagation criteria used to decide when the phonon remains well defined.

Anharmonic Phonon Interactions

Cubic and quartic vertices

After substituting the mode expansion, the leading corrections have the schematic form

$$\hat X_j := \hat b_j+\hat b_j^\dagger,$$ $$\hat H_3 := \sum_{123} V^{(3)}_{123} \hat X_1\hat X_2\hat X_3,$$

and

$$\hat H_4 := \sum_{1234} V^{(4)}_{1234} \hat X_1\hat X_2\hat X_3\hat X_4.$$

Compact labels such as $1=(\mathbf q_1,s_1)$ include momentum and branch. Translation symmetry restricts each vertex by crystal-momentum conservation modulo a reciprocal vector.

Phonon number is emergent

The harmonic number operator is

$$\hat N_{\mathrm{ph}} := \sum_{\mathbf q s} \hat b_{\mathbf q s}^\dagger \hat b_{\mathbf q s}.$$

Generally,

$$[ \hat H_3, \hat N_{\mathrm{ph}} ] \ne0.$$

Terms can describe:

• one phonon decaying into two;
• two phonons combining into one;
• three-phonon virtual processes;
• source-like terms if the reference structure was not truly stationary.

Nonconservation of phonon number does not violate energy conservation or microscopic atom-number conservation. Phonon number is an occupation count in an approximate normal-mode basis.

Observable consequences

Anharmonicity can produce:

• temperature-dependent frequency shifts;
• finite linewidths and lifetimes;
• thermal expansion;
• mode mixing and avoided crossings;
• multiphonon continua;
• finite thermal resistance through normal, umklapp, defect, and boundary processes;
• stabilization or destabilization of structures beyond the bare harmonic Hessian.

The harmonic approximation is controlled when typical displacements remain small enough that higher-order terms are perturbative over the observables and timescales of interest.

Soft Modes and Structural Instability

Softening

A mode softens when its renormalized frequency decreases as a parameter is varied:

$$\Omega_{\mathbf q_\star s_\star} \longrightarrow 0.$$

The associated polarization identifies the displacement pattern becoming inexpensive. Condensation of that pattern can lead to a new equilibrium structure.

Harmonic instability versus renormalized stability

A negative eigenvalue of the zero-temperature harmonic Hessian means the chosen reference point is a saddle at that level of approximation. At finite temperature, anharmonic fluctuations can renormalize the free-energy curvature, so the physically stable phase need not be inferred from the bare Hessian alone.

Conversely, a small positive harmonic frequency is not proof of a continuous phase transition. Coupling to strain, other modes, disorder, or higher-order terms can alter the transition.

Do not quantize an imaginary branch as usual

For

$$\omega^2=-\Omega_I^2<0,$$

the quadratic Hamiltonian is an inverted oscillator. Ladder operators built with $\sqrt{\omega}$ do not define a stable Fock vacuum.

The correct next step is to locate a stable configuration or treat the unstable coordinate with an appropriate nonlinear or time-dependent theory.

Phonons Coupled to Other Degrees of Freedom

Generic linear coupling

An electronic density can couple to displacement through

$$\begin{aligned} \hat H_{\mathrm{e-ph}} :={}& \sum_{\mathbf k\mathbf q} \sum_{mn s} g_{mn}^{s}( \mathbf k,\mathbf q ) \\ & \times \hat c_{\mathbf k+\mathbf q,m}^\dagger \hat c_{\mathbf k,n} \left( \hat b_{\mathbf q s} + \hat b_{-\mathbf q s}^\dagger \right). \end{aligned}$$

The same pattern applies to spins, defects, excitons, cavity fields, and order parameters with different operators and selection rules.

Integrating out a harmonic phonon

Formally eliminating a Gaussian phonon coordinate generates a retarded interaction

$$V_{\mathrm{eff}}( \mathbf q,\omega ) \sim |g(\mathbf q)|^2 D^R(\mathbf q,\omega),$$

up to sign and normalization conventions fixed by the action and coupling.

This interaction is frequency dependent. Calling it simply “an attraction” without stating the frequency window, electronic repulsion, screening, and retardation is incomplete.

BCS Mean-Field Theory owns the reduced pairing model. A full electron–phonon theory belongs in Quantum Matter.

Hybrid modes

When a phonon couples strongly to another excitation of the same symmetry, the eigenmodes are mixtures. Near resonance they can exchange polarization character and show avoided crossings.

It can still be useful to say “phonon-like” when the displacement spectral weight dominates, but the exact normal mode belongs to the coupled system rather than to either uncoupled sector alone.

Relation to Quantum Field Theory

The shared oscillator architecture

A free bosonic field is decomposed into independent momentum modes:

$$\hat H_{\mathrm{free}} := \sum_{\mathbf k\lambda} \hbar\omega_{\mathbf k\lambda} \left( \hat a_{\mathbf k\lambda}^\dagger \hat a_{\mathbf k\lambda} + \frac12 \right).$$

The harmonic phonon Hamiltonian has the same algebra:

$$\hat H_2 := \sum_{\mathbf q s} \hbar\omega_{\mathbf q s} \left( \hat b_{\mathbf q s}^\dagger \hat b_{\mathbf q s} + \frac12 \right).$$

This is why phonons are a natural bridge from many-body quantum mechanics to quantum field theory.

Continuum elastic field

At wavelengths much longer than the lattice spacing, acoustic displacements can be represented by a field $\mathbf u(\mathbf x,t)$. A harmonic elastic Lagrangian has the form

$$\mathcal L := \int d^d x \left[ \frac{\rho_m}{2} \dot u_i\dot u_i - \frac12 C_{ijkl} u_{ij}u_{kl} \right],$$

with strain

$$u_{ij} := \frac12 \left( \partial_i u_j+\partial_j u_i \right).$$

Fourier decomposition again produces oscillator modes. Quantizing them gives long-wavelength phonons.

Important differences from fundamental fields

The shared algebra does not erase physical distinctions:

• a phonon is emergent and medium dependent;
• the Brillouin zone supplies a microscopic momentum cutoff;
• several branches and polarizations can occur;
• Lorentz invariance is generally absent;
• the dispersion need not remain linear;
• phonon number is not a fundamental conserved charge;
• anharmonic interactions arise from the medium’s nonlinear energy landscape;
• the description can fail near melting, strong disorder, or wavelengths comparable with microscopic structure.

The lesson is structural: quantizing normal modes produces particles. It is not that every phonon is a fundamental particle.

When the Phonon Picture Works

Harmonic normal modes

The strongest idealization assumes:

1. a well-defined reference configuration;
2. small displacements;
3. a positive semidefinite harmonic Hessian;
4. sufficiently weak coupling among normal modes;
5. observation over times short compared with strong decay or rearrangement.

Then $(\mathbf q,s)$ is a useful mode label and phonon Fock states organize the spectrum.

Renormalized quasiparticles

The bare harmonic approximation may be inaccurate while a renormalized phonon remains sharp. In that case the measured pole frequency and polarization differ from the bare Hessian prediction, but the excitation is still adiabatically connected to a lattice vibrational mode.

Breakdown

The particle-like phonon description becomes unreliable when:

• a peak broadens until no isolated pole remains;
• the mean free path approaches a wavelength or microscopic spacing;
• strong anharmonicity mixes many number sectors nonperturbatively;
• structural rearrangements invalidate the reference lattice;
• disorder produces strongly localized or diffusive vibrational states;
• the mode lies inside a dense continuum and loses identifiable spectral weight.

A vibrational density of states can remain meaningful after a sharp phonon quasiparticle has failed.

A Practical Calculation Workflow

Step 1: Specify the reference structure

State:

• Bravais lattice and basis;
• masses and equilibrium positions;
• boundary conditions;
• whether translations are exact, pinned, or externally imposed;
• the approximation defining the potential-energy surface.

Step 2: Construct and audit force constants

Check:

$$\Phi_{ij}=\Phi_{ji}$$

for conservative real coordinates, translation constraints, point-group constraints, and numerical convergence.

Step 3: Build the dynamical matrix

State the Fourier phase convention and mass weighting. Verify

$$D(\mathbf q)^\dagger=D(\mathbf q)$$

at representative momenta.

Step 4: Diagonalize and track modes

Record:

• $\omega_{\mathbf q s}^2$ rather than only signed plotting conventions;
• normalized polarization vectors;
• degeneracies and symmetry labels;
• acoustic sum-rule residuals;
• continuity of mode projectors across momentum.

Step 5: Quantize only stable modes

Use the same eigenvector and Fourier convention in $\hat{\mathbf u}$, $\hat{\mathbf p}$, scattering amplitudes, and interaction vertices. Treat exact zero coordinates separately.

Step 6: Connect to an observable

Specify the probe operator. A dispersion curve alone does not determine intensity, selection rules, linewidth, or whether a mode is visible.

Step 7: Test the approximation

Compare characteristic linewidths, displacement amplitudes, anharmonic shifts, and experimental resolution with the scales relevant to the claim.

Common Mistakes

Calling a mode a phonon

A mode is an oscillator degree of freedom; a phonon is a quantum of it. The mode exists even in its vacuum state.

Treating one phonon as one displaced atom

One phonon changes an extended normal coordinate. Its local displacement expectation still vanishes in a number state.

Equating a one-phonon state with a classical wave

A classical mean displacement requires phase coherence, usually represented by a coherent state or another superposition of number states.

Double-counting opposite momenta

The displacement field contains both positive- and negative-frequency pieces. One may sum over the full Brillouin zone with independent $\hat b_{\mathbf q s}$ operators or use real standing-wave coordinates. Combining half-zone and full-zone conventions creates factor-of-two errors.

Forgetting mass factors

The polarization normalization and displacement prefactor depend on whether coordinates are mass weighted. A formula copied from a different convention can violate the microscopic canonical commutator.

Reading crystal momentum as literal mechanical momentum

Crystal momentum is defined modulo reciprocal vectors. Mechanical momentum balance includes the lattice, supports, and fields.

Assuming every optical branch couples to light

Optical means relative basis motion near the zone center. Infrared and Raman activity require separate symmetry and coupling conditions.

Treating every acoustic branch as audible

“Acoustic” describes the long-wavelength branch approaching zero frequency. Most of its Brillouin-zone frequencies can be far above the audible range.

Calling an imaginary frequency a negative-energy phonon

An imaginary harmonic frequency signals an unstable reference structure. The usual stable-oscillator Fock basis does not exist there.

Inferring absence from a dark spectrum

A mode can have zero matrix element for one probe geometry. Check polarization, basis interference, symmetry, and resolution.

Assigning a conserved chemical potential

Equilibrium phonon number is ordinarily not conserved. A nonzero effective chemical potential needs a stated nonequilibrium preparation and lifetime hierarchy.

Confusing a bare and a dressed phonon

The harmonic Hessian defines bare modes within an approximation. Experiments observe excitations dressed by anharmonicity and coupling to other sectors.

Exercises

Exercise 1: Chain dispersion and sound speed

For the monatomic chain

$$H := \sum_n \left[ \frac{p_n^2}{2M} + \frac{K}{2} (u_{n+1}-u_n)^2 \right],$$

derive the dispersion and its long-wavelength sound speed. Identify the physical meaning of $q=0$.

Solution

The equation of motion is

$$M\ddot u_n := K( u_{n+1}+u_{n-1}-2u_n ).$$

Insert

$$u_n := u_qe^{i(qna-\omega t)}.$$

Then

$$\begin{aligned} M\omega^2 &= 2K( 1-\cos qa ) \\ &= 4K \sin^2\frac{qa}{2}. \end{aligned}$$

Hence

$$\omega(q) := 2\sqrt{\frac{K}{M}} \left| \sin\frac{qa}{2} \right|.$$

For $|q|a\ll1$,

$$\omega(q) \approx a\sqrt{\frac{K}{M}}\,|q|,$$

so

$$c=a\sqrt{\frac{K}{M}}.$$

At $q=0$, every mass moves by the same amount. Neighbor separations do not change, so the spring energy vanishes. It is the uniform translation coordinate.

Exercise 2: Recover the canonical commutator

For a single atom per cell and one polarization, use

$$\begin{aligned} \hat u_l ={}& \frac{1}{\sqrt N} \sum_q \sqrt{\frac{\hbar}{2M\omega_q}} \\ & \times \left( e^{iqR_l}\hat b_q + e^{-iqR_l}\hat b_q^\dagger \right), \\ \hat p_l ={}& \frac{-i}{\sqrt N} \sum_q \sqrt{\frac{\hbar M\omega_q}{2}} \\ & \times \left( e^{iqR_l}\hat b_q - e^{-iqR_l}\hat b_q^\dagger \right) \end{aligned}$$

to show $[\hat u_l,\hat p_{l'}]=i\hbar\delta_{ll'}$.

Solution

Only annihilator–creator cross terms survive. For each $q$ their sum gives

$$[ \hat u_l,\hat p_{l'} ] := \frac{i\hbar}{N} \sum_q e^{iq(R_l-R_{l'})}.$$

The discrete Fourier completeness relation is

$$\frac1N \sum_q e^{iq(R_l-R_{l'})} := \delta_{ll'}.$$

Therefore

$$[ \hat u_l,\hat p_{l'} ] := i\hbar\delta_{ll'}.$$

The square roots of $M$ and $\omega_q$ cancel. Changing one prefactor without changing the other would fail this test.

Exercise 3: One phonon without a mean wave

For one oscillator mode, compute $\langle1|\hat Q|1\rangle$ and $\langle1|\hat Q^2|1\rangle$ when

$$\hat Q := \sqrt{\frac{\hbar}{2\omega}} ( \hat b+\hat b^\dagger ).$$

Explain why the result does not contradict the physical reality of a one-phonon excitation.

Solution

Because $\hat b$ and $\hat b^\dagger$ change occupation by one,

$$\langle1|\hat Q|1\rangle=0.$$

For the square,

$$\hat Q^2 := \frac{\hbar}{2\omega} \left( \hat b^2 + \hat b^{\dagger2} + \hat b\hat b^\dagger + \hat b^\dagger\hat b \right).$$

The first two terms have zero diagonal matrix element, while

$$\begin{aligned} \langle1| \hat b^\dagger\hat b |1\rangle &=1, \\ \langle1| \hat b\hat b^\dagger |1\rangle &=2. \end{aligned}$$

Thus

$$\langle1|\hat Q^2|1\rangle := \frac{3\hbar}{2\omega}.$$

A number state has no definite oscillator phase, so its mean coordinate vanishes. It is detected through energy, variance, transition matrix elements, and correlations rather than a classical mean wave.

Exercise 4: Acoustic sum rule

Suppose the harmonic potential is unchanged by

$$u_{l\kappa\alpha} \longrightarrow u_{l\kappa\alpha}+a_\alpha.$$

Show that a uniform displacement is a zero eigenvector of the harmonic Hessian and identify the corresponding mass-weighted polarization.

Solution

Differentiate translation invariance once with respect to $a_\beta$ and once with respect to a displacement. At equilibrium this gives

$$\sum_{l'\kappa'} \Phi_{l\kappa\alpha,l'\kappa'\beta} =0.$$

Acting on a uniform physical displacement $u_{l'\kappa'\beta}=a_\beta$ therefore gives zero restoring force.

In mass-weighted coordinates,

$$U_{\kappa\alpha} := \sqrt{M_\kappa}\, a_\alpha.$$

Hence the zero-mode polarization in direction $\beta$ is proportional to

$$e_{\kappa\alpha}^{(\beta)}(\mathbf0) := \frac{ \sqrt{M_\kappa}\, \delta_{\alpha\beta} } { \sqrt{\sum_{\kappa'}M_{\kappa'}} }.$$

The denominator normalizes the polarization vector.

Exercise 5: Coherent state as a classical lattice wave

Let only one mode be in a coherent state with

$$\hat b|\beta\rangle=\beta|\beta\rangle.$$

Show that its expected normal coordinate follows the classical oscillator equation.

Solution

In the Heisenberg picture,

$$\hat b(t) := \hat b(0)e^{-i\omega t}.$$

Therefore

$$\begin{aligned} \langle\hat Q(t)\rangle :={}& \sqrt{\frac{\hbar}{2\omega}} \left( \beta e^{-i\omega t} + \beta^*e^{i\omega t} \right) \\ = {}& \sqrt{\frac{2\hbar}{\omega}} |\beta| \cos( \omega t-\arg\beta ). \end{aligned}$$

It obeys

$$\frac{d^2}{dt^2} \langle\hat Q(t)\rangle + \omega^2 \langle\hat Q(t)\rangle =0.$$

The mean occupation is $|\beta|^2$. The classical-looking wave is therefore a coherent superposition of number states, not a one-phonon number state.

Exercise 6: Creation and annihilation intensity

Using

$$\overline n := \frac{1}{e^{\beta\hbar\omega}-1},$$

show that the ratio of phonon-annihilation to phonon-creation thermal factors is $e^{-\beta\hbar\omega}$.

Solution

The creation factor is $\overline n+1$ and the annihilation factor is $\overline n$. Their ratio is

$$\frac{\overline n}{\overline n+1}.$$

Since

$$\overline n+1 := \frac{ e^{\beta\hbar\omega} } { e^{\beta\hbar\omega}-1 },$$

we obtain

$$\frac{\overline n}{\overline n+1} := e^{-\beta\hbar\omega}.$$

This is the mode-level origin of the detailed-balance asymmetry between energy-gain and energy-loss sides of a thermal spectrum.

Exercise 7: Limits of the diatomic chain

For

$$\begin{aligned} A &:= \frac{1}{M_1}+\frac{1}{M_2}, \\ B(q) &:= \frac{4}{M_1M_2} \sin^2\frac{qa}{2} , \\ \omega_\pm^2(q) &:= K \left[ A \pm \sqrt{ A^2-B(q) } \right], \end{aligned}$$

evaluate both branches at $q=0$ and at $q=\pi/a$. Assume $M_1>M_2$.

Solution

At $q=0$, the square root equals $1/M_1+1/M_2$. Hence

$$\begin{aligned} \omega_-^2(0)&=0, \\ \omega_+^2(0) &= 2K \left( \frac{1}{M_1}+\frac{1}{M_2} \right). \end{aligned}$$

At the zone boundary,

$$\sin^2\frac{qa}{2}=1.$$

The square root becomes

$$\left| \frac1{M_1}-\frac1{M_2} \right| := \frac1{M_2}-\frac1{M_1}.$$

Therefore

$$\begin{aligned} \omega_-^2\left(\frac{\pi}{a}\right) &= \frac{2K}{M_1}, \\ \omega_+^2\left(\frac{\pi}{a}\right) &= \frac{2K}{M_2}. \end{aligned}$$

The heavier sublattice sets the lower zone-boundary frequency and the lighter sublattice the upper one.

Exercise 8: Audit an anharmonic decay claim

A calculation reports a cubic process

$$(\mathbf q,s) \longrightarrow (\mathbf q_1,s_1) + (\mathbf q_2,s_2).$$

List the minimum checks needed before interpreting it as a physical phonon lifetime.

Solution

At minimum, verify:

1. Stable external modes: the initial and final excitations have well-defined positive renormalized frequencies.
2. Crystal momentum: $\mathbf q=\mathbf q_1+\mathbf q_2+\mathbf G$ for some reciprocal vector $\mathbf G$.
3. Energy: $\Omega_{\mathbf q s}=\Omega_{\mathbf q_1s_1}+\Omega_{\mathbf q_2s_2}$ within the linewidth and approximation used.
4. Vertex: the cubic matrix element is nonzero after symmetry and polarization selection rules.
5. Occupation factors: spontaneous and stimulated terms are included consistently at the stated temperature.
6. Normalization: eigenvectors, masses, cell count, and Brillouin-zone weights use one convention.
7. Other channels: isotope, boundary, electron, defect, and higher-order processes are not silently attributed to the same rate.
8. Lifetime convention: amplitude decay, population decay, half width, and full width are distinguished.
9. Quasiparticle validity: the resulting width remains small enough that an isolated phonon pole is meaningful.

Passing momentum and energy conservation alone is necessary but not sufficient.

Summary

• Coupled atomic displacements become independent harmonic normal coordinates after diagonalizing the mass-weighted dynamical matrix.
• Quantizing each stable normal coordinate gives bosonic operators $\hat b_{\mathbf q s}$ and $\hat b_{\mathbf q s}^\dagger$.
• A normal mode is an available oscillator; a phonon is one quantum occupying it; a classical lattice wave is generally a coherent many-phonon state.
• A one-phonon state has zero mean displacement but nonzero energy, variance, and displacement transition matrix elements.
• There are $dr$ branches per momentum for $r$ basis atoms in $d$ dimensions; ordinary crystals have $d$ acoustic branches and $d(r-1)$ optical branches near the zone center.
• Acoustic sum rules express translation invariance. Exact zero coordinates must be treated separately from positive-frequency oscillators.
• Phonons carry crystal momentum modulo reciprocal vectors, not automatically literal mechanical momentum.
• Zero-point and thermal motion follow from the same displacement covariance, with the factor $\coth(\beta\hbar\omega/2)$.
• One-phonon scattering intensities depend on polarization, masses, basis interference, Debye–Waller factors, and Bose occupation.
• Anharmonicity makes phonons interacting, shifts frequencies, permits number-changing processes, and generates finite lifetimes.
• A phonon is an emergent oscillator quantum, providing a concrete bridge from many-body quantum mechanics to field quantization.

References and Further Reading

1. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, 1954). The classic systematic treatment of harmonic lattice dynamics, force constants, long-wavelength limits, and crystal stability.
2. P. Debye, “Zur Theorie der spezifischen Wärmen,” Annalen der Physik 344, 789–839 (1912), doi:10.1002/andp.19123441404. Continuum mode counting and the low-temperature heat-capacity law.
3. I. E. Tamm, “Über die Quantentheorie der molekularen Lichtzerstreuung in festen Körpern,” Zeitschrift für Physik 60, 345–363 (1930). An early quantum treatment of light scattering by crystal vibrations.
4. A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, 2nd ed. (Academic Press, 1971). A detailed reference for dynamical matrices, Green functions, and lattice models.
5. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976), Chapters 22–25. A standard treatment of phonons, thermodynamics, scattering, and anharmonic processes.
6. W. Cochran, “Crystal stability and the theory of ferroelectricity,” Advances in Physics 9, 387–423 (1960), doi:10.1080/00018736000101229. Soft-mode lattice instability and structural transitions.
7. R. A. Cowley, “Anharmonic crystals,” Reports on Progress in Physics 31, 123–166 (1968), doi:10.1088/0034-4885/31/1/303. A broad review of anharmonic thermodynamics, scattering, and transport.
8. A. A. Maradudin and A. E. Fein, “Scattering of neutrons by an anharmonic crystal,” Physical Review 128, 2589–2608 (1962), doi:10.1103/PhysRev.128.2589. Frequency shifts and linewidths from anharmonic interactions.
9. P. G. Klemens, “Anharmonic decay of optical phonons,” Physical Review 148, 845–848 (1966), doi:10.1103/PhysRev.148.845. A canonical analysis of optical-phonon decay into acoustic modes.
10. G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 3rd ed. (Cambridge University Press, 2012). Probe conventions, one-phonon structure factors, and multiphonon scattering.
11. G. D. Mahan, Many-Particle Physics, 3rd ed. (Springer, 2000), doi:10.1007/978-1-4757-5714-9. Phonon Green functions, self-energies, and electron–phonon coupling.
12. S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, “Phonons and related crystal properties from density-functional perturbation theory,” Reviews of Modern Physics 73, 515–562 (2001), doi:10.1103/RevModPhys.73.515. First-principles force constants, dynamical matrices, polar materials, and strain response.
13. B. Fultz, “Vibrational thermodynamics of materials,” Progress in Materials Science 55, 247–352 (2010), doi:10.1016/j.pmatsci.2009.05.002. Vibrational entropy, quasiharmonicity, and anharmonic effects in materials.
14. F. Giustino, “Electron–phonon interactions from first principles,” Reviews of Modern Physics 89, 015003 (2017), doi:10.1103/RevModPhys.89.015003. Modern conventions and methods for electron–phonon matrix elements, renormalization, and lifetimes.