Harmonic Chain

The harmonic chain is a periodic array of equal masses coupled by quadratic nearest-neighbor springs, exactly reducible to independent phonon modes plus a free center-of-mass coordinate whose zero frequency must not be quantized as an oscillator.

The four-site example below makes the oscillator count explicit. A uniform displacement has no restoring force; the three relative displacements oscillate.

Helpful background. Coupled oscillators explains normal coordinates; phonons connects their quantization to collective excitations.

A Periodic Chain of Masses and Springs

The calculation uses the following physical choices.

Field	Baseline choice
sites	$N\geq3$ equal masses labeled $n=0,\ldots,N-1$
geometry	one-dimensional periodic chain of reference spacing $a$
coordinate	one scalar displacement $u_n$ per site
momentum	canonical momentum $p_n$
mass	$M>0$
coupling	nearest-neighbor spring constant $K>0$
boundary condition	$u_{n+N}=u_n$ and $p_{n+N}=p_n$
onsite pinning	absent
anharmonicity	absent
equilibrium ensemble	fixed $N,a,M,K$ and temperature $T$
thermodynamic limit	$N\to\infty$ at fixed $a,M,K$
zero-mode convention	center of mass retained unless explicitly fixed

The site label is periodic, but the scalar displacement itself is not silently compactified. Uniformly shifting every $u_n$ costs no spring energy and produces a free center-of-mass degree of freedom. A pinned chain, an open chain, a diatomic chain, and an anharmonic chain are different physical models.

Degrees of Freedom and Hilbert Space

The microscopic canonical algebra is

$$\begin{aligned} [u_n,p_m] &= i\hbar\delta_{nm}, \\ [u_n,u_m] &= 0, \\ [p_n,p_m] &= 0. \end{aligned}$$

Before separating the uniform translation, the coordinate representation is

$$\mathcal H = L^2(\mathbb R^N).$$

Define the center-of-mass displacement and total momentum by

$$X = \frac1N \sum_{n=0}^{N-1}u_n, \qquad P = \sum_{n=0}^{N-1}p_n.$$

They satisfy

$$[X,P] = i\hbar.$$

After an orthogonal normal-coordinate transformation, the Hilbert space factors as

$$\mathcal H \simeq L^2(\mathbb R_X) \otimes \bigotimes_{\lambda=1}^{N-1} L^2(\mathbb R_{Q_\lambda}).$$

The first factor is a free coordinate. The remaining $N-1$ factors are harmonic oscillators. The relative-coordinate theory omits the first factor. This is a reduction of the degrees of freedom, not a normalizable state with both $X=0$ and $P=0$: the commutator $[X,P]=i\hbar$ forbids simultaneous sharp values. With

$$u_n^{\mathrm{rel}}=u_n-X,\qquad p_n^{\mathrm{rel}}=p_n-P/N,$$

the reduced variables obey

$$[u_n^{\mathrm{rel}},p_m^{\mathrm{rel}}] =i\hbar(\delta_{nm}-1/N).$$

There are only $N-1$ independent canonical pairs. Below, “fixed center of mass” refers to this reduced theory; pinning instead adds a physical restoring force.

Hamiltonian and Stiffness Matrix

The baseline Hamiltonian is

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

Writing $\mathbf u=(u_0,\ldots,u_{N-1})^{\mathsf T}$ and similarly for $\mathbf p$ gives

$$H = \frac{1}{2M} \mathbf p^{\mathsf T}\mathbf p + \frac12 \mathbf u^{\mathsf T}C\mathbf u,$$

with the periodic stiffness matrix

$$C_{nm} = K \left( 2\delta_{nm} - \delta_{n,m+1} - \delta_{n,m-1} \right).$$

All indices in this expression are understood modulo $N$. Every row sums to zero:

$$\sum_m C_{nm} = 0.$$

Consequently, the uniform vector

$$\mathbf e_0 = \frac{1}{\sqrt N} (1,1,\ldots,1)^{\mathsf T}$$

is an exact zero eigenvector. For $K>0$, $C$ is positive semidefinite and this is its only zero direction. For $K<0$, nonuniform directions are unstable, so the usual oscillator quantization does not define a lower-bounded Hamiltonian.

Fourier and Normal-Mode Conventions

Allowed wave numbers are

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

chosen once in a first Brillouin zone. Define

$$\begin{aligned} u_q &= \frac{1}{\sqrt N} \sum_n e^{-iqna}u_n, \\ p_q &= \frac{1}{\sqrt N} \sum_n e^{-iqna}p_n. \end{aligned}$$

Hermiticity gives

$$u_q^{\dagger} = u_{-q}, \qquad p_q^{\dagger} = p_{-q},$$

and the transformed commutator is

$$[u_q,p_{q'}] = i\hbar \delta_{q+q',0}.$$

The Hamiltonian becomes

$$H = \frac12 \sum_q \left[ \frac{p_qp_{-q}}{M} + M\omega_q^2u_qu_{-q} \right],$$

where

$$\omega_q^2 = \frac{4K}{M} \sin^2\left( \frac{qa}{2} \right).$$

Thus

$$\omega_q = 2\sqrt{\frac KM} \left| \sin\frac{qa}{2} \right|.$$

For $q\ne0$, introduce bosonic mode operators. One full-zone convention is

$$u_n ={} X + \frac{1}{\sqrt N} \sum_{q\ne0} \sqrt{ \frac{\hbar}{2M\omega_q} } \times \left( b_qe^{iqna} + b_q^{\dagger}e^{-iqna} \right),$$

and

$$p_n ={} \frac PN - \frac{i}{\sqrt N} \sum_{q\ne0} \sqrt{ \frac{\hbar M\omega_q}{2} } \times \left( b_qe^{iqna} - b_q^{\dagger}e^{-iqna} \right).$$

The mode algebra is

$$[b_q,b_{q'}^{\dagger}] = \delta_{qq'}.$$

With the zero coordinate separated,

$$H = \frac{P^2}{2NM} + \sum_{q\ne0} \hbar\omega_q \left( b_q^{\dagger}b_q + \frac12 \right).$$

There are $N-1$ positive-frequency oscillator modes, not $2(N-1)$. Summing $q$ over the full discrete Brillouin zone gives one traveling-wave oscillator for each allowed nonzero $q$. A real standing-wave convention instead combines each generic pair $q$ and $-q$ into cosine and sine oscillators. Mixing the two conventions causes double counting.

Symmetries and Conserved Quantities

The baseline model has:

• continuous uniform-shift symmetry $u_n\mapsto u_n+c$, generated by $P$;
• cyclic lattice translation $n\mapsto n+1$, giving crystal momentum modulo $2\pi\hbar/a$;
• reflection symmetry $n\mapsto-n$;
• time-reversal symmetry with $p_n\mapsto-p_n$;
• conservation of the energy in every nonzero normal mode;
• conservation of total momentum $P$.

For the exactly harmonic Hamiltonian,

$$n_q = b_q^{\dagger}b_q$$

commutes with $H$ for every $q\ne0$. The total phonon occupation also commutes with this quadratic Hamiltonian, but it is not a microscopic conserved charge protected by a fundamental $U(1)$ symmetry. Anharmonic terms can create and destroy phonons while preserving the actual energy and crystal-momentum constraints.

The degeneracy $\omega_q=\omega_{-q}$ follows from inversion and time-reversal symmetry. A perturbation that breaks those symmetries can alter that relation without changing the number of microscopic coordinates.

Natural Scales and Control Parameters

The spring timescale and band-edge frequency are

$$t_* = \sqrt{\frac MK}, \qquad \omega_{\max} = 2\sqrt{\frac KM}.$$

At long wavelength,

$$\omega_q = c|q| \left[ 1- \frac{(qa)^2}{24} +O((qa)^4) \right],$$

with sound speed

$$c = a\sqrt{\frac KM}.$$

The smallest nonzero frequency is

$$\omega_{\min} = 2\sqrt{\frac KM} \sin\left( \frac{\pi}{N} \right) \simeq \frac{2\pi c}{Na}.$$

Useful dimensionless quantities are

$$qa, \qquad \frac{k_{\mathrm B}T}{\hbar\omega_{\max}}, \qquad N,$$

together with a strain criterion that tests the harmonic approximation. For example, the model should not be extrapolated to a microscopic lattice unless

$$\sqrt{ \left\langle (u_{n+1}-u_n)^2 \right\rangle } \ll a.$$

This inequality is a validity test, not a consequence of writing a quadratic Hamiltonian. The harmonic model itself has no parameter that describes the neglected cubic and quartic force constants.

What the Normal Modes Solve

The harmonic chain is exactly solvable after its stiffness matrix is diagonalized, but the claim has boundaries.

Quantity	Status	Caveat
finite-chain normal frequencies and eigenvectors	exact discrete Fourier solution	requires the stated periodic monatomic spring model
center-of-mass motion	exact free-particle solution	it is not a harmonic oscillator and has no normalizable ground state on $\mathbb R$
relative-coordinate spectrum	exact product of $N-1$ oscillator spectra	zero mode must be fixed, pinned, or retained separately
relative ground-state and thermal Gaussian correlations	exact within the harmonic Hamiltonian	a general non-Gaussian initial state does not become Gaussian under quadratic evolution
equilibrium oscillator occupations	exact Bose factors for positive-frequency modes	equilibrium phonon chemical potential is zero
real-time harmonic dynamics	exact coherent mode propagation	no damping, mode mixing, or thermalization occurs
low-wave-number continuum theory	controlled asymptotic limit	lattice dispersion and Brillouin-zone periodicity are discarded
material phonon spectrum	not predicted by the toy model alone	force constants, basis atoms, dimensions, and electronic effects are missing
transport coefficients	not generically finite in the ideal chain	integrability and absent scattering make ordinary relaxation assumptions fail

For a fixed center of mass, an exact eigenstate is labeled by occupations $n_q=0,1,2,\ldots$:

$$E_{\{n_q\}} = \sum_{q\ne0} \hbar\omega_q \left( n_q+\frac{1}{2} \right).$$

The free dynamics is

$$b_q(t) = e^{-i\omega_qt} b_q(0).$$

Exact propagation does not imply irreversible equilibration. It instead preserves every $n_q$ indefinitely.

Thermodynamics of the Relative Modes

With the center of mass fixed, the relative partition function is

$$Z_{\mathrm{rel}} = \prod_{q\ne0} \frac{1} {2\sinh(\beta\hbar\omega_q/2)}.$$

The Bose occupation of a positive-frequency mode is

$$\overline n_q = \frac{1} {e^{\beta\hbar\omega_q}-1}.$$

The relative internal energy and heat capacity are

$$U_{\mathrm{rel}} = \sum_{q\ne0} \hbar\omega_q \left( \overline n_q+\frac{1}{2} \right),$$

and

$$C_{V,\mathrm{rel}} ={} k_{\mathrm B} \sum_{q\ne0} (\beta\hbar\omega_q)^2 \times \frac{e^{\beta\hbar\omega_q}} {(e^{\beta\hbar\omega_q}-1)^2}.$$

In the thermodynamic limit, the vibrational density of states per chain length $L=Na$ is

$$\frac{g(\omega)}{L} = \frac{2} {\pi a\omega_{\max}} \frac{1} {\sqrt{1-(\omega/\omega_{\max})^2}},$$

for $0<\omega<\omega_{\max}$. It approaches

$$\frac{g(\omega)}{L} \longrightarrow \frac{1}{\pi c}$$

at low frequency and has a one-dimensional van Hove divergence at the band edge.

The thermodynamic-limit zero-point energy per site is

$$\frac{E_{\mathrm{zp}}}{N} = \frac{\hbar\omega_{\max}}{\pi} = \frac{2\hbar}{\pi} \sqrt{\frac KM}.$$

In the acoustic quantum window

$$\hbar\omega_{\min} \ll k_{\mathrm B}T \ll \hbar\omega_{\max},$$

the thermal energy density and heat capacity are

$$\frac{U(T)-U(0)}{L} = \frac{\pi}{6} \frac{(k_{\mathrm B}T)^2} {\hbar c},$$

and

$$\frac{C_V}{L} = \frac{\pi k_{\mathrm B}^2T} {3\hbar c}.$$

At high temperature, every fixed-center oscillator approaches classical equipartition:

$$C_{V,\mathrm{rel}} \longrightarrow (N-1)k_{\mathrm B}.$$

For a finite chain at temperatures below $\hbar\omega_{\min}/k_{\mathrm B}$, the nonzero modes freeze out exponentially. The linear-$T$ law therefore requires a thermodynamic or sufficiently large-system window.

Infrared Fluctuations

For a fixed center of mass, the absolute displacement variance is

$$\left\langle (u_n-X)^2 \right\rangle_T ={} \frac{\hbar}{2MN} \sum_{q\ne0} \frac{1}{\omega_q} \times \coth\left( \frac{\beta\hbar\omega_q}{2} \right).$$

Because $\omega_q\sim c|q|$, this quantity grows as $\log N$ at $T=0$ and as $N$ at any fixed $T>0$. The unpinned one-dimensional chain therefore does not support size-independent absolute localization around the chosen reference sites in the thermodynamic limit.

Relative displacements behave better. The variance of

$$B_n = u_{n+1}-u_n$$

contains an additional factor $4\sin^2(qa/2)$ that removes the acoustic infrared divergence. Local bond strain can remain finite even when absolute displacements wander. In the classical thermal limit the finite-ring sums can be evaluated exactly:

$$\left\langle(u_n-X)^2\right\rangle =\frac{k_{\mathrm B}T}{12K}\frac{N^2-1}{N}, \qquad \langle B_n^2\rangle=\frac{k_{\mathrm B}T}{K}\left(1-\frac1N\right).$$

The first grows with $N$; the second tends to a constant. The factor $1-1/N$ in the bond variance records the periodic constraint $\sum_n B_n=0$. These classical expressions require temperature above the phonon band; at lower temperature use the full quantum covariance before taking the large-$N$ limit.

Important Limits

Limit or modification	Result	Main caution
$q\to0$	$\omega_q\simeq c\lvert q\rvert$	exact $q=0$ remains a free coordinate
$N\to\infty$	continuous acoustic band	absolute displacement fluctuations diverge in one dimension
$K\to0^+$	all frequencies collapse	the system approaches $N$ free masses, not $N$ soft oscillators with fixed Fock normalization
$M\to\infty$	vibrational frequencies vanish	thermal and quantum limits must be reordered carefully
weak onsite pinning	acoustic branch gains a gap	uniform-shift symmetry is explicitly broken
continuum limit	free elastic scalar field	lattice cutoff and Brillouin-zone structure are lost
high temperature	$(N-1)k_{\mathrm B}$ relative heat capacity	the free center of mass requires its own ensemble
weak anharmonicity	finite shifts, scattering, and lifetimes	mode occupations cease to be exact constants

For the continuum limit, define mass density and elastic coefficient

$$\rho = \frac Ma, \qquad \kappa = Ka.$$

Then $c^2=\kappa/\rho$, and the relative Hamiltonian approaches

$$H = \int dx \left[ \frac{\pi(x)^2}{2\rho} + \frac{\kappa}{2} (\partial_xu)^2 \right].$$

Harmonic Oscillator to Fields explains the continuum-field correspondence.

The Free Translation Mode

The $q=0$ sector is

$$H_0 = \frac{P^2}{2NM}.$$

On $X\in\mathbb R$, it has continuous spectrum and no normalizable ground state. Its unconstrained canonical position integral is proportional to the available translation length and diverges if no finite translation range or reduction to relative variables is specified. The uniform translation is a physical symmetry, not automatically a gauge redundancy.

Three common choices define different finite systems:

1. Retain the center of mass. Quantize $H_0$ as a free particle and specify the allowed range or wave-packet state of $X$.
2. Use relative coordinates only. Remove the free factor and retain $N-1$ independent oscillator pairs with the projected commutator above. This does not impose simultaneous eigenvalue constraints on the original $X$ and $P$.
3. Pin the chain. Add an onsite potential

$$H_{\mathrm{pin}} = \frac{M\Omega^2}{2} \sum_n u_n^2,$$

which changes the dispersion to

$$\omega_q^2 = \Omega^2 + \frac{4K}{M} \sin^2\left( \frac{qa}{2} \right).$$

The limits $N\to\infty$, $\Omega\to0$, and $T\to0$ need not commute for infrared-sensitive observables. An accidental small eigenvalue from rounding is a numerical error, not a physical pinning model. Classify the known translation vector as zero within a justified tolerance. If a positive regulator is deliberately added, state its operator: pinning only the center of mass differs from the onsite term above, which shifts every mode.

For even $N$, the zone-boundary momentum $q=\pi/a$ is its own negative modulo a reciprocal vector. Generic $q$ and $-q$ form degenerate pairs, but this self-conjugate point contributes only one oscillator. Correct mode counting always returns $N-1$ positive-frequency modes after the center of mass is fixed.

Typical Observables

Natural observables include:

• displacement and momentum covariances;
• bond strain $B_n=u_{n+1}-u_n$;
• normal-mode occupations $n_q$;
• vibrational energy and heat capacity;
• group velocity and wave-packet propagation;
• dynamical structure factors and spectral functions;
• local energy density and energy current.

Inside $0<q<\pi/a$, the group velocity is

$$v_g(q) = \frac{d\omega_q}{dq} = c\cos\left( \frac{qa}{2} \right).$$

It approaches $c$ near the zone center and vanishes at the band edge. The existence of a normal mode does not guarantee that a particular probe sees it: scattering intensity also depends on polarization, form factors, and operator selection rules. Structure Factors explains those response conventions.

Minimal Worked Example

Take $N=4$. The allowed momenta may be represented by

$$q \in \left\{ 0, \frac{\pi}{2a}, -\frac{\pi}{2a}, \frac{\pi}{a} \right\}.$$

The frequencies are

$$\begin{aligned} \omega_0 &= 0, \\ \omega_{\pm\pi/(2a)} &= \sqrt{\frac{2K}{M}}, \\ \omega_{\pi/a} &= 2\sqrt{\frac KM}. \end{aligned}$$

The $q=0$ eigenvector is uniform. The zone-boundary eigenvector is proportional to $(1,-1,1,-1)$. The $q=\pm\pi/(2a)$ traveling waves are degenerate and can be replaced by real sine and cosine standing waves.

After fixing the center of mass, the ground energy is

$$E_{0,\mathrm{rel}} = \hbar\sqrt{\frac KM} \left( 1+\sqrt2 \right).$$

The exact relative partition function is

$$Z_{\mathrm{rel}} ={} \left[ 2\sinh\left( \frac{\beta\hbar}{2} \sqrt{\frac{2K}{M}} \right) \right]^{-2} \times \left[ 2\sinh\left( \beta\hbar \sqrt{\frac KM} \right) \right]^{-1}.$$

The real normal vectors can be chosen proportional to $(1,1,1,1)$, $(1,0,-1,0)$, $(0,1,0,-1)$, and $(1,-1,1,-1)$. The middle two span the same degenerate space as the traveling waves at $q=\pm\pi/(2a)$.

Each arrow gives a site’s scalar displacement; a dot without an arrow has zero displacement. Vector amplitudes are schematic, not a common normalization. The uniform pattern is a free coordinate. The other three are oscillator directions; the two middle patterns are independent despite sharing a frequency.

Numerical Benchmark

For a compact matrix calculation, use

$$N=8, \qquad M=K=a=\hbar=1.$$

Diagonalize the mass-weighted stiffness matrix

$$D_{nm} = 2\delta_{nm} - \delta_{n,m+1} - \delta_{n,m-1}.$$

The analytic eigenvalues and frequencies are

Mode labels $m$	Degeneracy	$\omega_m^2$	$\omega_m$
$0$	$1$	$0$	$0$
$1,7$	$2$	$2-\sqrt2$	$\sqrt{2-\sqrt2}$
$2,6$	$2$	$2$	$\sqrt2$
$3,5$	$2$	$2+\sqrt2$	$\sqrt{2+\sqrt2}$
$4$	$1$	$4$	$2$

Useful decimal frequencies are

$$\begin{aligned} \sqrt{2-\sqrt2} &= 0.7653668647301795\ldots, \\ \sqrt2 &= 1.4142135623730951\ldots, \\ \sqrt{2+\sqrt2} &= 1.8477590650225735\ldots. \end{aligned}$$

Independent matrix invariants are

$$D\mathbf1 = 0, \qquad \operatorname{tr}D = 16,$$

and

$$\operatorname{pdet}D = \prod_{\lambda_j>0}\lambda_j = 64.$$

The fixed-center zero-point energy is

$$\begin{aligned} E_{\mathrm{zp}} ={}& 1+\sqrt2 +\sqrt{2-\sqrt2} +\sqrt{2+\sqrt2} \\ =& 5.027339492125848\ldots. \end{aligned}$$

For the lowest nonzero mode in a sequence of chain lengths,

$$\frac{\omega_1}{c|q_1|} = \frac{\sin(\pi/N)}{\pi/N} = 1- \frac{\pi^2}{6N^2} +O(N^{-4}).$$

A valid implementation should:

1. construct periodic neighbors without duplicating or omitting the wraparound spring;
2. verify row sums and the exact null vector before diagonalization;
3. compare eigenvalues, degeneracies, trace, and pseudodeterminant;
4. form frequencies only after classifying negative, zero, and positive eigenvalues;
5. compare eigenvector subspaces rather than individual vectors inside degenerate pairs;
6. state whether the zero mode is retained, fixed, or pinned;
7. test the $N^{-2}$ acoustic convergence rather than only one chain length.

Boundaries, Pinning, and Anharmonicity

Boundary conditions

Open, free-end, and fixed-end chains have standing-wave spectra and different zero-mode structure. Reusing periodic wave numbers for them gives the wrong mode count and endpoint behavior.

Onsite pinning

The onsite term proportional to $\Omega^2$ gaps the acoustic branch and renders absolute displacement fluctuations infrared finite. It also explicitly breaks uniform-shift symmetry, so it is not a harmless numerical regulator unless the $\Omega\to0$ limit is demonstrated for the observable being calculated.

More atoms per cell

A diatomic or multi-atom unit cell produces several branches. Acoustic and optical labels refer to long-wavelength displacement patterns, while exact frequencies require a matrix-valued dynamical problem. The general treatment belongs to Phonons as Many-Body Excitations.

Longer-range and disordered couplings

Translation-invariant longer-range springs change $\omega_q$ but preserve Fourier diagonalization. Random masses or spring constants destroy momentum as an exact label and can localize vibrational modes.

Anharmonicity

Cubic and quartic displacement terms couple normal modes. They generate thermal expansion, frequency shifts, finite lifetimes, and scattering. The harmonic occupations $n_q$ then cease to be exact constants of motion.

Higher dimensions and fields

Several displacement polarizations and lattice directions produce a dynamical matrix at every crystal momentum. The long-wavelength continuum becomes an elastic field theory rather than a single scalar field.

Exercises

Exercise 1: Derive the dispersion

Insert a plane wave $u_n(t)=u_qe^{i(qna-\omega t)}$ into the classical equations of motion and derive the exact lattice dispersion.

Solution

Hamilton’s equations give

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

For the plane wave,

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

Using $2-2\cos qa=4\sin^2(qa/2)$ yields

$$\omega_q^2 = \frac{4K}{M} \sin^2\left( \frac{qa}{2} \right).$$

The nonnegative frequency is therefore

$$\omega_q = 2\sqrt{\frac KM} \left| \sin\frac{qa}{2} \right|.$$

Exercise 2: Isolate the center of mass

Show that the uniform Fourier coordinate gives $H_0=P^2/(2NM)$ and explain why its oscillator ground-state formula is invalid.

Solution

The normalized zero-mode variables are

$$u_0 = \frac1{\sqrt N} \sum_nu_n = \sqrt N X,$$

and

$$p_0 = \frac1{\sqrt N} \sum_np_n = \frac P{\sqrt N}.$$

The spring term vanishes for a uniform displacement. The kinetic contribution is

$$H_0 = \frac{p_0p_0}{2M} = \frac{P^2}{2NM}.$$

This is a free-particle Hamiltonian in $X$. Its spectrum is continuous on $\mathbb R$, and it has no normalizable state with the oscillator energy $\hbar\omega_0/2=0$. The factors $1/\sqrt{\omega_0}$ in oscillator coordinates also diverge. One must retain, fix, or pin this coordinate explicitly.

Exercise 3: Count the four-site modes

For $N=4$, find the allowed momenta, frequencies, and fixed-center ground energy. Explain why the two modes at $q=\pm\pi/(2a)$ are not double counting.

Solution

The allowed momenta are $0$, $\pi/(2a)$, $-\pi/(2a)$, and $\pi/a$. Substitution into the dispersion gives

$$\omega = \left\{ 0, \sqrt{\frac{2K}{M}}, \sqrt{\frac{2K}{M}}, 2\sqrt{\frac KM} \right\}.$$

After fixing the zero coordinate, there are three oscillators. Their zero-point energies sum to

$$\begin{aligned} E_{0,\mathrm{rel}} &= \frac{\hbar}{2} \left[ 2\sqrt{\frac{2K}{M}} + 2\sqrt{\frac KM} \right] \\ &= \hbar\sqrt{\frac KM} (1+\sqrt2). \end{aligned}$$

The $q$ and $-q$ traveling waves are independent directions of propagation. Equivalently, their two-dimensional degenerate subspace has independent cosine and sine standing waves. Either basis contains two oscillators, not four.

Exercise 4: Derive the low-temperature heat capacity

Use the acoustic dispersion and thermodynamic-limit mode density to derive the leading thermal energy and heat capacity per length.

Solution

Both signs of $q$ contribute. With $\omega=c|q|$,

$$\frac{U(T)-U(0)}{L} = \frac1\pi \int_0^\infty dq\, \frac{\hbar cq} {e^{\beta\hbar cq}-1}.$$

Set $x=\beta\hbar cq$. Then

$$\frac{U(T)-U(0)}{L} = \frac{(k_{\mathrm B}T)^2} {\pi\hbar c} \int_0^\infty dx\, \frac{x}{e^x-1}.$$

Using

$$\int_0^\infty dx\, \frac{x}{e^x-1} = \frac{\pi^2}{6}$$

gives

$$\frac{U(T)-U(0)}{L} = \frac{\pi}{6} \frac{(k_{\mathrm B}T)^2} {\hbar c}.$$

Differentiating at fixed length yields

$$\frac{C_V}{L} = \frac{\pi k_{\mathrm B}^2T} {3\hbar c}.$$

The derivation requires $\hbar\omega_{\min}\ll k_{\mathrm B}T$ so that the momentum sum may be replaced by an integral.

Exercise 5: Compare absolute and relative fluctuations

Use the low-$q$ form of the covariance to determine the infrared scaling of $\langle(u_n-X)^2\rangle$ at zero and nonzero temperature. Explain why the bond variance does not have the same divergence.

Solution

At zero temperature, the long-wavelength contribution is proportional to

$$\int_{q_{\min}} \frac{dq}{\omega_q} \sim \frac1c \int_{q_{\min}} \frac{dq}{q} \sim \log N.$$

At fixed $T>0$ and sufficiently small $q$,

$$\coth\left( \frac{\beta\hbar cq}{2} \right) \sim \frac{2k_{\mathrm B}T} {\hbar cq}.$$

The integrand is then proportional to $1/q^2$, so

$$\int_{q_{\min}} \frac{dq}{q^2} \sim \frac1{q_{\min}} \propto N.$$

For a bond displacement, Fourier transformation supplies

$$|e^{iqa}-1|^2 = 4\sin^2\left( \frac{qa}{2} \right) \sim q^2a^2.$$

This factor cancels enough of the small-$q$ singularity to make the local bond variance infrared finite. Absolute positional order and local elastic strain therefore have different thermodynamic behavior.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston (1976).
• 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).
• M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford University Press (1954).
• P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press (1995).
• P. Debye, “Zur Theorie der spezifischen Wärmen”, Annalen der Physik 344, 789–839 (1912).
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2004).
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley (2004).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• 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).
• Z. Rieder, J. L. Lebowitz, and E. Lieb, “Properties of a harmonic crystal in a stationary nonequilibrium state”, Journal of Mathematical Physics 8, 1073–1078 (1967).