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Fill the lower energy band of the dimerized tight-binding chain with both spin states. Take zero on-site energy and measure ground-state energy per doubled primitive unit cell. State counting in a Bloch band and $q=ka$ giveEquivalently, for $|\delta |\le t$, $\mathcal{E}(\delta )=-(8t/\pi )E(\sqrt{1-{\delta }^2/t^2})$, where $E(k)$ is the complete elliptic integral of the second kind in the modulus convention.

For fixed $t>0$ and $|\delta |/t\to 0$, the change isThe logarithm comes from states near the edge of the Brillouin zone. Put $u=\pi /2-q$. To leading order the square root there is $\sqrt{t^2{u}^2+{\delta }^2}$. For a fixed small cutoff $\eta >0$, direct integration givesThe rest of the band, and the higher-order corrections near its edge, contribute only $O({\delta }^2/t)$; multiplying by $-8/\pi$ gives the displayed asymptotic expansion. Here the remainder uses Big O notation. The negative logarithmic term exceeds any fixed quadratic elastic cost $K{\delta }^2$, $K>0$, for sufficiently small nonzero $\delta$. This explains the zero-temperature instability to a Peierls distortion in the ideal chain model.

A Peierls transition is a lattice distortion coupled to a gap in an initially partly filled quasi-one-dimensional electronic band. A modulation at $2k_F$ couples the opposite Fermi points. At half filling with one orbital per atom, it doubles the cell and alternates strong and weak bonds; the dimerized tight-binding chain shows explicitly how the gap opens at the Fermi energy. In the ideal weak-coupling zero-temperature model, the Peierls energy gain has a negative term proportional to $-{\delta }^2\ln (t/|\delta |)/t$, which can outweigh a positive quadratic elastic cost.

Observation should connect electronic and structural evidence: cooling through the transition can reduce conductivity as the gap develops, while diffraction satellites identify the ordering wavevector. A softened phonon near $2k_F$ tests the precursor electron-lattice coupling. In inelastic neutron scattering of TTF-TCNQ, measurements from $60$ to $300\mathrm{K}$ found an increasing Kohn anomaly near $2k_F$ as the $54\mathrm{K}$ transition was approached. Those particular measurements were above the transition, so they probe the precursor dynamics rather than a static low-temperature diffraction pattern.^[ref]

A real finite-temperature transition also depends on interchain coupling and fluctuations. The ideal one-dimensional energy argument by itself does not prove finite-temperature long-range order.

A partly filled one-dimensional energy band is especially sensitive to a lattice modulation at $2k_F$. At half filling, $2k_F=\pi /a$, so the Peierls distortion alternates the bond lengths and doubles the primitive unit cell. Mixing states at the Fermi points opens a band gap, lowering the energy of occupied states. In the ideal weak-coupling, zero-temperature chain, the electronic gain behaves as $-u^2\ln (1/|u|)$ while the elastic cost is proportional to $u^2$, favouring a small distortion. The dimerized tight-binding chain gives an explicit band calculation.

The structural signature is extra diffraction intensity at the modulation wavevector; optical absorption can detect the gap and transport can detect loss of metallic conduction. A dynamic precursor is a Kohn anomaly, with phonon softening near $2k_F$. Inelastic neutron scattering of TTF-TCNQ, using a deuterated sample over $60$–$300\mathrm{K}$, found an anomaly near $2k_F$ developing below about $150\mathrm{K}$ as the $54\mathrm{K}$ transition was approached. This probes a lattice mode coupled to the electronic instability rather than inferring distortion from resistance alone.^[ref]

Away from half filling, the modulation need not simply double the lattice. Finite-temperature ordering in real quasi-one-dimensional solids also involves interchain coupling and fluctuations; the ideal isolated-chain argument alone does not establish a finite-temperature transition.

A Peierls distortion is a spontaneous lattice modulation that can lower the energy of a one-dimensional metal by opening a gap at its Fermi points. The associated change from metallic to gapped electronic behavior is a Peierls transition. The key wavevector is $2k_F$: it couples states near $+k_F$ and $-k_F$, which otherwise have the same energy.

Consider a spin-degenerate chain with one orbital per atom at spacing $a$ and one electron per atom. The band is half filled and $k_F=\pi /(2a)$. A distortion of period $2a$ alternates short and long bonds, doubles the primitive cell, and folds the original Fermi points onto the new zone boundary. The distortion can therefore mix the degenerate states precisely where the occupied and empty states meet.

An explicit dimerized tight-binding chain has alternating hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, where $t>0$. In the reduced zone $|k|\le \pi /(2a)$,There are two electrons per doubled cell, including spin, so the lower band is filled and the upper one empty. Splitting the degenerate states pushes occupied levels downward and empty ones upward, giving an electronic energy gain.

In weak coupling the electronic gain per length has the characteristic form $-C{\delta }^2\ln (t/|\delta |)$, with $C>0$. The elastic cost of moving the ions is proportional to $+K{\delta }^2$. The logarithm makes the undistorted ideal one-dimensional zero-temperature metal unstable to a small distortion even when the electron-lattice coupling is weak. A self-consistent balance selects a finite distortion amplitude. Two opposite dimerization patterns have the same energy for a translation-symmetric isolated chain.

The basic picture links structural modulation to electronic bond order or charge-density modulation, and can open an insulating gap. It does not assert that every real chain has exactly one electron per atom or period doubling: away from half filling, a modulation at $2k_F$ need not be commensurate with two lattice spacings. Temperature, finite chain length and fluctuations cut off the ideal logarithmic response. Interchain coupling and the real lattice dynamics determine whether a macroscopic finite-temperature ordered phase actually forms. The strictly one-dimensional zero-temperature instability alone is not a proof of finite-temperature long-range order.