Tight-Binding Chain

One-Sentence Description

The tight-binding chain is the one-dimensional lattice model in which a particle hops between neighboring localized orbitals, producing a cosine energy band.

Physical Setup

Choose one localized orbital per unit cell on a chain with lattice spacing $a$. The model is a minimal single-band description of band formation from hopping between neighboring sites. It is the many-site continuation of the Tight-Binding Dimer. The Tight-Binding Chain dossier supplies the convention-complete record and validation targets; the full derivation and boundary comparison live in Tight-Binding Model.

The model can be used in a one-particle Hilbert space or in a many-particle Fock space. The Hamiltonian form is the same, but filling, statistics, and interactions require extra specification.

Hamiltonian

For a chain with $N$ sites and periodic boundary conditions,

$$H = \epsilon_0\sum_{j=1}^N c_j^\dagger c_j - t\sum_{j=1}^N \left( c_{j+1}^\dagger c_j + c_j^\dagger c_{j+1} \right),$$

with $c_{N+1}=c_1$. The hopping $t$ is taken real in this convention.

The Fourier modes are

$$c_j = \frac{1}{\sqrt N} \sum_k e^{ikja}c_k, \qquad k=\frac{2\pi m}{Na}.$$

The band dispersion is

$$E(k) = \epsilon_0-2t\cos(ka), \qquad -\frac{\pi}{a}<k\le\frac{\pi}{a}.$$

Parameters

Symbol	Meaning
$a$	lattice spacing
$N$	number of sites
$\epsilon_0$	onsite energy
$t$	nearest-neighbor hopping amplitude
$k$	crystal momentum in the first Brillouin zone

What It Teaches

The chain shows how local hopping turns a set of localized orbitals into a band. The same hopping amplitude controls bandwidth, group velocity, and curvature near the band extrema:

$$W=4\lvert t\rvert.$$

Near a band minimum, the dispersion can be approximated by an effective mass. At the band edges in one dimension, the density of states has characteristic singular behavior after the continuum limit.

Canonical Links

• Tight-Binding Chain dossier
• Tight-Binding Model
• Tight-Binding Dimer
• Peierls Phase Preview
• Bloch Theorem
• Density of States
• Effective Mass
• Many-Particle Hamiltonians

Common Mistakes

• Forgetting to specify open or periodic boundary conditions.
• Treating the site basis as the energy eigenbasis when $t\ne0$.
• Mixing sign conventions for the hopping without tracking the resulting band minimum.
• Calling the one-particle chain a metal or insulator before specifying filling and statistics.
• Applying the nearest-neighbor cosine band where longer-range hopping is important.

Quick Check

For $t>0$, where is the bottom of the nearest-neighbor band?

Solution

The dispersion is $E(k)=\epsilon_0-2t\cos(ka)$. For $t>0$, this is minimized when $\cos(ka)=1$, so the band bottom is at $k=0$ with energy $\epsilon_0-2t$.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976.
• C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2005.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013.