Tight-Binding Dimer

The tight-binding dimer is the simplest lattice Hamiltonian: one particle can occupy either of two localized site orbitals and can hop between them. It is the two-site version of the same two-level algebra used for coupled wells, but the language is that of lattice sites, hopping amplitudes, and bonding or antibonding orbitals.

The dimer is important because it is the smallest model in which a localized basis and an energy eigenbasis are visibly different. Longer chains and their bands are developed in Tight-Binding Model; interacting many-particle versions lead toward Hubbard models. This page keeps the one-particle two-site problem as the wave-mechanics first encounter.

Site Basis

Choose two orthonormal localized orbitals

$$\lvert1\rangle, \qquad \lvert2\rangle.$$

They may represent two atoms in a molecule, two wells in a double-well device, two quantum dots, or two localized Wannier-like orbitals in a larger lattice. In the one-particle sector, the state is

H₂⁺ Ion gives a microscopic molecular realization. Its gerade–ungerade pair becomes an orthonormal localized pair under a basis rotation, while the raw atom-centered $1s$ functions overlap and must first be treated with a generalized eigenproblem.

$$\lvert\psi\rangle = c_1\lvert1\rangle+c_2\lvert2\rangle.$$

The tight-binding approximation keeps only a small number of localized orbitals and represents motion between them by matrix elements. For the dimer, the Hamiltonian is

$$H = \epsilon_1\lvert1\rangle\langle1\rvert + \epsilon_2\lvert2\rangle\langle2\rvert - J \left( \lvert1\rangle\langle2\rvert + \lvert2\rangle\langle1\rvert \right),$$

with $J\geq0$ in the standard sign convention. In matrix form,

$$H = \begin{pmatrix} \epsilon_1 & -J\\ -J & \epsilon_2 \end{pmatrix}.$$

The diagonal entries are onsite energies. The off-diagonal entries are hopping amplitudes. Hopping is coherent Hamiltonian coupling, not a classical random jump.

The tight-binding dimer has two localized site orbitals with onsite energies $\epsilon_1,\epsilon_2$ and hopping amplitude $J$. For equal onsite energies, the eigenstates are bonding and antibonding superpositions.

Symmetric Dimer

For equal onsite energies,

$$\epsilon_1=\epsilon_2=\epsilon_0,$$

the Hamiltonian is

$$H=\epsilon_0 I-J\sigma_x.$$

The eigenstates are the bonding state

$$\lvert B\rangle = \frac{1}{\sqrt2} \left( \lvert1\rangle+\lvert2\rangle \right),$$

and the antibonding state

$$\lvert A\rangle = \frac{1}{\sqrt2} \left( \lvert1\rangle-\lvert2\rangle \right).$$

Their energies are

$$E_B=\epsilon_0-J, \qquad E_A=\epsilon_0+J.$$

Thus the splitting is

$$E_A-E_B=2J.$$

With this sign convention, the bonding state is lower. Changing the sign convention for the hopping changes which displayed superposition is called bonding, but not the physical spectrum once the convention is used consistently.

Biased Dimer

If the two sites have different onsite energies, define

$$\bar\epsilon=\frac{\epsilon_1+\epsilon_2}{2}, \qquad \delta=\frac{\epsilon_1-\epsilon_2}{2}.$$

Then

$$H = \bar\epsilon I+\delta\sigma_z-J\sigma_x.$$

The energies are

$$E_\pm = \bar\epsilon \pm \sqrt{\delta^2+J^2}.$$

This is the same avoided-crossing formula as the coupled-well model, with the hopping $J$ playing the role of the tunneling matrix element. When $J=0$, the onsite energies cross as $\delta$ changes sign. When $J\ne0$, the minimum gap is $2J$.

The lower eigenstate is mostly on the lower onsite-energy site far from the crossing. Near $\delta=0$, the eigenstates are strongly delocalized over both sites.

Coherent Site Oscillation

For the symmetric dimer, suppose the particle starts on site $1$:

$$\lvert\psi(0)\rangle=\lvert1\rangle.$$

Because

$$\lvert1\rangle = \frac{1}{\sqrt2} \left( \lvert B\rangle+\lvert A\rangle \right),$$

the bonding and antibonding components acquire different phases. The probability to find the particle on site $2$ at time $\tau$ is

$$P_2(\tau) = \sin^2 \left( \frac{J\tau}{\hbar} \right).$$

This is the same coherent oscillation as a symmetric double well. Tight-binding language calls the off-diagonal coupling hopping; wave-mechanics language calls it tunneling. In a closed two-site Hamiltonian, both words describe coherent unitary evolution.

Second-Quantized Form

The same one-particle Hamiltonian is often written with creation and annihilation operators:

$$H = \epsilon_1 c_1^\dagger c_1 + \epsilon_2 c_2^\dagger c_2 - J \left( c_1^\dagger c_2 + c_2^\dagger c_1 \right).$$

In the one-particle sector, the basis states are

$$\lvert1\rangle=c_1^\dagger\lvert0\rangle, \qquad \lvert2\rangle=c_2^\dagger\lvert0\rangle.$$

Acting on this sector, the second-quantized Hamiltonian has exactly the same two-by-two matrix as above. For one particle, bosonic or fermionic statistics do not change this matrix. Statistics and interactions become essential when more particles can occupy the two sites.

The many-particle notation is developed in Occupation-Number Basis and Many-Particle Hamiltonians.

Complex Hopping And Gauge Phase

One may write a more general hopping term as

$$- \left( J e^{i\alpha}\lvert1\rangle\langle2\rvert + J e^{-i\alpha}\lvert2\rangle\langle1\rvert \right).$$

For an isolated two-site dimer, this phase can be removed by redefining the phase of one site orbital. Therefore the spectrum depends only on $J$.

In a lattice with closed loops, not all hopping phases can be removed at once. Gauge-invariant phases around loops encode magnetic flux and become physically meaningful. That physics belongs to larger tight-binding and quantum-matter models, not to the isolated dimer.

Relation To Coupled Wells

The tight-binding dimer and the coupled double-well model are the same effective Hamiltonian in different language:

$$\lvert1\rangle,\lvert2\rangle \quad\leftrightarrow\quad \lvert L\rangle,\lvert R\rangle,$$

and

$$J \quad\leftrightarrow\quad K.$$

The dimer terminology emphasizes lattice sites and hopping. The coupled-well terminology emphasizes localized wavefunctions and tunneling through a barrier. Both teach that energy eigenstates are generally superpositions of localized basis states.

Common Mistakes

• Treating the site basis as the energy eigenbasis when $J\ne0$.
• Interpreting hopping as irreversible motion rather than coherent Hamiltonian coupling.
• Forgetting that the sign of $J$ is partly a convention for a two-site isolated problem.
• Confusing the one-particle dimer with an interacting two-site Hubbard dimer.
• Applying the two-site spectrum to a long chain without accounting for boundary conditions and additional momentum states.
• Ignoring onsite-energy bias, which can localize eigenstates even when hopping is nonzero.

Where This Is Used

• Coupled Wells and Avoided Crossings gives the wave-mechanics tunneling version of the same matrix.
• Two-State Hamiltonians gives the general diagonalization.
• Pauli-Matrix Hamiltonians writes the dimer as $\bar\epsilon I+\delta\sigma_z-J\sigma_x$.
• Landau–Zener Problem: First Encounter sweeps a two-level detuning through an avoided crossing.
• Double-Well Potential explains how localized left-right states arise from a continuous potential.
• Many-Particle Hamiltonians introduces hopping terms in second quantization.
• Tight-Binding Model extends the dimer to chains, graphs, boundary conditions, and bands.
• Hubbard Model adds spin, onsite interaction, and many-particle sectors to the two-site hopping problem.

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.
• M. P. Marder, Condensed Matter Physics, 2nd ed., Wiley, 2010.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.

Exercises

1. Diagonalize the symmetric dimer Hamiltonian

$$H = \begin{pmatrix} \epsilon_0 & -J\\ -J & \epsilon_0 \end{pmatrix}.$$

Solution

The vector $(1,1)/\sqrt2$ has eigenvalue

$$E_B=\epsilon_0-J,$$

and the vector $(1,-1)/\sqrt2$ has eigenvalue

$$E_A=\epsilon_0+J.$$

Thus

$$\lvert B\rangle = \frac{\lvert1\rangle+\lvert2\rangle}{\sqrt2}, \qquad \lvert A\rangle = \frac{\lvert1\rangle-\lvert2\rangle}{\sqrt2}.$$

2. Show that the biased dimer has minimum gap $2J$.

Solution

The eigenvalues are

$$E_\pm = \bar\epsilon \pm \sqrt{\delta^2+J^2}.$$

The gap is

$$E_+-E_- = 2\sqrt{\delta^2+J^2}.$$

This is minimized at $\delta=0$, where

$$E_+-E_-=2J.$$

3. For the symmetric dimer, find the first time at which a particle initially on site $1$ is certainly on site $2$.

Solution

The site-$2$ probability is

$$P_2(\tau) = \sin^2 \left( \frac{J\tau}{\hbar} \right).$$

The first maximum occurs when

$$\frac{J\tau}{\hbar}=\frac{\pi}{2}.$$

Therefore

$$\tau=\frac{\pi\hbar}{2J}.$$

4. Verify the one-particle matrix from the second-quantized hopping term.

Solution

Use the one-particle basis

$$\lvert1\rangle=c_1^\dagger\lvert0\rangle, \qquad \lvert2\rangle=c_2^\dagger\lvert0\rangle.$$

The onsite terms give

$$\langle1\vert H\vert1\rangle=\epsilon_1, \qquad \langle2\vert H\vert2\rangle=\epsilon_2.$$

The hopping term gives

$$\langle1\vert H\vert2\rangle=-J, \qquad \langle2\vert H\vert1\rangle=-J.$$

Therefore the one-particle matrix is

$$\begin{pmatrix} \epsilon_1 & -J\\ -J & \epsilon_2 \end{pmatrix}.$$