Second Quantization and the Tight-Binding Hamiltonian

The Physical Picture: From Isolated Atoms to a Solid

Imagine a collection of atoms, say hydrogen atoms, initially spaced very far apart. Each electron is tightly bound to its proton, and the system is an insulator. Now, what happens as we bring these atoms closer together to form a regular crystal lattice? As the distance between atoms decreases, the wavefunction of an electron on one atom begins to overlap with the wavefunctions of its neighbors. This overlap provides a pathway for an electron to "hop" or "tunnel" from one atomic site to the next. When this hopping becomes significant, the electrons are no longer localized to individual atoms but become delocalized throughout the crystal, giving rise to metallic behavior.

This transition from an insulating to a metallic state is a profound many-body phenomenon. The tight-binding model is the simplest quantum mechanical description of the delocalized, or itinerant, nature of electrons arising from this wavefunction overlap.

The video below offers an excellent conceptual introduction to this idea, motivating the Hubbard model by considering this exact process.

Building the Hamiltonian

To translate this physical picture into a mathematical model, we need to define our basis and then write the Hamiltonian in terms of operators acting on that basis. Your background in quantum mechanics and previous work with second quantization will be very helpful here.

1. The Basis: Localized Wannier States

We consider a lattice of N sites. Instead of using delocalized plane waves (momentum states), we'll work in a basis of states localized at each lattice site. These states, called Wannier functions, are centered on specific lattice sites. We can denote the single-particle basis state for an electron with spin $\sigma$ localized at site $i$ as $|i,\sigma ⟩$.

These states are assumed to be orthonormal: $⟨i,\sigma |j,{\sigma }^{\prime }⟩={\delta }_{ij}{\delta }_{\sigma {\sigma }^{\prime }}$.

Associated with this basis, we define the fermionic creation operator $c_{i\sigma }^{†}$ that creates an electron in the state $|i,\sigma ⟩$, and its Hermitian conjugate, the annihilation operator $c_{i\sigma }$. As you'll recall from Module 1, these operators obey the canonical anticommutation relations:

The second relation mathematically enforces the Pauli exclusion principle: you cannot create two electrons with the same spin on the same site, as $(c_{i\sigma }^{†}{)}^2=0$.

2. The Hamiltonian Terms

The total Hamiltonian is the sum of kinetic and potential energy. In the tight-binding framework, we model these as follows:

• On-site energy (${ϵ}_0$): This is the energy of an electron when it is localized at a particular site, essentially the energy of the atomic orbital it occupies. If all lattice sites are identical, this is a constant value, ${ϵ}_0$. This is a one-body operator, and its second-quantized form is:

  $${\hat{H}}_{\text{site}}=\sum _{i,\sigma }{ϵ}_0{c}_{i\sigma }^{†}{c}_{i\sigma }={ϵ}_0\sum _{i,\sigma }{\hat{n}}_{i\sigma }$$

  where ${\hat{n}}_{i\sigma }=c_{i\sigma }^{†}{c}_{i\sigma }$ is the number operator. In many cases, this term is a simple energy offset and can be set to zero by redefining the energy reference.

• Hopping Energy ($-t$): This term represents the kinetic energy of the electrons. It quantifies the ability of an electron to tunnel between adjacent lattice sites due to the overlap of their wavefunctions. The process of an electron moving from site $j$ to site $i$ is represented by the operator term $c_{i\sigma }^{†}{c}_{j\sigma }$. The amplitude for this process is the hopping integral, or transfer integral, $t$. The kinetic part of the Hamiltonian sums over all possible hops between neighboring sites.

The following reading provides a detailed and formal construction of the tight-binding Hamiltonian.

3. The Final Hamiltonian

By combining these ideas and making the standard tight-binding approximation, we arrive at the final expression. The key approximations are:

1. We consider only one electronic orbital per atom.
2. We assume hopping only occurs between nearest-neighbor sites.
3. The hopping integral $t$ is the same for all nearest-neighbor pairs.

The resulting Hamiltonian is:

Let's break this down:

• $⟨i,j⟩$ denotes a sum over all pairs of nearest-neighbor sites. Each pair is counted only once.
• The term $c_{i\sigma }^{†}{c}_{j\sigma }$ describes an electron of spin $\sigma$ hopping from site $j$ to site $i$.
• The term $c_{j\sigma }^{†}{c}_{i\sigma }$ is its Hermitian conjugate, describing a hop from $i$ to $j$. Including it ensures the Hamiltonian is Hermitian, as any physical observable must be.
• The parameter $t$ is the hopping amplitude. It is positive by convention, and the negative sign indicates that the energy is lowered when electrons delocalize. A larger $t$ corresponds to stronger wavefunction overlap and a wider energy band.

The on-site energy term ${\sum }_i{ϵ}_0{\hat{n}}_i$ is often dropped, as it simply shifts the entire energy spectrum by a constant without changing the physics. The Hamiltonian above is the most common form of the tight-binding model.

For an alternative and very clear presentation of this construction, you can consult the following notes.

The resource also introduces the on-site interaction term U. The tight-binding model we have constructed is exactly the Hubbard model in the limit where $U=0$.

In the triangular lattice shown above, an electron at site 1 in the central unit cell (UC '0') can hop to six nearest neighbors: sites 2 and 3 in the same cell, as well as adjacent sites in neighboring unit cells. The Hamiltonian ${\hat{H}}_{\text{TB}}$ contains a term for each of these possible moves.

Conclusion

In this lesson, we have formally constructed the tight-binding Hamiltonian, a fundamental model in condensed matter physics. It provides a simple yet powerful description of how electrons behave in a crystalline solid when interactions between them are ignored.

Here are the key takeaways:

• Physical Basis: The model captures the quantum mechanical tunneling of electrons between adjacent atomic sites on a lattice.
• Mathematical Form: The tight-binding Hamiltonian is expressed in second quantization using creation and annihilation operators on a basis of localized Wannier states:
  ${\hat{H}}_{\text{TB}}=-t{\sum }_{⟨i,j⟩,\sigma }\left(c_{i\sigma }^{†}{c}_{j\sigma }+H.c.\right)$
• The Hopping Parameter $t$: This single parameter encapsulates the kinetic energy of the electrons. It determines the energy scale of electron delocalization and the width of the resulting energy band.

You have now successfully built the non-interacting part of the Hubbard model. In the next lesson, we will solve it. Using the Fourier transform technique you learned at the end of Module 1, we will diagonalize this Hamiltonian to find its eigenvalues—the famous band structure—and construct the many-body ground state.