Second Quanitized Hamiltonians in Tight-Binding Models of Materials

Alex Heilman

February 15, 2023

Mathematics of Physics

Tight-binding Hamiltonian heuristic diagram.

Creation and annihilation operators may be used to construct tight-binding models of crystalline systems. Below, this form is constructed intuitively in an attempt to facilitate conceptual understanding.

The Hamiltonian is the mathematical object that completely defines the time evolution of a physical system. For closed systems, it also gives us the total energy of a state. Assuming that the system is closed and that all forces acting on the system are conservative, we may then assume the Hamiltonian is simply the sum of the kinetic and potential energies of all constituent particles.

Recalling the form of classical kinetic energy for a particle of mass \(m\), and momentum \(p=mv\) we may give the slighlty more informative, yet still general form of the classical Hamiltonian:

\[ H=T+V = \frac{1}{2}mv^2 + V(r) = \frac{p^2}{2m}+V(r) \]

where \(V(r)\) represents a potential field which depends on position, and corresponds to an arbitrary (conservative) force or set of forces acting on the system.

Now that we have a general form in mind, we'll first consider what we care about in the modeling of crystalline systems. Then, we'll build a useful form for a Hamiltonian describing crystalline systems in which electrons are effectively confined to the lattice sites and only interact with their nearest neighbors.

Simplifications

When modeling material systems, we first need to determine what, exactly, we need to consider to capture the material properties of the system. Put more simply: what do we really care about when considering solid state systems?

In solid state, the most interesting steady-state dynamical properties, such as electrical and thermal conductivity, tend to depend mostly on the electrons. In fact, in many systems, we need only consider the behaviour of the electrons to decribe these phenomena to a good approximation. [Disclaimer: Other factors may also have large influence, but here we intend to simplify matters and approximate; see e.g. quasi-particles]

As such, we leave the ionic lattice, with nuclei at each site, in the background. We then only consider the electron's dynamics under the effect of some periodic potential, corresponding to the lattice's effects.

Below, we first give a review of the first and second quantized free-electron Hamiltonians, and then build the analogous Hamiltonians for electrons constrained to a background lattice.

Free Electron Hamiltonian

In the case of the free-electron, we have no potential energy and essentially have only a kinetic term in the Hamiltonian:

\[ H=\frac{1}{2}mv^2=\frac{p^2}{2m} \]

This is the classical form of the free electron's Hamiltonian. However, in modern considerations, we know we need to account for quantum mechanical effects to better desrcribe the dynamics of the system.

First-Quantization

Now in the framework of first quantization, we promote variables, such as position, \(x\), and momentum, \(p\), to operators which act on states of the system. In position space, they take the form below:

\[ p\rightarrow -i\hbar \vec{\nabla}\quad\quad\quad x\rightarrow \hat{x} \]

where \( \vec{\nabla}\) is the position-space gradient of the state being acted on, whereas \(\hat{x}\) returns the position vector of the state it acts on.

This gives us the well-known form for the quantum mechanical free-electron Hamiltonian (as an operator on position space) as follows:

\[\hat{H}=-\frac{\hbar^2}{2m}\nabla^2\]

Now, applying the Schrodinger Equation to the position-basis electron wavefunction, we may easily determine the eigenstates of this Hamiltonian, as sketched below:

\[ \Big( -\frac{\hbar^2}{2m}\nabla^2 \Big)\psi(\vec{r},t) = -i\hbar \frac{\partial}{\partial t} \psi(\vec{r},t) \]\[ \Rightarrow \quad \psi_{\vec{p}}(\vec{r},t)=Ae^{-i(\vec{r}\cdot \vec{p}-E t)/\hbar} \]

where the eigenvalues of eigenstates \(\psi_{\vec{p}}\) are determined by their momentum \(\vec{p}\), since each's energy is \(p^2/2m\).

Further setting \(\omega=E/\hbar\) and \(\vec{k}=\vec{p}/\hbar\), we may simplify a bit to arrive at the expression for a free electron's energy (now determined by it's \(k\) value):

\begin{align} H\psi_k &= E_k\psi_k\\ \Big( -\frac{\hbar^2}{2m}\nabla^2 \Big)e^{-i(\vec{r}\cdot \vec{k}-\omega t)} &= \Big( \frac{\hbar^2k^2}{2m}\Big)e^{-i(\vec{r}\cdot \vec{k}-\omega t)}\\ \\ \Rightarrow \quad E_k & = \frac{\hbar^2k^2}{2m} \end{align}

Now we have a nice form for the dispersion relation of a free electron. But, we still have a somewhat ugly Hamiltonian that requires us to take derivatives with respect to position (though these are admittedly easy in the basis we've presented). We now seek a more sophisticated form of the Hamiltonian which simply 'counts' the energy of the system, piece by piece.

Second-Quantization

In second-quantized formalisms or, equivalently, the occupation number representation, we represent particles as operators on the vacuum state (think creation and annihilation operators a la the QHO). In lieu of a deep discussion of second-quantization, we defer interested parties to other articles, and review only the most basic and relevant concepts here.

For some state defined by good quantum numbers (here \(\vec{k}\), but this may also include a spin state \(\sigma\), etc.), we have a unique pair of a creation and annihilation operator: (\(\hat{a}^{\dagger}_{\vec{k}},\hat{a}_{\vec{k}}\)), respectively. These act on the particle number basis (Fock) states as shown below:

\begin{align} \hat{a}^{\dagger}_{\vec{k}}\vert n_k\rangle & = \sqrt{n_k+1}\vert n_k +1\rangle\\ \hat{a}_{\vec{k}}\vert n_k\rangle & = \sqrt{n_k}\vert n_k -1\rangle\\ \end{align}

[Note that we consider all other \(\vec{k}\) states here to be empty; the antisymmetric behaviour of Fermionic systems may leave the above false up to a minus sign if different operators are concatenated]

We may compose these two operators of each pair to construct a number operator that also has an eigenbasis coinciding with the particle number basis, with eigenvalues representing the number of particles in each state. Explicity, we construct the number operator for a state defined by \(\vec{k}\) to be:

\[ \hat{n}_{\vec{k}} = \hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}} \]

Note that for Fermions, such as electrons, the number operator \( \hat{n}_{\vec{k}}\) may only return 0 or 1 for each \(\vec{k}\).

Considering this, we may now compose a new form of the free-electron Hamiltonian (or really any Hamiltonian in a good basis for it's eigenspace), as given below:

\[ \hat{H} = \int_k \frac{d\vec{k}}{(2\pi)^3} E_k \hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}} \]

This may look scary, but it's action really is more intuitive: for every possible energy state of the system we simply see if there's a particle in that state, and if there is, we add it's energy to the total energy.

Having now seen the second-quantized form of some Hamiltonian, we consider the second-quantized form of particles constrained to some lattice.

Lattice Hamiltonian

In the free-electron picture, we didn't account for any interactions between electrons or even transitions between states. Thus, from a certain perspective, the electrons were 'constrained' to their respective \(\vec{k}\) states (since they couldn't transition).

Now, for electrons tightly bound to their positions on the underlying lattice, we may imagine these electrons are no longer constrained in \(k\)-space, but now are constrained to certain lattice sites (or one of several orbitals localized at each lattice site) in real space.

In this scenario, momentum is no longer a good quantum number. However, the lattice position \(i\) (where \(\vec{r}_i\) is a lattice vector) now is a good quantum number, since each electron has a unique lattice position or orbital in real space.

In the most extreme case of localization (where electron's position-space wavefunctions, or Wannier functions, may be taken to not overlap at all between lattice positions), there will be no interaction between different sites. And hence, if all lattice sites are equivalent, and each site has a bond strength of \( u_{site} \), the Hamiltonian takes the following form:

\[ \hat{H}_{sites} = -u_{site}\sum_{i}\hat{a}^{\dagger}_{i}\hat{a}_{i} \]

where each lattice site \( i \) with lattice vector \(\vec{r}_i\), has it's own set of creation and annihilation operators (\(\hat{a}^{\dagger}_{i}, \hat{a}_{i}\)). This Hamiltonian is now just counting the number of electrons (either 0 or 1) at every site \(i\) and adding it's contributed energy \(u_{site}\) if there's one at that site.

Nearest Neighbors

In a somewhat less localized situation then, the interaction between electrons can be modeled, to some approximation, to only involve nearest neighbors.

Again, using the intuition gained from the construction of the second-quantized free-electron Hamiltonian, we may then consider the form of such a nearest neighbors term of the Hamiltonian, which naturally acts on a particle number representation corresponding to specfic lattice sites, as below.

\[ \hat{H}_{n.n.} = -t\sum_{\langle i, j \rangle}\big(\hat{a}^{\dagger}_{j}\hat{a}_{i}+\hat{a}^{\dagger}_{i}\hat{a}_{j}\big) \]

Here, \(n.n.\) denotes nearest neighbors, subscript \(i\) denotes a particle located at lattice position \(\vec{R}_i\), and \(\langle i,j\rangle\) denotes the set of nearest neighbor sites \(i,j\). Then, \(t\), often termed the hopping parameter, corresponds to the interaction energy between nearest neighbors. Note that here we've assumed interactions are isotropic and uniform (such that all sites are equivalent).

To understand this form further, let's consider the first term above, \(-t\hat{a}^{\dagger}_{j}\hat{a}_{i}\). This term corresponds to a particle at site \(i\) being destroyed and one at site \(j\) being created, effectively accounting for an electron that 'hops' from site \(i \) to site \(j\) and gaining energy \(t\). However, if one electron is already at site \(j\ (n_j=1)\), this cannot happen and the contribution from the term will be zero.

Guarantee of Hermiticity

One may wonder why we need to include the second term (\(\hat{a}^{\dagger}_{i}\hat{a}_{j}\)) when we have the first (\(\hat{a}^{\dagger}_{j}\hat{a}_{i}\)). Recall that Hamiltonians must be Hermitian. Also recall that any non-Hermitian operator may be transformed into a Hermitian operator by defining a new operator that is the sum of itself and it's Hermitian conjugate:

\[ A \neq A^{\dagger} \ \ \ \ \Rightarrow \quad \text{Define:} \ \ C = C^{\dagger}=A+A^{\dagger} \]

Hence, if we simply include, always, the Hermitian conjugate of the form of the term we care to model, we are guaranteed to have a Hermitian Hamiltonian. To be explicit, for the case of nearest neightbors, the term

\[ \hat{a}^{\dagger}_{j}\hat{a}_{i} \]

is not Hermitian. But if we add it's Hermitian conjugate, \(h.c. \), to the term, the resulting operator is guaranteed to be Hermitian. The addition of just the term \(h.c.\) to denote this scheme is standard practice in applications of the models discussed here.

\(H_{n.n.}\) accounts only for the energy of the exchange of particles between nearest neighbor sites. That is, we've assumed that each real-space lattice site has a unique state \(i\) and that the only interaction is between states directly next to each other, where the likelihood of transition/energy relevant to each interaction is parameterized by \(t\).

If \(t\) is large, the hopping becomes less likely and the sites become more isolated in terms of interaction. Conversely, if \(t\) is small, hopping is more likely and neighbors tend to interact more.

Tight-Binding Hamiltonian

For a tightly-bound system of electrons constrained to some underlying lattice, we may then approximate the Hamiltonian to be the sum of the two Hamiltonians considered above.

\[ \hat{H}_{tight-binding} = \hat{H}_{sites}+\hat{H}_{n.n.} \]

Thus, we can intuit that this tight-binding model succintly describes the energy and dynamics of a situation in which neighboring electrons interact by transitioning into other states or lattice sites. It also accounts for the site's energies themselves.

For isotropic cases, where all site are equivalent, the on-site term (\(\hat{H}_{sites}\)) just acts as an additive constant, and may be neglected in considerations of dynamics.

Example: Square Lattice

As a simple example of an application of this form, let's consider a uniform square lattice. Since each site is equivalent, we'll neglect the constant on-site term (effectively setting it to zero) and then treat only the nearest neighbors, leaving us with the second-quantized tight-binding Hamiltonian for the square lattice:

\[ \hat{H} = -t\sum_{\langle i, j \rangle}\big(\hat{a}^{\dagger}_{j}\hat{a}_{i}+\hat{a}^{\dagger}_{i}\hat{a}_{j}\big) \]

Now, there are four nearest neighbors for every site. Furthermore, these are all offset by the cartesian vectors \(\pm a\hat{x}\) or \(\pm a\hat{y}\). So, expanding our sum over nearest neighbors above into a sum over all lattice sites \(r_i\) and nearest-neighbor displacement vectors \(\delta\), we arrive at the following:

\[\small \sum_{\langle i,j\rangle}=\frac{1}{2}\sum_{r_i}\sum_{\delta}\quad \Rightarrow\quad\hat{H} = -\frac{t}{2}\sum_{r_i}\sum_{\delta}^{\lbrace\pm a\hat{x},\pm a\hat{y}\rbrace}\big(\hat{a}^{\dagger}_{r_i+\delta}\hat{a}_{r_i}+\hat{a}^{\dagger}_{r_i}\hat{a}_{r_i+\delta}\big) \]

While this isn't very useful in and of itself, it is an explicit form for the Hamiltonian of the square lattice system. So, in principle, we now have the unique operator dictating the evolution of such a system (we could also substitute the Fourier transforms of the creation and annihilation operators into this form to then solve for the system's dispersion relation).

Of course this truncation to nearest neighbor interactions is usually just an approximation (it's not impossible that the Wannier functions don't overlap on a larger scale). Some materials require us to include further range terms; we could also add a term representing next-nearest-neighbor interactions of the form below:

\[ \hat{H}_{n.n.n.} = -v\sum_{\langle\langle i, j \rangle\rangle}\big(\hat{a}^{\dagger}_{j}\hat{a}_{i}+\hat{a}^{\dagger}_{i}\hat{a}_{j}\big) \]

(where now \(\langle\langle i, j \rangle\rangle\) denotes the set of next nearest neighbors) but the essential concept is the same: each lattice site has a unique set of creation and annihilation operators; and these allow us to write the corresponding Hamiltonian succinctly in a form with an obvious interpretation.

We also haven't really accounted for interactions between electrons themselves. But, since we're considering a system that's localized, this may be a reasonable interaction to neglect.

The form of the tight-binding Hamiltonian given above tends to be a more appropriate model for insulating systems, as they tend to have more localized electrons.

Further Resources

Note that the point of this article was only to provide some rationale and justification of the real-space form of the tight-binding Hamiltonian in a second-quantized form. This real-space force can be converted into a momentum-space form, from which we may extract a dispersion relation, by inserting the Fourier transform of the creation and annihilation operators. In this case we would end up with a diagonal form in \(k\)-space of \(H\):

\[ \vec{H} = \sum_{\vec{k}}^{B.Z.}\epsilon (\vec{k})\hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}} \]

where \(\epsilon(\vec{k})\) is the dispersion relation, and details the band structure of the system (the sum is over Bloch wave vectors in the first Brillouin Zone).

For more detail pertaining to the mathematics, see these notes on second-quantized tight-binding models. Also, see these more in-depth lecture notes.

The second-quantized tight-binding model is often a first example for more contemporary treatments such as the t-J and Hubbard model, as well as the Hofstadter model. Interactions between more than two sites at a time, with terms of the form \(a^{\dagger}_ja^{\dagger}_ka_ia_j\) and larger, are also possible but neglected here.

Tags: Solid State Physics Intro

Alex Heilman

A physics graduate student with a B.S. in physics from Lebanon Valley College.