Electron Density Calculation Using the Number Operator

Welcome to the next lesson in our exploration of the tight-binding model. Last time, we took a significant step by constructing the many-body ground state for our system of non-interacting electrons. We saw how, at zero temperature, the electrons fill the available energy levels up to the Fermi energy, forming what is known as the Fermi sea. Our ground state, $∣ Ψ_{0} ⟩$ , is therefore a state where all momentum modes with $∣ k ∣ \leq k_{F}$ are occupied.

Having defined the ground state, our focus now shifts to using it to understand the physical properties of the system. In this lesson, we will address a fundamental question: what is the distribution of electrons in real space? We will accomplish this by calculating the expectation value of the number operator at an arbitrary site $j$ , which corresponds to the local electron density. This calculation will bridge the gap between our abstract momentum-space description and the concrete, physical lattice.

The Site Number Operator in Real and Momentum Space

Let's begin by defining the operator that counts the total number of electrons at a specific lattice site, say site $j$ . This operator, $n^_{j}$ , is simply the sum of the number operators for spin-up and spin-down electrons at that site:

n^_{j} = n^_{j ↑} + n^_{j ↓} = c_{j ↑}^{†} c_{j ↑} + c_{j ↓}^{†} c_{j ↓}

Our goal is to compute the expectation value of this operator in the non-interacting ground state, $⟨ Ψ_{0} ∣ n^_{j} ∣ Ψ_{0} ⟩$ .

A challenge immediately presents itself: the number operator $n^_{j}$ is expressed in terms of site-based creation and annihilation operators ( $c_{j σ}^{†}, c_{j σ}$ ), while our ground state $∣ Ψ_{0} ⟩$ is defined in the basis of momentum eigenstates ( $c_{k σ}^{†}, c_{k σ}$ ). To proceed, we must express $n^_{j}$ in the momentum basis.

Recall the Fourier transform relations we used to diagonalize the tight-binding Hamiltonian. These allow us to move between the site (real-space) basis and the momentum (k-space) basis. For a 1D chain with $N$ sites and lattice spacing $a$ , the position of site $j$ is $r_{j} = j a$ . The transformations are:

c_{j σ} = \frac{1}{N} k \sum e^{ik r_{j}} c_{k σ}

c_{j σ}^{†} = \frac{1}{N} k \sum e^{- ik r_{j}} c_{k σ}^{†}

Here, the sum is over the $N$ allowed values of momentum $k$ in the first Brillouin zone. The document "The Limits of the Hubbard Model" provides a good reference for these transformations.

The Limits of the Hubbard Model

This document provides a detailed mathematical treatment of the Hubbard model's limits. The section on the tight-binding model contains the explicit definitions of the momentum-space operators.

In section 6.2, "Tight Binding: Weak Interactions", please review the definitions for the momentum space operators and their inverses. These are the key relations we will use to transform our number operator.

Now, let's substitute these transformations into the expression for $n^_{j}$ :

n^_{j} = σ \sum c_{j σ}^{†} c_{j σ} = σ \sum (\frac{1}{N} k \sum e^{- ik r_{j}} c_{k σ}^{†}) (\frac{1}{N} k^{'} \sum e^{i k^{'} r_{j}} c_{k^{'} σ})

Combining the terms, we get:

n^_{j} = \frac{1}{N} σ \sum k, k^{'} \sum e^{- i (k - k^{'}) r_{j}} c_{k σ}^{†} c_{k^{'} σ}

This expression for the site number operator is now entirely in terms of momentum-space operators, which is exactly what we need to calculate its expectation value.

Calculating the Expectation Value

We are now ready to compute $⟨ n^_{j} ⟩ = ⟨ Ψ_{0} ∣ n^_{j} ∣ Ψ_{0} ⟩$ .

⟨ n^_{j} ⟩ = ⟨ Ψ_{0} \frac{1}{N} σ \sum k, k^{'} \sum e^{- i (k - k^{'}) r_{j}} c_{k σ}^{†} c_{k^{'} σ} Ψ_{0} ⟩

We can take the constant factors and the summation outside the expectation value:

⟨ n^_{j} ⟩ = \frac{1}{N} σ \sum k, k^{'} \sum e^{- i (k - k^{'}) r_{j}} ⟨ Ψ_{0} ∣ c_{k σ}^{†} c_{k^{'} σ} ∣ Ψ_{0} ⟩

The crucial part of this expression is the expectation value $⟨ Ψ_{0} ∣ c_{k σ}^{†} c_{k^{'} σ} ∣ Ψ_{0} ⟩$ . Let's analyze it. The ground state $∣ Ψ_{0} ⟩$ is built by occupying a specific set of momentum states. The operator $c_{k^{'} σ}$ annihilates a particle with momentum $k^{'}$ and spin $σ$ . If this state is empty in $∣ Ψ_{0} ⟩$ , the result is zero. If it's occupied, the result is a new state with one fewer particle. The operator $c_{k σ}^{†}$ then creates a particle with momentum $k$ .

Because the states with different momenta are orthogonal, the overall matrix element $⟨ Ψ_{0} ∣ c_{k σ}^{†} c_{k^{'} σ} ∣ Ψ_{0} ⟩$ will be non-zero only if $k = k^{'}$ . In this case, we are simply calculating the expectation value of the number operator for that momentum mode, $n^_{k σ} = c_{k σ}^{†} c_{k σ}$ .

Therefore, we have:

⟨ Ψ_{0} ∣ c_{k σ}^{†} c_{k^{'} σ} ∣ Ψ_{0} ⟩ = δ_{k, k^{'}} ⟨ Ψ_{0} ∣ n^_{k σ} ∣ Ψ_{0} ⟩ = δ_{k, k^{'}} ⟨ n^_{k σ} ⟩_{0}

where $⟨ n^_{k σ} ⟩_{0}$ is the average occupation number of the state $(k, σ)$ in the ground state.

Substituting this back into our main equation simplifies it dramatically. The sum over $k^{'}$ collapses due to the Kronecker delta $δ_{k, k^{'}}$ , and the exponential term becomes $e^{0} = 1$ :

⟨ n^_{j} ⟩ = \frac{1}{N} σ \sum k \sum ⟨ n^_{k σ} ⟩_{0}

Uniform Electron Density

The expression we've arrived at is quite revealing. The right-hand side, $\frac{1}{N} \sum_{σ, k} ⟨ n^_{k σ} ⟩_{0}$ , is simply the total number of electrons in the system, $N_{e}$ , divided by the total number of sites, $N$ . This is, by definition, the average electron filling, $n$ .

⟨ n^_{j} ⟩ = \frac{N _{e}}{N} = n

This result is independent of the site index $j$ . It tells us that in the non-interacting tight-binding model, the electron density is perfectly uniform across the entire lattice.

This should feel intuitively correct. The Hamiltonian possesses translational symmetry—there is nothing that distinguishes one site from another. The eigenstates we found are Bloch waves, which are completely delocalized plane waves modulated by the lattice periodicity. It is therefore natural that the charge distribution in these states is also uniform. Every site has an equal probability of being occupied.

This result holds for any filling fraction $n$ . For example, at half-filling ( $n = 1$ ), where we have one electron per site on average, the expectation value of the number operator at any site $j$ is exactly 1. This does not mean that every site is occupied by exactly one electron at all times; quantum fluctuations, driven by the hopping term $t$ , mean that electrons are constantly moving. At any instant, a site could be empty, singly occupied, or doubly occupied. However, the time-averaged (and spatially-averaged) density at every site is precisely $n$ .

The following video discusses the filling of bands and touches upon this outcome from a more physical, less mathematical perspective.

Introduction to Solid State Physics, Lecture 6: One-dimensional Tight Binding Model for Electrons

This lecture by Sergey Frolov provides an excellent physical picture of band filling in the tight-binding model.

Watch the segment on filling the bands. Notice the key point: for a system with N sites and N_e valence electrons, the average filling is n = N_e/N. The conclusion that a band is half-filled for N_e=N directly implies a uniform average density of 1 electron per site.

Conclusion

In this lesson, we have connected our momentum-space picture of the tight-binding ground state back to real space by calculating the local electron density.

Here are the key takeaways:

To calculate the expectation value of a real-space operator like $n^_{j}$ , it must be expressed in the momentum basis using Fourier transforms.
The expectation value $⟨ n^_{j} ⟩$ simplifies because the ground state is an eigenstate of the momentum-space number operators $n^_{k σ}$ .
The final result, $⟨ n^_{j} ⟩ = n$ , shows that the electron density is uniform across all sites and equal to the average filling. This is a direct consequence of the translational symmetry of the non-interacting system.

This uniform density provides a crucial baseline. When we introduce the on-site Coulomb interaction $U$ to form the Hubbard model, this simple picture will break down. The interaction term will penalize double occupancy, introducing correlations between electrons that can lead to non-trivial charge and spin ordering.

In our next lesson, we will complete our analysis of the non-interacting model by calculating another important ground-state property: the total kinetic energy as a function of band filling. This will provide us with a reference point for understanding how electron interactions later modify the system's energy.

Recommended resources

The Site Number Operator in Real and Momentum Space

Calculating the Expectation Value

Uniform Electron Density

Conclusion