Tight binding model on square lattice with stagger flux
Introduction: The Staggered Flux Model
Electrons moving in a two-dimensional crystal lattice subject to a magnetic field exhibit fascinating quantum phenomena. One particularly interesting setup is the square lattice with a staggered magnetic flux. In this configuration, the effective magnetic flux piercing through adjacent elementary plaquettes alternates in sign (e.g., $+\phi$ and $-\phi$). Because the net macroscopic magnetic flux is zero, the system preserves certain symmetries that are otherwise broken in a uniform magnetic field.
Staggered flux models are critical for engineering artificial gauge fields in cold atom optical lattices and solid-state systems. They naturally give rise to Dirac points—locations in momentum space where the energy dispersion is perfectly linear, much like in graphene. Furthermore, they serve as a fundamental building block for exploring topological phases of matter; by breaking specific symmetries, these models can be used to realize Chern insulators and analogies to the Haldane model.
To study this system, we use a tight-binding model characterized by two main parameters: $t$, the nearest-neighbor hopping amplitude of the electrons, and $\phi$, the magnetic flux per plaquette. Because the flux alternates, we must select a magnetic unit cell that encompasses two sites, which we will denote as sublattices $A$ and $B$. To construct the Hamiltonian, we also need to choose a gauge for the vector potential. While the choice of gauge affects the mathematical form of the equations, the observable physics—like the energy spectrum—remains identical. Below, we derive the properties of this model using two different gauge choices: a Landau-like gauge and a symmetric gauge.
Landau Gauge
We first adopt a transverse Landau-like gauge to study the system. As shown in Figure (#fig:1), the unit cell is selected to enclose two degrees of freedom, corresponding to the $A$ and $B$ sublattices.
The real-space tight-binding Hamiltonian describes electrons hopping between nearest neighbors, picking up a Peierls phase depending on the gauge field. By performing a Fourier transform into momentum space, the Bloch Hamiltonian can be written in the basis of the two-component spinor $\Psi_{\mathbf{k}} = (\psi_{A}(\mathbf{k}), \psi_{B}(\mathbf{k}))^T$ as:
\[H = \sum_{\mathbf{k}} \Psi_{\mathbf{k}}^\dagger H(\mathbf{k}) \Psi_{\mathbf{k}}\]where the Hamiltonian matrix $H(\mathbf{k})$ is given by:
\[H(\mathbf{k}) = t \begin{pmatrix} 0 & f(\mathbf{k}) \\ f^*(\mathbf{k}) & 0 \end{pmatrix}\]The off-diagonal term $f(\mathbf{k})$ encodes the sum of the complex hopping amplitudes from site $A$ to its neighboring $B$ sites. For our specific choice of gauge and unit cell, it takes the form:
\[f(\mathbf{k}) = 1 + e^{i(k_x-k_y)} + e^{i(-k_x-k_y)} + e^{-i(2k_y+\phi)}\]| To find the energy spectrum, we solve for the eigenvalues of $H(\mathbf{k})$, which are given by $\varepsilon(\mathbf{k}) = \pm t | f(\mathbf{k}) | $. By computing $ | f(\mathbf{k}) | = \sqrt{f(\mathbf{k})f^*(\mathbf{k})}$ and applying standard trigonometric identities, the energy dispersion simplifies to: |
We can easily locate the Dirac points of this dispersion relation by finding the momenta where the energy gap closes, namely where $\varepsilon(\mathbf{k}) = 0$. Setting the terms under the square root to zero yields the locations of the Dirac nodes:
\[\left( \pm \frac{\pi}{2}, \pm \frac{\pi}{2} - \frac{\phi}{2} \right), \quad \left( \mp \frac{\pi}{2}, \pm \frac{\pi}{2} - \frac{\phi}{2} \right)\]While the Dirac points are clear in this gauge, demonstrating the linear dispersion around them is mathematically simpler when using a symmetric gauge, which we will explore next.
Symmetric Gauge
An alternative approach is to use a symmetric gauge, which distributes the Peierls phase symmetrically around the perimeter of each plaquette. As illustrated in Figure (#fig:2), the unit cell is chosen as a single plaquette with positive flux $\phi$, which again contains the two sublattice degrees of freedom, $A$ and $B$.
In this gauge, the Bloch Hamiltonian retains the same bipartite structure:
\[H(\mathbf{k}) = t \begin{pmatrix} 0 & f(\mathbf{k}) \\ f^*(\mathbf{k}) & 0 \end{pmatrix}\]However, due to the symmetric distribution of the gauge field, the function $f(\mathbf{k})$ is modified to:
\[f(\mathbf{k}) = e^{i\frac{\phi}{4}} + e^{i\left(\frac{\phi}{4} - 2k_y\right)} + e^{i\left(k_x-k_y-\frac{\phi}{4}\right)} + e^{i\left(-k_x-k_y-\frac{\phi}{4}\right)}\]| The energy spectrum remains $\varepsilon(\mathbf{k}) = \pm t | f(\mathbf{k}) | $. Evaluating the magnitude of this new $f(\mathbf{k})$ yields a beautifully symmetric dispersion relation: |
The dispersion relation in Equation (#eq:14) shows that the zero-energy points are located symmetrically at $(\pm \frac{\pi}{2}, \pm \frac{\pi}{2})$ and $(\mp \frac{\pi}{2}, \pm \frac{\pi}{2})$.
To explicitly reveal the linear dispersion (the Dirac cone), we can perform a Taylor expansion of $f(\mathbf{k})$ around one of these Dirac points, for instance, $\mathbf{K} = (\frac{\pi}{2}, \frac{\pi}{2})$. Let us define a small momentum deviation $\mathbf{q} = (q_x, q_y) = \mathbf{k} - \mathbf{K}$, such that $q_x = k_x - \frac{\pi}{2}$ and $q_y = k_y - \frac{\pi}{2}$. Expanding to first order in $\mathbf{q}$, we find:
\[f(\mathbf{K}+\mathbf{q}) \approx 2t \left[ (q_x - q_y)\sin\frac{\phi}{4} + i(q_x + q_y)\cos\frac{\phi}{4} \right]\]This linearization confirms that the low-energy physics is governed by massless Dirac fermions. The effective low-energy Hamiltonian near the Dirac point can thus be written in terms of the Pauli matrices $\vec{\sigma}$:
\[H_{\text{eff}}(\mathbf{q}) \approx 2t (\mathbf{v} \cdot \mathbf{q}) \cdot \vec{\sigma}\]where the components of the generalized velocity $\mathbf{v}$ are directly determined by the flux $\phi$.
Conclusion
In summary, the tight-binding model on a square lattice with staggered magnetic flux provides a robust platform for engineering Dirac semimetals in two dimensions. Regardless of the chosen gauge—Landau or symmetric—the system consistently exhibits a vanishing energy gap at specific points in the Brillouin zone, surrounded by a perfectly linear, cone-like energy dispersion. These massless Dirac points are protected by the symmetries of the lattice and the alternating flux. By carefully tuning the flux $\phi$ or introducing additional symmetry-breaking terms (such as next-nearest-neighbor hoppings or staggered sublattice mass potentials), this foundational model can be transitioned into exotic topological insulating phases, making it an essential concept for understanding quantum anomalous Hall effects and topological states of matter.
