🚀 Deployed App

The app is available on Streamlit (activate the app if required): Click here to access

To run locally, follow these steps:

• install libraries via "pip install -r requirements.txt" or "conda install --file requirements.txt"
• streamlit run app.py

📚 Theory

Hofstadter's butterfly for 2D finite lattices $n\times n$ with Zeeman term:

$$ \hat{H} = \sum_{ij, \sigma \sigma^{\prime}} t^{ \sigma \sigma^{\prime}}_{ij} \hat{a}^{\dagger \sigma }_i \hat{a}^{ \sigma^{\prime}}_j - \frac{1}{2} g \mu_B B_z \sum \hat{a}^{\dagger \sigma }_i \sigma^z \hat{a}^{\sigma^{\prime}}_i. $$

Peierls substitutions is taken into account via phase factor:

$$ t_{ij} \rightarrow t_{ij} e^{\gamma_{ij}}, $$

$$ \gamma_{ij} = \Phi_{ij} / \Phi_0 = \frac{\int B_z dS_{ij}}{h/e} = -2 \pi i \frac{e}{h} B_z \frac{1}{2} (x_i + x_j) (y_i - y_j).$$

▶️ Usage

class HB_lattice contains the following methods:

• create_lattice(type, num_cells, bond_len) creates and plots the lattice with given properties
  • type (str): symmetry of lattice (triangular, square, honeycomb)
  • num_cells (int): number of cells
  • bond_len (float): bond length (in nm)
• create_custom_lattice(file_path) creates the lattice using file_path csv or txt file. File should contain two (x, y) columns (in nm) with sites coordinates.
• plot_hofstadter(b_max, b_steps, g_factor, ham_type, **params) calculate and plot eigenvalues varying magnetic field.
  • b_max (float): maximum magnetic field (in Tesla).
  • b_steps (int): Number of points in the magnetic field range (0, b_max)
  • g_factor (float): g-factor used for Zeeman splitting
  • ham_type (str): type of Hamiltonian construction method to use. There are two ways: "hopping" and "interpolation" with the following parameters
  • t (list): array of hoppings for different neighbor distances (in eV).
  • t_so (list): array of spin–orbit coupling hoppings for different neighbor distances (in eV).
  • a_param and b_param (float): parameter of interpolation (in eV) t_ij = a_param * exp ( - r_ij / b_param), r_ij - distance between sites
• plot_dos(b_value, e_min, e_max, e_step, smear) plot densities of states
  • b_value (list): magnetic field values to choose eigenvalue sets
  • e_min (float), e_max (float), e_step (int) energy range np.linspace(e_min, e_max, e_step) to plot DOS
  • smear (float): numerial smearing
• plot_map(b_value, num_eigvecs, mapRes, smear) plots a spatial map of the eigenstate probability density.
  • b_value (float): magnetic field values to choose eigenvector sets
  • num_eigvecs (list): indices of eigenstates to include in the map.
  • mapRes (int): resolution of the map grid.
  • smear (float): smearing parameter for the Gaussian function (default 10 nm).

📌 Examples

• Square, triangular and honeycomb lattices [1]. Parameters: t = 0.01 eV; num_cells = 1, 15; bond_len = 10 nm;

• 2x2 lattice square lattice with Peierls only (g = 0) Zeeman and Peierls & Zeeman terms:

• DOS and charge density analysis for square lattice with num_cells = 2; bond_len = 10 nm; t = 0.01 eV;