Quickstart

The pybott package provides tools for calculating the Bott index, topological invariant that can be used in real space to distinguish topological insulators from trivial insulators. This index measures the commutativity of projected position operators, and is based on the formalism described by T. A. Loring and M. B. Hastings. This package also allow to compute the spin Bott index as well as functions to assist in creating phase diagrams.

Installation

pip install pybott

Usage

Here are three examples of PyBott applications. Explanations are minimal, and the codes show only the use of PyBott, not how the Hamiltonian are defined, for that you can download the associated files by clicking on the section’s titles.

Haldane model

from pythtb import * 
from pybott import bott

# Define the parameters of the Haldane model
n_side = 16  # Grid size for the model
t1 = 1       # NN coupling
t2 = 0.3j    # NNN complex coupling
delta = 1    # On-site mass term
fermi_energy = 0  # Energy level in the gap where the Bott index is calculated

### Define the Haldane model using the `pythTB` library
### lat=[[1.0,0.0],[0.5,np.sqrt(3.0)/2.0]] ...

lattice = fin_model.get_orb()
ham = fin_model._gen_ham()

bott_index = bott(lattice, ham, fermi_energy)

print(f"The Bott index for the given parameters δ={delta} and {t2=} is: {bott_index}")

This code should output:

The Bott index for the given parameters δ=1 and t2=0.3j is: 0.9999999999999983

In this example, we use the pythTB library to create a finite piece of the Haldane model, which is a well-known model in condensed matter physics used to simulate topological insulators without an external magnetic field. The model is defined on a hexagonal lattice with both nearest-neighbor (NN) couplings and complex next-nearest-neighbor (NNN) couplings, as well as an on-site mass term. After constructing the model, we cut out a finite system from which we extract the coordinate lattice and the Hamiltonian. Finally, we use the bott function to compute the Bott index for a given fermi energy chosen in the bulk gap.

Photonic crystal

In this example, we model a photonic honeycomb crystal, which introduces additional complexity compared to electronic systems. Here, the interactions are mediated by the electromagnetic field, and the system can break time-reversal symmetry using an external magnetic field, represented by delta_b. Additionally, the inversion symmetry can be broken by the term delta_ab. For an extensive description of this system, you can read this paper.

Since the system involves light polarization, we need to account for the polarization effects when computing the Bott index.

Note that this system, unlike the Haldane model, is not Hermitian; therefore, this must be taken into account when computing the Bott index. Additionally, the frequencies of the system are not the eigenvalues $\lambda$ but $-\mathrm{Re}(\lambda)/2$. This requires special treatment, which is performed before using the provided function sorting_eigenvalues.

import numpy as np

from pybott import bott_vect,sorting_eigenvalues

ham = np.load("effective_hamiltonian_light_honeycomb_lattice.npy")
# The matrix is loaded directly because calculating it is not straightforward.
# For more details, refer to Antezza and Castin: https://arxiv.org/pdf/0903.0765
grid = np.load("honeycomb_grid.npy") # Honeycomb structure
omega = 7

delta_b = 12
delta_ab = 5

modified_ham = break_symmetries(ham, delta_b, delta_ab)

evals, evects = np.linalg.eig(modified_ham)

frequencies = -np.real(evals) / 2

frequencies, evects = sorting_eigenvalues(
    frequencies, evects, False
)

b_pol = bott_vect(
    grid,
    evects,
    frequencies,
    omega,
    orb=2,
    dagger=True,
)

print(f"The Bott index for the given parameters Δ_B={delta_b} and Δ_AB={delta_ab} is: {b_pol}")

This code should output:

The Bott index for the given parameters Δ_B=12 and Δ_AB=5 is: -0.9999999999999082

Kane-Mele Model

In this example, we calculate the spin Bott index for the Kane-Mele model, which is a fundamental model in condensed matter physics for studying quantum spin Hall insulators. The Kane-Mele model incorporates both spin-orbit coupling and Rashba interaction, leading to topological insulating phases with distinct spin properties.

The system is defined on a honeycomb lattice, and interactions are mediated through parameters like nearest-neighbor hopping (t1), next-nearest-neighbor spin-orbit coupling (t2), and Rashba coupling (rashba). Additionally, on-site energies (esite) introduce mass terms that can break certain symmetries in the system.

To compute the spin Bott index, we need to account for the spin of the system, which is done using the $\sigma_z$ spin operator.

Note that if ths Rashba term is too strong, differentiating between spin-up states and spin-down states might not be possible, resulting in a wrong computation of the index.

import numpy as np

from pybott import spin_bott
from kanemele import kane_mele_model

# Parameters for the finite Kane-Mele model
nx, ny = 10, 10
t1 = 1
esite = 0.1
t2 = 0.2j
rashba = 0.2

# Parameters for spin Bott
fermi_energy = 0
threshold_bott = -0.1

# Build the Kane-Mele model and solve for eigenvalues/eigenvectors
lattice, evals, evects = kane_mele_model(nx, ny, t1, esite, t2, rashba, pbc=True)

n_sites = evals.shape[0]


def get_sigma_bott(N):
    """Return the σ_z spin operator for Bott index calculation."""
    return np.kron(np.array([[1, 0], [0, -1]]), np.eye(N))


sigma = get_sigma_bott(n_sites // 2)

lattice_x2 = np.concatenate((lattice, lattice))

# Calculate and print the spin Bott index
name_psp = f"psp_spectrum_{n_sites=}_{esite=}_{t2=}_{rashba=}.pdf"

c_sb = spin_bott(
    lattice_x2,
    evals,
    evects,
    sigma,
    fermi_energy,
    threshold_bott,
    plot_psp=True,
    name_psp=name_psp,
)

print(
    f"The spin Bott index computed in the Kane-Mele model for the given parameters {esite=}, {t2=} and {rashba=} is: {c_sb}"
)

This code should output:

The spin Bott index computed in the Kane-Mele model for the given parameters esite=0.1, t2=0.2j and rashba=0.2 is: 1.0000000000000013