Alternative boundary conditionsΒΆ

The physical system in this example is similar to Evolution of a scattering state under a voltage pulse in a quantum dot.

tkwant features highlighted

  • Selecting boundary conditions manually

  • Impact of the boundary type on performance

import time as timer
import numpy as np
from matplotlib import pyplot as plt

import kwant
import tkwant

def make_system(a=1, gamma=1.0, radius=10):
    """Make a tight binding system on a single square lattice"""
    # `a` is the lattice constant and `gamma` the hopping integral
    # both set by default to 1 for simplicity.

    lat = kwant.lattice.square(a, norbs=1)

    syst = kwant.Builder()

    # Define the quantum dot
    def circle(pos):
        (x, y) = pos
        return x ** 2 + y ** 2 < radius ** 2

    syst[lat.shape(circle, (0, 0))] = 4 * gamma
    syst[lat.neighbors()] = -gamma

    lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)))
    lead[(lat(0, j) for j in range(-radius // 2 + 2, radius // 2 - 1))] = 4 * gamma
    lead[lat.neighbors()] = -gamma


    return syst

# for adding a time-dependent voltage on top of the leads
def faraday_flux(time):
    return 0.05 * (time - 10 * np.sin(0.1 * time))

def evolve(times, psi, operator):
    ops = []
    for time in times:
        expectation = psi.evaluate(operator)
    return ops

def main():
    syst = make_system()

    # add a time-dependent voltage to lead 0 -- this is implemented
    # by adding sites to the system at the interface with the lead and
    # multiplying the hoppings to these sites by exp(-1j * faraday_flux(time))
    extra_sites = tkwant.leads.add_voltage(syst, 0, faraday_flux)
    lead_syst_hoppings = [(s, site) for site in extra_sites
                          for s in syst.neighbors(site)
                          if s not in extra_sites]

    kwant.plot(syst, site_color='k', lead_color='grey', num_lead_cells=4,
               hop_lw=lambda a, b: 0.3 if (a, b) in lead_syst_hoppings else 0.1,
               hop_color=lambda a, b: 'r' if (a, b) in lead_syst_hoppings else 'k')

    syst = syst.finalized()

    # create an observable for calculating the current flowing from the left lead
    current_operator = kwant.operator.Current(syst, where=lead_syst_hoppings,

    energy = 1.
    tmax = 600

    # create initial scattering state
    scattering_states = kwant.wave_function(syst, energy=energy, params={'time': 0})
    psi_st = scattering_states(0)[0]

    # maximal velocity of a mode in the leads
    vmax = 2 * np.linalg.norm(syst.leads[0].inter_cell_hopping(), ord=2)
    num_buffer_cells = int(vmax * tmax / 2)

    # boundary conditions typically used by `tkwant.solve`
    simple_boundaries = [tkwant.leads.SimpleBoundary(num_buffer_cells)
                         for l in syst.leads]

    # boundary conditions with an absorbing potential that increases
    # according to x**n where `x` is the distance into the lead
    absorbing_boundaries = [tkwant.leads.MonomialAbsorbingBoundary(num_absorb_cells=100,
                            for l in syst.leads]

    # create time-dependent wavefunctions that starts in a scattering state
    # originating from the left lead, using two different types of boundary
    # conditions
    psi_simple = tkwant.onebody.WaveFunction.from_kwant(syst=syst, psi_init=psi_st,

    psi_alt = tkwant.onebody.WaveFunction.from_kwant(syst=syst, psi_init=psi_st,

    # evolve forward in time, calculating the current
    times = np.arange(0, tmax)

    # Simple boundary conditions
    start = timer.process_time()
    current_simple = evolve(times, psi_simple, current_operator)
    stop = timer.process_time()
    print('Simple Boundary conditions elapsed time: {:.2f}s'.format(stop - start))

    # Absorbing boundary conditions
    start = timer.process_time()
    current_alt = evolve(times, psi_alt, current_operator)
    stop = timer.process_time()
    print('Absorbing Boundary conditions elapsed time: {:.2f}s'.format(stop - start))

    plt.plot(times, current_simple, lw=2, label='SimpleBoundary')
    plt.plot(times, current_alt, '--', lw=2, label='MonomialAbsorbingBoundary')
    plt.xlabel(r'time $t$')
    plt.ylabel(r'current $I(t)$')

if __name__ == '__main__':
Simple Boundary conditions elapsed time: 8.92s
Absorbing Boundary conditions elapsed time: 5.34s

See also

The complete source code of this example can be found in