Error in the calculation of Z2 invariant of Kane-Mele Model

[Dominik Gresch]

Dear Sayan Mondal,


I would suspect that the values for which you are seeing these errors
are those that are close to a topological phase transition. In these
cases, the hybrid Wannier charge center evolution changes more abruptly,
and the calculation would take longer to converge.


To increase the number of steps the calculation will perform before
aborting, you can decrease the 'min_neighour_dist' value, and increase
the upper limit of the 'iterator' (these keywords are described in the
surface.run reference
<https://z2pack.greschd.ch/en/latest/reference/surface.html#z2pack.surface.run>).


For values of lambda_R and lambda_v which are exactly at a phase
transition, the calculation will not converge, because the topological
indices are not well-defined.


Best regards,

Dominik Gresch


On 17.09.21 08:28, SAYAN MONDAL wrote:
> Dear Developer,
>                              I am trying to get the phase diagram of
> the Kane-Mele model as described in Fig. 1 of 
> (https://doi.org/10.1103/PhysRevLett.95.146802
> <https://doi.org/10.1103/PhysRevLett.95.146802>). The Z2 invariant is
> 1 inside the curve while it vanishes outside. So I tried to get the Z2
> invariant for several values of lambda_R and lambda_v. The code is
> giving correct results for some values of lambda_R and lambda_v, but
> for some other values it is showing some errors as follows:
>
> WARNING: Iterator stopped before the calculation could converge.
> WARNING: Iterator stopped before the calculation could converge.
> WARNING: 'min_neighbour_dist' reached: cannot add line at t =
> 0.6562500000000001
> WARNING: Iterator stopped before the calculation could converge.
> WARNING: Iterator stopped before the calculation could converge.
> WARNING: 'min_neighbour_dist' reached: cannot add line at t =
> 0.6562500000000001
> WARNING: Iterator stopped before the calculation could converge.
> WARNING: Iterator stopped before the calculation could converge.
> WARNING: 'min_neighbour_dist' reached: cannot add line at t =
> 0.6562500000000001
>
> My code is attached below:
> *************************************
> import logging
>
> import z2pack
> import numpy as np
> from numpy import cos, sin, kron, sqrt
> import matplotlib.pyplot as plt
>
>
> logging.getLogger('z2pack').setLevel(logging.WARNING)
>
> # defining pauli matrices
> identity = np.identity(2, dtype=complex)
> pauli_x = np.array([[0, 1], [1, 0]], dtype=complex)
> pauli_y = np.array([[0, -1j], [1j, 0]], dtype=complex)
> pauli_z = np.array([[1, 0], [0, -1]], dtype=complex)
>
>
> def get_kane_mele_hamiltonian(t, lambda_v, lambda_R, lambda_SO):
>     def inner(k):
>         k = np.array(k) * 2 * np.pi
>         kx, ky = k
>         # change to reduced coordinates
>         x = (kx - ky) / 2
>         y = (kx + ky) / 2
>         return (
>             t * (1 + 2 * cos(x) * cos(y)) * kron(pauli_x, identity) +
>             lambda_v * kron(pauli_z, identity) + lambda_R *
>             (1 - cos(x) * cos(y)) * kron(pauli_y, pauli_x) +
>             -sqrt(3) * lambda_R * sin(x) * sin(y) * kron(pauli_y,
> pauli_y) +
>             2 * t * cos(x) * sin(y) * kron(pauli_y, identity) +
> lambda_SO *
>             (2 * sin(2 * x) - 4 * sin(x) * cos(y)) * kron(pauli_z,
> pauli_z) +
>             lambda_R * cos(x) * sin(y) * kron(pauli_x, pauli_x) +
>             -sqrt(3) * lambda_R * sin(x) * cos(y) * kron(pauli_x, pauli_y)
>         )
>
>     return inner
>
>
> if __name__ == '__main__':
>     arr = []
>     lambda_R_list = np.linspace(0, 0.31, 20)
>     lambda_v_list = np.linspace(0, 0.24, 20)
>     for lR in lambda_R_list:
>         for lv in lambda_v_list:
>             system = z2pack.hm.System(
>                 get_kane_mele_hamiltonian(
>                     t=1, lambda_v = lv, lambda_R=lR, lambda_SO=0.06
>                 ),
>                 dim=2,
>                 check_periodic=True
>             )
>             res = z2pack.surface.run(system=system, surface=lambda s,
> t: [s / 2, t])
>         #    print('Z2 invariant: {}'.format(z2pack.invariant.z2(res)))
>             val = z2pack.invariant.z2(res)
>             arr.append([lR, lv, val])
>     arr = np.asarray(arr)
>     np.savetxt('pd.txt', arr)
>     print(arr)
> #    z2pack.plot.wcc(res, axis=ax)
> #    ax.set_xticks([0, 1])
> #    ax.set_xticklabels(['0', '0.5'])
> #    ax.set_xlabel(r'$k_y$')
> #    ax.set_yticks([0, 1])
> #    ax.set_ylabel(r'$\bar{y}$', rotation='horizontal')
> #    plt.savefig('plot.pdf', bbox_inches='tight')
>
> **************************************************
>
>
> Is there any way to correct those errors? I shall look forward to your
> kind cooperation.
>
> Thank you
>
> Best Regards,
> Sayan Mondal,
> Research Scholar,
> Dept. of Physics,
> Roll No. -   186121022,
> E-mail:      mondal18 at iitg.ac.in
> Indian Institute of Technology Guwahati
> Guwahati - 781039
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