Help regarding chern number calculation

[Dominik Gresch]

Dear Priyanka,


The reason for this problem could be the fact that the Hamiltonian is 
not periodic across the Brillouin zone. As such, the Chern number is not 
well-defined (because the wave functions do not form a smooth fiber bundle).


Not having studied the reference in detail, I believe the reason for 
this is that the Hamiltonian of eq. (14) is expanded only around a 
single point, whereas you would need a Hamiltonian which extends 
correctly across the BZ to compute the Chern number.


Best regards,

Dominik


On 31.05.19 16:41, MS. PRIYANKA SINHA wrote:
>
> Dear Sir,
>
>                 I am a new user of z2 pack and trying to calculate 
> chern number following the journal 
> https://journals.aps.org/prb/pdf/10.1103/PhysRevB.85.115439 
> <https://journals.aps.org/prb/pdf/10.1103/PhysRevB.85.115439> . I have 
> defined the Hamiltonian as given in equation 14. I should get non-zero 
> chern number. But instead of that I am getting zero. I am attaching my 
> code below. Kindly help me in this regard.
>
>
>
> *********************************************************************
>
> #!/usr/bin/env python
> # -*- coding: utf-8 -*-
> #
> # Author:  Dominik Gresch <greschd at gmx.ch>
> # Date:    16.08.2016 14:25:39 CEST
> # File:    haldane.py
>
> import json
> import logging
>
> import z2pack
> import numpy as np
> import matplotlib.pyplot as plt
> from numpy import linalg as LA
>
> 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)
> Sx= 1./2. * np.array([[0, 1], [1, 0]], dtype=complex)
> Sy= 1./2. * np.array([[0, -1j], [1j, 0]], dtype=complex)
> Sz= 1./2. * np.array([[1, 0], [0, -1]], dtype=complex)
>
> def Hamilton(k, m, t1, VR):
>     kx, ky,_= k
>     H  = (3. *t1 /2. * (  pauli_x * kx + pauli_y * ky)) * identity
>     H += (3. *t1 /2. * (- pauli_x * kx + pauli_y * ky)) * identity
>     H +=  3. *VR /2. * (  pauli_x * Sy - pauli_y * Sx)
>     H += 3. *VR /2. *  (- pauli_x * Sy - pauli_y * Sx)
>     #H3 = n * 3. *sqrt3 * t2 * pauli_z *Sz
>     H += (m * Sz) * identity
>     #H5 = u * pauli_z * identity
>     return H
>
>
> def get_chern( m, t1, VR, **settings):
>     system = z2pack.hm.System(lambda k: Hamilton(k, m, t1, VR), bands=1)
>
>     result = z2pack.surface.run(system=system, surface=lambda t1, t2: 
> [t1, t2, 1], **settings)
>     return z2pack.invariant.chern(result)
>
>
> if __name__ == "__main__":
>     # Task a)
>     print(get_chern(0.3, 1., 0.5 ))
>     #print(get_chern(0.5, 1., 1. / 3., -0.5 * np.pi))
>     #print(Hamilton(0.1, 1, 0.3,0.5))
> ********************************************************************
>
> Thanking You,
> Priyanka Sinha
> IIT Guwahati
> Research Scholar
>
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