Dear Alexandre,
When you define your site potential you call your function once. For
hopping, it is deferent: when you define it for (site1, site2)
kwant tries to ensure that the hamiltonian is Hermitian so he calls again
the function and takes its Hermitian conjugate to set the hopping for
(site2,site1). Your function being random gives a different value when you
call it the second time.
check the function test_hermitian_conjugation() taken from the file
test_builder.py
(some modifications are done to the following function)
import kwant
import tinyarray as ta
import random
def test_hermitian_conjugation():
def f(i, j, arg):
i, j = i.tag, j.tag
if j[0] == i[0] + 1:
return arg * (1+1j)*(random.random() - 0.5)*2 *ta.array([[1,
2j], [3 + 1j, 4j]])
else:
raise ValueError
syst = kwant.builder.Builder()
fam = kwant.builder.SimpleSiteFamily()
syst[fam(0)] = syst[fam(1)] = ta.identity(2)
syst[fam(0), fam(1)] = f
assert syst[fam(0), fam(1)] is f
assert isinstance(syst[fam(1), fam(0)], kwant.builder.HermConjOfFunc)
assert (syst[fam(1), fam(0)](fam(1), fam(0), 2) == syst[fam(0),
fam(1)](fam(0), fam(1), 2).conjugate().transpose())
test_hermitian_conjugation()
I hope this helps.
Adel
On Tue, May 22, 2018 at 11:01 AM, <alexandre.berna...@u-psud.fr> wrote:
> Hi,
>
> I don't quite understand what 'having the same symmetry' means in this
> context.
> For example, everything works fine for on-site random potential, which
> doesn't have the 1D translationnal symmetry in the unit cell, but Kwant
> makes the hamiltonian such that the symmetry with respect to the cells
> separated by a symmetry vector is valid.
>
> So, this does work for on-site random term but not for the hoppings ? (it
> also doesn't work for random real nearest neighbors hopping terms)
>
> I will try to fix it by hand, by specifying the hopping terms between the
> cells, and see if it works.
>
> Best regards,
> Alexandre
>
> ------------------------------
> *De: *"Abbout Adel" <abbout.a...@gmail.com>
> *À: *"alexandre bernard1" <alexandre.berna...@u-psud.fr>
> *Cc: *"kwant-discuss" <kwant-discuss@kwant-project.org>
> *Envoyé: *Samedi 19 Mai 2018 23:21:02
> *Objet: *Re: [Kwant] Hamiltonian not hermician for an infinite wire with
> random complex hopping
>
>
> Dear Alexandre,
>
> In the documentation, it is mentioned that the function defining the
> potential in the lead should have the same symmetry as the lead which is
> obviously not fulfilled with your random function.
>
> Another thing, you need to be careful when you define your unit cell in
> the lead (or in your system with the translational symmetry). If you define
> a larger unit cell, the translational symmetry will make the sites doubled.
>
> The last thing I would like to comment is that when you have a system with
> random sites, you can not pretend to have a continuous spectrum (or
> disjoint bands). In fact, from random matrix theory, we know that usually
> the eigenvalues are repelling each other and the probability to get two
> eigenvalues close to each other tends to zero. So, I suspect that, more
> likely you will have a spectrum composed by disjoint points rather than
> bands.
>
> I hope this helps.
> Adel
>
> On Fri, May 18, 2018 at 4:21 PM, <alexandre.berna...@u-psud.fr> wrote:
>
>> Hello (again),
>>
>> Everything is in the title.
>>
>> I want to implement random (and complex) next nearest neighbors hopping
>> in a Haldane infinite wire.
>>
>> Without the random part, 'kwant.plotter.bands()' correctly plot the band
>> structure of my system.
>> However, when I add a random contribution on the modulus of the NNN
>> hopping term, I get an error from 'kwant.plotter.bands()' :
>>
>> 'The cell Hamiltonian is not Hermitian'
>>
>> Which is strange because I red everywhere that the hermicity constraint
>> is always fullfilled using Kwant.
>> The function for random NNN hopping works perfectly fine for a finite
>> system.
>>
>> So, is this comming from the inter-cell hopping terms ? If so, how to
>> correct it ?
>> Kwant does not consider the hopping terms of both ends of the cell as
>> being equal ? It should be because u_k(r) = u_k(r+R) is like periodic
>> boundary conditions for u_k.
>>
>> I add my script at the end so you can see for yourself (the NNN hopping
>> randomness parameter is 'rand_nnn').
>>
>> Best regards,
>>
>> Alexandre BERNARD
>>
>> ----------------- my script ----------------------
>>
>> from types import SimpleNamespace
>>
>> import matplotlib
>> import math
>> from matplotlib import pyplot
>> from mpl_toolkits import mplot3d
>> import numpy as np
>> import random
>> import kwant
>>
>>
>> def random_uni(param):
>> return param*(random.random() - 0.5)*2 # returns a random number
>> between +param and -param
>>
>> def onsite(site, p):
>> # Here we affect a potential whose sign depends on which sublattice
>> the atom lives in (breaking sublattice symmetry)
>> if site.family == A: # sublattice A
>> return p.m + random_uni(p.rand_os)
>> else: # sublattice B
>> return -p.m + random_uni(p.rand_os)
>>
>> def nn_hopping(site1, site2, params):
>> return params.t_1 + random_uni(params.rand_nn)
>>
>> def nnn_hopping(site1, site2, params):
>> return (math.cos(params.phi) + 1j*math.sin(params.phi)) * (params.t_2
>> + random_uni(params.rand_nnn))
>>
>> def ribbon_shape_armchair(pos):
>> return (-1 <= pos[0] < w)
>>
>> def hopping_color(site1, site2):
>> if site1.family == A and site2.family == A:
>> return 'b'
>> elif site1.family == B and site2.family == B:
>> return 'r'
>> else:
>> return '0.5'
>>
>> def site_color(site):
>> if site.family == A:
>> return 'b'
>> else:
>> return 'r'
>>
>> np.random.seed(271828) # Fixing the RNG seed for reproducibility (or not,
>> because of how Kwant works)
>>
>> # Graphene lattice "pointing up", with two sublattices
>> graphene = kwant.lattice.general([[1, 0], [1/2, np.sqrt(3)/2]], # Here
>> are the vectors of the Bravais lattice
>> [[0, 0], [0, 1/np.sqrt(3)]]) # Here
>> are the positions of the atoms in the unit cell
>> A, B = graphene.sublattices
>>
>> w = 29 # width of the ribbon
>> p = SimpleNamespace(t_1=1.0, t_2=0.1, m=0.0, phi=np.pi/2, rand_os=0.,
>> rand_nn=0., rand_nnn=0.1)
>>
>> # Creation of the list of (oriented) bonds that will have the complex
>> hopping term
>> nnn_hoppings_A = (((-1, 0), A, A), ((0, 1), A, A), ((1, -1), A, A)) #
>> Bonds within the A sublattice
>> nnn_hoppings_B = (((1, 0), B, B), ((0, -1), B, B), ((-1, 1), B, B)) #
>> Bonds within the B sublattice
>> nnn_hoppings = nnn_hoppings_A + nnn_hoppings_B
>>
>> haldane_armchair = kwant.Builder(kwant.TranslationalSymmetry((0,
>> 1*np.sqrt(3))))
>>
>> haldane_armchair[graphene.shape(ribbon_shape_armchair, (0, 0))] = onsite
>> haldane_armchair[graphene.neighbors(1)] = nn_hopping
>> haldane_armchair[[kwant.builder.HoppingKind(*hopping) for hopping in
>> nnn_hoppings]] = nnn_hopping
>>
>> kwant.plot(haldane_armchair, site_color=site_color,
>> hop_color=hopping_color, fig_size=(6, 9));
>>
>> kwant.plotter.bands(haldane_armchair.finalized(), args=[p],
>> fig_size=(12, 8));
>>
>
>
>
> --
> Abbout Adel
>
>
--
Abbout Adel
