Mailman 3 December 2017 - Kwant-discuss

honeycomb lattice scattering region with square lattice leads
by Khani Hosein 13 Dec '17

13 Dec '17

Dear developers, I am considering a system with honeycomb lattices (scattering region) connected to square lattices (leads). For the square lattices of the leads, I want to use the honeycomb lattice but with hoppings same as square lattices, so it works like square lattice leads. My question is: how can I find out the hoppings between the scattering region and the leads. The leads are attached to the scattering region using sys.attach_lead(lead). Regards, Hosein Khani

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regarding rashba term in kwant
by PRIYANKA SINHA 11 Dec '17

11 Dec '17

Sir, I was trying to implement kane-mele Hamiltonian including tight-binding,intrinsic ans rashba spin-orbit for conductance calculation.In Rashba term we have cross product of vector s and d_ij with z-component, where d_ij is the vector points from site i to site j and corresponds to the nearest-neighbor vectors. My question is that what is the possible way to implement that term ,especially d_1 and d_2 for different hopping.Here is my code only for tight-binding and intrinsic SOC. from __future__ import division # so that 1/2 == 0.5, and not 0 from math import pi, sqrt, tanh import numpy as np import kwant import random import math import tinyarray # For computing eigenvalues import scipy.sparse.linalg as sla # For plotting from matplotlib import pyplot # Define the graphene lattice graphene = kwant.lattice.honeycomb() a, b = graphene.sublattices #define pauli matrices sigma_0 = tinyarray.array([[1, 0], [0, 1]]) sigma_x = tinyarray.array([[0, 1], [1, 0]]) sigma_y = tinyarray.array([[0, -1j], [1j, 0]]) sigma_z = tinyarray.array([[1, 0], [0, -1]]) def make_system(L=20, W=20): t=1 t2= 0.1 lambda_R=0.1 v_ij= 1 nn_hopping= -t * sigma_0 nnn_hoppinga= -1j * t2 * v_ij * sigma_z nnn_hoppingb= -1j * t2 * v_ij * sigma_z #### Define the scattering region. #### def rec(pos): x, y = pos return 0 < x < L and 0 < y < W sys = kwant.Builder() sys[graphene.shape(rec, (1, 1))] = 0 * sigma_0 # specify the hoppings of the graphene lattice nn_hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) nnn_hoppings_a = (((-1, 0), a, a), ((0, 1), a, a), ((1, -1), a, a)) nnn_hoppings_b = (((-1, 0), b, b), ((0, 1), b, b), ((1, -1), b, b)) sys[[kwant.builder.HoppingKind(*hopping) for hopping in nn_hoppings]] = nn_hopping sys[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_a]] = -nnn_hoppinga sys[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_b]] = -nnn_hoppingb #### Define the leads. #### # left lead sym0 = kwant.TranslationalSymmetry(graphene.vec((-1, 0))) sym0.add_site_family(graphene.sublattices[0], other_vectors=[(-1, 2)]) sym0.add_site_family(graphene.sublattices[1], other_vectors=[(-1, 2)]) def lead0_shape(pos): x, y = pos return (0 < y < W) lead0 = kwant.Builder(sym0) lead0[graphene.shape(lead0_shape, (0, 0))] = 0 * sigma_0 lead0[[kwant.builder.HoppingKind(*hopping) for hopping in nn_hoppings]] = nn_hopping # lead0[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_a]] = -nnn_hoppinga # lead0[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_b]] = -nnn_hoppingb # right lead sym1 = kwant.TranslationalSymmetry(graphene.vec((1, 0))) sym1.add_site_family(graphene.sublattices[0], other_vectors=[(-1, 2)]) sym1.add_site_family(graphene.sublattices[1], other_vectors=[(-1, 2)]) def lead1_shape(pos): x, y = pos return (0 < y < W) lead1 = kwant.Builder(sym1) lead1[graphene.shape(lead1_shape, (0, 0))] = 0 * sigma_0 lead1[[kwant.builder.HoppingKind(*hopping) for hopping in nn_hoppings]] = nn_hopping # lead1[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_a]] = -nnn_hoppinga # lead1[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_b]] = -nnn_hoppingb sys.attach_lead(lead1) sys.attach_lead(lead0) return sys def family_color(site): if site.family == a: return 'red' else: return 'blue' def plot_conductance(sys, energies): # Compute transmission as a function of energy data = [] for energy in energies: smatrix = kwant.smatrix(sys, energy) data.append(smatrix.transmission(0, 1)) pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() def plot_bandstructure(flead, momenta): bands = kwant.physics.Bands(flead) energies = [bands(k) for k in momenta] pyplot.figure() pyplot.plot(momenta, energies) pyplot.xlabel("momentum [(lattice constant)^-1]") pyplot.ylabel("energy [t]") pyplot.show() def main(): sys = make_system() # Check that the system looks as intended. kwant.plot(sys, site_lw=0.1, site_color=family_color) # Finalize the system. sys = sys.finalized() # Compute the band structure of lead 0. momenta = np.linspace(-pi, pi , 200) plot_bandstructure(sys.leads[0], momenta) Es = np.linspace(-3, 3, 200) plot_conductance(sys, Es) # Call the main function if the script gets executed (as opposed to imported). # See <http://docs.python.org/library/__main__.html>. if __name__ == '__main__': main() -- INSERT -- -- Priyanka Sinha Junior research Fellow Department of Physics, IIT Guwahati, India

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regarding rashba term in kwant
by PRIYANKA SINHA 11 Dec '17

11 Dec '17

Sir, I was trying to implement kane-mele Hamiltonian including tight-binding,intrinsic ans rashba spin-orbit for conductance calculation.In Rashba term we have cross product of vector s and d_ij with z-component, where d_ij is the vector points from site i to site j and corresponds to the nearest-neighbor vectors. My question is that what is the possible way to implement that term ,especially d_1 and d_2 for different hopping.Here is my code only for tight-binding and intrinsic SOC. from __future__ import division # so that 1/2 == 0.5, and not 0 from math import pi, sqrt, tanh import numpy as np import kwant import random import math import tinyarray # For computing eigenvalues import scipy.sparse.linalg as sla # For plotting from matplotlib import pyplot # Define the graphene lattice graphene = kwant.lattice.honeycomb() a, b = graphene.sublattices #define pauli matrices sigma_0 = tinyarray.array([[1, 0], [0, 1]]) sigma_x = tinyarray.array([[0, 1], [1, 0]]) sigma_y = tinyarray.array([[0, -1j], [1j, 0]]) sigma_z = tinyarray.array([[1, 0], [0, -1]]) def make_system(L=20, W=20): t=1 t2= 0.1 lambda_R=0.1 v_ij= 1 nn_hopping= -t * sigma_0 nnn_hoppinga= -1j * t2 * v_ij * sigma_z nnn_hoppingb= -1j * t2 * v_ij * sigma_z #### Define the scattering region. #### def rec(pos): x, y = pos return 0 < x < L and 0 < y < W sys = kwant.Builder() sys[graphene.shape(rec, (1, 1))] = 0 * sigma_0 # specify the hoppings of the graphene lattice nn_hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) nnn_hoppings_a = (((-1, 0), a, a), ((0, 1), a, a), ((1, -1), a, a)) nnn_hoppings_b = (((-1, 0), b, b), ((0, 1), b, b), ((1, -1), b, b)) sys[[kwant.builder.HoppingKind(*hopping) for hopping in nn_hoppings]] = nn_hopping sys[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_a]] = -nnn_hoppinga sys[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_b]] = -nnn_hoppingb #### Define the leads. #### # left lead sym0 = kwant.TranslationalSymmetry(graphene.vec((-1, 0))) sym0.add_site_family(graphene.sublattices[0], other_vectors=[(-1, 2)]) sym0.add_site_family(graphene.sublattices[1], other_vectors=[(-1, 2)]) def lead0_shape(pos): x, y = pos return (0 < y < W) lead0 = kwant.Builder(sym0) lead0[graphene.shape(lead0_shape, (0, 0))] = 0 * sigma_0 lead0[[kwant.builder.HoppingKind(*hopping) for hopping in nn_hoppings]] = nn_hopping # lead0[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_a]] = -nnn_hoppinga # lead0[[kwant.builder.HoppingKind(*hopping) for hopping in nnn_hoppings_b]] = -nnn_hoppingb # right lead sym1 = kwant.TranslationalSymmetry(graphene.vec((1, 0))) sym1.add_site_family(graphene.sublattices[0], other_vectors=[(-1, 2)]) sym1.add_site_family(graphene.sublattices[1], other_vectors=[(-1, 2)]) def lead1_shape(pos): else: return 'blue' def plot_conductance(sys, energies): # Compute transmission as a function of energy data = [] for energy in energies: smatrix = kwant.smatrix(sys, energy) data.append(smatrix.transmission(0, 1)) pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() def plot_bandstructure(flead, momenta): bands = kwant.physics.Bands(flead) energies = [bands(k) for k in momenta] pyplot.figure() pyplot.plot(momenta, energies) pyplot.xlabel("momentum [(lattice constant)^-1]") pyplot.ylabel("energy [t]") pyplot.show() def main(): sys = make_system() # Check that the system looks as intended. kwant.plot(sys, site_lw=0.1, site_color=family_color) # Finalize the system. sys = sys.finalized() # Compute the band structure of lead 0. momenta = np.linspace(-pi, pi , 200) plot_bandstructure(sys.leads[0], momenta) Es = np.linspace(-3, 3, 200) plot_conductance(sys, Es) # Call the main function if the script gets executed (as opposed to imported). # See <http://docs.python.org/library/__main__.html>. if __name__ == '__main__': main() -- INSERT -- -- Priyanka Sinha Junior research Fellow Department of Physics, IIT Guwahati, India

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conservation law in the case of a spinfull Nambu basis
by Tibor Sekera 08 Dec '17

08 Dec '17

Dear all, as is explained in https://kwant-project.org/doc/dev/tutorial/superconductors the conservation law argument in Builder (e.g. a lead) is conservation_law=-tau_z with eigenvalues -1 and 1 yields scattering states of electron and hole type, respectively. This leads to a block form of the scattering matrix. One can also get the electron and hole projectors as (projector_e, projector_h) = lead.discrete_symmetry().projectors In presence of spin (described by sigma matrices) and a magnetic field (giving rise to a Zeeman term) we could have a conservation law of the following form conservation_law=np.kron(-tau_z,-sigma_z) where the Hamiltonian is written in the Nambu basis (e up, e down, h up, -h down). Eigenvalues of this conservation law are now 1,-1,1,-1, which are degenerate. Question 1: How can one resolve the four-by-four block structure of the scattering matrix in the electron-hole ⊗ spin up-down space? Question 2: How can one get the four projectors? I am looking for something like: (projector_e_up, projector_e_down, projector_h_up, projector_h_down) = lead.discrete_symmetry().projectors

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Disorder with kwant.digest.uniform
by Antonio Lucas Rigotti Manesco 07 Dec '17

07 Dec '17

Hi everyone, I wish to know if the parameter "salt" on the function kwant.digest.uniform have a physical meaning. Varying it, it seems to me that when I use it on a system, the lower the value of salt is, the local disorder implemented becomes more smooth, but I didn't found any information suggesting it at the documentation. Is that correct? Thanks in advance! -- Antônio Lucas Rigotti Manesco PhD fellow - University of São Paulo, Brazil

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