Mailman 3 - Kwant-discuss

Fwd: Query on Conductance Quantization in Kwant Simulations within the Integer Quantum Hall effect
by Henrique Veiga 15 Aug '24

15 Aug '24

Dear Kwant community, I have been studying the integer quantum Hall effect using the Kwant software, through which I computed the longitudinal and transverse conductance of a central sample (without disorder) in a four-terminal setup. I model the central device, which is subject to a strong magnetic field, as a square of lateral length, L. Both the sample and the four ideal leads are described by a square-lattice tight-binding model with nearest-neighbor hoppings. By computing the transverse conductance as a function of the Fermi energy, so that the first two Landau levels are crossed, I observe the expected behavior: the conductance appears quantized in integer steps of e²/h. However, upon zooming in on each of the two plateaus, I notice that the conductance does not converge immediately to the expected value. Instead, it exhibits a rippling effect, which seems to be dependent on the leads' cross-section, L_{Lead}. This phenomenon is illustrated in the attached .pdf file, and I have also included the code used to evaluate the conductance. My question is whether this behavior is a numerical issue, or if it is expected within the context of charge transport simulations in four-terminal setups. Thank you in advance, Henrique Veiga

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Alternative procedure to calculate the quantum Hall conductivity?
by adiptapal7＠gmail.com 26 Jul '24

26 Jul '24

Hello Kwant users, I have been trying to calculate the Hall conductivity in a time reversal broken Weyl semimetal (Weyl axis z) by using a 6-terminal Hall bar geometry with two current leads along z and 4 voltage leads perpendicular to z and then injecting unit current through the left current lead and calculate the p.d. across the voltage leads by inverting the conductance matrix as shown in qhe.py. However, it seems for some cases in thin film systems (thin film along x), the conductance matrix is singular and I cannot get results. Is there an alternative way to calculate the Hall conductivity in this case where one does not need to invert the conductance matrix? Thanks in advance.

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Matching the eigenmodes in scattering region and leads
by YIGITHAN GEDIZ 23 Jul '24

23 Jul '24

Dear KWANT Community, I am defining a scattering region and its corresponding leads by specifying how many unit cells are there in transversal and longitudinal direction. SR and leads have same number of unit cells in transversal direction but leads have only one unit cell in longitudinal direction. Here is my code that does that: def scattering_region_shape(pos): pos = np.array(pos) # Get the primitive vector coordinates of the position l, w = np.dot(self.Pinv, pos.reshape(-1, 1)).flatten() if l >= -0.1 and l <= n_cell_l + self.ucell_eps and w >= -(self.armchair_edge_eps + self.ucell_eps) and w <= n_cell_w + self.ucell_eps: return True return False def leads_shape(pos): pos = np.array(pos) x,y = pos # Get the primitive vector coordinates of the position l, w = np.dot(self.Pinv, pos.reshape(-1, 1)).flatten() if l >= -0.1 and l <= 1 + self.ucell_eps and w >= -(self.armchair_edge_eps + self.ucell_eps) and w <= n_cell_w + self.ucell_eps: return True return False 1) First of all, what is the difference between these two ways of calculating wavefunctions of scattering region?: wf = kwant.wave_function(system.sys, -2.4)(0) I would expect wf(0) and wf(1) returns wavefunctions that has different shapes because one of them is for SR and the other is for leads where I put less sites to the leads. But it does not, both wavefunctions have 444 elements. Why? Second way: H = system.sys.hamiltonian_submatrix() eigvals, eigvecs = np.linalg.eigh(H) idx = np.argsort(eigvals) eigvals = eigvals[idx] eigvecs = eigvecs[:,idx] E = -2.4 # Find the 10 closest eigenenergies to E closest_indices = np.argsort(np.abs(eigvals - E))[:10] # Get the 10 closest eigenenergies and their corresponding eigenvectors closest_eigvals = eigvals[closest_indices] closest_eigvecs = eigvecs[:, closest_indices] 2) Secondly, scattering region has no translational symmetry and it has discrete energy levels. On the other hand, leads have translational symmetry and their energy levels form continous bands (we can sample as many momentum space points as we want). How can we obtain a solution to Schrödinger Equation for any energy that crosses at least one band? Maybe the spectrum of the scattering region does not contain that energy and we cannot match a wavefunction of lead to the wavefunction in the scattering region to get a solution for the whole system with a definite energy? I hope I made my questions clear. I am curiously waiting for your answers Thank you -- *Yiğithan Gediz* *Junior Undergraduate Student* *Department of Physics* *Koç University*

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Unexpected Landau Level Discrepancies in Multilayer Graphene Compared to Published Results
by yl 22 Jun '24

22 Jun '24

Dear Kwant Group, I attempted to replicate the Landau levels of ABA-trilayer graphene as described in the paper "Landau level spectra and the quantum Hall effect of multilayer graphene," Physical Review B 83, 165443 (2011), Figure 2, using Kwant. The results obtained with Kwant match the paper's results approximately well; however, there is an issue. The number of Landau levels (LLs) under each magnetic field does not match the results presented in the paper. For example (see the attached figure): At a magnetic field strength of ( B = 10 ) Tesla: The left figure shows a total of 46 LLs (18 + 28). The right figure shows a total of 34 LLs (16 + 18). For the specific cases labeled as 'a' to 'd', each of these cases has 2 more LLs compared to a reference. For the case labeled 'e', this case has 4 more LLs compared to a reference. I believe the reference is correct as I was able to reproduce the completely identical LLs results using WannierTools(an open source code which can calculate LLs by Peierls substitution method). I am currently unsure where the problem lies or if it is a bug. I have attached the code (test.py) and a small modification to the spectrum function in plotter.py. Thank you very much to the Kwant development team for your efforts. I hope to get an answer, as it would be very helpful to me. Versions used: 1.4.3 + landau_levels.py Best regards, ac76888 ac76888(a)sjtu.edu.cn

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[kwant] How to make a finite lead?
by 김봉수 12 Jun '24

12 Jun '24

Dear kwant users, I am trying to reproduce the paper following. https://arxiv.org/abs/1110.4280 (in this paper, fig5, 6 will help to understand my questions.) What I’m struggling with now is to build a finite(not infinite, and not having any translational symmetry) lead as below. I wonder 1. I think "kwant.builder.SelfEnergyLead()” function might be proper thing to build a finite lead without any translational symmetry. Is it correct? 2. If it's right, when I run this code using SelfEnergyLead(), the following error occurs. How can I solve it? ------------------------------------------------------------------------------------------------ lat = kwant.lattice.chain(norbs=1) syst = kwant.Builder() syst[(lat(i) for i in range(4))] = 4 syst[lat.neighbors()] = -1 def self_energy(energy, args=(), *, params=None): return -1j left_lead = kwant.builder.SelfEnergyLead(self_energy, interface=[lat(0)], parameters = []) right_lead = kwant.builder.SelfEnergyLead(self_energy, interface=[lat(3)], parameters = []) syst.leads.append(left_lead) syst.leads.append(right_lead) syst = syst.finalized() H = syst.leads[0].kwant.hamiltonian_submatrix kwant.plot(syst) ------------------------------------------------------------------------------------------------ AttributeError: 'SelfEnergyLead' object has no attribute ‘kwant' Any help is appreciated! Best, Bongsu Kim

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Error in calculating conductance of 3D system with periodic boundaries
by Naveen Yadav 07 Jun '24

07 Jun '24

Dear Kwant Developers, I am trying to calculate the conductance of a 3D system with dimension L × W × H, by imposing periodic boundary conditions in all three directions. I am able to calculate the band structure but unable to calculate the corresponding conductance using kwant.smatrix. I am not sure whether kwant is able to do transport calculations on systems with periodic boundaries using smatrix. Please advise. The code is following import kwant import scipy.sparse.linalg as sla import matplotlib.pyplot as plt import tinyarray import numpy as np from numpy import cos, sin, pi import cmath from cmath import exp from matplotlib import rc, rcParams rc('text', usetex=True) rc('axes', linewidth=2) rc('font', weight='bold') #rcParams['text.latex.preamble'] = [r'\usepackage{sfmath} \boldmath'] #rcParams['text.latex.preamble'] = [r'\usepackage{sfmath} \boldmath'] 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_sys(a=1, m_0 = 2, L=1, W=100, H=1, t=1.0, t_z=0.0, phi=0.01): #L=10; W=80; H=60; def onsite(site): return 2 * m_0 * sigma_x def hoppingx(site0, site1): y = site1.pos[1] return (t * sigma_x) def hoppingy(site0, site1): return -1j * t * sigma_y - (m_0/2) * sigma_x def hoppingz(site0, site1): y = site1.pos[1] return (-1j * t * sigma_z - (m_0/2) * sigma_x - 1j * t_z * sigma_0) * exp(2 * pi * 1j * (y) * phi) lat =kwant.lattice.cubic(norbs=2) #sys = kwant.Builder(kwant.TranslationalSymmetry(lat.vec((0, 0, H)))) #sys[(lat(x, y, z) for x in range(L) for y in range(W) for z in range(H))] = onsite #sys = kwant.Builder(kwant.TranslationalSymmetry((L, 0, 0),(0, 0, H))) #sys[(lat(0, y, 0) for y in range(W))] = onsite sys = kwant.Builder(kwant.TranslationalSymmetry((L, 0, 0),(0, W, 0),(0, 0, H))) sys[lat.shape(lambda p: True, (0,0,0))] = onsite sys[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingx sys[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingy sys[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingz sys = kwant.wraparound.wraparound(sys, keep=None) leadL = kwant.Builder(kwant.TranslationalSymmetry((-a, 0, 0),lat.vec((0, W, 0)),lat.vec((0, 0, H)))) leadL[(lat(0, y, z) for y in range(W) for z in range(H))] = onsite leadL[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingx leadL[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingy leadL[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingz leadR = kwant.Builder(kwant.TranslationalSymmetry((a, 0, 0),lat.vec((0, W, 0)),lat.vec((0, 0, H)))) leadR[(lat(0, y, z) for y in range(W) for z in range(H))] = onsite leadR[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingx leadR[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingy leadR[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingz leadL = kwant.wraparound.wraparound(leadL, keep=0) leadR = kwant.wraparound.wraparound(leadR, keep=0) sys.attach_lead(leadL) sys.attach_lead(leadR) sys = sys.finalized() return sys syst=make_sys() def plot_bands_2d(syst, momenta=(101, 101, 101), ky=0, kz=0): if not isinstance(syst, kwant.system.FiniteSystem): raise TypeError("Need a system without symmetries.") fig = plt.figure(figsize=(5, 5)) #kwant.plotter.bands(syst.leads[0], momenta=np.linspace(-np.pi, np.pi, 101), show=False) kxs = np.linspace(-np.pi, np.pi, momenta[0]) def get_energies(ky=ky, kz=kz): energies = [np.linalg.eigvalsh(syst.hamiltonian_submatrix ((kx, ky, kz))) for kx in kxs] return energies energies = np.array(get_energies(ky=ky, kz=kz)) # Plot the band structure for band in np.array(energies).T: plt.plot(kxs, band) plt.ylim(-2.5, 2.5) plt.xlim(-pi, pi) plt.xlabel("kx") plt.ylabel("Energy") plot_bands_2d(syst) kzs = np.linspace(-pi,pi,101) kys = np.linspace(-pi,pi,101) energies = np.linspace(0.0, 0.5, 30) data=[] for energy in energies: Tkx = np.zeros(101) for i in np.arange(len(kzs)): kz=kzs[i] ky=kys[i] smatrix = kwant.smatrix(syst, energy, [kz], [ky]) Tkx[i]=smatrix.transmission(1, 0) data.append(sum(Tkx)) plt.plot(energies, data) plt.show() Also as a separate question, I am trying to run the notebook Kubo_Bastin_conductivity_with_periodic_boundary_conditions.ipynb from https://gitlab.com/kpm-tools/bloch/-/tree/master/examples?ref_type=heads to understand the kubo conductivity calcualton on a system with periodic boundaries but various errors are there in importing bloch.py. I am using kwant.__version__ = '1.4.4'. Please help in this regard. Thanks in advance. -- Best Regards, Naveen Yadav Department of Physics & Astrophysics University Of Delhi New Delhi-110007

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ValueError: Value functions must not take *args or **kwargs
by wilson2048＠outlook.com 04 Jun '24

04 Jun '24

Dear kwant community: I have encountered this error in my code. Basically I don't want to redefine the hop in the y direction, so I use the existing function `hop`. However, the code throw me an error which I don't know how to fix. Below is a simplest code example, the real situation is more complex but the spirit is the same. ``` lat = kwant.lattice.square() def make_system(r=10, t=1): def circle(pos): x, y = pos return x**2+y**2<r**2 syst = kwant.Builder() def onsite(site): return 0 syst[lat.shape(circle, (0, 0))] = onsite def hop(site1, site2, some_param): # some complex calculations not shown. return t def modified_hop(alpha, hop): # I don't want to define a hopy function, just use the exist hop return lambda *a, **kw: alpha*hop(*a, **kw) alpha = 0.5 syst[kwant.builder.HoppingKind(lat.prim_vecs[0], lat, lat)] = hop syst[kwant.builder.HoppingKind(lat.prim_vecs[1], lat, lat)] = modified_hop(alpha, hop) # alpha*hop would be very nice, but it won't work! syst.eradicate_dangling() return syst syst = make_system() fsyst = syst.finalized() H = fsyst.hamiltonian_submatrix(params=dict(some_param=1)) ``` Best, Wilson

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Leads self-energy matrix elements not ordered as system's hamiltonian_submatrix?
by wilson2048＠outlook.com 03 Jun '24

03 Jun '24

Dear kwant community: I am investigating the `kwant.greens_function`。 I would like to get the lead self-energies and add to the original system Hamiltonian and then inverse it, thus obtaining the retarded green's function. I tested, but the resulted green's function is not identical to the one kwant calculated. I want to know the reason. Is it because what I describe in the title? Below is a minimal code example: ``` lat = kwant.lattice.square() def make_system(r=20, t=-1): def circle(pos): x, y = pos return x**2 + y**2 < r**2 syst = kwant.Builder() syst[lat.shape(circle, (0, 0))] = 0 syst[lat.neighbors()] = t syst.eradicate_dangling() def lead_shape(pos): return -12 < pos[0] < 12 lead = kwant.Builder( kwant.TranslationalSymmetry((0, 1))) lead[lat.shape(lead_shape, (0, 0))] = 0 lead[lat.neighbors()] = -t syst.attach_lead(lead) return syst syst = make_system() fsyst = syst.finalized() kwant.plot(fsyst) greens_function=kwant.greens_function(fsyst) Hc = fsyst.hamiltonian_submatrix(sparse=False) index = fsyst.lead_interfaces[0] Hc[np.ix_(index, index)] += greens_function.lead_info[0] Gr = np.linalg.inv(-Hc) Gr[np.ix_(index, index)]-greens_function.submatrix(0, 0) # should be a zero matrix ``` Bests, Wilson

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Kwant-compatible Hartree-Fock Package
by Johanna 03 Jun '24

03 Jun '24

Hi all, We're excited to share a new Python package we've developed at the Quantum Tinkerer group: MeanFi. MeanFi is a self-consistent Hartree-Fock solver for tight-binding models designed to work smoothly with Kwant. Our algorithm is optimized to easily and extensibly perform all your mean-field calculations. We made it such that you can: 1) define the initial Hamiltonian and the density-density interaction as Kwant models, 2) turn it into a tight-binding format, 3) find the groundstate solution, and 4) finally turn it back to a Kwant system to do all the analysis you want! You can find the documentation, tutorials, and installation instructions here: <https://meanfi.readthedocs.io/en/latest/>. We welcome your feedback and contributions. If you have any questions or run into issues, feel free to reach out or open an issue on GitHub here: <https://github.com/quantum-tinkerer/MeanFi>. All the best, MeanFi Developers Kostas, Johanna, Anton and Antonio

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Questions About Magnetic Field Units in Landau_level.py for Landau Level Calculations
by ac76888＠sjtu.edu.cn 30 May '24

30 May '24

Dear Kwant Community, I am writing to seek assistance with two issues I have encountered while using Kwant for solving transport problems. Your guidance would be greatly appreciated. 1. I am trying to calculate the Landau levels of monolayer graphene using the following Hamiltonian (python): ``` hamiltonian = """ + hbar_vF * (k_x * kron(sigma_0, sigma_x) + k_y * kron(sigma_0, sigma_y)) """ params = dict(hbar_vF=6.582) ``` In the documentation for the `discretize_landau` function, it is stated that "The units of magnetic field are :math:`ϕ₀ / 2 π a²` with :math:`ϕ₀ = h/e` the magnetic flux quantum and :math:`a` the unit length." Is a^2 here the area (A) of a primitive cell of graphene (A = 5.24 Å²)? How can I convert this unit into Tesla? 2. Can I use this code to calculate the Landau levels of multilayer stacking graphene with a sequences of hoppings ? Are there specific considerations or modifications I should be aware of when extending the calculation to multilayer graphene? Thank you in advance for your help. Best regards, ac76888

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