Your band structure is correct. When you add magnetic field you change your momentum p to (p-eA), or let's say you push your spectrum. Try it with with a chain with a width of 2 sites and you can see it better.
On 25 August 2016 at 11:23, <kwant-discuss-requ...@kwant-project.org> wrote: > Send Kwant-discuss mailing list submissions to > kwant-discuss@kwant-project.org > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman-mail5.webfaction.com/listinfo/kwant-discuss > or, via email, send a message with subject or body 'help' to > kwant-discuss-requ...@kwant-project.org > > You can reach the person managing the list at > kwant-discuss-ow...@kwant-project.org > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of Kwant-discuss digest..." > > Today's Topics: > > 1. Re: Band structure under B-field (T.C. Wu) > > > ---------- Forwarded message ---------- > From: "T.C. Wu" <t...@connect.ust.hk> > To: Michael Wimmer <m.t.wim...@tudelft.nl>, < > kwant-discuss@kwant-project.org> > Cc: > Date: Wed, 24 Aug 2016 19:50:49 +0800 > Subject: Re: [Kwant] Band structure under B-field > Hi Michael, > > Thanks for your help. I modified the hopping of the vertical leads from -t > to hopping_Ax. Some LL-like flat bands are observed. However, they are > asymmetry about k = 0 (As shown in the attached figures) and it looks a bit > strange to me. Is it making sense? I suspect something is still wrong with > my code. Thanks for your time. > > Best, > Johnny > > On 24 August 2016 at 18:43, Michael Wimmer <m.t.wim...@tudelft.nl> wrote: > >> Dear Johnny, >> >> you simply did not include any magnetic field in the vertical leads, see >> lead_hop_vert(). Passing the parameters as you do is fine. >> >> Best, >> >> Michael >> >> >> On 24-08-16 07:02, T.C. Wu wrote: >> > Dear Kwant users, >> > >> > I would like to plot the band structure of my system under a magnetic >> > field. Based on the code demostrated >> > in http://topocondmat.org/w3_pump_QHE/Laughlinargument.html, I plot the >> > band structure but I cannot observe any Landau level like spectrum. I >> > suspect it has something to do with the line >> > >> > p = SimpleNamespace(t=1.0, mu=0.3, mu_lead=0.3, B=0.5) >> > bands = kwant.physics.Bands(flead,args = [p]) >> > >> > Can the B-field be passed to the Hamiltonian in this way? The complete >> > code is attached below. Thanks for your help. >> > >> > Best, >> > Johnny >> > >> > #------My code-----------# >> > from __future__ import division >> > import numpy as np >> > import kwant >> > from matplotlib import pyplot as plt >> > from math import pi >> > >> > class SimpleNamespace(object): >> > def __init__(self, **kwargs): >> > self.__dict__.update(kwargs) >> > >> > >> > def qhe_corbino(r_out=100, r_in=65, w_lead=10): >> > """Create corbino disk. >> > >> > Square lattice, one orbital per site. >> > Returns kwant system. >> > >> > Arguments required in onsite/hoppings: >> > t, mu, mu_lead, B, phi >> > """ >> > # ring shape >> > def ring(pos): >> > (x, y) = pos >> > rsq = x ** 2 + y ** 2 >> > return (r_in ** 2 < rsq < r_out ** 2) >> > >> > # Onsite and hoppings >> > def onsite(site, p): >> > (x, y) = site.pos >> > return 4 * p.t - p.mu <http://p.mu> >> > >> > def crosses_branchcut(hop): >> > x1, y1 = hop[0].pos >> > x2, y2 = hop[1].pos >> > return y1 < 0 and x1 > 0.5 and x2 < 0.5 >> > >> > def hopping(site1, site2, p): >> > x1, y1 = site1.pos >> > x2, y2 = site2.pos >> > # Check for correctness! >> > return -p.t * np.exp(-0.5j * p.B * (x1 - x2) * (y1 + y2)) >> > >> > def branchcut_hopping(site1, site2, p): >> > return hopping(site1, site2, p) * np.exp(1j * p.phi) >> > >> > # Building system >> > lat = kwant.lattice.square() >> > sys = kwant.Builder() >> > >> > sys[lat.shape(ring, (0, r_in + 1))] = onsite >> > sys[lat.neighbors()] = hopping >> > >> > # adding special hoppings >> > def hops_across_cut(sys): >> > for hop in kwant.builder.HoppingKind((1, 0), lat, lat)(sys): >> > if crosses_branchcut(hop): >> > yield hop >> > sys[hops_across_cut] = branchcut_hopping >> > >> > # Attaching leads >> > sym_lead = kwant.TranslationalSymmetry((-1, 0)) >> > lead = kwant.Builder(sym_lead) >> > >> > def lead_shape(pos): >> > (x, y) = pos >> > return (-w_lead / 2 < y < w_lead / 2) >> > >> > lead[lat.shape(lead_shape, (0, 0))] = lambda site, p: 4 * p.t - >> > p.mu_lead >> > lead[lat.neighbors()] = lambda site1, site2, p: -p.t >> > >> > #### Attach the leads and return the system. #### >> > sys.attach_lead(lead) >> > sys.attach_lead(lead, origin=lat(0, 0)) >> > >> > return sys >> > >> > >> > def total_charge(value_array): >> > determinants = [np.linalg.det(s) for s in value_array] >> > charge = np.cumsum(np.angle(determinants / np.roll(determinants, >> 1))) >> > return charge / (2 * np.pi) >> > >> > >> > def qhe_ribbon(W, periodic=False): >> > """ Creates ribbon system >> > >> > If we have periodic boundary conditions, the flux through a single >> > unit cell is quantized. >> > """ >> > W = 2 * (W // 2) >> > >> > def ribbon_shape(pos): >> > (x, y) = pos >> > return (-W / 2 <= y <= W / 2) >> > >> > def onsite(site, p): >> > (x, y) = site.pos >> > return 4 * p.t - p.mu <http://p.mu> >> > >> > def hopping(site1, site2, p): >> > x1, y1 = site1.pos >> > x2, y2 = site2.pos >> > return -p.t * np.exp(-0.5j * p.B * (x1 - x2) * (y1 + y2)) >> > >> > def hopping_periodic(site1, site2, p): >> > x1, y1 = site1.pos >> > x2, y2 = site2.pos >> > return -p.t * np.exp(-1j * np.pi / (W + 1) * np.round((W + 1) * >> > p.B / (2 * np.pi)) * (x1 - x2) * (y1 + y2)) >> > >> > lat = kwant.lattice.square() >> > sym_sys = kwant.TranslationalSymmetry((-1, 0)) >> > sys = kwant.Builder(sym_sys) >> > >> > sys[lat.shape(ribbon_shape, (0, 0))] = onsite >> > >> > if periodic: >> > sys[lat.neighbors()] = hopping_periodic >> > sys[lat(0, - W / 2), lat(0, + W / 2)] = hopping_periodic >> > else: >> > sys[lat.neighbors()] = hopping >> > >> > return sys >> > >> > # Quantum hall bar codes >> > >> > >> > def qhe_hall_bar(L=50, W=10, w_lead=10, w_vert_lead=None): >> > """Create a hall bar system. >> > >> > Square lattice, one orbital per site. >> > Returns kwant system. >> > >> > Arguments required in onsite/hoppings: >> > t, mu, mu_lead >> > """ >> > >> > L = 2 * (L // 2) >> > W = 2 * (W // 2) >> > w_lead = 2 * (w_lead // 2) >> > if w_vert_lead is None: >> > w_vert_lead = w_lead >> > else: >> > w_vert_lead = 2 * (w_vert_lead // 2) >> > >> > # bar shape >> > def bar(pos): >> > (x, y) = pos >> > return (x >= -L / 2 and x <= L / 2) and (y >= -W / 2 and y <= W >> / 2) >> > >> > # Onsite and hoppings >> > def onsite(site, p): >> > (x, y) = site.pos >> > return 4 * p.t - p.mu <http://p.mu> >> > >> > def hopping_Ax(site1, site2, p): >> > x1, y1 = site1.pos >> > x2, y2 = site2.pos >> > return -p.t * np.exp(-0.5j * p.B * (x1 + x2) * (y1 - y2)) >> > >> > def make_lead_hop_y(x0): >> > def hopping_Ay(site1, site2, p): >> > x1, y1 = site1.pos >> > x2, y2 = site2.pos >> > return -p.t * np.exp(-1j * p.B * x0 * (y1 - y2)) >> > return hopping_Ay >> > >> > def lead_hop_vert(site1, site2, p): >> > return -p.t >> > >> > # Building system >> > lat = kwant.lattice.square() >> > sys = kwant.Builder() >> > >> > sys[lat.shape(bar, (0, 0))] = onsite >> > sys[lat.neighbors()] = hopping_Ax >> > >> > # Attaching leads >> > sym_lead = kwant.TranslationalSymmetry((-1, 0)) >> > lead = kwant.Builder(sym_lead) >> > >> > def lead_shape(pos): >> > (x, y) = pos >> > return (-w_lead / 2 <= y <= w_lead / 2) >> > >> > lead_onsite = lambda site, p: 4 * p.t - p.mu_lead >> > >> > sym_lead_vertical = kwant.TranslationalSymmetry((0, 1)) >> > lead_vertical1 = kwant.Builder(sym_lead_vertical) >> > lead_vertical2 = kwant.Builder(sym_lead_vertical) >> > >> > def lead_shape_vertical1(pos): >> > (x, y) = pos >> > return (-L / 4 - w_vert_lead / 2 <= x <= -L / 4 + w_vert_lead / >> 2) >> > >> > def lead_shape_vertical2(pos): >> > (x, y) = pos >> > return (+L / 4 - w_vert_lead / 2 <= x <= +L / 4 + w_vert_lead / >> 2) >> > >> > lead_vertical1[lat.shape(lead_shape_vertical1, (-L / 4, 0))] = >> > lead_onsite >> > lead_vertical1[lat.neighbors()] = lead_hop_vert >> > lead_vertical2[lat.shape(lead_shape_vertical2, (L / 4, 0))] = >> > lead_onsite >> > lead_vertical2[lat.neighbors()] = lead_hop_vert >> > >> > sys.attach_lead(lead_vertical1) >> > sys.attach_lead(lead_vertical2) >> > >> > sys.attach_lead(lead_vertical1.reversed()) >> > sys.attach_lead(lead_vertical2.reversed()) >> > >> > lead[lat.shape(lead_shape, (-1, 0))] = lead_onsite >> > lead[lat.neighbors()] = make_lead_hop_y(-L / 2) >> > >> > sys.attach_lead(lead) >> > >> > lead = kwant.Builder(sym_lead) >> > lead[lat.shape(lead_shape, (-1, 0))] = lead_onsite >> > lead[lat.neighbors()] = make_lead_hop_y(L / 2) >> > >> > sys.attach_lead(lead.reversed()) >> > >> > return sys >> > >> > >> > def calculate_sigmas(G): >> > # reduce by one dimension G -> G[temp, temp] >> > temp = [0, 1, 3, 4, 5] >> > G = G[temp, :] >> > G = G[:, temp] >> > # invert R = G^-1 >> > # find out whether it is a numpy object >> > r = np.linalg.inv(G) >> > # Voltages follow: V = R I[temp] >> > V = r.dot(np.array([0, 0, 0, 1, -1])) >> > # Completely solved the six terminal system. >> > # Consider the 2x2 conductance now: Use I = sigma U >> > E_x = V[1] - V[0] >> > E_y = V[1] - V[3] >> > # formula above >> > sigma_xx = E_x / (E_x**2 + E_y**2) >> > sigma_xy = E_y / (E_x**2 + E_y**2) >> > return sigma_xx, sigma_xy >> > >> > def plot_bandstructure(flead, momenta): >> > >> > bands = kwant.physics.Bands(flead,args = [p]) >> > energies = [bands(k) for k in momenta] >> > >> > plt.figure() >> > plt.plot(momenta, energies) >> > plt.xlabel(r"$k[a^{-1}]$") >> > plt.ylabel(r"$E [t]$") >> > plt.show() >> > >> > >> > sys = qhe_hall_bar(L=60, W=100, w_lead=90, w_vert_lead=28).finalized() >> > kwant.plot(sys) >> > p = SimpleNamespace(t=1.0, mu=0.3, mu_lead=0.3, B=0.5) >> > Bs = np.linspace(0.02, 0.15, 200) >> > num_leads = len(sys.leads) >> > >> > momenta = [-pi + 0.02 * pi * i for i in range(101)] >> > plot_bandstructure(sys.leads[2], momenta) >> > >> > > > _______________________________________________ > Kwant-discuss mailing list > Kwant-discuss@kwant-project.org > https://mailman-mail5.webfaction.com/listinfo/kwant-discuss > > -- Sumit Ghosh Spintronics Theory Group PSE Division KAUST -- ------------------------------ This message and its contents, including attachments are intended solely for the original recipient. 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