How to realize local Chern marker in kwant.kpm?
Hello Kwant community, I want to calculate the local Chern marker in kwant, roughly a realspace approach to topological numbers such as the Chern number. Basically, one needs to calculate the expectation value of the commutator [x, y] of position operators projected to the occupied states. The main reference is Eqs.(69) in [ http://dx.doi.org/10.1103/PhysRevB.84.241106 ], which has been applied in many other cases.
My understanding is that kwant can define x,y as ‘density operators’ and then this task should fit into the kwant.kpm.correlator framework, because it is a correlation function at zero temperature between the two operators. So I tried in the following code a few ways (some commented) for the simplest squarelattice quantum Hall effect (QHE). But it does not seem to give a sign of quantization, although I'm sure that the same system gives reasonable QHE in conductivity and conductance calculation. E.g., for 1/B in [3, 12], it should show the C=1, C=2 plateaus. I feel that I probably missed or misunderstood something basic here. Is my use of operator and kpm.correlator correct here? Thank you for your help!
from cmath import exp import numpy as np import matplotlib.pyplot as plt import kwant import scipy.sparse.linalg as sla import scipy.sparse as sparse import tinyarray
Nx, Ny = 101, 101 Lx, Ly = Nx1, Ny1 edge=18 eps = 1e7 L_sys = [Lx, Ly]
Ef=0.4
def make_system(a=1, t=1): lat = kwant.lattice.square(a, norbs=1) syst = kwant.Builder() def fluxx(site1, site2, Bvec): pos = [(site1.pos[i]+site2.pos[i]L_sys[i])/2.0 for i in range(2)] return exp(1j * Bvec * pos[1] ) def fluxy(site1, site2, Bvec): pos = [(site1.pos[i]+site2.pos[i]L_sys[i])/2.0 for i in range(2)] return 1 #exp(1j * (Bvec * pos[0]))
def hopx(site1, site2, Bvec): # The magnetic field is controlled by the parameter Bvec return fluxx(site1, site2, Bvec) * t def hopy(site1, site2, Bvec): # The magnetic field is controlled by the parameter Bvec return fluxy(site1, site2, Bvec) * t def onsite(site): return 4*t
#### Define the scattering region. #### syst[(lat(x, y) for x in range(L_sys[0]) for y in range(L_sys[1]))] = onsite # hoppings in xdirection syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = hopx # hoppings in ydirections syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = hopy
# Finalize the system. fsyst = syst.finalized() return lat, fsyst
lat, fsyst = make_system()
sites = fsyst.sites X = sparse.diags([site.pos[0] for site in sites]) Y = sparse.diags([site.pos[1] for site in sites])
def where(site): pos=site.pos x0_min, y0_min = edgeeps, edgeeps x0_max, y0_max= Lxedge+eps, Lyedge+eps return x0_min < pos[0] < x0_max and y0_min < pos[1] < y0_max
def op0(site): return site.pos[0]
def op1(site): return site.pos[1]
def chern(lat, fsyst): # correlator=kwant.kpm.Correlator(fsyst, params=params, operator1=kwant.operator.Density(fsyst, onsite=op0, where=where),\ # operator2=kwant.operator.Density(fsyst, onsite=op1, where=where),\ # num_vectors=10, num_moments=400, vector_factory=kwant.kpm.RandomVectors(fsyst, where=where),\ # bounds=(0, 8.), eps=0.01, rng=0, kernel=None, mean=True, accumulate_vectors=False)
correlator=kwant.kpm.Correlator(fsyst, params=params, operator1=kwant.operator.Density(fsyst, onsite=op0),\ operator2=kwant.operator.Density(fsyst, onsite=op1),\ num_vectors=10, num_moments=400, vector_factory=kwant.kpm.RandomVectors(fsyst, where=where),\ bounds=(0, 8.), eps=0.01, rng=0, mean=True, accumulate_vectors=False)
# correlator=kwant.kpm.Correlator(fsyst, params=params, operator1=X,\ # operator2=Y,\ # num_vectors=10, num_moments=400, vector_factory=kwant.kpm.RandomVectors(fsyst, where=where),\ # bounds=(0, 8.), eps=0.01, rng=0, kernel=None, mean=True, accumulate_vectors=False)
c = correlator(mu=Ef, temperature=0.0) / ((Lx2*edge)*(Ly2*edge)) print(2*np.pi*c) print("Chern number: ", np.real(np.pi*1j*(cc.conj()))) # Chern number
Bfields = [] for BInv in np.arange(3., 12., 2): Bfields.append(1/BInv) NB = len(Bfields) params = dict(Bvec=0) for B in Bfields: params['Bvec'] = B print("B: ", B) chern(lat, fsyst)
Hello Kwant community, After posting the question, I got no reply but came across another paper by some of Kwant's authors [PHYSICAL REVIEW RESEARCH 2, 013229 (2020), Computation of topological phase diagram of disordered Pb1−xSnxTe using the kernel polynomial method]. This paper uses kwant to calculate the Chern marker I asked although with mirror symmetry and disorder. These additional features aside, its code still seems to be rather different from just applying kwant.kpm.correlator as in my code. I am a bit confused now.
I initially thought the topological marker could be directly calculated using kwant.kpm.correlator. Is it true at all? If yes, what's the relation with this 2020 paper? If no, so this 2020 paper is the correct and probably optimized calculation of topological marker in kwant?
Any comment will be appreciated!
Hi Jerry,
The approach of PRR is what you should use if you want the Chern marker at a single energy. Note, however, that explicit methods will scale similarly in 2D—those are often competitive with KPM, and that's why we chose a 3D application. I'm not sure what's the most efficient way if you want the answer at all energies. Off the top of my head, the KPM correlator will scale as the resolution squared, which is something you almost never want.
Best, Anton
On Sun, 7 Aug 2022 at 09:16, Jerry xhm xiaoxiao.zhang@riken.jp wrote:
Hello Kwant community, After posting the question, I got no reply but came across another paper by some of Kwant's authors [PHYSICAL REVIEW RESEARCH 2, 013229 (2020), Computation of topological phase diagram of disordered Pb1−xSnxTe using the kernel polynomial method]. This paper uses kwant to calculate the Chern marker I asked although with mirror symmetry and disorder. These additional features aside, its code still seems to be rather different from just applying kwant.kpm.correlator as in my code. I am a bit confused now.
I initially thought the topological marker could be directly calculated using kwant.kpm.correlator. Is it true at all? If yes, what's the relation with this 2020 paper? If no, so this 2020 paper is the correct and probably optimized calculation of topological marker in kwant?
Any comment will be appreciated!
Dear Anton, Thank you for your reply. I’m probably more interested in 3D and will scan magnetic fields. Could you please help clarify whether the following understandings are correct?
1. Both the PRR method and the builtin kwant.kpm.correlator can calculate the Chern marker. The former is more efficient in 3D.
2. Although the code in my first post for a 2D QHE is probably not yet enough tuned, conceptually it is correct: define x,y as kwant density operators and then feed to kwant.kpm.correlator. (I'm not sure since there is no example in the documentation other than the special case of kpm.conductivity.)
3. The main difference between the two methods lies in how they use kpm expansion. kwant.kpm.correlator uses the socalled twodimensional kpm expansion. The PRR method uses the conventional kpm expansion to approximate the projection below Fermi energy.
4. Actually, I wonder what makes the PRR method special as I want to code it myself. Is it only 2.? Or anything else?
Thank you very much!
participants (2)

Anton Akhmerov

Jerry xhm