Edge States on Ribbons

Cutting a periodic 2D model into a ribbon (periodic in one direction, finite/open in the other – see tbkit.kspace.ribbon()) is the standard way to see edge physics in a tight-binding band structure. Two classic examples:

1. A zigzag graphene ribbon has a partially-flat band pinned at E=0, built entirely from states localized on the two edges.

2. Adding Kane-Mele intrinsic spin-orbit coupling (see The Haldane Model: Berry Curvature and a Topological Phase Transition for the closely related Haldane model) opens a bulk gap and produces helical edge states: gapless, spin-momentum-locked modes that cross the bulk gap, one Kramers pair per edge. This is the original model of a 2D (Z2) topological insulator.

import numpy as np
import matplotlib.pyplot as plt

from tbkit.lattice import Lattice
from tbkit.kspace import ribbon, PAULI
from tbkit.system import System
from tbkit.plot import Plot


DX, DY = 0.5 * 3 ** 0.5, 0.5
unit_cell = [{'tag': 'a', 'r0': (0., 0.)}, {'tag': 'b', 'r0': (DX, DY)}]
prim_vec = [(2*DX, 0.), (DX, 1.5)]
lat = Lattice(unit_cell=unit_cell, prim_vec=prim_vec)
t1 = 1.
graphene_hop = [{'i': 0, 'j': 1, 'R': (0, 0), 't': t1},
                        {'i': 0, 'j': 1, 'R': (-1, 0), 't': t1},
                        {'i': 0, 'j': 1, 'R': (0, -1), 't': t1}]
width = 30

Zigzag graphene ribbon: the E=0 edge flat band

# A narrow version of the same ribbon, drawn: periodic left-to-right and
# open top and bottom. It is those two zigzag edges that carry the states
# found below.
rib_small = ribbon(lat, graphene_hop, width=6, direction=1)
# ribbon() stacks its rows along the other primitive vector, which here has
# a component along the periodic direction too, so tiling its cell as-is
# draws a slanted parallelogram. Sliding each site back by whole multiples
# of a1 is the same lattice, drawn as the strip one pictures.
a1 = np.array(rib_small.lat.prim_vec[0])
folded = [{'tag': d['tag'],
               'r0': tuple(np.array(d['r0']) - round(np.dot(d['r0'], a1)/(a1 @ a1))*a1)}
              for d in rib_small.lat.unit_cell]
lat_draw = Lattice(unit_cell=folded, prim_vec=rib_small.lat.prim_vec)
lat_draw.get_lattice(n1=9)
vis = System(lat_draw)
vis.set_hopping([{'n': 1, 't': t1}])
fig_rib = Plot(vis).lattice(plt_hop=True, ms=9, figsize=(8, 4.5))

The ribbon actually diagonalized is 30 cells wide:

rib = ribbon(lat, graphene_hop, width=width, direction=1)
ks = np.linspace(-np.pi, np.pi, 400)
en = rib.get_bands(ks[:, None])

tol = 1e-2  # a finite-width ribbon splits the edge-state degeneracy by an
              # amount exponentially small in the width, so "zero energy"
              # needs a little tolerance away from the zone boundary.
n_zero_modes = np.sum(np.any(np.abs(en) < tol, axis=1))
print('Zigzag ribbon: {} k-points host a (near-)zero mode (out of {}), '
           'vs. the 1/3 expected analytically.'.format(n_zero_modes, len(ks)))
assert n_zero_modes > len(ks) // 4

fig, ax = plt.subplots()
ax.plot(ks, en, 'b', lw=0.7)
ax.set_xlabel('$k$')
ax.set_ylabel('$E$')
ax.set_title('Zigzag graphene ribbon ({} unit cells wide)'.format(width))
ax.set_xlim([ks[0], ks[-1]])

Zigzag ribbon: 138 k-points host a (near-)zero mode (out of 400), vs. the 1/3 expected analytically.

(-3.141592653589793, 3.141592653589793)

Kane-Mele ribbon: helical edge states inside a spin-orbit gap

The intrinsic (sz-conserving) spin-orbit term opens a bulk gap, but a Kramers pair of edge states still crosses zero energy somewhere in the Brillouin zone (its momentum is set by where the bulk Dirac points K, K’ project onto this ribbon’s 1D Brillouin zone) – Kramers’ theorem forbids gapping out a single pair of counter-propagating edge modes, which is the hallmark of a Z2 topological insulator (contrast with the zigzag ribbon above, where nothing protects the flat band from a generic perturbation).

lam = 0.1
kane_mele_hop = list(graphene_hop)
for R in [(1, 0), (0, 1), (1, -1)]:
    kane_mele_hop.append({'i': 0, 'j': 0, 'R': R, 't': 1j*lam*PAULI['z']})
    kane_mele_hop.append({'i': 1, 'j': 1, 'R': R, 't': -1j*lam*PAULI['z']})

rib_km = ribbon(lat, kane_mele_hop, width=width, direction=1, spin=True)
en_km = rib_km.get_bands(ks[:, None])

min_abs_e_per_k = np.min(np.abs(en_km), axis=1)
i_cross = np.argmin(min_abs_e_per_k)
print('Kane-Mele ribbon: edge-state crossing near k={:.4f}, E={:.5f}.'
           .format(ks[i_cross], en_km[i_cross, np.argmin(np.abs(en_km[i_cross]))]))
assert min_abs_e_per_k[i_cross] < 0.01
# deep in the "bulk" region of k (far from the edge-state crossing), the
# gap must be fully open.
bulk_gap = min_abs_e_per_k[np.argmin(np.abs(ks))]  # at k=0
print('Bulk gap at k=0 (far from the edge-state crossing): {:.4f}.'.format(bulk_gap))
assert bulk_gap > 0.5

fig2, ax2 = plt.subplots()
ax2.plot(ks, en_km, 'b', lw=0.7)
ax2.set_xlabel('$k$')
ax2.set_ylabel('$E$')
ax2.set_ylim([-1., 1.])
ax2.set_title('Kane-Mele ribbon: helical edge states in the bulk gap')
ax2.set_xlim([ks[0], ks[-1]])

Kane-Mele ribbon: edge-state crossing near k=-1.8188, E=0.00161.
Bulk gap at k=0 (far from the edge-state crossing): 1.0095.

(-3.141592653589793, 3.141592653589793)