We present a mathematically simple procedure for explaining and visualizing the dynamics of quantized transport in topological insulators. The procedure serves to illustrate and clarify the dynamics of topological transport in general, but for the sake of concreteness, it is phrased here in terms of electron transport in a charge-ordered chain, which may be mapped exactly onto transport between edge channels in the Integer Quantum Hall Effect. This approach has the advantage that it allows a direct visualization of the real-space and real-time evolution of the electronic charges throughout the topological pumping cycle, thus demystifying how charge flows between remote edges separated by an insulating bulk, why the amount of transported charge is given by a topological invariant, and how continuous driving yields a discrete, quantized amount of transported charge.

1.
K.
von Klitzing
,
G.
Dorda
, and
M.
Pepper
,
Phys. Rev. Lett.
45
,
494
497
(
1980
).
2.
D. J.
Thouless
,
M.
Kohmoto
,
M. P.
Nightingale
, and
M.
den Nijs
,
Phys. Rev. Lett.
49
,
405
408
(
1982
).
3.
J. E.
Avron
and
R.
Seiler
,
Phys. Rev. Lett.
54
,
259
262
(
1985
).
4.
A.
Tzalenchuk
,
S.
Lara-Avila
,
A.
Kalaboukhov
,
S.
Paolillo
,
M.
Syväjärvi
,
R.
Yakimova
,
O.
Kazakova
,
T. J. B. M.
Janssen
,
V.
Fal'ko
, and
S.
Kubatkin
,
Nat. Nanotechnol.
5
,
186
189
(
2010
).
5.
R. B.
Laughlin
,
Phys. Rev. Lett.
50
,
1395
1398
(
1983
).
6.
D.
Tsui
,
H.
Stormer
, and
A.
Gossard
,
Phys. Rev. Lett.
48
,
1559
1562
(
1982
).
7.
F. D. M.
Haldane
,
Phys. Rev. Lett.
61
,
2015
2018
(
1988
).
8.
C. L.
Kane
and
E. J.
Mele
,
Phys. Rev. Lett.
95
,
226081
(
2005
).
9.
C. L.
Kane
and
E. J.
Mele
,
Phys. Rev. Lett.
95
,
146802
(
2005
).
10.
B. A.
Bernevig
and
S.-C.
Zhang
,
Phys. Rev. Lett.
96
,
106802
(
2006
).
11.
L.
Fu
,
C. L.
Kane
, and
E. J.
Mele
,
Phys. Rev. Lett.
98
,
106803
(
2007
).
12.
N. P.
Armitage
,
E. J.
Mele
, and
A.
Vishwanath
,
Rev. Mod. Phys.
90
,
015001
(
2018
).
13.
A. C.
Potter
and
P. A.
Lee
,
Phys. Rev. B
85
,
094516
(
2012
).
14.
M. Z.
Hasan
and
C. L.
Kane
,
Rev. Mod. Phys.
82
,
3045
3067
(
2010
).
15.
C.
Chiu
,
J.
Teo
,
A.
Schnyder
, and
S.
Ryu
,
Rev. Mod. Phys.
88
,
035005
(
2016
).
16.
Q.
Niu
,
Phys. Rev. B
34
,
5093
5100
(
1986
).
17.
D. J.
Thouless
,
Phys. Rev. B
27
,
6083
6087
(
1983
).
18.
L.
Wang
,
M.
Troyer
, and
X.
Dai
,
Phys. Rev. Lett.
111
,
026802
(
2013
).
19.
J.
Kruthoff
,
J.
de Boer
,
J.
van Wezel
,
C. L.
Kane
, and
R.-J.
Slager
,
Phys. Rev. X
7
,
041069
(
2017
).
20.
R. B.
Laughlin
,
Phys. Rev. B
23
,
5632
5633
(
1981
).
21.
F.
Flicker
and
J.
van Wezel
,
EPL
111
,
37008
(
2015
).
22.
M. A.
Lizunova
,
F.
Schreck
,
C. M.
Smith
, and
J.
van Wezel
,
Phys. Rev. B
99
,
115114
(
2019
).
23.
J. P.
Eisenstein
,
Am. J. Phys.
61
,
179
183
(
1993
).
24.
M. T.
Ahari
,
G.
Ortiz
, and
B.
Seradjeh
,
Am. J. Phys.
84
,
858
868
(
2016
).
25.
D. F.
Holcomb
,
Am. J. Phys.
67
,
278
297
(
1999
).
26.
B. R.
Holstein
,
Am. J. Phys.
57
,
1079
1084
(
1989
).
27.
T. F.
Rosenbaum
and
D. L.
Stein
,
Am. J. Phys.
49
,
128
133
(
1981
).
28.
R.
Peierls
,
More Surprises in Theoretical Physics
(
Princeton U.P
.,
Princeton
,
1991
).
29.
H.
Fröhlich
,
Proc. R. Soc. A
223
,
296
305
(
1954
).
30.
W.
Kohn
,
Phys. Rev. Lett.
2
,
393
394
(
1959
).
31.
W. A.
Little
,
Phys. Rev. A
134
,
1416
1424
(
1964
).
32.
C.
Brun
,
Z.-Z.
Wang
,
P.
Monceau
, and
S.
Brazovskii
,
Phys. Rev. Lett.
104
,
256403
(
2010
).
33.
Y. E.
Kraus
,
Y.
Lahini
,
Z.
Ringel
,
M.
Verbin
, and
O.
Zilberberg
,
Phys. Rev. Lett.
109
,
106402
(
2012
).
34.
G.
Grüner
,
Rev. Mod. Phys.
60
,
1129
1181
(
1988
).
35.
36.
H.
Fukuyama
,
J. Phys. Soc. Jpn.
41
,
513
520
(
1976
).
37.
P. A.
Lee
,
T. M.
Rice
, and
P. W.
Anderson
,
Solid State Commun.
14
,
703
709
(
1974
).
38.
See supplemental material at https://doi.org/10.1119/10.0001656.
39.
D. R.
Hofstadter
,
Phys. Rev. B
14
,
2239
2249
(
1976
).
40.
M.-C.
Chang
and
Q.
Niu
,
Phys. Rev. Lett.
75
,
1348
1351
(
1995
).
41.
J.
Klinovaja
and
D.
Loss
,
Phys. Rev. Lett.
111
,
196401
(
2013
).
42.
J.
Kruthoff
,
J.
de Boer
, and
J.
van Wezel
,
Phys. Rev. B
100
,
075116
(
2019
).
43.
J.
Zak
,
Phys. Rev. B
32
,
2218
2226
(
1985
).
44.
J.
Zak
,
Phys. Rev. B
23
,
2824
2835
(
1981
).
45.
A.
Silva
, “
Crystal symmetries in charge-orbital order and topological band theory
,” Ph.D. thesis,
University of Amsterdam
,
2019
, https://hdl.handle.net/11245.1/e7ed8e5d-2eb7-4dd6-9f28-bb70acb08895.

Supplementary Material

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