Fluid mechanics and some novel surface waves Part I Baile Zhang Nanyang Technological University, Singapore 1
Fluid mechanics and some novel surface waves
Part I
Baile Zhang
Nanyang Technological University, Singapore
1
4Kundu and Cohen, Fluid Mechanics, 3rd edition
2v ds rv C
Circulation
2
Cv
r >R
Bathtub VortexCurl-free bathtub vortex
0v
5Physics Today 62, 38 (2009)/AIP
Aharonov-Bohm effect
2v ds rv C
Circulation
2
Cv
r >R
Bathtub Vortex vs AB EffectCurl-free bathtub vortex
0v
6
Eur. J. Phys. 1, 154-162 (1980)
M. V. Berry
Bathtub Vortex vs AB Effect
wave
Wavefront dislocations
7
r
ek i A
c
In the continuumrp i pr i
rk i kr i
For a charge in EM field
“Minimal coupling”
In a lattice
kr i + ?
‘Vector Potential’ in k-Space
8
In the continuumkr i
( ) ( ) ik rr dk k e ( ) ( ) ik rr r dkr k e
( )[ ]ik r
kdk k i e ( ) ik r
kdki k e
kr i
‘Vector Potential’ in k-Space
Kai Sun, Univ. of Michigan
‘Vector Potential’ in k-Space
9
In a lattice
,( ) ( ) ( )n n k
n
r dk k r , ,( ) ( ) ik r
n k n kr u r e
,( ) ( ) ( )( )ik r
n n k k
n
r r dk k u r i e
, ,[ ( ) ( ) ( ) ( )] ik r
k n n k n k n k
n
dk i k u r k i u r e
' ' '
, ,[ ( ) ( ) ( ) ( ) ( )] ik r
k n n k n k n k
n
dk i k u r k dr r r i u r e
' * ' '
, , , ,[ ( ) ( ) ( ) ( ) ( ) ( )] ik r
k n n k n m k m k k n k
n m
dk i k u r k dr u r u r i u r e
, ,[ ] ( ) ( )k m n n n k
n m
dk i k r
' * ' '
, , ,( ) ( )m k k n k m ni dr u r u r Define
kr i + ?
( )
Kai Sun, Univ. of Michigan
10
, ,( ) [ ] ( ) ( )k m n n n k
n m
r r dk i k r ' * ' '
, , ,( ) ( )m n m k k n ki dr u r u r where
If only a single band is considered
, ,( ) [ ] ( ) ( )k n n n n kr r dk i k r
Sok nr i
Berry connection
r
ek i A
c Vector potential
, ,| |n n k k n ki u u
‘Vector Potential’ in k-Space
Kai Sun, Univ. of Michigan
Vector Potential vs ‘Vector Potential’ in k-Space
11
Berry connection
n
Vector potential
A
Magnetic field
B A
Berry curvature
n nk
Magnetic monopole
2M
cB ds n
e
Chern number
2BZ
dk C
14Kundu and Cohen, Fluid Mechanics, 3rd edition
22
1 12 1 1 22
1 2 2
1
1 ( / )
R Rv r
R R R r
When2 0
22
1 11 12
1 2 2
1
1 ( / )
R Rv r
R R R r
Circular Couette Flow
Analytic solution from Navier-Stokes equations
15
1tv v v p
Euler’s equation:
0v v v
1r r0( )tp v
0 02
1 1( ) ( ) 0t tv v
c
22 2
2
1 1( , ) ln ln
4 2V x y
c
Schrodinger-type
where
J. Acoust. Soc. Am. 87 (6), 2292 (1990).
2[( ) ( , )] 0effiA V x y
0n
0
2
( , )eff
v x yA
c
At0p p p
Master equation
Let
Continuity:
( ) 0t v
Acoustic Waves with ‘Magnetic Flux’
16
Haldane model, PRL 61, 2015-2018 (1988)
Complex next-nearest-neighbor (NNN) hopping
The first theoretical model of topological insulator beyond quantum Hall effect
Haldane Model with Staggered Magnetic Field
19
Topology and Topological Insulator
=
=
= =
Hole num.: 0 Hole num.: 1 Chern num.: 0 Chern num.: 1
=
20
Trivial insulator
C=0
C≠0Topological insulator
insulating
insulating
conductive
Topology and Topological Insulator