topological mechanics
TRANSCRIPT
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Topological Mechanics
I. Topology and the Wave Equation
II. Topological Phonons and Photons:
- Topological band gaps at finite frequency
III. Classical Mechanical Modes in Isostatic Lattices
- Floppy modes and Maxwell’s counting rule
- Twisted Kagome lattice Model
- Topological boundary modes.
- Index theorem
- Analog SSH model
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Topological Electronic Phases
2D Integer quantum Hall effectno symmetry
Bulk: Integer Chern invariant
Boundary: Chiral edge states
2D topological insulatortime reversal symmetry
Bulk: Z2 invariant
Boundary: Helical edge states
3D topological insulatortime reversal symmetry
Bulk: Z2 invariant
Boundary: Helical surface state
1D Topological Superconductorparticle-hole symmetry
Bulk: Z2 invariant
Boundary: Majorana zero mode
Quantum Hall state
n=1
Quantum spin Hall
insulator
1D topo. SC
0
E
k
3D TI
E
Bulk Topological Invariant Boundary Topological Modes
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Wave Equations
i ij ji H
i ij jmu D u
e.g. Schrodinger Equation
e.g. Newton’s Laws
Quantum:
Classical:
Common features:
Normal modes define an eigenvalue problem.
Role of symmetry and topology for bulk and boundary modes
Search for periodic “metamaterials” with topological band structures
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Periodic Table of Topological Insulators and Superconductors
- Time Reversal :
- Particle - Hole :
Unitary (chiral) symmetry :
1( ) ( ) 12 ; H H k k
1( ) ( ) 12 ; H H k k
1( ) ( )H H k k ;
Kitaev, 2008
Schnyder, Ryu, Furusaki, Ludwig 2008
Even richer topological classes when accounting for crystalline space group symmetries :
“weak topological insulators”, “topological crystalline insulators”, ……
Insulator: no symmetry
BdG superconductor
T invariant BdG superconductor
T invariant insulator
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Periodic Table of Topological Insulators and Superconductors
- Time Reversal :
- Particle - Hole :
Unitary (chiral) symmetry :
1( ) ( ) 12 ; H H k k
1( ) ( ) 12 ; H H k k
1( ) ( )H H k k ;
Kitaev, 2008
Schnyder, Ryu, Furusaki, Ludwig 2008
Even richer topological classes when accounting for crystalline space group symmetries :
“weak topological insulators”, “topological crystalline insulators”, ……
no symmetry
T - invariant
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Classical Topological Band Phenomena
Topological bandgaps and chiral edge modes at finite frequency in classical systems
In two dimensions with broken time reversal symmetry.
Photonic
Phononic
Gyroscopic Metamaterial
Haldane, Raghu PRL 2008
Wang, Chong, Joannopoulos,
Soljacic, PRL 2008
…..
Fre
qu
en
cy (
GH
z)
wavevector (2p/a)
Prodhan, Prodhan, PRL 2009
Nash, Kleckner, Read, Vitelli,
Turner, Irvine, PNAS 2015
……..
Microwave Waveguide
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Nfm ≥ d ns - nbNfm – Nss = d ns - nb
Maxwell Problem
Is a “frame” or configuration of masses
and springs mechanically stable ?
ns = # sites
nb = # bonds, d = dimension
Nfm = # zero frequency “floppy modes”
Maxwell Counting Rule:
JC Maxwell 1865
* (Calladine ’78)
Nss = # states of self-stress
State of self-stress
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Proof of Maxwell-Calladine Counting Rule:
2
1
1 1 1
2 2 2
bnT
m
m
U u D u kx u QQ u
1
sdnT
m mi i
i
x Q u
elastic energy :
bond extension xm: site displacement ui
1
bn
i im m
mi
Uf Q t
u
site force fi: bond tension tm=kxm
dns x nb “Equilibrium Matrix” Qim
floppy mode : 0TQ u Nfm = # zero eigenvectors of QT
self-stress state: 0Q t Nss = # zero eigenvectors of Q
Rank Nullity Theorem of Linear Algebra :
n = Nfm - Nss = #rows - #columns of Qim = dns - nb
n = “index” of Q : simplest version of an index theorem.
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Periodic Isostatic Lattice
A periodic structure with dns – nb = 0
Coordination number (# neighbors): z = 2d
d=2 square lattice (z=4) d=3 pyrochlore (z=6)
On the verge of
mechanical instability
d=2 kagome lattice (z=4)
A model system for problems in soft matter and statistical physics
• Rigidity percolation
• Random closed packing, Jamming
• Network glasses
isostatic on
the average
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Kagome Lattice Model Sun, Souslov, Mao and Lubensky 2012
floppy
mode (fm)
state of
self stress
(ss)
Untwisted
Twisted
periodic boundary condition
Periodic b.c. : 2 nsite - nbond = 0
NFM = NSS ~ L/a
Periodic b.c. : twisting eliminates
both FM and SS
floppy modes
w(k) = 0
kx
ky
kx
ky
Open b.c. : 2 nsite - nbond ~ L/a
NFM ~ L/a; NSS = 0
Open b.c. : NFM ~ L/a; NSS = 0
Localized on Boundary
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Floppy Modes on a Free Boundary
For twisted Kagome, floppy modes required by Maxwell’s count are localized on boundary
Fixed Boundary
Free Boundary 0 p/ap/akx
wn
x
bulk acoustic
modes
zero frequency mode
localized at boundary
Normal Mode SpectrumStrip Geometry
0
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Tom Lubensky
2012 Tom: Are my boundary modes related
to your boundary modes?
CLK: I don’t think so
i ij jmu D u i ij ji H
2013 Tom: Are you sure ?
2nd order in time
Dynamical matrix D has only
positive eigenvalues mw2
1st order in time
Hamiltonian H has positive
or negative eigenvalues E
Newton’s LawsSchrodinger Equation
w
k
E
k
No “valence band” Topologically classify valence band
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Dirac’s Problem :
Klein Gordon Equation
2 2 2 2( )t x yp p m
Paul Dirac: “I’m trying to take
the square root of something”2 2 2( )x y x x y y zp p m I p p m
2 2 2
2 2 2
0
0
x y x y x y
x y x y x y
m p ip m p ip p p m
p ip m p ip m p p m
( )p i
Dirac’s Square Root predicted
the anti-electron (= positron)
e-
e+
E
p
Our Problem :2
t u D u
TD QQ 21 1 1
2 2 2
T
n
n
U u D u k x u QQ u
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TD QQ TD Q Q
“Supersymmetric partners”
0
0
T
T
QQ u
Q Q t
floppy mode
state of self stress
20 0
0 0 ;
T
T T
Q QQH H
Q Q Q
Equivalent “Quantum Hamiltonian”eigenvalues of H :
n nE w
Symmetries
Time reversal ( H=H* )
Particle – Hole (H tz = -tz H) Class “BDI” (same as SSH model)
2nD D w and have same eigenvalues:
2 2 ( ) ( )T T T T
n n n n n nQQ u u Q Q Q u Q uw w
Except for zero modes
+ both kinds of
zero modes
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no symmetry
T - invariant
( ) ( , )Q k GL n C
Integer Topological Invariant:
0 det[ ]Q C
11 1Tr[ ( ) ( )] log det[ ( )]
2 2k
BZ BZn Q k dQ k dk Q k
i ip p
= winding number of
phase of det[Q]d=1 :
d=3 : 3
1
2
1Tr ( ) ( )
24 BZn Q k dQ k
p
invertable complex
n x n matrix
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D=2: “Weak Topological Invariants”
kx
ky
p/a
p/a
p/a
p/a
BZ
C1
C2
Two independent (1D) winding numbers
1
1,2
1Tr[ ( ) ( )]
2 jj
Cn Q k dQ k
ip
The two invariants define a lattice vector
1 1 2 2T n n R a a
where Cj is along reciprocal lattice generator bj with 2i j ijp a b
Twisted Kagome lattice model : 0T R
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New Topological Phases and Domain Walls
Z x Z topological invariant: RT = n1 a1 + n2 a2 (lattice vector)
“Deformed” Kagome lattice model can have : RT ≠ 0
RT= 0
RT =
RT = 0
periodic boundary condition
floppy
modes
self-
stress
0 p/ap/akx
Spectrum of ~H D
En
x
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Two Kinds of Zero Modes?
1. Edge modes due to mismatch of # sites and # bonds
Global count of zero modes:
Maxwell/Calladine rule
Nfm – Nss = d ns - nb
2. Topological boundary modes
No mismatch in sites
and bonds
Are they related?
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Index Theorem
A “local” generalization of the Maxwell-Caladine counting rule
Variant of a famous theorem in mathematics
fm ss
S S S S
L TN N n n
S S S
L s bdn nn “Local count” of sites
and bonds in S
1
ˆcell
dS
T TS
d Sn
Vn
R
# floppy modes and
states of self stress
in region S
“Topological count” on
boundary of S
RT1 RT
2
S1
S2
Depends on edge
termination
Depends on topological
class(es) of bulk
Attiyah and Singer ‘63
Callias, Bott and Seeley ‘78
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Sketch of proof of Index Theorem
0
0
TQH
Q
1 0
0 1s
b
dnz
n
t
'
'
0ˆ
0
i ii
m mm
rr
r
Region S defined by “support function”: 1 for
( )0 for
S
S
S
rr
rˆ ˆ( )S S r
0ˆTr ( ) limS S z L T
S fm ss S S S
iN N
i H
t n n
r
projection onto zero modes
ˆTr ( )L z S S
S S s bdn nn t r
0 0
1 1ˆ ˆlimTr ( ) limTr [ ( ), ]
2
T L z z
S S S S S
HH
i H i H n n n t t
r r
Since H is local, contribution comes only from boundary of S. Further manipulation
relates the result to the topological polarization integrated around boundary.
, 0zH t
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Boundary modes for different edge terminations
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Recall the Su Schrieffer Heeger Model
“A phase” : u<0
“B phase” : u>0
Polyacetylene: A 1D conducting polymer
u<0
u>0
≠ 0
B phase
≠0
A phase
u0
=0: Topological Quantum
Critical Point
Insulator
Gap ~ |u| ≠ 0
• Dimerized : u = ± u0
• Undimerized : Conductor
Gap =0
CH
E
E
c.b.
v.b.
EF0
0
2 band model: ( ) | ( ) |E k k h
( )H k `0
0
x y
x y
h ih
h ih
( )k h kp/ap/a
( ), 0zH k Class BDI
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A and B phases are topologically distinct
Bulk Boundary Correspondence :
At the boundary between topologically distinct insulating phases,
there exist topologically protected low energy states.
0
E=0zero mode
conduction band
valence band
Jackiw and Rebbi 76,
Su Schrieffer, Heeger 79
u(x)0(x)
A B
Distinguished by integer* topological invariant
h(k) h(k)
hx
hy
hx
hy
Nw=0 Nw=1
Nw = winding number characterizing h(k)
* Assuming ‘particle-hole’ symmetry
0
z z
z
z
E E
H H h
spectrum symmetric
under E → -E
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Mechanical Analog of SSH Model
A B
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A model of the modelB. Chen, N. Upadhyaya, V. Vitelli, PNAS 111, 13004 (2014).
Vincenzo Vitelli Bryan Chen
University of Leiden
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Conclusion
Topological boundary modes are an elegant consequence of a
mathematical structure that has applications in diverse areas
• Topological Electronic Phases
• Mechanical Modes of isostatic systems
Much more to do:
• New materials and experiments on electronic systems
• Experiments on metamaterials?
- mechanical systems
- optical, electronic, plasmonic systems?
• Role of interactions and nonlinearities