protected edge modes without symmetry michael levin university of maryland
TRANSCRIPT
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Protected edge modes without symmetry
Michael LevinUniversity of Maryland
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Topological phases
2D quantum many body system with:
Finite energy gap in bulk
Fractional statistics
ei 1
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Examples Fractional quantum Hall liquids
Many other examples in principle: Quantum spin systems Cold atomic gases, etc
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Edge physics
= 1/3
Some topological phases have robust gapless edge modes:
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Edge physics
Fully gapped edge
“Toric code” model
Some phases don’t:
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Main question
Which topological phases have protected
gapless edge modes and which do not?
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Two types of protected edges
1. Protection based on symmetry(top. insulators, etc.)
2. Protection does not depend on symmetry
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Two types of protected edges
1. Protection based on symmetry(top. insulators, etc.)
2. Protection does not depend on symmetry
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Edge protection without symmetry
nR – nL 0 protected edge mode
nR = 2nL = 1
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Edge protection without symmetry
nR – nL 0 protected edge mode
nR = 2nL = 1
~ KH , “Thermal Hall conductance”
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Edge protection without symmetry
What if nR - nL = 0? Can edge ever be
protected?
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Edge protection without symmetry
What if nR - nL = 0? Can edge ever be
protected?
Yes!
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Examples
= 2/3
nR – nL = 0
Protected edge
Superconductor
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Examples
= 8/9
No protected edge
= 2/3
Protected edge
nR – nL = 0nR – nL = 0
SuperconductorSuperconductor
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General criterion
An abelian phase with nR = nL can have a gappededge if and only if there exists a subset of quasiparticle types, S = {s1,s2,…} satisfying:
(a) eiss’ = 1 for any s, s’ S “trivial statistics”
(b) If tS, then there exists sS with eist 1 “maximal”
eist
s
t
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Examples
= 8/9: Quasiparticle types = {0, e/9, 2e/9,…, 8e/9}
Find S = {0, 3e/9, 6e/9} works
edge can be gapped
2/3: Quasiparticle types = {0, e/3, 2e/3}
Find no subset S works
edge is protected
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Microscopic analysis: = 8/9
= 8/9
L = 1/4 x1(t1 – v1 x1) -9/4 x2(t2 – v2 x2)
Electron operators: 1 = ei1
2 = e-9i2
12
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Microscopic analysis: = 8/9
L = 1/4 xT (K t- V x)
1 0 v1 0
2 0 –9 0 v2
V =K ==
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Microscopic analysis: = 8/9
Simplest scattering terms:
“U 1m 2
n + h.c.” = U Cos(m - 9n)
Will this term gap the edge?
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Null vector criterion
Can gap the edge if
(m n) 1 0 = 0 0 -9
Guarantees that we can chg. variables to = m1 - 9n2, = n1 + m2 with:
L x t – v2 (x)2 – v2 (x)2 + U cos()
mn
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Null vector criterion
(m n) 1 0 = 0 0 -9
m2 – 9 n2 = 0
Solution: (m n) = (3 -1)
U cos(31 + 92) = “13 2
* + h.c” can gap edge.
mn
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Microscopic analysis: = 2/3
= 2/3
L = 1/4 x1(t1 – v1 x1) -3/4 x2(t2 – v2 x2)
Electron operators: 1 = ei1
2 = e-3i2
12
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Null vector condition
(m n) 1 0 = 0 0 -3
m2 – 3 n2 = 0
No integer solutions.
(simple) scattering terms cannot gap edge!
mn
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General case
Edge can be gapped iff there exist {1,…,N}
satisfying:
iT K j = 0 for all i,j (*)
Can show (*) is equivalent to original criterion
2N
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Problems with derivation
1. Only considered simplest kind of backscattering terms
proof that = 2/3 edge is protected is not complete
2. Physical interpretation is unclear
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Annihilating particles at a gapped edge
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Annihilating particles at a gapped edge
ss
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Annihilating particles at a gapped edge
ss
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Annihilating particles at a gapped edge
ss
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Annihilating particles at a gapped edge
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Annihilating particles at a gapped edge
a b
ss
“s, s can be annihilated at the edge”
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Annihilating particles at a gapped edge
Define:
S = {s : s can be annihilated at edge}
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Constraints from braid statistics
Ws
Have: Ws |0> = |0>
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Constraints from braid statistics
Ws Ws’
Have: Ws’ Ws |0> = |0>
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Constraints from braid statistics
Ws Ws’
Similarly: Ws Ws’ |0> = |0>
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Constraints from braid statistics
On other hand: Ws Ws’ |0> = eiss’ Ws’Ws |0>
Ws Ws’
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Constraints from braid statistics
On other hand: Ws Ws’ |0> = eiss’ Ws’Ws |0>
Ws Ws’
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Constraints from braid statistics
On other hand: Ws Ws’ |0> = eiss’ Ws’Ws |0>
Ws Ws’
eiss’ = 1 for any s, s’ that can be annihilated at edge
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Braiding non-degeneracy in bulk
t
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Braiding non-degeneracy in bulk
If t can’t be annihilated (in bulk) thenthere exists s with eist 1
t s
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Braiding non-degeneracy at a gapped edge
t
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Braiding non-degeneracy at a gapped edge
If t can’t be annihilated at edge then there exists s with eist 1 which CAN be annihilated at edge
t
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Braiding non-degeneracy at edge
Have:
(a). eiss’ = 1 for s,s’ S
(b). If t S then there exists s S with eist 1
Proves the criterion
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Summary Phases with nL– nR = 0 can have protected
edge
Edge protection originates from braiding statistics
Derived general criterion for when an abelian topological phase has a protected edge mode