Protected edge modes without symmetry

Michael LevinUniversity of Maryland

Topological phases

2D quantum many body system with:

Finite energy gap in bulk

Fractional statistics

ei 1

Examples Fractional quantum Hall liquids

Many other examples in principle: Quantum spin systems Cold atomic gases, etc

Edge physics

= 1/3

Some topological phases have robust gapless edge modes:

Edge physics

Fully gapped edge

“Toric code” model

Some phases don’t:

Main question

Which topological phases have protected

gapless edge modes and which do not?

Two types of protected edges

1. Protection based on symmetry(top. insulators, etc.)

2. Protection does not depend on symmetry

Two types of protected edges

1. Protection based on symmetry(top. insulators, etc.)

2. Protection does not depend on symmetry

Edge protection without symmetry

nR – nL 0 protected edge mode

nR = 2nL = 1

Edge protection without symmetry

nR – nL 0 protected edge mode

nR = 2nL = 1

~ KH , “Thermal Hall conductance”

Edge protection without symmetry

What if nR - nL = 0? Can edge ever be

protected?

Edge protection without symmetry

What if nR - nL = 0? Can edge ever be

protected?

Yes!

Examples

= 2/3

nR – nL = 0

Protected edge

Superconductor

Examples

= 8/9

No protected edge

= 2/3

Protected edge

nR – nL = 0nR – nL = 0

SuperconductorSuperconductor

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

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

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

Microscopic analysis: = 8/9

L = 1/4 xT (K t- V x)

1 0 v1 0

2 0 –9 0 v2

V =K ==

Microscopic analysis: = 8/9

Simplest scattering terms:

“U 1m 2

n + h.c.” = U Cos(m - 9n)

Will this term gap the edge?

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

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

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

Null vector condition

(m n) 1 0 = 0 0 -3

m2 – 3 n2 = 0

No integer solutions.

(simple) scattering terms cannot gap edge!

mn

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

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

Annihilating particles at a gapped edge

Annihilating particles at a gapped edge

ss

Annihilating particles at a gapped edge

ss

Annihilating particles at a gapped edge

ss

Annihilating particles at a gapped edge

Annihilating particles at a gapped edge

a b

ss

“s, s can be annihilated at the edge”

Annihilating particles at a gapped edge

Define:

S = {s : s can be annihilated at edge}

Constraints from braid statistics

Ws

Have: Ws |0> = |0>

Constraints from braid statistics

Ws Ws’

Have: Ws’ Ws |0> = |0>

Constraints from braid statistics

Ws Ws’

Similarly: Ws Ws’ |0> = |0>

Constraints from braid statistics

On other hand: Ws Ws’ |0> = eiss’ Ws’Ws |0>

Ws Ws’

Constraints from braid statistics

On other hand: Ws Ws’ |0> = eiss’ Ws’Ws |0>

Ws Ws’

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

Braiding non-degeneracy in bulk

t

Braiding non-degeneracy in bulk

If t can’t be annihilated (in bulk) thenthere exists s with eist 1

t s

Braiding non-degeneracy at a gapped edge

t

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

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

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