Layer construction of f three- dim imensional topological states - - PowerPoint PPT Presentation

Layer construction of f three- dim imensional topological states and Stri ring-String braiding statis istics

Xiao-Liang Qi Stanford University Vienna, Aug 29th, 2014

Outline

Part I

• 2D topological states and layer construction
• Generalization to 3D: a simplest example
• Layer construction of 3D topological states: general

setting and examples

• Field theory description

Part II

• Some general results on string-string braiding.
• Ref: Chao-Ming Jian & XLQ, arXiv:1405.6688

Supported by

(arriving here tonight…)

• Topological ground state degeneracy; quasiparticles with

fractional quantum numbers and fractional statistics.

𝐹 𝐹𝑕𝑏𝑞 𝐹 𝐹𝑕𝑏𝑞

𝑕 = 0 1 ground state 𝑕 = 1 𝑛 ground states

Topologically ordered states

𝑏 𝑐 𝑑 𝑑

𝐶 ⊗ 𝐶 ⊗

integer quantum Hall fractional quantum Hall

• Quasiparticles have no knowledge about
• distance. Only topology matters.
• Fusion 𝑏 × 𝑐 = 𝑂𝑏𝑐

𝑑 𝑑

• Braiding
• Braiding 𝑏, 𝑐 and spinning 𝑏, 𝑐 is equivalent to spinning

𝑑. Topological spin of particles ℎ𝑏

Key properties of topologically ordered states

𝑏 𝑐 𝑑 𝑏 𝑐 𝑏 𝑐 𝑑 𝑑

= 𝑆𝑏𝑐

𝑑 ⋅ 𝑏

= 𝑓𝑗2𝜌ℎ𝑏 𝑆𝑏𝑐

𝑑 𝑆𝑐𝑏 𝑑

= 𝑓𝑗2𝜌 ℎ𝑏+ℎ𝑐−ℎ𝑑

• 1. Laughlin state Ψ

𝑨𝑗 = 𝑗<𝑘 𝑨𝑗 − 𝑨

𝑘 𝑛𝑓− 𝑗 𝑨𝑗 2

• Quasiparticles labeled by 𝑟 = 0,

1 𝑛 , 2 𝑛 , … , 1 − 1 𝑛

• Fusion rule braiding 𝑆𝑟1𝑟2

𝑟1+𝑟2 = exp 𝑗𝜌 𝑟1𝑟2 𝑛

• Spin ℎ𝑟 =

𝑟2 2𝑛

• 2. 𝑎2 gauge theory (toric code)
• Quasiparticles include charge 𝑓, flux 𝑛 and their

boundstate 𝜔 = 𝑓 × 𝑛.

• Nontrivial braiding 𝑆𝑓𝑛

𝜔 = 𝑗

• Goal of this work: understanding 3D topological

states from 2D ones

Examples of topologically ordered states

𝑟1 𝑟2 𝑟1 + 𝑟2

Part I: Layer construction

• 2D topological states can be constructed from

coupled 1D chains (Sondhi&Yang ‘01, Kane et al ‘02, Teo&Kane, ‘10)

• Weakly coupled chains as a controlled limit that can

realize these topological states.

• Both integer and fractional quantum Hall states can

be realized. 2D topological

• rder

cut glue

• Example 1: integer quantum Hall

(Sondhi&Yang ‘01)

• Electron tunneling between edge

states of each strip: 〈𝑑𝑜𝑀

+ 𝑑𝑜+1,𝑆〉 ≠ 0,

• Electron tunneling can be

equivalently viewed as exciton condensation

• Condensation of the exciton (particle-hole pair) leads

to coherent tunneling between quasi-1D strips

• The strips are glued to a quantum Hall state

Layer construction of 2D topological states

𝑓 −𝑓

• Example 2: Laughlin 1/3 state

(Kane et al ‘02)

• Electron tunneling between

𝜉 =1/3 edges of chiral Luttinger liquids 𝑓 = 3 ×

𝑓 3 ,

𝑑𝑜𝑀

+ 𝑑𝑜+1,𝑆 = 𝑓𝑗 𝜚𝑜𝑀−𝜚𝑜+1,𝑆 3

,

• Electron tunneling effectively generates coherent

quasiparticle tunneling 2D topological order.

• The coherent tunneling can be understood as a

“boson condensation” of the quasiparticle exciton with charge

𝑓 3 , − 𝑓 3

Layer construction of 2D topological states

𝑓/3 −𝑓/3

Generalization of the layer construction to 3D

• General principle: Inter-layer coupling by boson

condensation Wang&Senthil ‘2013

• Abelian states: Chern-Simons theory and 𝐿 matrix (Wen)

ℒ = 1 4𝜌 𝐿𝐽𝐾𝑏𝐽𝜈𝜗𝜈𝜉𝜐𝜖𝜉𝑏𝐾𝜐 − 𝑚𝐽𝑏𝐽𝜈𝑘𝜈

• Quasiparticles labeled by integer vectors 𝑚
• Equation of motion 𝑘𝜈𝑚𝐽 =

1 2𝜌 𝐿𝐽𝐾𝜗𝜈𝜉𝜐𝜖𝜉𝑏𝐾𝜐

• A quasiparticle carries flux 𝛼 × 𝑏I = 2𝜌 𝐿−1𝑚 I

𝑞𝑗 𝑟𝑗

boson 𝑚𝑗

𝑚2 𝑚1

=

𝑚2 𝑚1

• Mutual statistics of 𝑚1, 𝑚2 given by 𝜄12 = 2𝜌𝑚1

𝑈𝐿−1𝑚2

• Local particles given by 𝜇 = 𝐿𝑚 (bosons or fermions)
• Examples:
• Laughlin 1/𝑛 state 𝐿 = 𝑛. Quasiparticle braiding

𝜄12 =

2𝜌𝑟1𝑟2 𝑛

. Local particle (electron) 𝑟 = 𝑛

• 𝑎𝑂 gauge theory 𝐿 = 0 𝑂

𝑂 0

• Charge 𝑓 = 1

0 , 𝑛 = 0 1 . Quasiparticle braiding 𝜄𝑓𝑛 = 2𝜌 𝐿−1 12 =

2𝜌 𝑂

Examples of K-matrix theory

General setting of the layer construction

• 𝑀 layers of 2D Abelian states, each with a 𝐿 matrix
• Find quasiparticles 𝑞𝑗, 𝑟𝑗 in each layer, so that the bound state

are bosonic and mutually bosonic.

• In 2D language,
• Requirements

𝑞𝑗

𝑈𝐿−1𝑞𝑘 + 𝑟𝑗 𝑈𝐿−1𝑟𝑘 = 0,

𝑞𝑗

𝑈𝐿−1𝑟𝑘 = 0.

• Number of condensed particles: 𝑗 = 1,2, … , 𝑂 when dim 𝐿 =

2𝑂.

• This is an “almost complete” set of null vectors. (Haldane ‘95, Levin

‘13, Barkeshli et al ‘13) There may be remaining particles, responsible

for the topological order.

• With open boundary, 𝑟𝑗 at top surface is always deconfined.

𝑞𝑗

(𝑜)

𝑟𝑗

(𝑜+1)

Example 1: 3D 𝑎𝑞 gauge theories

• Starting from layers of 2D 𝑎𝑞 gauge theories

𝐿 = 0 𝑞 𝑞 0 , ℒ =

𝑞 2𝜌 𝑏𝜈𝜗𝜈𝜉𝜐𝜖𝜉𝑐𝜐 + 𝑏𝜈𝑘𝑓 𝜈 + 𝑐𝜈𝑘𝑛 𝜈 ,

• Coupling the neighbor layers by

condensation of

𝑓 −𝑓 pair

• 𝑞 = 1

0 , 𝑟 = −1

• Particles with nontrivial braiding

with the condensed particle are confined.

• Particles different by a

condensed particle are identified

• Deconfined particles: 𝑓 in 3D, and

𝑛 string (flux tube) 3D 𝑎𝑞 gauge theory

1 0 = 𝑓 𝜄𝑓𝑛 = 2𝜌 𝑞 1 = 𝑛

e

• e

e

• e

e m m m m m m

e

• e

e

• 𝑎𝑞 toric code with tri-layer coupling
• A variation of the construction

in Wang&Senthil ’13

• 𝑞 ≠ 3𝑜

All bulk particles are confined. purely 2D topological order

• Surface central charge 𝑑 = 4 for

𝑞 = 3𝑜 − 1. (𝑞 = 2: surface theory

• f a 3D bosonic TI Vishwanath&Senthil ‘13)
• 𝑞 = 3𝑜

Bulk deconfined particles coexisting with surface particles. 𝑎3 bulk topological order

• Surface central charge 𝑑 = 2

Example 2: Surface and bulk topological order

e+m e e-m e+m e e-m e-m e e-m

e+m e e-m e+m e e-m e-m e e-m n(e+m) n(-e+m) e e e e

• 𝑞𝑗, 𝑟𝑗 expand all quasiparticles in a layer

𝑞𝑗 𝑟𝑗 condensation leads to surface-only topological order.

Surface particles are 𝑟𝑗 at top surface, 𝑞𝑗 at bottom surface

• Surface 𝐿 matrix 𝐿𝑇 = 𝑟𝑗

𝑈𝐿−1𝑟𝑘 −1

• The same topological order at the side surfaces
• Bulk has nontrivial particle when 𝑞𝑗 ∩ 𝑟𝑗 ≠ 𝜚
• Relation to Walker-Wang model (K Walker & Z Wang, ‘12):

modular tensor category  Surface-only topological order Pre-modular tensor category Bulk nontrivial topological order

General criteria of surface-only topological order

Example 3: String-String braiding

• 𝑎4𝑜 toric code theories

with 4-layer coupling

• Condensed particles:

hybridization of the red and blue layers

• Bulk deconfined

particles: 2 point particles, 2 strings

• String-particle braiding
• String-string braiding

phase 𝜕𝑓𝑛 =

2𝜌𝑀 4𝑜

proportional to the number of layers

2e2

• e1
• 2e2

e1

• m2

e1-2m1 m2 e1+2m1 2e2

• e1
• 2e2

e1

• m2

e1-2m1 m2 e1+2m1 2e2 e1 2n⋅m2 m2 m2 m2 m2 e2 e2 e2 e2

String-String braiding and dislocations

• Strings wraping around z direction have braiding

proportional to system size

• Contractible strings have

trivial braiding

• The more fundamental

process of string braiding can be defined at presence

• f an edge dislocation
• Braiding at presence of

the dislocation 𝜕𝑓𝑛

𝑒

=

2𝜌 4𝑜 𝑐𝑨, proportional

to the Burgers vector 𝑐𝑨

Topological field theory description

• A generalized BF theory can be written down to characterize

the string-particle braiding and string-string braiding ℒ = 𝑅𝐽𝐾 2𝜌 𝜗𝜈𝜉𝜏𝜐𝑐𝜈𝜉

𝐽 𝜖𝜏𝑏𝜐 𝐾 + Θ

8𝜌2 𝑆𝐽𝐾𝜗𝜈𝜉𝜏𝜐𝜖𝜈𝑏𝜉

𝐽 𝜖𝜇𝑏𝜏 𝐾 + 𝑘𝜈 𝐽 𝑏𝐽 𝜈

+ 𝐾𝜈𝜉

𝐽 𝑐𝐽 𝜈𝜉

• 𝑘𝜈

𝐽 : particle current; 𝐾𝜈𝜉 𝐽 : string current

• 𝑅𝐽𝐾: string-particle braiding
• 𝑆𝐽𝐾: string-string braiding when strings

are linked with Θ vortex loop.

• Difference from BF theory for TI (Cho&Moore ‘11, Vishwanath&Senthil ’12,

Keyserlingk et al ‘13): Θ is a dynamical field

• Winding number 2𝜌𝑜 of ΘChern-Simons term of 𝑏 with

𝐿 = 𝑜𝑆. String braiding 𝜕𝐽𝐾

𝑜 = 2𝜌𝑜 𝑅−1 𝑈𝑆𝑅−1 𝐽𝐾

• Ordinary 𝑎𝑞 gauge theory: 𝑅 = 𝑞, 𝑆 = 0
• Example 3: 𝑅 = 2𝑜

2 , 𝑆 = 0 1 1

• General structure of string braiding: two strings braid

nontrivially only if they are not contractible.

• Consistent with other recent works on 3-string

braiding (Wang&Levin 1403.7435, Jiang et al 1404.1062, Wang&Wen 1404.7854,

Moradi&Wen 1404.4618)

• The dislocation is described by a Θ vortex string, which

is an extrinsic defect.

• Intrinsic 3-string braiding can possibly be realized by

deconfinement of the dislocations.

Topological field theory description

Part II: General results on string-string braiding

• General structure of 3D topologically ordered states

are not understood yet.

• In 2D, we know the braiding phase 𝑆𝑏𝑐

𝑒

is not arbitrary. There are some identities satisfied by braiding and fusion, such as the hexagon identity.

• In 3D, some similar identities may exist as a property
• f the general structure of topologically ordered states

• Wang&Levin 1403.7435 proposed an identity of the 3-string

braiding in twisted 𝑎𝑞 gauge theories 𝑞 𝜕𝑏𝑐

𝑑 + 𝜕𝑐𝑑 𝑏 + 𝜕𝑑𝑏 𝑐

= 0 (mod 2𝜌),

• Here we give a more general proof

to a stronger identity

𝜕𝑏𝑐

𝑑 + 𝜕𝑐𝑑 𝑏 + 𝜕𝑑𝑏 𝑐 = 0 (mod 2𝜌)

with the general conditions 1) Strings can fuse and split without additional phase; 2) Strings are Abelian; 3) Strings are not marked.

𝑏 𝑐 𝑑

General results on string-string braiding

Step 1 of the proof: 𝜕𝑏𝑐

𝑑

= Ω𝑏𝑐

𝑑

String braiding 𝜕𝑏𝑐

𝑑

𝑏 𝑐 𝑑

String-particle braiding Ω𝑏𝑐

𝑑

between link of 𝑏, 𝑐 and string 𝑑

𝑀𝑏𝑐

Step 2 of the proof: 𝜕𝑏𝑐

𝑑

= Ω𝑏𝑐

𝑑

“linked” string braiding 𝜕𝑏𝑐

𝑑 , for 3 mutually-linked strings

String-particle braiding Ω𝑏𝑐

𝑑

between link of 𝑏, 𝑐 and string 𝑑

𝑀𝑏𝑐 𝑏 𝑐 𝑑

• 𝜕𝑏𝑐

𝑑 : 2𝜌 rotation of 𝑏 and 𝑐 around 𝑑

• 

𝜕𝑏𝑐

𝑑 +

𝜕𝑐𝑑

𝑏 +

𝜕𝑑𝑏

𝑐 ≃ global 4𝜌 rotation ≃ trivial

Step 3 of the proof: 𝜕𝑏𝑐

𝑑 +

𝜕𝑐𝑑

𝑏 +

𝜕𝑑𝑏

𝑐 = 0

≃

𝑏 𝑐 𝑑 𝑏 𝑐 𝑑

• Using this proof we obtain three

identities 𝜕𝑏𝑐

𝑑 +

𝜕𝑐𝑑

𝑏 +

𝜕𝑑𝑏

𝑐 = 0

𝜕𝑏𝑐

𝑑 + 𝜕𝑐𝑑 𝑏 + 𝜕𝑑𝑏 𝑐 = 0

Ω𝑏𝑐

𝑑

+ Ω𝑐𝑑

𝑏 + Ω𝑑𝑏 𝑐 = 0

• A new feature of 3D topological order

that is qualitatively distinct from 2D case

• Open question: In general, is it always

possible to require the strings to be unmarked, i.e., translation invariant along the string direction?

String braiding identities

𝑏 𝑐 𝑑 𝑏 𝑐 𝑑

A non-Abelian example of string-string braiding

• Little is known about non-Abelian strings.
• However, an example can be found in 3D topological

superconductors

left, 𝐷 = 1 right, 𝐷 = −1 SC pairing Majorana mass Δ𝑀𝑓𝑗𝜄𝑀 Δ𝑆𝑓𝑗𝜄𝑆

Weyl fermions H = 𝑒3𝑦 𝑤 𝜔𝑀

+𝜏 ⋅ 𝑞𝜔𝑀 − 𝜔𝑆 +𝜏 ⋅ 𝑞𝜔𝑆

Superconducting pairing ∫ 𝑒3𝑦 Δ𝑓𝑗𝜄𝑀𝜔𝑀

+𝜏𝑧𝜔𝑀 + + Δ𝑓𝑗𝜄𝑆𝜔𝑆 +𝜏𝑧𝜔𝑆 +

• Chiral vortex strings: vortex loops of 𝜄𝑀 or 𝜄𝑆
• Each vortex string is an axion string, carrying a 1+1

Majorana-Weyl fermion (Callan&Harvey ’85, XLQ&Witten&Zhang ‘12)

• Majorana zero modes carried by vortices with odd

linking number.

• Non-Abelian braiding of 𝑏, 𝑐 similar to 𝑞 + 𝑗𝑞

vortices (Read&Green ‘2000)

A non-Abelian example of string-string braiding

𝑏 𝑐 𝑑 𝑙 𝐹𝑙 𝑙 𝐹𝑙

(see also M Sato, Physics Letters B 575 (2003) 126–130)

• Key difference from Abelian string: splitting/fusion of

string is not adiabatic.

• Non-Abelian strings can fuse to

Abelian strings.

• Braiding depends on the fusion

channel.

A non-Abelian example of string-string braiding

𝜏 𝜏 1 or 𝜔 𝑏 𝑐 𝑑 𝑏 𝑐 𝑑 2 zero modes

• n 𝑏, 𝑐

no zero mode

≠

Summary

• Layer construction provides an explicit approach to

3D topological states.

• Different types of 3D topological states can be

generated, with surface-only topological order and/or bulk topological order

• String-string braiding can be induced in system with

periodic boundary condition or dislocations

• General identity proved for Abelian string-string

braiding

• Non-Abelian 3D topological order: An example can

be found in topological superconductors. There are a lot of open questions for more general cases.