Layer construction of f three- dim imensional topological states and Stri ring-String braiding statis istics Xiao-Liang Qi Stanford University Vienna, Aug 29 th , 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…)
𝐶 Topologically ordered states ⊗ 𝐶 ⊗ fractional quantum Hall integer quantum Hall • Topological ground state degeneracy; quasiparticles with fractional quantum numbers and fractional statistics. 𝐹 𝐹 𝑑 𝐹 𝑏𝑞 𝐹 𝑏𝑞 𝑐 𝑏 = 0 = 1 𝑑 1 ground state 𝑛 ground states
Key properties of topologically ordered states 𝑑 • Quasiparticles have no knowledge about distance. Only topology matters. 𝑑 𝑑 • Fusion 𝑏 × 𝑐 = 𝑂 𝑏𝑐 𝑐 𝑏 • Braiding 𝑑 𝑑 𝑑 ⋅ = 𝑆 𝑏𝑐 𝑐 𝑏 𝑐 𝑏 • Braiding 𝑏, 𝑐 and spinning 𝑏, 𝑐 is equivalent to spinning 𝑑 . Topological spin of particles ℎ 𝑏 𝑑 𝑆 𝑐𝑏 = 𝑓 𝑗2𝜌ℎ 𝑏 𝑑 𝑏 = 𝑓 𝑗2𝜌 ℎ 𝑏 +ℎ 𝑐 −ℎ 𝑑 𝑆 𝑏𝑐
Examples of topologically ordered states 𝑛 𝑓 − 𝑗 𝑨 𝑗 2 • 1. Laughlin state Ψ = 𝑗<𝑘 𝑨 𝑗 − 𝑨 𝑨 𝑗 𝑘 1 2 1 • Quasiparticles labeled by 𝑟 = 0, 𝑛 , 𝑛 , … , 1 − 𝑛 𝑟 1 𝑟 2 𝑟 1 +𝑟 2 = exp 𝑗𝜌 • Fusion rule braiding 𝑆 𝑟 1 𝑟 2 𝑟 1 + 𝑟 2 𝑛 𝑟 2 • Spin ℎ 𝑟 = 2𝑛 𝑟 2 𝑟 1 • 2. 𝑎 2 gauge theory (toric code) • Quasiparticles include charge 𝑓 , flux 𝑛 and their boundstate 𝜔 = 𝑓 × 𝑛 . 𝜔 = 𝑗 • Nontrivial braiding 𝑆 𝑓𝑛 • Goal of this work: understanding 3D topological states from 2D ones
Part I: Layer construction cut 2D topological order glue • 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.
Layer construction of 2D topological states • Example 1: integer quantum Hall (Sondhi&Yang ‘01) • Electron tunneling between edge states of each strip: + 𝑑 𝑜+1,𝑆 〉 ≠ 0 , 〈𝑑 𝑜𝑀 𝑓 • Electron tunneling can be −𝑓 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 • Example 2: Laughlin 1/3 state (Kane et al ‘02) • Electron tunneling between 𝜉 = 1/3 edges of chiral Luttinger 𝑓 𝑓/3 liquids 𝑓 = 3 × 3 , 3 + 𝑑 𝑜+1,𝑆 = 𝑓 𝑗 𝜚 𝑜𝑀 −𝜚 𝑜+1,𝑆 𝑑 𝑜𝑀 , −𝑓/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 𝑓 𝑓 3 , − with charge 3
Generalization of the layer construction to 3D • General principle: Inter-layer coupling by boson condensation Wang&Senthil ‘2013 𝑞 𝑗 = 𝑟 𝑗 boson 𝑚 𝑗 𝑚 1 𝑚 2 𝑚 1 𝑚 2 • Abelian states: Chern-Simons theory and 𝐿 matrix (Wen) ℒ = 1 4𝜌 𝐿 𝐽𝐾 𝑏 𝐽𝜈 𝜗 𝜈𝜉𝜐 𝜖 𝜉 𝑏 𝐾𝜐 − 𝑚 𝐽 𝑏 𝐽𝜈 𝑘 𝜈 • Quasiparticles labeled by integer vectors 𝑚 • Equation of motion 𝑘 𝜈 𝑚 𝐽 = 1 2𝜌 𝐿 𝐽𝐾 𝜗 𝜈𝜉𝜐 𝜖 𝜉 𝑏 𝐾𝜐 • A quasiparticle carries flux 𝛼 × 𝑏 I = 2𝜌 𝐿 −1 𝑚 I
Examples of K-matrix theory 𝑈 𝐿 −1 𝑚 2 • Mutual statistics of 𝑚 1 , 𝑚 2 given by 𝜄 12 = 2𝜌𝑚 1 • Local particles given by 𝜇 = 𝐿𝑚 (bosons or fermions) • Examples: • Laughlin 1/𝑛 state 𝐿 = 𝑛 . Quasiparticle braiding 2𝜌𝑟 1 𝑟 2 𝜄 12 = . Local particle (electron) 𝑟 = 𝑛 𝑛 • 𝑎 𝑂 gauge theory 𝐿 = 0 𝑂 𝑂 0 • Charge 𝑓 = 1 0 , 𝑛 = 0 1 . Quasiparticle braiding 2𝜌 𝜄 𝑓𝑛 = 2𝜌 𝐿 −1 12 = 𝑂
General setting of the layer construction • 𝑀 layers of 2D Abelian states, each with a 𝐿 matrix • Find quasiparticles 𝑞 𝑗 , 𝑟 𝑗 in each layer, so that the bound state are bosonic and mutually bosonic. • In 2D language, (𝑜) 𝑞 𝑗 • Requirements (𝑜+1) 𝑟 𝑗 𝑈 𝐿 −1 𝑞 𝑘 + 𝑟 𝑗 𝑈 𝐿 −1 𝑟 𝑘 = 0 , 𝑞 𝑗 𝑈 𝐿 −1 𝑟 𝑘 = 0 . 𝑞 𝑗 • 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, 𝑟 𝑗 at top surface is always deconfined.
Example 1: 3D 𝑎 𝑞 gauge theories • Starting from layers of 2D 𝑎 𝑞 gauge theories 𝐿 = 0 𝑞 𝜈 + 𝑐 𝜈 𝑘 𝑛 𝜈 , 𝑞 2𝜌 𝑏 𝜈 𝜗 𝜈𝜉𝜐 𝜖 𝜉 𝑐 𝜐 + 𝑏 𝜈 𝑘 𝑓 0 , ℒ = 𝜄 𝑓𝑛 = 2𝜌 𝑞 𝑞 • Coupling the neighbor layers by 𝑓 condensation of −𝑓 pair • 𝑞 = 1 0 , 𝑟 = −1 1 0 0 0 = 𝑓 1 = 𝑛 • Particles with nontrivial braiding with the condensed particle e are confined. m e -e m • Particles different by a -e m condensed particle are identified m • Deconfined particles: 𝑓 in 3D, and e m e 𝑛 string (flux tube) 3D 𝑎 𝑞 m e -e gauge theory
Example 2: Surface and bulk topological order e-m e • 𝑎 𝑞 toric code with tri-layer coupling e-m • A variation of the construction in Wang&Senthil ’13 e+m e e+m • 𝑞 ≠ 3𝑜 e-m e All bulk particles are confined. e-m purely 2D topological order • Surface central charge 𝑑 = 4 for 𝑞 = 3𝑜 − 1 . ( 𝑞 = 2: surface theory e-m e e of a 3D bosonic TI Vishwanath&Senthil ‘13 ) e-m e • 𝑞 = 3𝑜 n(e+m) e+m Bulk deconfined particles coexisting n(-e+m) e e e+m with surface particles. 𝑎 3 bulk e-m e topological order e e-m • Surface central charge 𝑑 = 2
General criteria of surface-only topological order • 𝑞 𝑗 , 𝑟 𝑗 expand all quasiparticles in a layer 𝑞 𝑗 𝑟 𝑗 condensation leads to surface-only topological order. Surface particles are 𝑟 𝑗 at top surface, 𝑞 𝑗 at bottom surface −1 𝑈 𝐿 −1 𝑟 𝑘 • Surface 𝐿 matrix 𝐿 𝑇 = 𝑟 𝑗 • The same topological order at the side surfaces • Bulk has nontrivial particle when 𝑞 𝑗 ∩ 𝑟 𝑗 ≠ 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
Example 3: String-String braiding • 𝑎 4𝑜 toric code theories with 4-layer coupling e 1 e 1 +2m 1 • Condensed particles: 2e 2 -m 2 e 1 +2m 1 -e 1 e 1 -2m 1 e 1 hybridization of the red -m 2 -2e 2 m 2 2e 2 and blue layers e 1 -2m 1 -e 1 • Bulk deconfined m 2 -2e 2 particles: 2 point particles, 2 strings e 2 m 2 • String-particle braiding e 1 e 2 m 2 • String-string braiding 2e 2 2n ⋅ m 2 2𝜌𝑀 phase 𝜕 𝑓𝑛 = m 2 e 2 4𝑜 proportional to the e 2 m 2 number of layers
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 of an edge dislocation • Braiding at presence of the dislocation 2𝜌 𝑒 𝜕 𝑓𝑛 = 4𝑜 𝑐 𝑨 , proportional to the Burgers vector 𝑐 𝑨
Topological field theory description • A generalized BF theory can be written down to characterize the string-particle braiding and string-string braiding ℒ = 𝑅 𝐽𝐾 𝐾 + Θ 𝐾 + 𝑘 𝜈 𝐽 𝜖 𝜏 𝑏 𝜐 𝐽 𝜖 𝜇 𝑏 𝜏 𝐽 𝑏 𝐽 𝜈 2𝜌 𝜗 𝜈𝜉𝜏𝜐 𝑐 𝜈𝜉 8𝜌 2 𝑆 𝐽𝐾 𝜗 𝜈𝜉𝜏𝜐 𝜖 𝜈 𝑏 𝜉 𝐽 𝑐 𝐽 𝜈𝜉 + 𝐾 𝜈𝜉 𝐽 : particle current; 𝐾 𝜈𝜉 𝐽 : string current • 𝑘 𝜈 • 𝑅 𝐽𝐾 : string-particle braiding • 𝑆 𝐽𝐾 : 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 𝑜 = 2𝜌𝑜 𝑅 −1 𝑈 𝑆𝑅 −1 𝐿 = 𝑜𝑆 . String braiding 𝜕 𝐽𝐾 𝐽𝐾
Topological field theory description • Ordinary 𝑎 𝑞 gauge theory: 𝑅 = 𝑞, 𝑆 = 0 • Example 3: 𝑅 = 2𝑜 0 2 , 𝑆 = 0 1 0 1 0 • 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.
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