From Luttinger Liquid to Non-Abelian Quantum Hall States

Transcription

1 From Luttinger Liquid to Non-Abelian Quantum Hall States Jeffrey Teo and C.L. Kane KITP workshop, Nov 11 arxiv: v1

2 Outline Introduction to FQHE Bulk-edge correspondence Abelian Quantum Hall States Coupled wires Laughlin and hierarchy states Non-Abelian Quantum Hall States Coupled bundles of wires Moore Read and Read Rezayi states

3 Abelian FQH States Gapped (2+1)D-bulk Topological field theory Gapless (1+1)D-Edge Chiral Luttinger liquid e(p/q) FQH Bulk quasihole excitation Fractional charge Abelian statistics Chiral multi-component Luttinger liquid Wen, Zee, 92

4 Non-Abelian FQH States Gapped (2+1)D-bulk Ground state wave function Gapless (1+1)D-Edge Chiral Conformal field theory Vertex operator Luttinger liquid e(p/q) Moore Read state Ising non-abelian statistics Charge + neutral mode c = 1 + 1/2 Kac-Moody algebra FQH Majorana mode Read Rezayi state Zk non-abelian statistics Fibonacci anyons Charge + neutral mode c = 1 + 2(k-1)/(k+2) Kac-Moody algebra Moore, Read, 91; Read, Rezayi, 99

5 Coupled Wires Construction 1D Luttinger liquid simple description of interaction via abelian bosonization Interwire many-body tunneling => FQH states solvable intermediate between microscopic electronic model and effective field theory Representation of chiral edge CFT and quasiparticle excitations

6 Integer Quantum Hall i-1 i i+1 i+2 Kane, Mukhopadhyay, Lubensky, 02

7 Integer Quantum Hall B i-1 i i+1 i+2 Kane, Mukhopadhyay, Lubensky, 02

8 Integer Quantum Hall B i-1 i i+1 i+2 Egap Kane, Mukhopadhyay, Lubensky, 02

9 Laughlin States n = 1/m m m m B m m odd for fermions Kane, Mukhopadhyay, Lubensky, 02

10 Laughlin States n = 1/m i i+1 m m m m B m even for bosons Kane, Mukhopadhyay, Lubensky, 02

11 Laughlin States n = 1/m i m i+1 m m B m New electron operators Edge K-M algebra => fractional charge e/m, fraction statistics Kane, Mukhopadhyay, Lubensky, 02

12 Hierarchy States i i+1 B i+2 Kane, Mukhopadhyay, Lubensky, 02

13 Hierarchy States i i+1 B i+2 Change of variables K-M algebra Kane, Mukhopadhyay, Lubensky, 02

14 Moore Read State (n = 1 boson) i i+1 B i+2 (1) boson hopping in field (2) phase locking (3) crystal locking JYT, Kane, 11 (inspired by Fradkin, Nayak, Schoutens, 99)

15 Moore Read State (n = 1 boson) (1) boson hopping in field (2) phase locking (3) crystal locking Change of variables JYT, Kane, 11

16 Moore Read State (n = 1 boson) (1) boson hopping in field (2) phase locking (3) crystal locking Fermionize JYT, Kane, 11

17 Moore Read State (n = 1 boson) c = 1 + 1/2 Fermionize (1) boson hopping in field (2) phase locking (3) crystal locking t > u v => Moore Read State JYT, Kane, 11

18 Coset Construction i i+1 i+2 (1) boson hopping in field B SU(2)2 current algebra gapped by JYT, Kane, 11

19 Coset Construction i i+1 i+2 (2) phase locking (3) crystal locking B Remaining gapped by JYT, Kane, 11

20 Coset Construction Chiral CFT i i+1 i+2 c = 1 + 1/2 B JYT, Kane, 11

21 q-pfaffian States Chiral CFT q even for boson q odd for fermion i i+1 i+2 c = 1 + 1/2 JYT, Kane, 11

22 Read Rezayi States i,...,k i+1,...,k i+2,...,k Central charge JYT, Kane, 11

23 Read Rezayi States i,...,k i+1,...,k i+2,...,k (k - 1) - vector JYT, Kane, 11

24 Read Rezayi States Chiral CFT on edge: charge + Zk neutral mode Gap out Gap out JYT, Kane, 11

25 Conclusion Abelian bosonization of coupled electron wires leads to: Abelian FQH: nearest wire interaction => Laughlin states next nearest interaction => Hierarchy states Non-Abelian FQH: Outlook: conformal sectors gapped out separately by inter-bundle and intra-bundle coupling => Moore Read, Read Rezayi states Other FQH states Fractional Chern insulator, Fractional topological insulator

26 Read Rezayi States Gap out Gap out Zk - model JYT, Kane, 11

27 Read Rezayi States Gap out Gap out JYT, Kane, 11