Layer construction of threedimensional. String-String braiding statistics. Xiao-Liang Qi Stanford University Vienna, Aug 29 th, 2014

Transcription

1 Layr onstrution of thrdimnsional topologial stats and String-String braiding statistis Xiao-Liang Qi Stanford Univrsity Vinna, Aug 29 th, 2014

2 Outlin Part I 2D topologial stats and layr onstrution Gnralization to 3D: a simplst xampl Layr onstrution of 3D topologial stats: gnral stting and xampls Fild thory dsription Part II Som gnral rsults on string-string braiding. Rf: Chao-Ming Jian & XLQ, arxiv: Supportd by (arriving hr tonight )

3 Topologially ordrd stats B B intgr quantum Hall frational quantum Hall Topologial ground stat dgnray; quasipartils with frational quantum numbrs and frational statistis. E E a b E gap g = 0 1 ground stat E gap g = 1 m ground stats

4 Ky proprtis of topologially ordrd stats Quasipartils hav no knowldg about distan. Only topology mattrs. Fusion a b = N ab Braiding a b = R ab a b a b Braiding a, b and spinning a, b is quivalnt to spinning. Topologial spin of partils h a a = i2πh a R ab R ba = i2π h a+h b h

5 Exampls of topologially ordrd stats 1. Laughlin stat Ψ z i = i<j z i z j m i z i 2 Quasipartils labld by q = 0, 1, 2,, 1 1 m m m q Fusion rul braiding R 1 +q 2 q1 q 2 = xp iπ q 1q 2 q 1 + q 2 Spin h q = q2 q 1 q 2 2m 2. Z 2 gaug thory (tori od) Quasipartils inlud harg, flux m and thir boundstat ψ = m. ψ Nontrivial braiding R m = i Goal of this work: undrstanding 3D topologial stats from 2D ons m

6 Part I: Layr onstrution 2D topologial ordr ut glu 2D topologial stats an b onstrutd from oupld 1D hains (Sondhi&Yang 01, Kan t al 02, To&Kan, 10) Wakly oupld hains as a ontrolld limit that an raliz ths topologial stats. Both intgr and frational quantum Hall stats an b ralizd.

7 Layr onstrution of 2D topologial stats Exampl 1: intgr quantum Hall (Sondhi&Yang 01) Eltron tunnling btwn dg stats of ah strip: + nl n+1,r 0, Eltron tunnling an b quivalntly viwd as xiton ondnsation Condnsation of th xiton (partil-hol pair) lads to ohrnt tunnling btwn quasi-1d strips Th strips ar glud to a quantum Hall stat

8 Layr onstrution of 2D topologial stats Exampl 2: Laughlin 1/3 stat (Kan t al 02) Eltron tunnling btwn ν =1/3 dgs of hiral Luttingr liquids = 3 3, /3 + nl n+1,r = i φ 3 nl φ n+1,r, /3 Eltron tunnling fftivly gnrats ohrnt quasipartil tunnling 2D topologial ordr. Th ohrnt tunnling an b undrstood as a boson ondnsation of th quasipartil xiton with harg, 3 3

9 Gnralization of th layr onstrution to 3D Gnral prinipl: Intr-layr oupling by boson ondnsation Wang&Snthil 2013 p i q i boson l i l 1 = l 2 l 1 l 2 Ablian stats: Chrn-Simons thory and K matrix (Wn) L = 1 4π KIJ a Iμ ε μντ ν a Jτ l I a Iμ j μ Quasipartils labld by intgr vtors l Equation of motion j μ l I = 1 2π KIJ ε μντ ν a Jτ A quasipartil arris flux a I = 2π K 1 l I

10 Exampls of K-matrix thory Mutual statistis of l 1, l 2 givn by θ 12 = 2πl 1 T K 1 l 2 Loal partils givn by λ = Kl (bosons or frmions) Exampls: Laughlin 1/m stat K = m. Quasipartil braiding θ 12 = 2πq 1q 2. Loal partil (ltron) q = m m Z N gaug thory K = Charg = 1 0, m = 0 1 θ m = 2π K 1 12 = 2π N 0 N N 0. Quasipartil braiding

11 Gnral stting of th layr onstrution L layrs of 2D Ablian stats, ah with a K matrix Find quasipartils p i, q i in ah layr, so that th bound stat ar bosoni and mutually bosoni. In 2D languag, Rquirmnts p i T K 1 p j + q i T K 1 q j = 0, p i T K 1 q j = 0. Numbr of ondnsd partils: i = 1,2,, N whn dim K = 2N. This is an almost omplt st of null vtors. (Haldan 95, Lvin 13, Barkshli t al 13) Thr may b rmaining partils, rsponsibl for th topologial ordr. With opn boundary, q i at top surfa is always donfind. p i (n) q i (n+1)

12 Exampl 1: 3D Z p gaug thoris Starting from layrs of 2D Z p gaug thoris K = 0 p p 0, L = p a 2π με μντ ν b τ + a μ j μ + b μ j μ m, Coupling th nighbor layrs by ondnsation of pair p = 1 0, q = 1 0 Partils with nontrivial braiding with th ondnsd partil ar onfind. Partils diffrnt by a ondnsd partil ar idntifid Donfind partils: in 3D, and m string (flux tub) 3D Z p gaug thory 1 0 = θ m = 2π p 0 1 = m m m m m m m

13 Exampl 2: Surfa and bulk topologial ordr Z p tori od with tri-layr oupling A variation of th onstrution in Wang&Snthil 13 p 3n All bulk partils ar onfind. purly 2D topologial ordr Surfa ntral harg = 4 for p = 3n 1. (p = 2: surfa thory of a 3D bosoni TI Vishwanath&Snthil 13) p = 3n Bulk donfind partils oxisting with surfa partils. Z 3 bulk topologial ordr Surfa ntral harg = 2 +m -m +m -m -m -m +m -m +m -m -m -m n(+m) n(-+m)

14 Gnral ritria of surfa-only topologial ordr p i, q i xpand all quasipartils in a layr p i q i ondnsation lads to surfa-only topologial ordr. Surfa partils ar q i at top surfa, p i at bottom surfa Surfa K matrix K S = q T i K 1 1 q j Th sam topologial ordr at th sid surfas Bulk has nontrivial partil whn p i q i φ Rlation to Walkr-Wang modl (K Walkr & Z Wang, 12): modular tnsor atgory Surfa-only topologial ordr Pr-modular tnsor atgory Bulk nontrivial topologial ordr

15 Exampl 3: String-String braiding Z 4n tori od thoris with 4-layr oupling Condnsd partils: hybridization of th rd and blu layrs Bulk donfind partils: 2 point partils, 2 strings String-partil braiding String-string braiding phas ω m = 2πL 4n proportional to th numbr of layrs m m m m m m m m 2 2n m m 2 m m 2 m 2

16 String-String braiding and disloations Strings wraping around z dirtion hav braiding proportional to systm siz Contratibl strings hav trivial braiding Th mor fundamntal pross of string braiding an b dfind at prsn of an dg disloation Braiding at prsn of th disloation ω d m = 2π b 4n z, proportional to th Burgrs vtor b z

17 Topologial fild thory dsription A gnralizd BF thory an b writtn down to haratriz th string-partil braiding and string-string braiding L = Q IJ 2π εμνστ b I μν σ a J τ + Θ 8π 2 R IJε μνστ μ a I ν λ a J σ + j I μ μ a I I μν + J μν b I j I μ : partil urrnt; JI μν : string urrnt Q IJ : string-partil braiding R IJ : string-string braiding whn strings ar linkd with Θ vortx loop. Diffrn from BF thory for TI (Cho&Moor 11, Vishwanath&Snthil 12, Kysrlingk t al 13): Θ is a dynamial fild Winding numbr 2πn of Θ Chrn-Simons trm of a with K = nr. String braiding ω n IJ = 2πn Q 1 T RQ 1 IJ

18 Topologial fild thory dsription Ordinary Z p gaug thory: Q = p, R = 0 Exampl 3: Q = 2n 0 0 2, R = Gnral strutur of string braiding: two strings braid nontrivially only if thy ar not ontratibl. Consistnt with othr rnt works on 3-string braiding (Wang&Lvin , Jiang t al , Wang&Wn , Moradi&Wn ) Th disloation is dsribd by a Θ vortx string, whih is an xtrinsi dft. Intrinsi 3-string braiding an possibly b ralizd by donfinmnt of th disloations.

19 Part II: Gnral rsults on string-string braiding Gnral strutur of 3D topologially ordrd stats ar not undrstood yt. In 2D, w know th braiding phas R ab is not arbitrary. Thr ar som idntitis satisfid by braiding and fusion, suh as th hxagon idntity. d In 3D, som similar idntitis may xist as a proprty of th gnral strutur of topologially ordrd stats

20 Gnral rsults on string-string braiding Wang&Lvin proposd an idntity of th 3-string braiding in twistd Z p gaug thoris p ω ab + ω a b + ωb a = 0 (mod 2π), Hr w giv a mor gnral proof to a strongr idntity ω ab a + ω b b + ω a = 0 (mod 2π) with th gnral onditions a b 1) Strings an fus and split without additional phas; 2) Strings ar Ablian; 3) Strings ar not markd.

21 Stp 1 of th proof: ω ab = Ω ab String braiding ω ab a b L ab String-partil braiding Ω ab btwn link of a, b and string

22 Stp 2 of th proof: ω ab = Ω ab linkd string braiding ω ab, for 3 mutually-linkd strings a b L ab String-partil braiding Ω ab btwn link of a, b and string

23 Stp 3 of th proof: ω ab + ω a b + ω b a = 0 a b a b ω ab : 2π rotation of a and b around ω ab + ω a b + ω b a global 4π rotation trivial

24 String braiding idntitis Using this proof w obtain thr idntitis + ω a b + ω b a = 0 + ω a b + ω b a = 0 ω ab ω ab Ω ab a + Ω b + Ω b a = 0 A nw fatur of 3D topologial ordr that is qualitativly distint from 2D as Opn qustion: In gnral, is it always possibl to rquir th strings to b unmarkd, i.., translation invariant along th string dirtion? a a b b

25 A non-ablian xampl of string-string braiding Littl is known about non-ablian strings. Howvr, an xampl an b found in 3D topologial suprondutors lft, C = 1 SC pairing Majorana mass Δ L iθ L right, C = 1 Δ R iθ R Supronduting pairing Wyl frmions d 3 x Δ iθ Lψ + L σ y ψ + L + Δ iθ Rψ + + R σ y ψ R H = d 3 x v ψ + L σ pψ L ψ + R σ pψ R

26 A non-ablian xampl of string-string braiding Chiral vortx strings: vortx loops of θ L or θ R Eah vortx string is an axion string, arrying a 1+1 Majorana-Wyl frmion (Callan&Harvy 85, XLQ&Wittn&Zhang 12) Majorana zro mods arrid by vortis with odd linking numbr. a b E k E k k k Non-Ablian braiding of a, b similar to p + ip vortis (Rad&Grn 2000) (s also M Sato, Physis Lttrs B 575 (2003) )

27 A non-ablian xampl of string-string braiding Ky diffrn from Ablian string: splitting/fusion of string is not adiabati. Non-Ablian strings an fus to Ablian strings. Braiding dpnds on th fusion hannl. 1 or ψ σ σ a b a b 2 zro mods on a, b no zro mod

28 Summary Layr onstrution provids an xpliit approah to 3D topologial stats. Diffrnt typs of 3D topologial stats an b gnratd, with surfa-only topologial ordr and/or bulk topologial ordr String-string braiding an b indud in systm with priodi boundary ondition or disloations Gnral idntity provd for Ablian string-string braiding Non-Ablian 3D topologial ordr: An xampl an b found in topologial suprondutors. Thr ar a lot of opn qustions for mor gnral ass.