Introduction to topological aspects in! condensed matter physics. Andreas P. Schnyder. Max-Planck-Institut für Festkörperforschung, Stuttgart
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1 Irouco o opolocal apc co mar phc ra P. chr Ma-Plac-Iu für örprforchu, uar Ju 11-13, 14 Uvré Lorra
2 T-fol clafcao of opolocal ulaor uprcoucor 1 lcur: - Topolocal b hor - Topolocal ulaor 1D polacl) - Topolocal ulaor D IQHE, QHE) lcur: - Topolocal ulaor w/ TR D & 3D vara) - BG hor for uprcoucor - Topolocal uprcoucor 1D D - Majoraa bou a 3r lcur: - Topolocal uprcoucor D 3D w/ TR - Proc abl of opolocal ulaor uprcoucor 4h lcur: - Topolocal crall ulaor - Gapl opolocal maral
3 Boo rvw arcl Rvw arcl: - M.. Haa.L. Ka, Rv. Mo. Ph. 8, X.L. Q.. ha, Rv. Mo. Ph. 83, ) -. Ru,. P. chr,. urua,. Luw, Nw J. Ph. 1, Bar, ual Rvw of o. Ma. Ph. 4, ) - J. lca, Rp. Pro. Ph. 75, Y. o, J. Ph. oc. Jp. 8, 11 13) Boo: - hu-q h, Topolocal ulaor, prr r ol-a cc, Volum B. r Brv, "Topolocal Iulaor Topolocal uprcoucor, Prco Uvr Pr 13) - Mo Naahara, "Gomr, Topolo Phc", Talor & rac 3) -. Bohm,. Moafazah, H. Kozum, Q. Nu, J. wazr, Th omrc pha quaum m, prr 3) - M. raz L. Molamp, Topolocal Iulaor, omporar ocp of o Mar cc, Elvr 13)
4 1 lcur: Topolocal b hor 1. Irouco - Wha opolo? - Topolocal b hor. Topolocal ulaor 1D - Brr pha - mpl ampl: Two-lvl m - Polacl u-chrffr-hr mol) - Doma wall a 3. Topolocal ulaor D - Ir quaum Hall ffc - Bul bouar corrpoc - hr ulaor o quar lac
5 Wha opolo? Th u of omrc propr ha ar v o mooh formao or ampl, cor wo-moal urfac hr-moal pac lo urfac characrz b u # hol, 1 a r opolocal vara Gau-Bo Thorm Gu ca b pr rm of a ral of h Gau curvaur ovr h urfac I co mar phc: κ 4π1 ) opolocal vara Topolo of ula maral, opolo of b rucur
6 B hor of ol opolo Bloch horm: Elcro wavfuco cral cor lcro wavfuco proc cral poal cral momum ψ r ) Bloch wavfuco ha proc of poal Bloch Hamloa H) r H r H) ) E ) ) Brllou o B rucur f a mapp: Brllou zo Topolocal quvalc: H) Hamloa wh r ap B rucur ar quval f h ca b cououl form o o aohr whou clo h r ap Er Momum ap
7 o rprph M. r, W5/6 Mu hor M. r, W5/6 u rc B opolo φ o rprph o o rprph o u ) ) u rch h M. r, W5/6 M. r, W5/6 M. r, W5/6 φ ) o rprph o u Brr pha: U ) o rprph o u κ πχ π ) 1.1) φ h ) o rprph ) Pha ambu of wavfuco 3) h o u E o rprph o u M. r, W5/6 M. r, W5/6 M j q E h 1/ U fbr bul: o ach aach fbr ) U Bloch horm M. r, W5/6 ) U 1 M. r, W5/6 u rch γ 4) E E.6) r.6) h M. r, W5/6 f Brr coco: l EM vcor poal) h [T R), H] ψ ) E h 5/6 u rch " j j j 1/ q E/ q E/ 1/ q E/ ) o u o rprph γ u 1.) 1/ 5) z mpl ampl z r r φ φ H) ) H U ) ) ) ) ur au raformao: o u E ) U ) o rprph h o rprph ) h 1.3).6) φ ) φ φ 6) ) ) E.6) E.6)3) W5/6 u rch j om mor H) ) E ) ) z z z z 1/ q E/ z Brr pha: au vara qua) z j Bloch horm jw5/6 mloa for a BW5/6 uprcoucor am.o-crommrc cral. ) ) q E/ r, ) 4) M. r, u rch γ 1/ q 1/ E/ w hav Brr curvaur or pha )o ) r loop o rprph 3) o u 11 cha a clo T R), H] ψ ) ) r [T R), H], hfollow P3 B om m. I h {, } o h Paul marc ψ ) q E, ) z om mor mor H) Brllou o Bloch horm h ) ) ) µ µ mpl rampl Brr curvaur or: au p) 3) mpl ampl R), H] ψ ) 7) 1 µ h ) µ µ r r {τ, τmajoraa Paul marc h parcl-hol ra. W r E, q) Tr G E, τz } o h z z H) H 4) [T R), H] ψ ) 5) r r M. r, W5/6 z ) curvaur H) H o rprph o u o rprph o u z 8) or 3D: Brr ) o: z µ µ j E, z 4) q E, )3} 6) ) 5) Hamloa ) q ), j {1,, ral vara uprcoucor wh h γ ψ ψ, ) 1 r ) z ) 3) ) ) r r,"j r z j curvaur H) H 7) 1 H) H 9) Brr q Topolocal E, ), j {1,, 3} 5) H) ) E ) ) 6) vara ofm.b 1 M., {1,, 3} ) r, r, W5/6 u rch rucur:, ) ) j γ H) Tr ψ ψ W5/6 3) u rch ) ) E ) E, q) Tr G E, ) E, q) G E, q)v G E, ) mor "4) $ ψom 11 8) ) ) ) ) ) h) ), ) mor ) om maral Topolocal propr of ula hr umbr h : H) Tr G E, q)v G E, ) 6) 4) 9), H) 11, H) 1.4) j hr umbr ) E ). ) w hav q q v b : T π, j ) ) T hr umbr or w umbr) fll a ) ) ψ ) h H) ) E ) 11) Brllou o 7) φ π 1, {1,, 3} fll a or {1,, 3} )1 {1, γ,3} γ) ),ψ
8 o rprph /) M. r, W5/6o u u W5/6 5) u rch M. r, co/) Brr pha for wo-lvl m o rprph o u µ 3) µ 6) h o rprph o 5/6mpl ampl u rch 4) u E±co/) ± 7) /) 1 ) z o rprph II Two-lvl Hamloa: H) ) ET M. r, W5/6 z /) u M. r, W5/6 5) o rprph o u o rprph co/) ) M o rprph o u ) co,, co ) 8) rph o u param. b phrcal coor.: Brr 6) curvaur or µ M. µr, 3) W5/6 co/) zu wh r E± ± orh pol au) ) wo vcor 7) 4) 9) M. r, W5/6 H) ) σ /) ˆ ) ) ) M. r, z W5/6 M. r, W5/6 µ µ µ) U ) 5) " co/) /) u rch o rprph o u 3) u u 6) E.6) Brr curvaur co/) /) π) 8) Brro u vcor poal ) co,, co γ al wp j φol 4) γ ol al w /), ) 1/ q E/ ) ) γ ol al ˆ wp ou b ) o rprph au o u z or p) u Brr vcor poal: H), j u u ) Brr curvaur 9) E ± 5) co/), j ) ± u rch φ ˆ H) z o rprph H) ) σ W5/6 ) U 11) o u " ph o u h u rch u u /) u u µ ) ) ) 7) πo µ ) o rprph E.6) # o u µ o rprph π u /) W5/6 u rch ) E ± Brr u ) ± curvaur: au p) z z γ φ j Brr curvaur ) ) 1/ q E/ ) co,, co ) 6) M. r, W5/6 " M. u rch ) ) r, W5/6 ) h φ, Brr vcor poal ) ) Ifz p Jacobaγ mar o M.: r, W5/6 paramr ) Uj 11) 8), 7) ), j # mpl zmor u γ co/) om u ampl E.6) mpl ampl: ) ) co, ) 3) u ) 9) ) γ /) z u j z u 1 /) ˆγ 1/ q E/ co/) " z γ ol al wp ou b 1 ) u H) 4) z ) moopol fl) ˆ /) Bloch horm al ) ) 13) γ ol al wp ou b ) γ z /) ) σ j H) u u E, ) σ, j ˆ{1, q3),"3} co/) 8) Brr vcor poal H) ) r )
9 Polacl u-chrffr-hr mol) u-chrffr-hr mol crb polacl [ H ] [u, chrffr, Hr 79] Hamloa: H X h )c c B )c 1 c B h.c. Gap phoo la o Prl abl wo ra rou a: momum pac: ) ), B,,1 f a h) H) ) h ) ) co ) ) z ) Er Momum ublac mmr: zh)h) z z r pcrum mmrc) Er pcrum: E ± ± ± p ) [ ) ] co
10 M. r, W5/6 chrffr Hr Mol Polaclu u-chrffr-hr mol) mol for polacal mpl wo b mol mpl ampl 1 ] N Tr [ ) E) Gap u-chrffr-hr o rprph o u ) ).. H π mol c c c c h c o u B polacl o rprph B H ] 1 [ crb Eo rprph o rprph o u 11 M ± ± ral mmr: 5/6 achl: UT H )UT Gap 4 1 M. r, W5/6 r, W5/6 M. u rch,1 a, M. r, W5/6 H ), UT σ hol mmr: T σ B, w hav homoop Er /a w hav µ µ /a 3) Prl abl h) νmomum # majoraa majoraa H) ) ) U H )U h H ), U σ 4) ) ± > : ψ ψ pha ± γ1 ψ ψ Brr ) γ1 > P" for: > ±1 > ) γ ν γ ψ ψ ψ ) whch ) h H) Brr pha Off-aoal form ) )co ) co a ) [ H ] ) 1 $ % ) ) ) a ε )z ) or Brr curvaur ψ γ1 ) ψ q) H ) 1 5) γ γ : U) π3 [U)] 1 < : Brr pha ε 3) ψ γ1 z ) 1 ψ γ1 γ q BG q µ ) µ ) P< / W o: 1 ) Lac H BG : µ ) 4) Brr pha h) ε σ α σ γ h) Brr curvaur Prov mmr rqur h a wh > γ< ar16) opolocall c. z), q) q) : π ) q) : 1 h µmmr, )aropolocall σ5) quval. 1 σ)1σ {γ, γj } ) Whou µ ra all 1D b rucur,{γ), γj } δj,j ma fl 6) % ", j ) $ L / ma fl, # # Prov z ) rqur b ublac mmr) a 7) µ µ γ E ) I H ρ1, ) λ )?wh 5.4 > < h c ar opolocall γ γ, π E E γ,e µ 8) H ) )
11 Doma Wall a Polacl Doma wall bw ffr opolocal a ha opolocall proc zro-r mo zro-r a a oma wall [u, chrffr, Hr 79] [Jacw, Rbb] Effcv low-r couum hor: p arou ) m) m) Drac Hamloa wh a ma: Eq) ± p q m ublac mmr chral mmr ): { z,h} z E E or oma wall: m> 1 zro-r a a oma wall m< az for boua: H ) R m ) 1 Bul-bouar corrpoc: R L #zromo opolocal vara characrz oma wall)
12 Th Ir Quaum Hall a Ir Quaum Hall a: r ampl of D opolocal maral - D lcro a lar mac fl, a low T bouar. B r Er E ap ω c [vo Klz 8] D ccloro moo Lau lvl - Thr a r ap, bu o a ulaor Quaz Hall coucv: J σ E J σ h B E σ h - Plaau rv ρ 1 h
13 Th Ir Quaum Hall a Wha cau h prc quazao IQHE? Eplaao O: E a rapor IQHE ha a r ap h bul: bouar. B r char cao flow bul; ol alo 1D chal a chral a) chral a cao b localz b orr o baccar) a ar prfc char coucor Eplaao Two: Topolocal b hor Dco bw h r quaum Hall a a covoal ulaor a opolocal propr of h b rucur [Thoul al, 84] H) : Brllou zo Hamloa wh r ap laf b hr umbr: opolocal vara) π Kubo formula: fll a h X fll a o o cha ur mooh formao, a lo a bul r ap o clo
14 Bul-bouar corrpoc opolocal vara π fll a ro-r a a rfac Bul-bouar corrpoc: ro-r a mu a h rfac bw wo ffr opolocal pha ollow from h quazao of h opolocal vara. L R abl apl a: umbr or mo robu o mooh formao rpc mmr of h m) v o orr, mpobl o localz cao a purl 1D m rmo oubl horm) 1 mooh rao IQHE: chral Drac rmo E
15 hr ulaor o quar lac hr ulaor r quaum Hall a o a lac mlar o Hala hocomb mol [D. Hala PRL 88] ) hr ulaor o quar lac: H I ) ~ ) [Brv, Huh, ha] wo orbal mol: p. Ir-orbal coupl ra-orbal pro) bra m-rvral mmr) ) ) z ) M co co ) E ± ± ) pcrum fla: ˆ) ) ) M> M< 4 hr umbr: w o) Mapp rval pha z o a 1 8 B 4 <M< 1 µ ˆ h µ ˆ ˆ Brllou zo o-rval pha z chral a <M< 1 o-rval pha z quaz Hall ffc chral a ˆ) : ˆ) ) h
16 hr ulaor o quar lac hr ulaor o quar lac: H I ) ~ ) ) ) z ) M co co ) Effcv low-r couum hor for M: p arou ; rm ca b lc) H I M z E ± ± ± p wo fuco wh r: M u 1 p 1 u M) M p M) M Brr curvaur: M 3 v ozro hr umbr 1 1 Hall coucac ) M) zro-r a a bouar NB: hr umbr mu b r for ral ovr compac mafol. Propr rularzao of Drac Hamloa wll la o hral a a bouar bw wo hr ulaor wh ffr p R M ) 1
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Mor Topc ol-a Thor: Topolocal ulaor uprcoucor ra P. chr Ma-Plac-Iu für örprforchu, uar Uvrä uar Jauar 6 Topolocal ulaor uprcoucor. Topolocal b hor - Wha opolo? - H mol polacl) - hr ulaor IQH. Topolocal
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