TKNN formula for general Hamiltonian D. J. Thouless, M. Kohmoto, M. P. NighYngale, and M. den Nijs T. Onogi (Osaka Univ.) arXiv:1903.11852 with H. Fukaya, S. Yamaguchi (Osaka U), X. Wu (Ariel U) April 22, 2019, FLQCD 2019 @ YITP 2019/4/22 1
Cau$on Change of the topic Previous speakers (Fukaya, Furuta) Anomaly in four dimensions with boundary This talk Topological insulator in odd dimensions without boundary (D=2+1, D=4+1) 2019/4/22 2
1. Introduction 2019/4/22 3
Topological insulator Ø Interesting physics from non-trivial topology Bulk: insulator Surface: metal Topology guarantees edge modes (Bulk-Edge correspondence) Figure from Tokura et al. Nature Reviews Physics vol 1 , 126 (2019) Ø Close rela7onship to domain-wall New knowledge of topological ma@er è new hints to laAce fermions by Domain-wall fermion example: Gapped symmetric phase by 4-fermi interac7on (Talk by Kikukawa) 2019/4/22 4
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Two approaches to topology Ø Microscopic approach: Study the wavefunc9on of the free electron Hamiltonian Applied to various different free systems ( higher dim, higher symmetry) Classifica9on of topology is highly developed Looks rather technical (at least to me) Applicable only to free fermion systems TKNN formula (D. J. Thouless, M. Kohmoto, M. P. Nigh9ngale, and M. den Nijs) Ø Field theory approach: Introduce gauge field and study the effec9ve ac9on S e ff ( A ) Conceptually simple Applicable also to interac9ng fermion systems 2019/4/22 5
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> What characterizes topology? Ø Microscopic approach: Topology of Berry connection of single particle wavefunctions ⌘ � i h n, p | ∂ Z A ( n ) X Top. # = d 2 p c 1 ( A ( n ) ) | n, p i µ ∂ p µ n TKNN formula: ConducLvity çè Top. # Ø Field theory approach: Top # = Chern-Simons level of the 3-dim effecLve gauge acLon K. Ishikawa : ConducLvity ç è Top. # 2019/4/22 6
Ques%on Two topological characterizations are identical? In some specific cases, yes. How generally identical and why ? We try to answer this ques%on in this work. 2019/4/22 7
Outline 1. Introduc/on � 2. Review of TKNN formula 3. Review of field theory approach 4. Equivalence for general Hamiltonian 1. Chern-Simons level à Winding number 2. Winding number à TKNN formula 5. Summary 2019/4/22 8
2. Review of TKNN formula 2019/4/22 9
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Anomalous Hall effect 2+1 dim system with Parity Violation Hall current perpendicular to Electric field h j x i E = σ xy E y Hall conduc;vity can be expressed by topological quan;ty using 1) Kubo formula from perturba;on theory 2) Formulae in quantum mechanics 2019/4/22 10
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> Electron states under electric field (perturba9on theory) | m ih m | eE y y | n i X | n i E = | n i + | n i : eigenstate in free theory E n � E m | n i E : perturbed state m 6 = n � ev x Hall current under the electric field X h j x i E ⌘ h n | E L 2 | n i E n,E n < 0 Kubo formula � xy = � ie 2 ✏ ij h a, ~ p | v i | b, ~ p ih b, ~ p | v j | a, ~ p i X X X p )) 2 L 2 ( E a ( ~ p ) � E b ( ~ a ~ b 6 = a p where we have used a : band label , n ⇒ ( a, ~ p ) l Transla9onal invariance: p : bloch momentum ~ l Heisenberg equa9on: [ y, H ] = iv y 2019/4/22 11
<latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> <latexit sha1_base64="(nul)">(nul)</latexit> @ v i = @ p i H ( ~ p ) Derivation of Useful formula from H ( ~ p ) | a, ~ p i = E a ( ~ p ) | a, ~ p i h a, ~ p | b, ~ p i = 0 ( a 6 = b ) p | @ h a, ~ p | v i | b, ~ p i = ( E a ( ~ p ) � E b ( ~ p ) h a, ~ @ p i | b, ~ p i ( a 6 = b ) p | @ Combining with Kubo formula and defining A ( a ) ( ~ p ) ⌘ � i h a, ~ @ p i | a, ~ p i i Berry connecDon � xy = e 2 ✏ ij @ @ p i A ( a ) X X j ( ~ p ) L 2 a ~ p Z d 2 p = e 2 ✏ ij @ @ p i A ( a ) X Chern number c_1 ! j ( ~ p ) 2 ⇡ 2 ⇡ a TKNN formula 2019/4/22 12
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