The Localization-Topology Correspondence: Periodic Systems and - PowerPoint PPT Presentation

The Localization-Topology Correspondence: Periodic Systems and Beyond Gianluca Panati Dipartimento di Matematica ”G. Castelnuovo” based on joint papers with G. Marcelli, D. Monaco, M. Moscolari, A. Pisante, S. Teufel Quantissima in the Serenissima III Venezia, August 19-24, 2019

The predecessor: Transport-Topology Correspondence ◮ Idea originated in [TKNN 82] in the context of the Quantum Hall effect. G. Panati La Sapienza The Localization-Topology Correspondence 2 / 28

The predecessor: Transport-Topology Correspondence ◮ Idea originated in [TKNN 82] in the context of the Quantum Hall effect. ◮ Retrospectively, one can rephrase the idea as follows: BF periodic* Hamiltonian − → Fermi projector { P ( k ) } k ∈ T d − → Bloch bundle E P � � � i d k = 1 σ ( Kubo ) = − T 2 Tr P ( k )[ ∂ k 1 P ( k ) , ∂ k 2 P ( k )] 2 π C 1 ( E P ) xy (2 π ) 2 � �� � � �� � Topology Transport *) Periodic either with respect to magnetic translations (QHE) or with respect to ordinary translations (Chern insulators) G. Panati La Sapienza The Localization-Topology Correspondence 2 / 28

The predecessor: Transport-Topology Correspondence ◮ Idea originated in [TKNN 82] in the context of the Quantum Hall effect. ◮ Retrospectively, one can rephrase the idea as follows: BF periodic* Hamiltonian − → Fermi projector { P ( k ) } k ∈ T d − → Bloch bundle E P � � � i d k = 1 σ ( Kubo ) = − T 2 Tr P ( k )[ ∂ k 1 P ( k ) , ∂ k 2 P ( k )] 2 π C 1 ( E P ) xy (2 π ) 2 � �� � � �� � Topology Transport ◮ Mathematical results & generalizations: [Avron Seiler Simon; Bellissard, van Elst, Schulz-Baldes; Aizenman, Graf; Kellendonc; . . . ]. *) Periodic either with respect to magnetic translations (QHE) or with respect to ordinary translations (Chern insulators) G. Panati La Sapienza The Localization-Topology Correspondence 2 / 28

The predecessor: Transport-Topology Correspondence ◮ Idea originated in [TKNN 82] in the context of the Quantum Hall effect. ◮ Retrospectively, one can rephrase the idea as follows: BF periodic* Hamiltonian − → Fermi projector { P ( k ) } k ∈ T d − → Bloch bundle E P � � � i d k = 1 σ ( Kubo ) = − T 2 Tr P ( k )[ ∂ k 1 P ( k ) , ∂ k 2 P ( k )] 2 π C 1 ( E P ) xy (2 π ) 2 � �� � � �� � Topology Transport ◮ Mathematical results & generalizations: [Avron Seiler Simon; Bellissard, van Elst, Schulz-Baldes; Aizenman, Graf; Kellendonc; . . . ]. ◮ Recent generalizations to interacting electrons [Giuliani Mastropietro Porta; Bachmann, De Roeck, Fraas; Monaco, Teufel]. *) Periodic either with respect to magnetic translations (QHE) or with respect to ordinary translations (Chern insulators) G. Panati La Sapienza The Localization-Topology Correspondence 2 / 28

The predecessor: Transport-Topology Correspondence G. Panati La Sapienza The Localization-Topology Correspondence 3 / 28

Chern insulators and the Quantum Anomalous Hall effect Left panel: transverse and direct resistivity in a Quantum Hall experiment Right panel: transverse and direct resistivity in a Chern insulator (histeresis cycle) Picture: � A. J. Bestwick (2015) c G. Panati La Sapienza The Localization-Topology Correspondence 4 / 28

G. Panati La Sapienza The Localization-Topology Correspondence 5 / 28

Main question of this talk Ordinary Chern insulator insulator Vanishing anomalous Transport Non-zero anomalous ← → Hall conductivity Hall conductivity G. Panati La Sapienza The Localization-Topology Correspondence 6 / 28

Main question of this talk Ordinary Chern insulator insulator Topology Trivial Non-trivial ← → Bloch bundle Bloch bundle Vanishing anomalous Transport Non-zero anomalous ← → Hall conductivity Hall conductivity G. Panati La Sapienza The Localization-Topology Correspondence 6 / 28

Main question of this talk Q: How do we distinguish the topological phases in position space? Ordinary Chern insulator insulator Topology Trivial Non-trivial ← → Bloch bundle Bloch bundle Localization ← → ?? ?? Vanishing anomalous Transport Non-zero anomalous ← → Hall conductivity Hall conductivity G. Panati La Sapienza The Localization-Topology Correspondence 6 / 28

Part I The Localization-Topology Correspondence: the periodic case Joint paper with D. Monaco, A. Pisante and S. Teufel (Commun. Math. Phys. 359 (2018)) G. Panati La Sapienza The Localization-Topology Correspondence 7 / 28

How to measure localization of extended states? ◮ Our question: Relation between dissipationless transport, topology of the Bloch bundle and localization of electronic states. G. Panati La Sapienza The Localization-Topology Correspondence 8 / 28

How to measure localization of extended states? ◮ Our question: Relation between dissipationless transport, topology of the Bloch bundle and localization of electronic states. What is the relevant notion of localization for periodic quantum systems? G. Panati La Sapienza The Localization-Topology Correspondence 8 / 28

How to measure localization of extended states? ◮ Our question: Relation between dissipationless transport, topology of the Bloch bundle and localization of electronic states. ◮ Spectral type?? For ergodic random Schr¨ odinger operators localization is measured by the spectral type ( σ pp , σ ac , σ sc ) [AW]. However, for periodic systems the spectrum is generically purely absolutely continuous. G. Panati La Sapienza The Localization-Topology Correspondence 8 / 28

How to measure localization of extended states? ◮ Our question: Relation between dissipationless transport, topology of the Bloch bundle and localization of electronic states. ◮ Spectral type?? For ergodic random Schr¨ odinger operators localization is measured by the spectral type ( σ pp , σ ac , σ sc ) [AW]. However, for periodic systems the spectrum is generically purely absolutely continuous. G. Panati La Sapienza The Localization-Topology Correspondence 8 / 28

How to measure localization of extended states? ◮ Our question: Relation between dissipationless transport, topology of the Bloch bundle and localization of electronic states. ◮ Spectral type?? For ergodic random Schr¨ odinger operators localization is measured by the spectral type ( σ pp , σ ac , σ sc ) [AW]. However, for periodic systems the spectrum is generically purely absolutely continuous. ◮ Kernel of the Fermi projector?? For gapped periodic 1-body Hamiltonians one has | P µ ( x , y ) | ≃ e − λ gap | x − y | as a consequence of Combes-Thomas theory [AS 2 , NN]. [AW] M. Aizenman, S. Warzel : Random operators , AMS (2015). [AS 2 ] J. Avron, R. Seiler, B. Simon : Commun. Math. Phys. 159 (1994). [NN] A. Nenciu, G. Nenciu : Phys. Rev B 47 (1993). G. Panati La Sapienza The Localization-Topology Correspondence 8 / 28

The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) G. Panati La Sapienza The Localization-Topology Correspondence 9 / 28

The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) The Fermi projector P µ reads, for { w a ,γ } γ ∈ Γ , 1 ≤ a ≤ m a system of CWFs, m � � P µ = | w a ,γ � � w a ,γ | . a =1 γ ∈ Γ G. Panati La Sapienza The Localization-Topology Correspondence 9 / 28

The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) The Fermi projector P µ reads, for { w a ,γ } γ ∈ Γ , 1 ≤ a ≤ m a system of CWFs, m � � P µ = | w a ,γ � � w a ,γ | . a =1 γ ∈ Γ Notice that crucially � exist CWFs such that | P µ ( x , y ) | ≃ e − λ gap | x − y | � | w a ,γ ( x ) | ≃ e − c | x − γ | G. Panati La Sapienza The Localization-Topology Correspondence 9 / 28

The localization of electrons is conveniently expressed by Composite Wannier functions (CWFs) The Fermi projector P µ reads, for { w a ,γ } γ ∈ Γ , 1 ≤ a ≤ m a system of CWFs, m � � P µ = | w a ,γ � � w a ,γ | . a =1 γ ∈ Γ Notice that crucially � exist CWFs such that | P µ ( x , y ) | ≃ e − λ gap | x − y | � | w a ,γ ( x ) | ≃ e − c | x − γ | (GAP CONDITION) (TOPOLOGICAL TRIVIALITY) � G. Panati La Sapienza The Localization-Topology Correspondence 9 / 28

Theorem [MPPT]: The Localization Dichotomy for periodic systems Under assumptions specified later, the following holds: ◮ either there exists α > 0 and a choice of CWFs � w = ( � w 1 , . . . , � w m ) satisfying � m � R d e 2 β | x | | � w a ( x ) | 2 d x < + ∞ for every β ∈ [0 , α ); a =1 Notice that w a ,γ = T γ w a where T γ is the translation operator. [MPPT] Monaco, Panati, Pisante, Teufel : Commun. Math. Phys. 359 (2018). G. Panati La Sapienza The Localization-Topology Correspondence 10 / 28