Tuning between Weyl semimetals and fractional Chern insulators in frustrated materials Emil J. Bergholtz FU Berlin LDQMC workshop, Amsterdam, July 2, 2015
In this talk, I will Combine geometrical frustration and band topology + = Argue that frustrated systems naturally encompass novel fractional Chern insulators as well as Fermi arcs and Weyl semimetals Briefly tell you about a novel and controversial phase transition ?? Pseudoballistic Diffusive metal Disorder semimetal
First: My collaborators In Berlin External Jörg Behrmann Zhao Liu , Princeton -> Berlin Piet Brouwer Andreas Läuchli, Innsbruck Flore Kunst Roderich Moessner, Dresden Gregor Pohl Masafumi Udagawa, Tokyo Björn Sbierski Maximilian Trescher
Materials inspiration: Oxide interfaces & geometrically frustrated systems with strong spin-orbit coupling Perovskite materials, ABO 3 , routinely grown in a b c sandwich structures in the [100] direction a 0 A - Instead (111) slabs would be b 4 ~ B a Y C=-1 good for topological physics z 3 C=0 O 2 X (relatively flat C=1 bands). C=0 y x 1 C=1 e d B’ B B AO 3 λ λ&∆ ∆ 0 e g Epitaxial growth of (111)-oriented LaAlO 3 /LaNiO 3 ultra-thin superlattices C=1 -1 2 nd order SOC S. Middey, 1, a) D. Meyers, 1 M. Kareev, 1 E. J. Moon, 1 B. A. Gray, 1 X. Liu, 1 J. W. Freeland, 2 and J. Chakhalian 1 C=0 1) Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, 10Dq -2 j=1/2 USA C=0 a 1g 2) Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, -3 USA C=-1 The epitaxial stabilization of a single layer or superlattice structures composed of complex oxide materials on -4 t 2g arXiv:1212.0590v1 [cond-mat.mtrl-sci] 4 Dec 2012 e g ’ j=3/2 AB’O 3 ABO 3 ABO 3 AB’O 3 Γ Γ K M Γ D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto, - Fractional Chern insulators!? Nature Commun. 2 , 596 (2011). - But [111] is not a natural cleavage/growth direction... Suggestion: Consider (111) slabs of pyrochlore transition metal oxides, in particular A 2 Ir 2 O 7 iridate thin films - Natural cleavage/growth direction! - Provides useful general insights M. Trescher and E.J. Bergholtz, - Even richer physics… Phys. Rev. B 86, 241111(R) (2012) E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Phys. Rev. Lett. 114, 016806 (2015)
Localized modes on frustrated lattices References: D. L. Bergman, C. Wu, and L. Balents Band touching from real space topology in frustrated hopping models Phys. Rev. B 78, 125104 (2008) M. Trescher and E.J. Bergholtz, Flat bands with higher Chern number in pyrochlore slabs Phys. Rev. B 86, 241111(R) (2012) E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to C > 1 Fractional Chern Insulators Phys. Rev. Lett. 114, 016806 (2015)
Flat bands and localized See e.g., modes on frustrated lattices D. L. Bergman, C. Wu, and L. Balents Phys. Rev. B 78, 125104 (2008) Example: nearest neighbor hopping on a kagome lattice X c † H = t 1 i c j h i,j i 1 + e ik 1 1 + e ik 2 0 0 1 Bloch Hamiltonian: 1 + e − ik 1 1 + e − ik 3 H k = t 1 0 @ A 1 + e − ik 2 1 + e ik 3 0 0 E k /t 1 Localized modes explain the flat band “Graphene + a flat band” But these states are neither topological nor Wannier functions! - Quadratic touching point - Look for a slightly refined concept...
� � Idea: attempt to localize M. Trescher and E.J. Bergholtz, in the third dimension Phys. Rev. B 86, 241111(R) (2012) Consider frustrated layered systems, e.g. [111]-grown pyrochlore with kagome layers connected via local hopping to the intermediate on triangular layers Crucial insight: surface bands localized to the kagome 15 13 layers iff the total hopping amplitude to the triangular 14 layer vanish. 12 10 - Local constraint , destructive interference 9 - Unique solution , independent of details! 11 8 N ⌘ m ⇣ X | ψ i ( k ) i = N ( k ) | φ i ( k ) i m 6 r ( k ) 5 m =1 7 φ i 1 ( k ) + φ i 2 ( k ) + φ i 3 ( k ) 4 r ( k ) = − e − ik 2 φ i 1 ( k ) + e i ( k 1 − k 2 ) φ i 2 ( k ) + φ i 3 ( k ) 2 1 a 1 3 components of the single-layer Bloch spinor a 2 - Inherits the dispersion of the single layer model | r ( k ) | - Localized to top or bottom layer, depending on - Reminiscent of Fermi arcs…..
Illuminating, in color… M. Trescher and E.J. Bergholtz, Phys. Rev. B 86, 241111(R) (2012) | r ( k ) | > 1 state localized to the top r 2 ( k ) φ 1 ( k ) r 2 ( k ) φ 2 ( k ) r 2 ( k ) φ 3 ( k ) 0 r ( k ) φ 1 ( k ) Ψ ( k ) = N ( k ) r ( k ) φ 2 ( k ) r ( k ) φ 3 ( k ) 0 φ 1 ( k ) φ 2 ( k ) | r ( k ) | < 1 state localized to the bottom φ 3 ( k ) | r ( k ) | = 1 state delocalized! Non-trivial due to the r ( k ) twisted layer structure top view φ i 1 ( k ) + φ i 2 ( k ) + φ i 3 ( k ) r ( k ) = − e − ik 2 φ i 1 ( k ) + e i ( k 1 − k 2 ) φ i 2 ( k ) + φ i 3 ( k ) But in absence of spin- | r ( k ) | = 1 orbit coupling…
+ = Topology meets frustration References: M. Trescher and E.J. Bergholtz, Flat bands with higher Chern number in pyrochlore slabs Phys. Rev. B 86, 241111(R) (2012) Z. Liu, E.J. Bergholtz, H. Fan, and A. M. Läuchli, Fractional Chern Insulators in Topological Flat bands with Higher Chern Number Phys. Rev. Lett. 109, 186805 (2012) E.J. Bergholtz, Z. Liu, M. Trescher, R. Moessner, and M. Udagawa, Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to C > 1 Fractional Chern Insulators Phys. Rev. Lett. 114, 016806 (2015)
Add spin-orbit coupling Each separate layer becomes a Chern insulator C = − 1 C = 0 C = 1 becomes non-trivial | r ( k ) | localization to top layer delocalized localization to bottom layer
Bulk dispersion and Chern numbers M. Trescher and E.J. Bergholtz, Phys. Rev. B 86, 241111(R) (2012) Dispersion for one layer Dispersion with two layers C=1 (c) (a) (b) 4 4 3 3 2 2 M 1 1 0 0 Γ E ( k ) E ( k ) K − 1 − 1 − 2 − 2 C=2 − 3 − 3 − 4 − 4 − 5 − 5 − 6 − 6 Γ K M Γ Γ K M Γ For N kagome layers we find an almost flat band with C=N! (d) (e) (f) 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 E ( k ) E ( k ) E ( k ) − 1 − 1 − 1 − 2 − 2 − 2 C=3 C=8 − 3 − 3 − 3 − 4 − 4 − 4 − 5 − 5 − 5 − 6 − 6 − 6 Γ Γ Γ Γ Γ Γ K M K M K M (g) (h) (i) 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 E ( k ) E ( k ) E ( k ) − 1 − 1 − 1 − 2 − 2 − 2 C=100 C=12 − 3 − 3 − 3 − 4 − 4 − 4 − 5 − 5 − 5 − 6 − 6 − 6 Γ K M Γ Γ K M Γ Γ K M Γ
Chiral edge states: revealed in cylinder geometry Example: the C=2 band has 2 gapless chiral edge states at each end − 0 . 5 − 1 . 0 (Microscopically different − 1 . 5 E ( k x ) edges to avoid deceiving − 2 . 0 degeneracies) − 2 . 5 − 3 . 0 0 1 2 3 4 5 6 k x 1 . 0 y Edge state low y at π / 2 0 . 8 Edge state high y at π / 2 Edge state low y at π x 0 . 6 Edge state high y at π P Edge state low y at 3 / 2 π 0 . 4 Edge state high y at 3 / 2 π 0 . 2 0 . 0 5 10 15 20 25 30 y (in unit cells)
� � The (almost) flat bands are the surface bands constructed earlier N bands, each with C=1, hybridize so that the surface band absorbs all the topology (C=N) while the others become trivial (C=0) - Simple way of generating (flat) bands with any Chern number N ⌘ m ⇣ X | ψ i ( k ) i = N ( k ) | φ i ( k ) i m r ( k ) m =1 r 2 ( k ) φ 1 ( k ) r 2 ( k ) φ 2 ( k ) r 2 ( k ) φ 3 ( k ) 0 r ( k ) φ 1 ( k ) Ψ( k ) = N ( k ) r ( k ) φ 2 ( k ) r ( k ) φ 3 ( k ) 0 φ 1 ( k ) φ 2 ( k ) φ 3 ( k )
Are local interactions giving new FCI phases within the C>1 bands? Z. Liu, E.J. Bergholtz, H. Fan, A. M. Läuchli Phys. Rev. Lett. 109, 186805 (2012) Yes, we have convincing evidence for a series of bosonic FCI states at ν b = 1 / ( C + 1) C = 2 , ν b = 1 / 3 : (absent at higher filling fractions Fermionic states at ν f = 1 / (2 C + 1) for local interactions) A. Sterdyniak, C. Repellin, E.J. Bergholtz, Z. Liu, M. Strong evidence also for C>1 generalizations of B.A. Bernevig, and N. Trescher, R. Moessner, and M. Regnault, Phys. Rev. B non-Abelian FQH states found in this model! Udagawa, Phys. Rev. Lett. 114, 87, 205137 (2013) 016806 (2015) Different also from standard multi-layer systems (entanglement spectra, boundary conditions, layer-momentum correlation,…)
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