Characterising band topology out-of-equilibrium using Chern marker - PowerPoint PPT Presentation

Characterising band topology 
 out-of-equilibrium 
 using Chern marker currents Dr Gunnar Möller 
 Royal Society University Research Fellow School of Physical Sciences, University of Kent, Canterbury UK Visitor, Cavendish Laboratory, University of Cambridge M Caio, GM, NR Cooper & MJ Bhaseen, Nature Physics 15, 257 (2019). B. Andrews & G. Möller, Phys. Rev. B 97, 035159 (2018). L. Schoonderwoerd, F. Pollmann, G. Möller, arxiv:1908.00988 Conference on Signatures of Topology in Condensed Matter | (smr 3330) ICTP Trieste, Oct. 21-25, 2019

Overview Motivation & Background • Topological order in the fractional quantum Hall effect • Emulating the effects of magnetic fields without external fields. Main Part : New Probes for Topological Order - non-interacting systems Chern marker: • diagnosis of critical properties • propagation of Chern marker currents • measurement of the Chern marker Brief appetiser : Novel Phases - Fractional Chern Insulators • new fractional Chern insulator state in higher Chern r bands at filling fractions ν = r | Ck | + 1 • Direct quantum Hall plateau transition between |C|>1 Chern insulators and fractional Chern insulators Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Topological Order in the Quantum Hall Effect ‣ a macroscopic quantum phenomenon: magnetoresistance in 2D electron gases 2D electron gas Where? ‣ in semiconductor hetero- structures with clean two- dimensional electron gases ‣ at low temperatures (~0.1K) and in strong magnetic fields k B T ⌧ ~ ω c = ~ eB/m e What? ‣ plateaus in Hall conductance σ xy = ν e 2 h ‣ simultaneously: (near) zero longitudinal resistance ‣ different topologically ordered phases at each Hall plateau ‣ supporting fractionalized abelian and non-Abelian excitations Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Semiclassical picture of IQHE: skipping orbits J at edge of sample, ‘skipping orbits’ several LLs contribute a uni-directional current filled: E cyclotron motion produces no net current in bulk of sample picture for quantum transport: absence of backscattering ⇒ dissipationless current 
 ⇒ no voltage drop along lead! J low-energy or ‘gapless’ excitations present σ xy = ν e 2 near boundary h Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

<latexit sha1_base64="RwptRIO/JGOSD3e5yEor54MHMfo=">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</latexit> Quantum Hall conductance as a topological invariant Geometrical phase analogous to Aharonov-Bohm effect Z k | d Z k i d ~ A ( ~ k ) d ~ ~ � ( C ) = i h u ~ | u ~ k ⌘ k d ~ k C C S C Effective ‘vector potential’ called Berry connection Z A ( ~ ~ r ) ∗ ~ k y r ) d 2 r k ) = i k ( ~ k ( ~ u ~ r k u ~ UC Using Stokes’ theorem: k x Z Z A ( ~ k ) d ~ A ( ~ ~ r k ⇥ ~ ~ � ( C ) = k = k ) d ~ � C S C = ∂ S A ( ~ B = ~ ~ r k ⇥ ~ Berry curvature : is a property of the band eigenfunctions, only! k ) 1 R BZ d 2 k B ( k ) C = Chern number : takes only integer values! 2 π σ xy = e 2 • C determines the Hall conductance - TKNN (1981): X C n h n occ. Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Berry curvature and Chern number Geometrical phase analogous to Aharonov-Bohm effect Z k | d Z k i d ~ A ( ~ k ) d ~ ~ � ( C ) = i h u ~ | u ~ k ⌘ k d ~ k C C S C Effective ‘vector potential’ called Berry connection Z A ( ~ ~ r ) ∗ ~ k y r ) d 2 r k ) = i k ( ~ k ( ~ u ~ r k u ~ UC Using Stokes’ theorem: k x integral over real-space, so it does not matter in what physical space the system ‘lives’ Z Z A ( ~ k ) d ~ A ( ~ ~ r k ⇥ ~ ~ � ( C ) = k = k ) d ~ � in reciprocal space, only the change of the scalar product on that space matters C S C = ∂ S A ( ~ B = ~ ~ r k ⇥ ~ Berry curvature : is a property of the band eigenfunctions, only! k ) 1 R BZ d 2 k B ( k ) C = Chern number : topological: takes only integer values! 2 π • Chern number provides universal classification of all possible single-particle bands (cl. A) Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Quantum Hall effect without magnetic fields The fractional quantum Hall effect is observed under extreme conditions ‣ strong magnetic fields of several Tesla ‣ very low temperatures ‣ clean / high mobility semiconductor samples Many different opportunities for emulating magnetic fields: 1. Cold Atomic Gases 2. Solid State 3. Photons rotation spin orbit 
 coupling Fe 3 Sn 2 (proposed) strain Si waveguides laser assisted hopping Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019

Landau-levels vs lattice models with effective magnetic flux continuum lattice 1 h i b β e iA αβ + h.c. ˆ α ˆ X b † 2 m | p − e A | 2 + ˆ H = − J H = V h α , β i X V ij ˆ n i ˆ n j + A = B z x e y Berry curvature in a Landau-Level: flat Berry curvature in the Hofstadter model B = r ⇥ A B k y k y k x k x Gunnar Möller Signatures of Topology in Condensed Matter, ICTP Trieste, October 2019