1 Anyon workshop, Kaiserslautern, 12/15/2014 Measuring many-body topological invariants using polarons Fabian Grusdt Physics Department and Research Center OPTIMAS, University of Kaiserslautern, Germany Graduate School of Materials Science in Mainz, Kaiserslautern, Germany
Motivation Control over individual anyons? σ electrons ultracold atoms photons b 1 Topologically trivial +F exp 0 0 Hofstadter model 2 Detuning (J) 4 Stormer, RMP 71,4 (1999) Aidelsburger et al., arxiv:1407.4205 Hafezi et al., Nature Photonics 7 (2013) 2
3 Motivation ultra cold atoms photons 2D systems artificial gauge fields strong interactions coming low temperatures control of anyons quantum computer Outline: Following talk! Interferometric measurement of topological invariants Topological Polarons & many-body invariants
4 Interferometric approach for measuring topological invariants TKNN, Phys. Rev. Lett. 49 (1982) Xiao, Rev. Mod. Phys. 82 (2010) Berry (1984) Zak, Phys. Rev. Lett. 62 (1989) key idea: measure geometric phases!
5 1D Zak phase Zak phase of Bloch bands: I ' Zak = dk hu(k) i@ k u(k)i Bloch wavefunction quantized by inversion symmetry ' Zak =0, Zak, Phys. Rev. Lett. 62 (1989) Su-Schrieffer-Heeger model: Su et al., PRL 42 (1979) X Ĥ = t 1 â nâ n+1 +h.c. n odd t 2 X n even â nâ n+1 +h.c. D1 D2 ' Zak =0 ' Zak =
6 1D Zak phase Measurement of Zak phase: Atala et al., Nature Phys. 9, 2013 BEC in Su-Schrieffer-Heeger model: ' Zak =0.97(2)
2D Chern number Multi-band Chern number: C = 1 D 1 2 Z 2 0 dk y @ ky Im log det Ŵ (k y) D: number of bands Wilson loops multiband generalization of Berry phase apple Z 2 Ŵ (k y )=Pexp i dk x  x (k x ) 0 pedagogical overview: Makeenko, Phys.At.Nuc. 73 (2010) numerical implementation: Yu et al., PRB 84 (2011) Abelian case: ' Zak /2 Abanin et al., PRL 110, 2013 C = Z 2 0 dk y @ ky ' Zak (k y ) k y /2 units of 7
8 2D Z2 topological insulator (TI) Zak phases and Z 2 invariant Time-reversal I 1D: Bloch (TR) invariant oscillations TI + Ramsey interferometry I 1D: Bloch Zak oscillations phases can + Ramsey be measured. interferometry Z2 topological Zakinsulator: phases cankane be & measured. Mele, PRL 95 (2005) æ M. Atala et.al., following talk æ M. Atala et.al., following talk QHE QHE Zak phases and Z 2 invariant + SOC = I 2D TR invariant band structure Kane & Mele, 2005 I 2D TR invariant band structure Kane 3& Mele, 2005 1 2 Grusdt, Abanin & Demler, PRA 89 (2014) time reversal polarization: time reversal polarization: TR polarization: Fu & Kane, PRB 74 (2006) Fu Fu & & Kane, Kane, PRB PRB 74 (2006) 74 (2006) Ï (k y )=Ï I Zak(k y ) Ï II Ï (k y )=Ï I Zak(k y ) Ï II Z2 topological invariant: Zak(k Zak(k y ) y ) 4 0 0.2 0.4 0.6 0.8 1 0 Not sufficient to measure these Zak phases!! 0.5 2D = Ï Ï (fi) Ï (0) mod 4fi œ {0, 2fi} 2D = Ï Ï (fi) Ï (0) mod 4fi œ {0, 2fi}
9 2D Z2 topological insulator (TI) Grusdt, Abanin & Demler, PRA 89 (2014) 1 2 3 4 0 0.2 0.4 0.6 0.8 1 0 0.5 Not sufficient to measure these Zak phases!! Adiabatic & twist scheme Problem: measurement of ' Zak mod 2 only. I Ï Zak mod 2fi only! measure winding of TR polarization / TR polarization Ï (k y ) is discontinuous Solution: measure winding of TR polarization!
10 2D Z2 topological insulator (TI) Solution: Continuous TR polarization Grusdt, Abanin & Demler, PRA 89 (2014) Example: Kane-Mele model:? Interacting systems? 1 0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5
11 Summary Interferometric approach to measuring topological invariants Ramsey Interferometry + Bloch oscillations Many-body invariants can also be measured:
12 Collaborators: Eugene Demler Dmitry Abanin Norman Yao
Thanks for your attention! 13