Measuring many-body topological invariants using polarons

Transcription

1 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

2 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: Hafezi et al., Nature Photonics 7 (2013) 2

3 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 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 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 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)

7 2D Chern number Multi-band Chern number: C = 1 D 1 2 Z 2 0 dk 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 ky ' Zak (k y ) k y /2 units of 7

8 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, 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 ) 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 9 2D Z2 topological insulator (TI) Grusdt, Abanin & Demler, PRA 89 (2014) 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 10 2D Z2 topological insulator (TI) Solution: Continuous TR polarization Grusdt, Abanin & Demler, PRA 89 (2014) Example: Kane-Mele model:? Interacting systems?

11 11 Summary Interferometric approach to measuring topological invariants Ramsey Interferometry + Bloch oscillations Many-body invariants can also be measured:

12 12 Collaborators: Eugene Demler Dmitry Abanin Norman Yao

13 Thanks for your attention! 13