Synthetic Creutz-Hubbard model: interacting topological insulators with ultracold atoms Matteo Rizzi Johannes Gutenberg Universität Mainz J. Jünemann, et al., arXiv:1612.02996 -- accepted on PRX ICTP Trieste, 14 September 2017
Motivation • Topological phases of matter: academic & practical interests ! Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies with ultracold atoms
Motivation • Topological phases of matter: academic & practical interests ! i. paradigm models <==> real materials ? (e.g., Kitaev, Haldane, Kane-Mele, Harper-Hofstadter models, ...) • Two open issues: ii. role of interactions ==> correlation effects ? (a.k.a., can one get generalizations of fractional quantum Hall effect?) Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies with ultracold atoms
Motivation • Topological phases of matter: academic & practical interests ! i. paradigm models <==> real materials ? (e.g., Kitaev, Haldane, Kane-Mele, Harper-Hofstadter models, ...) • Two open issues: ii. role of interactions ==> correlation effects ? (a.k.a., can one get generalizations of fractional quantum Hall effect?) experiments: • Synthetic quantum matter (e.g., via cold atoms) could help! Bloch, Esslinger, Fallani, Spielman, & many more ! Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies with ultracold atoms
Motivation • Topological phases of matter: academic & practical interests ! i. paradigm models <==> real materials ? (e.g., Kitaev, Haldane, Kane-Mele, Harper-Hofstadter models, ...) • Two open issues: ii. role of interactions ==> correlation effects ? (a.k.a., can one get generalizations of fractional quantum Hall effect?) experiments: • Synthetic quantum matter (e.g., via cold atoms) could help! Bloch, Esslinger, Fallani, Spielman, • Quantum info. driven numerics (i.e., Tensor Networks), too! & many more ! Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies with ultracold atoms
Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character B z B x Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character B z Measurement of Zak phase via Ramsey interferometry B x in ultracold gases (SSH model) M. Atala, et al., Nat. Phys. 9, 795 (2013). Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling + Zeeman splitting ( undoubled ) Dirac point Topological character M. Creutz, PRL 83 2636 (1999) B z Measurement of Zak phase via Ramsey interferometry B x in ultracold gases (SSH model) M. Atala, et al., Nat. Phys. 9, 795 (2013). Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling + Zeeman splitting Discrete chiral symmetry only! Class AIII ( undoubled ) Dirac point Topological character M. Creutz, PRL 83 2636 (1999) B z Measurement of Zak phase via Ramsey interferometry B x in ultracold gases (SSH model) M. Atala, et al., Nat. Phys. 9, 795 (2013). Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Flat bands, AB cages & Edge States a c b + i˜ t 0 − ˜ − ˜ √ √ π t i / 2 i / 2 t π 1 / − 1 / 2 1 / 2 1 / 2 2 0 + i˜ t + i˜ t √ √ − i / 2 1 / 2 i / 2 − i / 2 i / 1 / 2 2 0 − ˜ ε = + 2˜ − ˜ ε = − 2˜ π ε l = 0 ε r = 0 t t π t t 0 + i˜ t Flat bands <==> basis of localized states (Aharanov-Bohm cages) Vidal, Mosseri, & Doucot, PRL 81, 5888 (1998) Topological <==> zero-energy (mid-gap) edge states Doubly degenerate ground at half filling (N p = N) Tovmasyan, van Nieuwenburg, & Huber, PRB 88, 220510(R) (2013) Takayoshi, Katsura, Watanabe, & Aoki, PRA 88, 063613 (2013) Huber & Altman, PRB 82, 184502 (2010) Tovmasyan, Peotta, Törmä ̈ , & Huber, arXiv:1608.00976 Sticlet, Seabra, Pollmann, & Cayssol, PRB 89, 115430 (2014) Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Flat bands, AB cages & Edge States a c b + i˜ t Bragg techniques 0 − ˜ − ˜ √ √ π t i / 2 i / 2 t π 1 / − 1 / 2 1 / 2 1 / to measure edge states 2 2 0 in ultracold cold atoms + i˜ t (with steep enough potential) + i˜ t √ √ − i / 2 1 / 2 i / 2 − i / 2 i / 1 / 2 2 0 − ˜ ε = + 2˜ − ˜ ε = − 2˜ π ε l = 0 ε r = 0 Goldman,Beugnon, Gerbier, t t π t t 0 PRL 108, 255303 (2012). + i˜ t Flat bands <==> basis of localized states (Aharanov-Bohm cages) Vidal, Mosseri, & Doucot, PRL 81, 5888 (1998) Topological <==> zero-energy (mid-gap) edge states Doubly degenerate ground at half filling (N p = N) Tovmasyan, van Nieuwenburg, & Huber, PRB 88, 220510(R) (2013) Takayoshi, Katsura, Watanabe, & Aoki, PRA 88, 063613 (2013) Huber & Altman, PRB 82, 184502 (2010) Tovmasyan, Peotta, Törmä ̈ , & Huber, arXiv:1608.00976 Sticlet, Seabra, Pollmann, & Cayssol, PRB 89, 115430 (2014) Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Flat bands, AB cages & Edge States a c b + i˜ t 0 ? − ˜ − ˜ √ √ π t i / 2 i / 2 t π 1 / − 1 / 2 1 / 2 1 / 2 2 0 + i˜ t + i˜ t √ √ − i / 2 1 / 2 i / 2 − i / 2 i / 1 / 2 2 0 − ˜ ε = + 2˜ − ˜ ε = − 2˜ π ε l = 0 ε r = 0 t t π t t 0 + i˜ t ? Flat bands <==> basis of localized states (Aharanov-Bohm cages) Vidal, Mosseri, & Doucot, PRL 81, 5888 (1998) Topological <==> zero-energy (mid-gap) edge states Doubly degenerate ground at half filling (N p = N) Zeeman Imbalance & Hubbard interactions bend bands / close gap ==> & cancel edge modes ... Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Flat bands, AB cages & Edge States a c b + i˜ t 0 − ˜ − ˜ √ √ π t i / 2 i / 2 t π 1 / − 1 / 2 1 / 2 1 / 2 2 0 + i˜ t + i˜ t √ √ − i / 2 1 / 2 i / 2 − i / 2 i / 1 / 2 2 0 − ˜ ε = + 2˜ − ˜ ε = − 2˜ π ε l = 0 ε r = 0 t t π t t 0 + i˜ t Flat bands <==> basis of localized states (Aharanov-Bohm cages) Vidal, Mosseri, & Doucot, PRL 81, 5888 (1998) Topological <==> zero-energy (mid-gap) edge states Doubly degenerate ground at half filling (N p = N) Zeeman Imbalance & Hubbard interactions bend bands / close gap ==> & cancel edge modes ... Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Flat bands, AB cages & Edge States a c b + i˜ t 0 − ˜ − ˜ √ √ π t i / 2 i / 2 t π 1 / − 1 / 2 1 / 2 1 / 2 2 0 + i˜ t + i˜ t √ √ − i / 2 1 / 2 i / 2 − i / 2 i / 1 / 2 2 0 − ˜ ε = + 2˜ − ˜ ε = − 2˜ π ε l = 0 ε r = 0 t t π t t 0 + i˜ t Outline of the attack plan: 1. Analytics: mappings onto effective Ising models for each transition 2. Numerics: matrix product states (MPS) & entanglement analysis 3. Experiments: sketch of proposal with assisted tunnelling in optical lattices Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
Weak interactions: gap behaviour Indicator 1: compressibility gap vs. degeneracy split TI: oPM: ? 4 3 δ , ∆ 2 1 0 0 . 5 1 . 5 2 . 5 3 . 5 0 1 2 3 4 ∆ ε / ˜ t Synthetic Creutz-Hubbard model: matteo.rizzi@uni-mainz.de ICTP Workshop 2017 interacting topol. insul. Quantum Technologies arXiv:1612.02996 with ultracold atoms
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