Synthetic Creutz-Hubbard model: interacting topological insulators with ultracold atoms
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1 interacting topological insulators Matteo Rizzi Johannes Gutenberg Universität Mainz J. Jünemann, et al., -- accepted on PRX ICTP Trieste, 14 September 2017
2 Motivation Topological phases of matter: academic & practical interests!
3 Motivation Topological phases of matter: academic & practical interests! Two open issues: i. paradigm models <==> real materials? (e.g., Kitaev, Haldane, Kane-Mele, Harper-Hofstadter models,...) ii. role of interactions ==> correlation effects? (a.k.a., can one get generalizations of fractional quantum Hall effect?)
4 Motivation Topological phases of matter: academic & practical interests! Two open issues: i. paradigm models <==> real materials? (e.g., Kitaev, Haldane, Kane-Mele, Harper-Hofstadter models,...) ii. role of interactions ==> correlation effects? (a.k.a., can one get generalizations of fractional quantum Hall effect?) Synthetic quantum matter (e.g., via cold atoms) could help! experiments: Bloch, Esslinger, Fallani, Spielman, & many more!
5 Motivation Topological phases of matter: academic & practical interests! Two open issues: i. paradigm models <==> real materials? (e.g., Kitaev, Haldane, Kane-Mele, Harper-Hofstadter models,...) ii. role of interactions ==> correlation effects? (a.k.a., can one get generalizations of fractional quantum Hall effect?) Synthetic quantum matter (e.g., via cold atoms) could help! Quantum info. driven numerics (i.e., Tensor Networks), too! experiments: Bloch, Esslinger, Fallani, Spielman, & many more!
6 Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling
7 Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling
8 Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character Bz Bx
9 Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character Bz Bx Measurement of Zak phase via Ramsey interferometry in ultracold gases (SSH model) M. Atala, et al., Nat. Phys. 9, 795 (2013).
10 Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling + Zeeman splitting Topological character (undoubled) Dirac point M. Creutz, PRL (1999) Bz Bx Measurement of Zak phase via Ramsey interferometry in ultracold gases (SSH model) M. Atala, et al., Nat. Phys. 9, 795 (2013).
11 Imbalanced Creutz Ladder Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling + Zeeman splitting Discrete chiral symmetry only! Class AIII Topological character (undoubled) Dirac point M. Creutz, PRL (1999) Bz Bx Measurement of Zak phase via Ramsey interferometry in ultracold gases (SSH model) M. Atala, et al., Nat. Phys. 9, 795 (2013).
12 Flat bands, AB cages & Edge States a +i t b 0 c t p t p 1/ p 2 i/2 1/2 i/2 1/2 1/ p 2 +i t 0 +i t 0 i/ p 2 1/2 i/2 1/2 i/2 i/ p 2 t p t p e l = 0 e =+2 t e = 2 t e r = 0 +i t 0 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 (Np = N) Tovmasyan, van Nieuwenburg, & Huber, PRB 88, (R) (2013) Takayoshi, Katsura, Watanabe, & Aoki, PRA 88, (2013) Huber & Altman, PRB 82, (2010) Tovmasyan, Peotta, Törmä, & Huber, arxiv: Sticlet, Seabra, Pollmann, & Cayssol, PRB 89, (2014)
13 Flat bands, AB cages & Edge States a +i t t p +i t +i t t p 0 0 b 0 c 1/ p 2 i/ p 2 i/2 1/2 1/2 i/2 i/2 1/2 1/2 i/2 1/ p 2 i/ p 2 Bragg techniques to measure edge states in ultracold cold atoms (with steep enough potential) t +i t p t p 0 e l = 0 e =+2 t e = 2 t e r = 0 Goldman,Beugnon, Gerbier, PRL 108, (2012). 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 (Np = N) Tovmasyan, van Nieuwenburg, & Huber, PRB 88, (R) (2013) Takayoshi, Katsura, Watanabe, & Aoki, PRA 88, (2013) Huber & Altman, PRB 82, (2010) Tovmasyan, Peotta, Törmä, & Huber, arxiv: Sticlet, Seabra, Pollmann, & Cayssol, PRB 89, (2014)
14 Flat bands, AB cages & Edge States a +i t b 0 c t p +i t t p 0 1/ p 2 i/2 1/2 i/2 1/2 1/ p 2? +i t 0 i/ p 2 1/2 i/2 1/2 i/2 i/ p 2 t p t p e l = 0 e =+2 t e = 2 t e r = 0 +i t 0 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 (Np = N) Zeeman Imbalance & Hubbard interactions ==> bend bands / close gap & cancel edge modes...
15 Flat bands, AB cages & Edge States a +i t b 0 c t p t p 1/ p 2 i/2 1/2 i/2 1/2 1/ p 2 +i t 0 +i t 0 i/ p 2 1/2 i/2 1/2 i/2 i/ p 2 t p t p e l = 0 e =+2 t e = 2 t e r = 0 +i t 0 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 (Np = N) Zeeman Imbalance & Hubbard interactions ==> bend bands / close gap & cancel edge modes...
16 Flat bands, AB cages & Edge States a +i t b 0 c t p t p 1/ p 2 i/2 1/2 i/2 1/2 1/ p 2 +i t 0 +i t 0 i/ p 2 1/2 i/2 1/2 i/2 i/ p 2 t p t p e l = 0 e =+2 t e = 2 t e r = 0 +i t 0 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
17 Weak interactions: gap behaviour Indicator 1: compressibility gap vs. degeneracy split TI: opm:? 4 3 δ, ε/ t
18 Weak interactions: coupled Ising chains Indicator 2: density imbalance between ladder legs?
19 Weak interactions: coupled Ising chains Indicator 2: density imbalance between ladder legs Bogolubov + Jordan Wigner trafos?
20 Weak interactions: coupled Ising chains Indicator 2: density imbalance between ladder legs self-consistent mean-field two simultaneous Ising transitions (c=1)?
21 Strong interactions: orbital Ising model Gutzwiller projector Super-exchange coupling single Ising critical line (c=1/2)? T i y ε/ t
22 Strong interactions: orbital Ising model Gutzwiller projector Super-exchange coupling single Ising critical line (c=1/2)? 0.5 T i y ε/ t ofm TI another transition at low imbalance!
23 AB-cages: exotic Hubbard NNN interactions + dens. dep. tunnelling without dipolar atoms or strange schemes!!? t imb p 2 t imb t imb V v /4 V v /4 J e + e l = e r T d V v /2 t imb e t imb Vv /4 J
24 AB-cages: exotic Hubbard NNN interactions + dens. dep. tunnelling without dipolar atoms or strange schemes!! CRUCIAL: we keep both bands! no projection!? t imb p 2 t imb t imb V v /4 V v /4 J e + e l = e r T d V v /2 t imb e t imb Vv /4 J
25 AB-cages: exotic Hubbard NNN interactions + dens. dep. tunnelling without dipolar atoms or strange schemes!! CRUCIAL: we keep both bands! no projection!? t imb p 2 t imb t imb V v /4 V v /4 J e + e l = e r T d V v /2 t imb e t imb Vv /4 J leads to bulk mediated edge-edge interaction à la Fano-Anderson...
26 Weak imbalance: yet another Ising restrict to singly occupied AB-c manifold... single Ising critical line (c=1/2)
27 Weak imbalance: yet another Ising restrict to singly occupied AB-c manifold... single Ising critical line (c=1/2) Jordan-Wigner gives effective bands dual to the original ones
28 Weak imbalance: yet another Ising restrict to singly occupied AB-c manifold... single Ising critical line (c=1/2) Jordan-Wigner gives effective bands dual to the original ones what happens then to the second Ising?
29 Weak imbalance: yet another Ising restrict to singly occupied AB-c manifold... single Ising critical line (c=1/2) impurity model <==> topological character i) bulk mediated edge-edge interaction shift energies but does not lift degeneracy ii) no dephasing if Bogoliubov modes gapped vs. dephasing if gapless ==> (i.e., not well defined edge modes!)
30 Matrix Product States Generic description of a many-body Hilbert space is exponentially expensive numbers
31 Matrix Product States Generic description of a many-body Hilbert space is exponentially expensive numbers Area-law for entanglement entropy generic state Physically accessible states Eisert, Cramer, Plenio RMP 82, 277 ( 10) Product states
32 Matrix Product States generic state Generic description of a many-body Hilbert space is exponentially expensive numbers Area-law for entanglement entropy Physically accessible states Eisert, Cramer, Plenio RMP 82, 277 ( 10) Product states
33 Matrix Product States generic state Generic description of a many-body Hilbert space is exponentially expensive numbers Economic description by Tensor Networks : (variational RG schemes, DMRG) Schollwock, Ann. Phys. 326, 96 (2011) Area-law for entanglement entropy numbers Physically accessible states Eisert, Cramer, Plenio RMP 82, 277 ( 10) Product states
34 Matrix Product States Generic description of a many-body Hilbert space is exponentially expensive numbers Economic description by Tensor Networks : (variational RG schemes, DMRG) Schollwock, Ann. Phys. 326, 96 (2011) numbers plenty of different decompositions in tensor products: MPS PEPS TTN MERA
35 ==> Entanglement signatures (MPS) entanglement entropy S(l) c=1 CFT central charge Vidal, Latorre, Rico & Kitaev, PRL 90, (2003); Calabrese & Cardy, J. Stat. Mech. P06002 (2004) N π sin ( ) πl N c=1/2 c=1/2
36 ==> Entanglement signatures (MPS) entanglement entropy S(l) c=1 CFT central charge Vidal, Latorre, Rico & Kitaev, PRL 90, (2003); Calabrese & Cardy, J. Stat. Mech. P06002 (2004) N π sin ( ) πl N c=1/2 c=1/2 MPS gives out more: degeneracy pattern in entanglement spectrum! H. Li and F. D. M. Haldane, PRL (2015) F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa PRB (2010)
37 Experimental ingredients state indep. gradient + intensity modulated OL Holthaus, PRL 69, 351 (1992) Jaksch & Zoller, NJP 5, 56 (2003) Gerbier & Dalibard, NJP 12, (2010) Eckardt, RMP 89, (2017)
38 Experimental ingredients state indep. gradient + intensity modulated OL Holthaus, PRL 69, 351 (1992) Jaksch & Zoller, NJP 5, 56 (2003) Gerbier & Dalibard, NJP 12, (2010) Eckardt, RMP 89, (2017) + Interactions away from driving-induced resonances Ma, et al. (Greiner), PRL 107, (2011) Chen, et al. (Bloch), PRL 107, (2011)
39 Experimental ingredients state indep. gradient + intensity modulated OL Holthaus, PRL 69, 351 (1992) Jaksch & Zoller, NJP 5, 56 (2003) Gerbier & Dalibard, NJP 12, (2010) Eckardt, RMP 89, (2017) a b c np 1 2 np 3 2 w 1 w 2 w3 w 1 + Interactions away from driving-induced resonances Ma, et al. (Greiner), PRL 107, (2011) Chen, et al. (Bloch), PRL 107, (2011) + Raman assisted tunnelling ns 1 2 e " e # e " e # w 1 w 2 = (e " e # )+D w 1 w 3 =(e " e # )+D Aidelsburger, et al. (Bloch), PRL 111, (2013) Miyake, et al. (Ketterle), PRL 111, (2013)
40 Experimental ingredients Alternative: Superlattice toolbox for topological insulators! a. b. c. d. e. f. L. Mazza, A. Bermudez, MR, et al., PRL 105, ( 10), NJP ( 12) MR, PoS - SISSA 193, 036 (2014)
41 Overall picture & perspectives Interaction driven split of a Dirac line (c=1) into two Majorana ones (c=1/2) ==> consequences for topological excitations!? nature of the tricritical point? Mappings onto impurity problems & broadening of edge modes (not shown) ==> new elements for understanding bulk-edge correspondence! High feasibility & detectability in current experimental setups :-) How stable is the picture for imperfect flux? (intuition: similar to imbalance) Away from half-filling: resonances? fractional effects? dualities? (among others: emergent local J. Jünemann A. Piga gauge symmetry, etc.) S.J. Ran M. Lewenstein A. Bermudez
42 Other related works
43 Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: Other related works
44 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 "i k U V µ( 0 ) O 2, 1 O 2,+1 2k F h 2k F =4k F j c
45 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 O 2, 1 O 2,+1 "i k U V µ( 0 ) 2k F h 2k F =4k F j c FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks y x N L m L M. Gerster, et al. (MR) arxiv: y x te i 2πx t L
46 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 O 2, 1 O 2,+1 "i k U V µ( 0 ) Pair Luttinger Liquid in Bosonic Flat Bands C. Cartwright, MR, G. dechiara, soon on arxiv 2k F h 2k F =4k F j c FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks y x N L m L M. Gerster, et al. (MR) arxiv: y x te i 2πx t L
47 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 O 2, 1 O 2,+1 "i k U V µ( 0 ) Pair Luttinger Liquid in Bosonic Flat Bands C. Cartwright, MR, G. dechiara, soon on arxiv 2k F h 2k F =4k F j c FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks y x N L m L M. Gerster, et al. (MR) arxiv: y x te i 2πx t L
48 Other related works
49 Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: Other related works
50 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 "i k U V µ( 0 ) O 2, 1 O 2,+1 2k F h 2k F =4k F j c
51 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 O 2, 1 O 2,+1 "i k U V µ( 0 ) 2k F h 2k F =4k F j c FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks y x N L m L M. Gerster, et al. (MR) arxiv: y x te i 2πx t L
52 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 O 2, 1 O 2,+1 "i k U V µ( 0 ) Pair Luttinger Liquid in Bosonic Flat Bands C. Cartwright, MR, G. dechiara, soon on arxiv 2k F h 2k F =4k F j c FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks y x N L m L M. Gerster, et al. (MR) arxiv: y x te i 2πx t L
53 Other related works Tunability of Drude Weight of 1D Dirac-Weyl Fermions M. Bischoff, J. Jünemann, M. Polini, MR, arxiv: $\nu = 1/2$ resonance in chiral fermionic ladders A. Haller, MR, M. Burrello, arxiv: "i #i E #i te i 2 te i 2 O 2, 1 O 2,+1 "i k U V µ( 0 ) Pair Luttinger Liquid in Bosonic Flat Bands C. Cartwright, MR, G. dechiara, soon on arxiv 2k F h 2k F =4k F j c FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks y x N L m L M. Gerster, et al. (MR) arxiv: THANKS FOR YOUR ATTENTION y x te i 2πx t L
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