Synthetic topology and manybody physics in synthetic lattices
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1 Synthetic topology and manybody physics in synthetic lattices Alessio Celi EU STREP EQuaM May 16th, 2017 Atomtronics - Benasque
2 Plan Integer Quantum Hall systems and Edge states Cold atom realizations: synthetic gauge field Real space Synthetic lattice (Extradimension) Topology in narrow strips Further applications: non-trivial topology interacting system. Dimerized interacting ladder Meissner/Vortex phase (in analogy to type II superconductors) Effect of the dimerization Prospects
3 Quantum Hall effect 1879: Classical Hall effect (consequence of Lorentz force) 1980: Quantum Hall effect: Electric conductivity quantized E. Hall K. Von Klitzing
4 Integer Quantum Hall effect on lattice IQH explained in terms of single particle physics (Landau level filling) Quantization determined by topology of filled bands (1-Chern number) Bulk-boundary correspondence (Topological insulator prototype) Edge states determined by spectrum of periodic system On square lattice simple formulation: Hofstadter model
5 Integer Quantum Hall effect on lattice IQH explained in terms of single particle physics (Landau level filling) Quantization determined by topology of filled bands (1-Chern number) Bulk-boundary correspondence (Topological insulator prototype) Edge states determined by spectrum of periodic system On square lattice simple formulation: Hofstadter model
6 How? Cold atoms in optical lattices as ideal electrons in metals V 0 Mott-superfluid phase transition (predicted observed 2002)
7 How? C. atoms in OL as charged electrons in external fields Increasing complexity OL simulator: Modify Hopping & connectivity Tune interaction (intensity, range) Distinguish internal states of atoms Hopping with phases Synthetic magnetic field for neutral atoms Synthetic Aharonov-Bohm effect = Σ i i = magnetic flux 1
8 How? C. atoms in OL as charged electrons in external fields Many ways of generating magnetic fluxes on the lattice Rotation (= constant magnetic field) Shaking (also non-abelian PRL (2012) ) Exp. collaboration with Hamburg: search for Spin liquid phases in frustrated antiferromagnets with Bosons
9 How? C. atoms in OL as charged electrons in external fields Many ways of generating magnetic fluxes on the lattice Rotation (= constant magnetic field) Shaking Raman laser + 2D superlattice Theory : Jaksch & Zoller NJP 5 56 (2003).. Experiments: PRL (2012), PRL (2013) (I.Bloch group) PRL (2013) (W.Ketterle group)
10 How? C. atoms in OL as charged electrons in external fields Many ways of generating magnetic fluxes on the lattice Rotation (= constant magnetic field) Shaking Raman laser + 2D superlattice Theory : Jaksch & Zoller NJP 5 56 (2003).. Experiments: PRL (2012), PRL (2013) (I.Bloch group) PRL (2013) (W.Ketterle group) Raman laser + Extradimension
11 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D?
12 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D? In a lattice Dimensionality Connectivity
13 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D? In a lattice Dimensionality Connectivity Hopping in D+1 hypercubic lattice as
14 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D? In a lattice Dimensionality Connectivity Hopping in D+1 hypercubic lattice as
15 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D? In a lattice Dimensionality Connectivity Hopping in D+1 hypercubic lattice as
16 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D? In a lattice Dimensionality Connectivity Coupled atomic states hopping in D-lattices
17 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics In OL 3D Hubbard model 1D,2D by tuning optical potential And > 3D? In a lattice Dimensionality Connectivity Coupled atomic states hopping in D-lattices Alternative to cold atoms e.g. photonic crystal Jukic & Buljan PRA (2013) cold molecules Wall, Maeda & Carr New J. Phys (2015). Ring resonators Opt. Lett. 41, (2016)
18 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics Observables 4D behavior: density of states, scaling of Entropy Example: Lens 173 Yb (F=5/2, 6 states) Applications: 4D-Q.Hall, S.-S.-C. Zhang and J. Hu, Science 294, 823 (2001) Edge,Tworzydlo, Beenakker PRL 109, (2012) Price,Zilberberg,Ozawa,Carusotto,Goldman PRL 115, (2015)
19 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics Observables 4D behavior: density of states, scaling of Entropy Example: Lens 173 Yb (F=5/2, 6 states) Applications: 4D-Q.Hall, S.-S.-C. Zhang and J. Hu, Science 294, 823 (2001) Topological observable: Linear and quadratic response
20 How? C. atoms with in OL as particles in Extradimension [Boada,AC, PRL 108, (2012)] Highlighted in Physics Observables 4D behavior: density of states, scaling of Entropy Example: Lens 173 Yb (F=5/2, 6 states) Applications: 4D-Q.Hall, S.-S.-C. Zhang and J. Hu, Science 294, 823 (2001) Topological observable: Linear and quadratic response Efficient algorithm for C 2 : M. Mochol-Grzelak, A. Dauphin, AC, M. Lewenstein, to appear
21 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] Or also
22 Synthetic dim. How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] X= n a 1d-lattice loaded e.g. with 87 Rb (F=1, m=-1,0,1) + Raman dressing J Constant magnetic flux! Ω Exp[i n] Sharp Boundaries Edge currents (hard to get in real 2d lattice) signal of Topological nature of Q. Hall (bulk-boundary corr.)
23 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] Spectrum Evolution Edge states
24 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] Spectrum Evolution Edge states Alternative realization: real-space ladder Hügel & Paredes PRA, Atala et al. Nat. Phys. (2014)
25 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] 4 Wrapping the synthetic dimension E/J Hofstadter butterfly spectrum π
![al PRL 112, 043001 (2014)] Experimental Realizations: I) NIST I.B.](/docs-images/89/99397593/images/26-1.jpg "Spielman group 87 Rb Edge & bulk states visualization [Science 349 1514-1518")
26 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] Experimental Realizations: I) NIST I.B. Spielman group 87 Rb Edge & bulk states visualization [Science (2015)] 2π/3 AC, Tarruell Probing the edge with cold atoms Science (2015) (Perspective)
27 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] Experimental Realizations: II) LENS Fallani group 173 Yb Chiral Edge in Fermionic gas [Science (2015)] 0.37 π AC, Tarruell Probing the edge with cold atoms Science (2015) (Perspective)
28 How? C. atoms in OL as charged electrons in external fields and synthetic dimension [AC et al PRL 112, (2014)] Experimental Realizations: II) LENS Fallani group 173 Yb Chiral Edge in Fermionic gas [Science (2015)]
29 Topology in narrow strips Narrow Hofstadter strips have edge states What about the bulk?
30 Topology in narrow strips Narrow Hofstadter strips have edge states What about the bulk? How big should it be to display topological properties? Is there some reminiscence of open/closed boundary correspondence? Is Chern number defined? / Can we measure it?
31 Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,.AC arxiv: Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument Cf. Thouless pump [Lohse et al Nat. Phys. (2015)] [Nakajima et al Nat. Phys (2016)]
32 Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,.AC arxiv: Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument Usual semiclassical approach Filled band: Ex. Thermal gas [Dauphin et al PRL (2013)] [Aidelsburger et al Nat. Phys (2014) We consider a wave packet localized in y (narrow dim) and extended in x (large dim) in the lowest band
33 Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,.AC arxiv: Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument Usual semiclassical approach Wave packet: State easy to prepare if the coupling Different than geometric pumping of Lu et al. PRL (2016)
34 Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,.AC arxiv: Scheme: Results:
35 Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,.AC arxiv: Why does it work? Perturbative argument also for edge states: Gap linear in Hybridization spin states (spreading in ) quadratic in Quadratic degradation of the measurement
36 Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,.AC arxiv: Higher possible for Ex: Better than Fukui-Hatsugai-Suzuki algorithm J. Phys. Soc. Jpn. (2005) Robust to disorder (< gap) and typical harmonic confinement Interactions? Gap small (although may hold thought adiabatic argument, see later)
37 How? C. atoms in OL as charged electrons in synthetic non-trivial topologies [Boada,AC et al NJP 17 (2015) ] Circle from a line on the lattice Laser induced hoppings Internal species
38 How? C. atoms in OL as charged electrons in synthetic non-trivial topologies [Boada,AC et al NJP 17 (2015) ] Möbius strip by twisting the cylinder
39 How? C. atoms in OL as charged electrons in synthetic non-trivial topologies [Boada,AC et al NJP 17 (2015) ] Observables: Single particle: ex. IQH in cylinder vs Möbius Edge current stops See Morais-Smith s group works PRB 89, (2014), SSC 215, 27 (2015) Many Body: interactions synthetic lattice long-range! Topology induced changes in Mott-superfluid transition in extended Hubbard model (& spin system)
40 How? C. atoms in OL as charged electrons in synthetic non-trivial topologies [Boada,AC et al NJP 17 (2015) ] Observables: Single particle: ex. IQH in cylinder vs Möbius Edge current stops See Morais-Smith s group works PRB 89, (2014), SSC 215, 27 (2015) Many Body: interactions synthetic lattice long-range! Topology induced changes in Mott-superfluid transition in extended Hubbard model (& spin system) General feature of synthetic lattices
41 Interactions in synthetic lattices Let s go back to the simple synthetic ladder Bosons with N internal states, e.g. 87 RB N=3, SU(3) inv. inter. synthetic hopping + density-density interactions + SU(N)-breaking terms = Novel effective Hamiltonians & phases Two possibilities: Grass, AC & Lewenstein PRA (2014) SU(N)-breaking from spatial hopping e.g using spin-dependent lattices + shaking (cf. PRL (2012)) -> Antiferromagnetic phase e.g. N=2 SU(N)-breaking from interactions Rich Mott structure! different effective Hamiltonians e.g. (for right chem. pot.) N-state Potts-like Hamiltonians
42 Majorana Fermions Majorana Quasi-Particles Protected by Z 2 Angular Momentum Conservation, arxiv: (Innsbruck-ICTP-LENS) Interactions in synthetic lattices Interesting experimentally motivated question: Synth. IQH in synthetic lattices + interactions -> Fractional QH effect?! Question addressed very recently by various collaborations for Fermions Magnetic crystals and helical liquids in alkaline-earth fermionic gases Nat. Commun. 6, 8134 (2015), (NJP 18, (2016)) (Fazio group) Charge Pumping of Interacting Fermion Atoms in the Synthetic Dimension Tian-Sheng Zeng, Ce Wang, Hui, PRL 115, (2015) Charge pumping as detection method Adiabatic Control of Atomic Dressed States for Transport and Sensing, N.R. Cooper and A.M. Rey, PRA (2015) Topological fractional pumping with alkaline-earth(-like) ultracold atoms, (Fazio group), arxiv:
43 Interactions in synthetic lattices Interesting experimentally motivated question: Synth. IQH in synthetic lattices + interactions -> Fractional QH effect?! and for Bosons mainly real space slabs Bosonic Mott insulator with Meissner currents, A. Petrescu and K.Le Hur, Phys. Rev. Lett (2013) Strongly interacting bosons on a three-leg ladder in the presence of a homogeneous flux, F. Kolley, M. Piraud, I.P. McCulloch, U. Schollwöck, F. Heidrich-Meisner, NJP (2015) (PRA A 94, (2016)) Persisting Meissner state and incommensurate phases of hard-core boson ladders in a flux, M.Di Dio, S. De Palo, E. Orignac, R. Citro, M.L. Chiofalo, Phys. Rev. B 92, (2015) (NJP (2016)) (arxiv: ) Laughlin-like states in bosonic and fermionic atomic synthetic ladders, M. Calvanese et al, arxiv: Precursor of Laughlin state of hard core bosons on a two leg ladder, A. Petrescu, M. Piraud, G. Roux, I. P. McCulloch, K. Le Hur, arxiv:
44 Interactions in synthetic lattices Interesting experimentally motivated question: Synth. IQH in synthetic lattices + interactions -> Fractional QH effect?! and for Bosons originally mainly for real space slabs Bosonic Mott insulator with Meissner currents, A. Petrescu and K.Le Hur, Phys. Rev. Lett (2013) Strongly interacting bosons on a three-leg ladder in the presence of a homogeneous flux, F. Kolley, M. Piraud, I.P. McCulloch, U. Schollwöck, F. Heidrich-Meisner, NJP (2015) (PRA A 94, (2016)) Persisting Meissner state and incommensurate phases of hard-core boson ladders in a flux, M.Di Dio, S. De Palo, E. Orignac, R. Citro, M.L. Chiofalo, Phys. Rev. B 92, (2015) (NJP (2016)) (arxiv: ) Laughlin-like states in bosonic and fermionic atomic synthetic ladders, M. Calvanese et al, arxiv: Precursor of Laughlin state of hard core bosons on a two leg ladder, A. Petrescu, M. Piraud, G. Roux, I. P. McCulloch, K. Le Hur, arxiv: Vortex Melting in dimerized synthetic ladder, E. Tirrito, R. Citro, M.Lewestein, AC, soon
45 Meissner/Vortex phase in flux ladder [Cf. Haug et al ] No interactions: real = synthetic ladder Weak interleg (Raman) coupling: Strong interleg (Raman) coupling: 2 minima, 1 minima, [Orignac,Giamarchi, PRB 2001] Analogous to type II, also in presence interactions Real ladder experiment [Atala et, Nature Phys. 2014]
46 Meissner/Vortex phase in flux ladder No interactions: real = synthetic ladder Weak interleg (Raman) coupling: 2 minima, Observables Strong interleg (Raman) coupling: 1 minima, [Orignac,Giamarchi, PRB 2001] Analogous to type II, also in presence interactions Real ladder experiment [Atala et, Nature Phys. 2014]
47 Meissner/Vortex phase in flux ladder No interactions: real = synthetic ladder Weak interleg (Raman) coupling: 2 minima, Observables Strong interleg (Raman) coupling: 1 minima, [Orignac,Giamarchi, PRB 2001] Analogous to type II, also in presence interactions Real ladder experiment [Atala et, Nature Phys. 2014]
48 Meissner/Vortex phase in flux ladder Effect of interactions: suppress vortex phase
49 Meissner/Vortex phase in flux ladder U Effect of interactions: suppress vortex phase Real ladder: vortex phase survives in the hard-cord limit for more phases at large see [Petrescu, Le Hur,PRL 2013] [Piraud et al, PRB 2015]
50 Meissner/Vortex phase in flux ladder U Effect of interactions: suppress vortex phase Real ladder: vortex phase survives in the hard-cord limit for more phases at Synthetic ladder: vortex phase disappears in the hard-cord limit large see [Petrescu, Le Hur,PRL 2013] [Piraud et al, PRB 2015] more phases at.
51 Meissner/Vortex phase in flux ladder U Effect of interactions: suppress vortex phase Real ladder: vortex phase survives in the hard-cord limit for more phases at Synthetic ladder: vortex phase disappears in the hard-cord limit large see [Petrescu, Le Hur,PRL 2013] [Piraud et al, PRB 2015] more phases at. Idea: enucleate vortices by dimerizing the lattice ( easy exp. handle)
52 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle
53 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle No interactions: 4 bands
54 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle No interactions: 4 bands Just band folding
55 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle No interactions: 4 bands Just band folding
56 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle No interactions: 4 bands Just band folding
57 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle No interactions: 4 bands Bands deform & mix
58 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Effect of dimerization: new handle No interactions: 4 bands Bands become flat Both regimes Meissner
59 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress U Effect of dimerization: new handle Interactions: 3 states per rang
60 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress U Effect of dimerization: new handle Interactions: 3 states per rang 9 states per plaquette
61 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress U Effect of dimerization: new handle Interactions: 3 states per rang 9 states per plaquette 1 n=0, 4 n=1, 4 n=2 Spectrum plaquette
62 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress U Effect of dimerization: new handle Interactions: 3 states per rang 9 states per plaquette 1 n=0, 4 n=1, 4 n=2 Spectrum plaquette Plaquette in n=2 Band insulator
63 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Interactions: U Effect of dimerization: new handle 3 states per rang 9 states per plaquette 1 n=0, 4 n=1, 4 n=2 Spectrum plaquette Plaquette in n=2 Plaquette in n=1 Band insulator Imprinted vortex
64 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress DMRG calculations confirm perturbative expectations Ex.
65 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress DMRG calculation confirms perturbative expectations Ex.
66 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress DMRG calculation confirms perturbative expectations Ex. Phase diagram through calculation of currents and structure factors
67 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Moderate Ex. Melted vortex Meissner charge density wave Meissner
68 Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress Moderate Ex. Melted vortex Meissner charge density wave Meissner Similar to attractive interleg interaction, [Orignac et al, arxiv: ] Working to see evidence of commensurate-incommensurate transition
69 Further steps No hard boson limit: bosons different fermions Study the accessible experimental parameters Search for visible Laughling-like states in such regimes.. Hopefully many more
70 Exciting time for Quantum Simulation Gravitino & Velo-Zwanziger Relativistic & Gravity Dirac fermions in FLRW spacetime Effect of trap, temp Interacting neutrinos Future plans Frustrated q. magnets Observables of spin-liquids Experiments 2d LENS Topological insulators Long-range interactions Extradimension Lattice gauge theories Ease simulator Quantum random walk Non-trivial topologies TN analysis Models landscape
71 New opportunities for Atomtronics Gravitino & Velo-Zwanziger Relativistic & Gravity Dirac fermions in FLRW spacetime Effect of trap, temp Interacting neutrinos Future plans Frustrated q. magnets Observables of spin-liquids Experiments 2d LENS Topological insulators Long-range interactions Extradimension Lattice gauge theories Ease simulator Quantum random walk Non-trivial topologies TN analysis Models landscape
72 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein
73 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass
74 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass N. Goldman G. Juzeliunas P. Massignan J. Ruseckas I.B. Spielman
75 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass N. Goldman G. Juzeliunas P. Massignan J. Ruseckas I.B. Spielman J. Rodriguez-Laguna
76 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass N. Goldman G. Juzeliunas P. Massignan J. Ruseckas I.B. Spielman J. Rodriguez-Laguna S. Mugel J. Asboth C. Lobo
77 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass N. Goldman G. Juzeliunas P. Massignan J. Ruseckas I.B. Spielman J. Rodriguez-Laguna S. Mugel J. Asboth C. Lobo A. Dauphin L. Tarruell
78 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass N. Goldman G. Juzeliunas P. Massignan J. Ruseckas I.B. Spielman J. Rodriguez-Laguna S. Mugel J. Asboth C. Lobo A. Dauphin L. Tarruell E. Tirrito R. Citro
79 Extradimensional collaborators J.I. Latorre O. Boada M. Lewenstein T. Grass N. Goldman G. Juzeliunas P. Massignan J. Ruseckas I.B. Spielman J. Rodriguez-Laguna S. Mugel J. Asboth C. Lobo. A. Dauphin L. Tarruell E. Tirrito R. Citro
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