body phases of synthetic Hofstadter strips EU STREP EQuaM Alessio - - PowerPoint PPT Presentation

▶
body phases of synthetic
SMART_READER_LITE
LIVE PREVIEW

body phases of synthetic Hofstadter strips EU STREP EQuaM Alessio - - PowerPoint PPT Presentation

Oct 14, 2023 467 likes •1.2k views

Topological properties and many- body phases of synthetic Hofstadter strips EU STREP EQuaM Alessio Celi Workshop on Quantum Science and Quantum Technologies ICTP Trieste 13/09/2017 Plan Integer Quantum Hall systems and Edge states

slide-1
SLIDE 1

Topological properties and many- body phases of synthetic Hofstadter strips

Alessio Celi

EU STREP EQuaM

Workshop on Quantum Science and Quantum Technologies – ICTP Trieste 13/09/2017

slide-2
SLIDE 2
  • Effect of the dimerization

Plan

  • Integer Quantum Hall systems and Edge states
  • Cold atom realizations: synthetic gauge field
  • Synthetic lattice (Extradimension)
  • Dimerized interacting ladder
  • Meissner/Vortex phase (in analogy to type II superconductors)
  • Prospects

Topology in narrow strips Reverse of chiral current (single particle) Commensurate-Incommensurate transition (strong interactions)

slide-3
SLIDE 3

Quantum Hall effect

1879: Classical Hall effect (consequence of Lorentz force)

  • E. Hall

1980: Quantum Hall effect: Electric conductivity quantized

  • K. Von Klitzing
slide-4
SLIDE 4

Integer Quantum Hall effect in a lattice

On square lattice simple formulation: Hofstadter model

IQH explained in terms of single particle physics (Landau level filling)

Quantization determined by topology of filled bands (1-Chern number) Edge states determined by spectrum of periodic system

Bulk-boundary correspondence (Topological insulator prototype)

slide-5
SLIDE 5

Integer Quantum Hall effect in a lattice

IQH explained in terms of single particle physics (Landau level filling)

Quantization determined by topology of filled bands (1-Chern number) On square lattice simple formulation: Hofstadter model Edge states determined by spectrum of periodic system

Bulk-boundary correspondence (Topological insulator prototype)

slide-6
SLIDE 6

Cold atoms in optical lattices as charged particles

V0

Hopping with phases

φ1 φ2 φ3 φ4

Synthetic Aharonov-Bohm effect φ = Σiφi = magnetic flux

Synthetic magnetic field for neutral atoms

slide-7
SLIDE 7

Cold atoms in optical lattices as charged particles

V0

Hopping with phases

φ1 φ2 φ3 φ4

Synthetic Aharonov-Bohm effect φ = Σiφi = magnetic flux

Synthetic magnetic field for neutral atoms

Several ways: here “Extradimension” + Raman laser = synthetic lattices

slide-8
SLIDE 8

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

In optical lattices 1D-3D Hubbard model by tuning optical potential And > 3D? In a lattice Dimensionality ≡ Connectivity

slide-9
SLIDE 9

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

And > 3D? In a lattice Dimensionality ≡ Connectivity

Hopping in D+1 hypercubic lattice as

In optical lattices 1D-3D Hubbard model by tuning optical potential

slide-10
SLIDE 10

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

And > 3D? In a lattice Dimensionality ≡ Connectivity

Hopping in D+1 hypercubic lattice as

In optical lattices 1D-3D Hubbard model by tuning optical potential

slide-11
SLIDE 11

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

And > 3D? In a lattice Dimensionality ≡ Connectivity

Hopping in D+1 hypercubic lattice as

In optical lattices 1D-3D Hubbard model by tuning optical potential

slide-12
SLIDE 12

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

And > 3D? In a lattice Dimensionality ≡ Connectivity

Coupled atomic states hopping in D-lattices

In optical lattices 1D-3D Hubbard model by tuning optical potential

slide-13
SLIDE 13

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

And > 3D? In a lattice Dimensionality ≡ Connectivity

Coupled atomic states hopping in D-lattices

Not only spin states Momentum states Trap modes…

In optical lattices 1D-3D Hubbard model by tuning optical potential

slide-14
SLIDE 14

Simulating an extra dimension

[Boada,AC,Latorre,Lewenstein, PRL 108, 133001 (2012)]

And > 3D? In a lattice Dimensionality ≡ Connectivity

Coupled atomic states hopping in D-lattices

Not only spin states Momentum states Trap modes… Not only atoms Cold molecules, Photonic crystal, Ring resonators…

In optical lattices 1D-3D Hubbard model by tuning optical potential

slide-15
SLIDE 15

φ φ φ φ φ φ φ

1d-lattice loaded e.g. with

87Rb (F=1, m=-1,0,1)

Synthetic gauge fields in synthetic dimension

[AC et al PRL 112 , 043001 (2014)] + Raman dressing

J

J’ Exp[i φ n]

X= n a Synthetic dim.

φ φ φ

Constant magnetic flux φ! Sharp Boundaries Edge currents (hard to get in real 2d lattice)

signal of Topological nature of quantum Hall (bulk-boundary correspondence)

slide-16
SLIDE 16

Synthetic gauge fields in synthetic dimension

[AC et al PRL 112 , 043001 (2014)] "Genuine" Edge states for small J’/J:

  • live in the gap,
  • have linear dispersion
  • have well defined spin

Spectrum

slide-17
SLIDE 17

Synthetic gauge fields in synthetic dimension

[AC et al PRL 112 , 043001 (2014)] Experimental Realizations:

Spectrum

"Genuine" Edge states for small J’/J:

  • live in the gap,
  • have linear dispersion
  • have well defined spin

I) Bosons: NIST Spielman group 87Rb [Science (2015)]

slide-18
SLIDE 18

Synthetic gauge fields in synthetic dimension

[AC et al PRL 112 , 043001 (2014)] Experimental Realizations:

II) Fermions: LENS Fallani group 173Yb [Science (2015)] I) Bosons: NIST Spielman group 87Rb [Science (2015)]

Spectrum

"Genuine" Edge states for small J’/J:

  • live in the gap,
  • have linear dispersion
  • have well defined spin
slide-19
SLIDE 19

Synthetic gauge fields in synthetic dimension

[AC et al PRL 112 , 043001 (2014)] Experimental Realizations:

II) Fermions: LENS Fallani group 173Yb [Science (2015)] I) Bosons: NIST Spielman group 87Rb [Science (2015)] Also with clock states (ladder) LENS: Livi et al. PRL 117, 220401 (2016) JILA: Kolkowitz et al. Nature 542 66 (2017)

Spectrum

"Genuine" Edge states for small J’/J:

  • live in the gap,
  • have linear dispersion
  • have well defined spin
slide-20
SLIDE 20

Topology in narrow strips

Narrow Hofstadter strips have edge states What about the “bulk”?

slide-21
SLIDE 21

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?

slide-22
SLIDE 22

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument

slide-23
SLIDE 23

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument Brillouin sketch

  • Large (periodic) system, a lowest band state well localized in y and spread in x
  • Apply a force along x
slide-24
SLIDE 24

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument

  • Large (periodic) system, a lowest band state well localized in y and spread in x
  • Apply a force along x

Brillouin sketch of semiclassical dynamics

slide-25
SLIDE 25

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument

  • Large (periodic) system, a lowest band state well localized in y and spread in x
  • Apply a force along x
  • After a Bloch oscillation observe the displacement

Brillouin sketch of semiclassical dynamics

Displacement in y due to anomalous velocity!

slide-26
SLIDE 26

Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument In formulae: semiclassical approach State easy to prepare if the coupling Wave packet:

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

slide-27
SLIDE 27

Pragmatic approach: measure transverse displacement to a force after a Bloch oscillation, Laughlin pump argument In formulae: semiclassical approach State easy to prepare if the coupling Wave packet: Applicable also to strips until we don’t reach the boundary…

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

slide-28
SLIDE 28

Scheme:

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

slide-29
SLIDE 29

Scheme: Results:

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

slide-30
SLIDE 30

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

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

slide-31
SLIDE 31

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)

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

slide-32
SLIDE 32

Higher possible for

Ex: “Better” than Fukui-Hatsugai-Suzuki algorithm J. Phys. Soc. Jpn. (2005)

Measuring Chern numbers in (narrow) Hofstadter strips, S.Mugel,….AC, SciPost 3, 012, (2017)

Narrow Hofstadter strips have also sym. prot. 1D topology [Barbarino et al., arXiv:1708:02929]

Robust to disorder (< gap) and typical harmonic confinement Interactions? Gap small (although may hold through adiabatic argument, see later)

slide-33
SLIDE 33

Synthetic lattices in interaction

  • Interesting route to interaction -> Fractional QH effect?!

No heating expected

  • Peculiarity: Interactions are naturally long range

in the synthetic dimension

  • Quasi 1D approach to 2D interesting both

theoretically & practically

slide-34
SLIDE 34

Synthetic lattices in interaction

  • Interesting route to interaction -> Fractional QH effect?!

No heating expected

  • Peculiarity: Interactions are naturally long range

in the synthetic dimension

  • Quasi 1D approach to 2D interesting both

theoretically & practically

Many studies: Meissner-vortex and commensurable incommensurable transitions, Fractional pumping, Laughlin like states, pseudo Majorana… [a lot here in Trieste!]

Here: effect of dimerization on synthetic Hofstadter ladder

slide-35
SLIDE 35

Meissner/Vortex phase in flux ladder

Weak interleg (Raman) coupling: 2 minima, Strong interleg (Raman) coupling: 1 minima,

[Orignac,Giamarchi, PRB 2001] Analogous to type II, also in presence interactions

φ φ φ φ φ

No interactions: real = synthetic ladder

Real ladder experiment [Atala et, Nature Phys. 2014]

slide-36
SLIDE 36

Meissner/Vortex phase in flux ladder

Weak interleg (Raman) coupling: 2 minima, Strong interleg (Raman) coupling: 1 minima,

[Orignac,Giamarchi, PRB 2001] Analogous to type II, also in presence interactions

φ φ φ φ φ

No interactions: real = synthetic ladder

Real ladder experiment [Atala et, Nature Phys. 2014]

Observables

slide-37
SLIDE 37

Meissner/Vortex phase in flux ladder

Weak interleg (Raman) coupling: 2 minima, Strong interleg (Raman) coupling: 1 minima,

[Orignac,Giamarchi, PRB 2001] Analogous to type II, also in presence interactions

φ φ φ φ φ

No interactions: real = synthetic ladder

Real ladder experiment [Atala et, Nature Phys. 2014]

Observables

slide-38
SLIDE 38

Meissner/Vortex phase in flux ladder

φ φ φ φ φ

Effect of interactions: suppress vortex phase

slide-39
SLIDE 39

Meissner/Vortex phase in flux ladder

φ φ φ φ φ

Effect of interactions: suppress vortex phase Real ladder: vortex phase survives in the hard-core limit for large

see [Petrescu, Le Hur,PRL 2013] [Piraud et al, PRB 2015]

more phases at U

slide-40
SLIDE 40

Meissner/Vortex phase in flux ladder

φ φ φ φ φ

Effect of interactions: suppress vortex phase Real ladder: vortex phase survives in the hard-core limit for large more phases at U Synthetic ladder: vortex phase disappears in the hard-core limit more phases at

see [Petrescu, Le Hur,PRL 2013] [Piraud et al, PRB 2015] ….

slide-41
SLIDE 41

Meissner/Vortex phase in flux ladder

φ φ φ φ φ

Effect of interactions: suppress vortex phase Real ladder: vortex phase survives in the hard-core limit for large more phases at U Synthetic ladder: vortex phase disappears in the hard-core limit more phases at

Idea: nucleate vortices by dimerizing the lattice (“easy” exp. handle)

see [Petrescu, Le Hur,PRL 2013] [Piraud et al, PRB 2015] ….

slide-42
SLIDE 42

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

φ φ φ φ φ

Effect of dimerization: new handle

slide-43
SLIDE 43

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: 4 bands

φ φ φ φ φ

Effect of dimerization: new handle

slide-44
SLIDE 44

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: 4 bands

Just band folding

φ φ φ φ φ

Effect of dimerization: new handle

slide-45
SLIDE 45

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: 4 bands

φ φ φ φ φ

Effect of dimerization: new handle

Just band folding

slide-46
SLIDE 46

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: 4 bands

φ φ φ φ φ

Effect of dimerization: new handle

Just band folding

slide-47
SLIDE 47

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: 4 bands

Bands deform & mix

φ φ φ φ φ

Effect of dimerization: new handle

slide-48
SLIDE 48

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: 4 bands Minima separate: dimerization enhances vortex phase!

φ φ φ φ φ

Effect of dimerization: new handle

Bands deform & mix

slide-49
SLIDE 49

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: Reverse of chiral current

φ φ φ φ φ

Effect of dimerization: new handle

slide-50
SLIDE 50

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

No interactions: Reverse of chiral current Current behavior confirms vortex enhancement!

φ φ φ φ φ

Effect of dimerization: new handle

slide-51
SLIDE 51

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

φ φ φ φ φ

Interactions: Effect of dimerization: new handle U 3 states per rung

slide-52
SLIDE 52

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

Interactions: 3 states per rung 9 states per plaquette

φ φ φ φ φ

Effect of dimerization: new handle U

slide-53
SLIDE 53

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

Interactions: 3 states per rung 9 states per plaquette 1 n=0, 4 n=1, 4 n=2

Spectrum plaquette

φ φ φ φ φ

Effect of dimerization: new handle U

slide-54
SLIDE 54

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

Interactions: 3 states per rung 9 states per plaquette 1 n=0, 4 n=1, 4 n=2

Spectrum plaquette

Plaquette in n=2 Band insulator

φ φ φ φ φ

Effect of dimerization: new handle U

slide-55
SLIDE 55

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

Interactions: 3 states per rung 9 states per plaquette 1 n=0, 4 n=1, 4 n=2

Spectrum plaquette

Plaquette in n=2 Band insulator Plaquette in n=1 Imprinted vortex

φ φ φ φ φ

Effect of dimerization: new handle U

slide-56
SLIDE 56

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

DMRG calculations confirm perturbative expectations

Ex.

slide-57
SLIDE 57

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

DMRG calculations confirm perturbative expectations

Ex.

slide-58
SLIDE 58

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

DMRG calculations confirm perturbative expectations

Ex.

Phase diagram through calculation of currents and structure factors

slide-59
SLIDE 59

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

Melted vortex Meissner charge density wave Meissner Ex.

slide-60
SLIDE 60

Vortex Nesting and Melting in synthetic ladders, E. Tirrito, R. Citro, M.Lewenstein, AC in progress

Melted vortex Meissner charge density wave Meissner Ex. Similar to attractive interleg interaction, [Orignac et al, PRB 96, 014518 (2017)]

Evidence of commensurate-incommensurate transition

slide-61
SLIDE 61

Further steps

  • No hard-core boson limit: bosons different fermions
  • Study the accessible experimental parameters

….. Hopefully many more

  • Search for “visible” Laughlin-like states in such regimes
  • cf. [Calvanese et al, PRX 7, 021033 (2017)],[Petrescu et al, PRB 96, 014524 (2017)]
slide-62
SLIDE 62

Summary

  • Synthetic edge state in synthetic Hofstadter strips
  • “Bulk topology” in synthetic Hofstadter strips
  • Effect of dimerization in synthetic Hofstadter ladder

w/o interactions

slide-63
SLIDE 63

Outlook

  • Better understanding decoupling argument

with interactions

  • Achieve different interaction patterns in

synthetic lattices than SU(N)

  • Comparison between dimerized ladders and 4-leg

strips

  • …
slide-64
SLIDE 64

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
slide-65
SLIDE 65

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
slide-66
SLIDE 66

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
  • G. Juzeliunas
  • P. Massignan
  • J. Ruseckas

I.B. Spielman

  • N. Goldman
slide-67
SLIDE 67

“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
slide-68
SLIDE 68

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
  • J. Rodriguez-Laguna
  • N. Goldman
  • G. Juzeliunas P. Massignan
  • J. Ruseckas

I.B. Spielman

  • C. Muschik

R.W. Chhajlany

slide-69
SLIDE 69

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
  • J. Rodriguez-Laguna
  • N. Goldman
  • G. Juzeliunas P. Massignan
  • J. Ruseckas

I.B. Spielman

  • C. Muschik

R.W. Chhajlany

  • S. Mugel
  • J. Asboth
  • C. Lobo
slide-70
SLIDE 70

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
  • J. Rodriguez-Laguna
  • N. Goldman
  • G. Juzeliunas P. Massignan
  • J. Ruseckas

I.B. Spielman

  • C. Muschik

R.W. Chhajlany

  • A. Dauphin
  • L. Tarruell
  • S. Mugel
  • J. Asboth
  • C. Lobo
slide-71
SLIDE 71

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
  • J. Rodriguez-Laguna
  • N. Goldman
  • G. Juzeliunas P. Massignan
  • J. Ruseckas

I.B. Spielman

  • C. Muschik

R.W. Chhajlany

  • S. Mugel
  • J. Asboth
  • C. Lobo
  • A. Dauphin
  • L. Tarruell
  • E. Tirrito
  • R. Citro
slide-72
SLIDE 72

“Extradimensional” collaborators

J.I. Latorre

  • O. Boada
  • M. Lewenstein
  • T. Grass
  • J. Rodriguez-Laguna
  • N. Goldman
  • G. Juzeliunas P. Massignan
  • J. Ruseckas

I.B. Spielman

  • C. Muschik

R.W. Chhajlany

  • S. Mugel
  • J. Asboth
  • C. Lobo
  • A. Dauphin
  • L. Tarruell
  • E. Tirrito
  • R. Citro

….