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Synthetic Creutz-Hubbard model: interacting topological insulators - - PowerPoint PPT Presentation

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Synthetic Creutz-Hubbard model: interacting topological insulators with ultracold atoms Matteo Rizzi Johannes Gutenberg Universitt Mainz J. Jnemann, et al., arXiv:1612.02996 -- accepted on PRX ICTP Trieste, 14 September 2017 Motivation

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Synthetic Creutz-Hubbard model: interacting topological insulators with ultracold atoms

Matteo Rizzi Johannes Gutenberg Universität Mainz ICTP Trieste, 14 September 2017

  • J. Jünemann, et al., arXiv:1612.02996 -- accepted on PRX
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ICTP Workshop 2017 Quantum Technologies Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms matteo.rizzi@uni-mainz.de

Motivation

  • Topological phases of matter: academic & practical interests !
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ICTP Workshop 2017 Quantum Technologies Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms matteo.rizzi@uni-mainz.de

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?)

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ICTP Workshop 2017 Quantum Technologies Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms matteo.rizzi@uni-mainz.de

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 !

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ICTP Workshop 2017 Quantum Technologies Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms matteo.rizzi@uni-mainz.de

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 !

  • Quantum info. driven numerics (i.e., Tensor Networks), too!
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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Imbalanced Creutz Ladder

Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Imbalanced Creutz Ladder

Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Imbalanced Creutz Ladder

Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character Bx Bz

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Imbalanced Creutz Ladder

Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character Bx Bz

  • M. Atala, et al., Nat. Phys. 9, 795 (2013).

Measurement of Zak phase via Ramsey interferometry in ultracold gases (SSH model)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Imbalanced Creutz Ladder

Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character Bx Bz

  • M. Atala, et al., Nat. Phys. 9, 795 (2013).

Measurement of Zak phase via Ramsey interferometry in ultracold gases (SSH model) Zeeman splitting + (undoubled) Dirac point

  • M. Creutz, PRL 83 2636 (1999)
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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Imbalanced Creutz Ladder

Spin-dep., complex, tunnelling + Spin-flipping, real, tunnelling Topological character Bx Bz

  • M. Atala, et al., Nat. Phys. 9, 795 (2013).

Measurement of Zak phase via Ramsey interferometry in ultracold gases (SSH model) Zeeman splitting + (undoubled) Dirac point

  • M. Creutz, PRL 83 2636 (1999)

Discrete chiral symmetry only! Class AIII

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Flat bands, AB cages & Edge States

π

a c

π π π

i/2 1/2 1/2

i/2

ε = −2˜ t ε = +2˜ t

i/2 −1/2 1/2

−i/2

εl = 0

εr = 0

1/ √ 2 i/ √ 2 1/ √ 2 −i/ √ 2

+i˜ t

−˜ t −˜ t

+i˜ t +i˜ t +i˜ t

−˜ t −˜ t

b

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, 220510(R) (2013) Takayoshi, Katsura, Watanabe, & Aoki, PRA 88, 063613 (2013) Huber & Altman, PRB 82, 184502 (2010) Tovmasyan, Peotta, Törmä̈, & Huber, arXiv:1608.00976 Sticlet, Seabra, Pollmann, & Cayssol, PRB 89, 115430 (2014)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Flat bands, AB cages & Edge States

π

a c

π π π

i/2 1/2 1/2

i/2

ε = −2˜ t ε = +2˜ t

i/2 −1/2 1/2

−i/2

εl = 0

εr = 0

1/ √ 2 i/ √ 2 1/ √ 2 −i/ √ 2

+i˜ t

−˜ t −˜ t

+i˜ t +i˜ t +i˜ t

−˜ t −˜ t

b

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, 220510(R) (2013) Takayoshi, Katsura, Watanabe, & Aoki, PRA 88, 063613 (2013) Huber & Altman, PRB 82, 184502 (2010) Tovmasyan, Peotta, Törmä̈, & Huber, arXiv:1608.00976 Sticlet, Seabra, Pollmann, & Cayssol, PRB 89, 115430 (2014)

Bragg techniques to measure edge states in ultracold cold atoms (with steep enough potential)

Goldman,Beugnon, Gerbier, PRL 108, 255303 (2012).

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Flat bands, AB cages & Edge States

π

a c

π π π

i/2 1/2 1/2

i/2

ε = −2˜ t ε = +2˜ t

i/2 −1/2 1/2

−i/2

εl = 0

εr = 0

1/ √ 2 i/ √ 2 1/ √ 2 −i/ √ 2

+i˜ t

−˜ t −˜ t

+i˜ t +i˜ t +i˜ t

−˜ t −˜ t

b

?

?

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 ... ==>

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Flat bands, AB cages & Edge States

π

a c

π π π

i/2 1/2 1/2

i/2

ε = −2˜ t ε = +2˜ t

i/2 −1/2 1/2

−i/2

εl = 0

εr = 0

1/ √ 2 i/ √ 2 1/ √ 2 −i/ √ 2

+i˜ t

−˜ t −˜ t

+i˜ t +i˜ t +i˜ t

−˜ t −˜ t

b

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 ... ==>

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Flat bands, AB cages & Edge States

π

a c

π π π

i/2 1/2 1/2

i/2

ε = −2˜ t ε = +2˜ t

i/2 −1/2 1/2

−i/2

εl = 0

εr = 0

1/ √ 2 i/ √ 2 1/ √ 2 −i/ √ 2

+i˜ t

−˜ t −˜ t

+i˜ t +i˜ t +i˜ t

−˜ t −˜ t

b

  • 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

Outline of the attack plan:

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies ?

Weak interactions: gap behaviour

Indicator 1: compressibility gap vs. degeneracy split

1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4 δ, ∆ ∆ε/˜ t

  • PM:

TI:

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak interactions: coupled Ising chains

? Indicator 2: density imbalance between ladder legs

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak interactions: coupled Ising chains

? Indicator 2: density imbalance between ladder legs Bogolubov + Jordan Wigner trafos

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak interactions: coupled Ising chains

? Indicator 2: density imbalance between ladder legs self-consistent mean-field two simultaneous Ising transitions (c=1)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Strong interactions: orbital Ising model

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 ⟨T y

i ⟩

∆ε/˜ t

? Gutzwiller projector Super-exchange coupling single Ising critical line (c=1/2)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Strong interactions: orbital Ising model

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 ⟨T y

i ⟩

∆ε/˜ t

? Gutzwiller projector Super-exchange coupling single Ising critical line (c=1/2)

  • FM ≠ TI

another transition at low imbalance !

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

AB-cages: exotic Hubbard

NNN interactions + dens. dep. tunnelling without dipolar atoms or strange schemes !!

ε+

ε−

εl = εr

timb −timb

Vv/4 Vv/2 Td Vv/4 Vv/4

−timb √ 2

−timb timb

˜ J ˜ J

?

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

AB-cages: exotic Hubbard

NNN interactions + dens. dep. tunnelling without dipolar atoms or strange schemes !!

ε+

ε−

εl = εr

timb −timb

Vv/4 Vv/2 Td Vv/4 Vv/4

−timb √ 2

−timb timb

˜ J ˜ J

CRUCIAL: we keep both bands! no projection!

?

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

AB-cages: exotic Hubbard

NNN interactions + dens. dep. tunnelling without dipolar atoms or strange schemes !!

ε+

ε−

εl = εr

timb −timb

Vv/4 Vv/2 Td Vv/4 Vv/4

−timb √ 2

−timb timb

˜ J ˜ J

leads to bulk mediated edge-edge interaction à la Fano-Anderson ... CRUCIAL: we keep both bands! no projection!

?

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak imbalance: yet another Ising

single Ising critical line (c=1/2) … restrict to singly occupied AB-c manifold ...

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak imbalance: yet another Ising

single Ising critical line (c=1/2) … restrict to singly occupied AB-c manifold ... Jordan-Wigner gives “effective” bands dual to the original ones …

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak imbalance: yet another Ising

single Ising critical line (c=1/2) … restrict to singly occupied AB-c manifold ... … what happens then to the second Ising? Jordan-Wigner gives “effective” bands dual to the original ones …

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Weak imbalance: yet another Ising

single Ising critical line (c=1/2) … restrict to singly occupied AB-c manifold ... 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!)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Matrix Product States

Generic description of a many-body Hilbert space is exponentially expensive numbers

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Matrix Product States

Generic description of a many-body Hilbert space is exponentially expensive numbers

generic state

Physically accessible states Product states Eisert, Cramer, Plenio RMP 82, 277 (’10)

Area-law for entanglement entropy

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Matrix Product States

Generic description of a many-body Hilbert space is exponentially expensive numbers

generic state

Physically accessible states Product states Eisert, Cramer, Plenio RMP 82, 277 (’10)

Area-law for entanglement entropy

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Matrix Product States

Generic description of a many-body Hilbert space is exponentially expensive numbers

generic state

Economic description by “Tensor Networks”: (variational RG schemes, DMRG) numbers

Schollwock, Ann. Phys. 326, 96 (2011) Physically accessible states Product states Eisert, Cramer, Plenio RMP 82, 277 (’10)

Area-law for entanglement entropy

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Generic description of a many-body Hilbert space is exponentially expensive numbers Economic description by “Tensor Networks”: (variational RG schemes, DMRG) numbers

Schollwock, Ann. Phys. 326, 96 (2011)

MPS PEPS MERA TTN

… plenty of different decompositions in tensor products:

Matrix Product States

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Entanglement signatures (MPS)

c=1/2 c=1/2 c=1 0.2 0.4 0.6 0.8 1 1.2 1 10 100 S(l)

2N π sin

πl

N

  • Vidal, Latorre, Rico & Kitaev,

PRL 90, 227902 (2003); Calabrese & Cardy, J. Stat. Mech. P06002 (2004)

entanglement entropy CFT central charge ==>

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Entanglement signatures (MPS)

c=1/2 c=1/2 c=1 0.2 0.4 0.6 0.8 1 1.2 1 10 100 S(l)

2N π sin

πl

N

  • Vidal, Latorre, Rico & Kitaev,

PRL 90, 227902 (2003); Calabrese & Cardy, J. Stat. Mech. P06002 (2004)

entanglement entropy CFT central charge ==> degeneracy pattern in entanglement spectrum ! MPS gives out more:

  • H. Li and F. D. M. Haldane, PRL 101 010504 (2015)
  • F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa PRB 81 064439 (2010)
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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Experimental ingredients

state indep. gradient + intensity modulated OL

Holthaus, PRL 69, 351 (1992) Jaksch & Zoller, NJP 5, 56 (2003) Gerbier & Dalibard, NJP 12, 033007 (2010) Eckardt, RMP 89, 011004 (2017)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Experimental ingredients

Ma, et al. (Greiner), PRL 107, 095301 (2011) Chen, et al. (Bloch), PRL 107, 210405 (2011)

+ Interactions away from driving-induced resonances state indep. gradient + intensity modulated OL

Holthaus, PRL 69, 351 (1992) Jaksch & Zoller, NJP 5, 56 (2003) Gerbier & Dalibard, NJP 12, 033007 (2010) Eckardt, RMP 89, 011004 (2017)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

a b c

nS 1

2

∆ nP 1

2

nP 3

2

∆ ε↑ ε↓ ε↑ ε↓

ω1 ω2

ω3

ω1

ω1 −ω3 = (ε↑ −ε↓)+∆ ω1 −ω2 = −(ε↑ −ε↓)+∆

+ Raman assisted tunnelling

Aidelsburger, et al. (Bloch), PRL 111, 185301 (2013) Miyake, et al. (Ketterle), PRL 111, 185302 (2013)

Experimental ingredients

Ma, et al. (Greiner), PRL 107, 095301 (2011) Chen, et al. (Bloch), PRL 107, 210405 (2011)

+ Interactions away from driving-induced resonances state indep. gradient + intensity modulated OL

Holthaus, PRL 69, 351 (1992) Jaksch & Zoller, NJP 5, 56 (2003) Gerbier & Dalibard, NJP 12, 033007 (2010) Eckardt, RMP 89, 011004 (2017)

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Experimental ingredients

  • L. Mazza, A. Bermudez, MR, et al., PRL 105, 190404 (‘10), NJP 14 015007 (‘12)

MR, PoS - SISSA 193, 036 (2014)

Alternative: Superlattice “toolbox” for topological insulators !

a. c. d. e. f. b.

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

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 gauge symmetry, etc.)

  • A. Bermudez
  • M. Lewenstein

S.J. Ran

  • A. Piga
  • J. Jünemann
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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions
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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

x

y

L L

m N

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ −t −t ei 2πφx x y L

FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks

  • M. Gerster, et al. (MR)

arXiv:1705.06515

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matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

Pair Luttinger Liquid in Bosonic Flat Bands

  • C. Cartwright, MR, G. deChiara,

soon on arXiv

x

y

L L

m N

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ −t −t ei 2πφx x y L

FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks

  • M. Gerster, et al. (MR)

arXiv:1705.06515

slide-47
SLIDE 47

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

Pair Luttinger Liquid in Bosonic Flat Bands

  • C. Cartwright, MR, G. deChiara,

soon on arXiv

x

y

L L

m N

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ −t −t ei 2πφx x y L

FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks

  • M. Gerster, et al. (MR)

arXiv:1705.06515

slide-48
SLIDE 48

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

slide-49
SLIDE 49

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions
slide-50
SLIDE 50

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

slide-51
SLIDE 51

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

x

y

L L

m N

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ −t −t ei 2πφx x y L

FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks

  • M. Gerster, et al. (MR)

arXiv:1705.06515

slide-52
SLIDE 52

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

Pair Luttinger Liquid in Bosonic Flat Bands

  • C. Cartwright, MR, G. deChiara,

soon on arXiv

x

y

L L

m N

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ −t −t ei 2πφx x y L

FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks

  • M. Gerster, et al. (MR)

arXiv:1705.06515

slide-53
SLIDE 53

matteo.rizzi@uni-mainz.de arXiv:1612.02996 Synthetic Creutz-Hubbard model: interacting topol. insul. with ultracold atoms ICTP Workshop 2017 Quantum Technologies

Other related works

  • M. Bischoff, J. Jünemann, M. Polini, MR,

arXiv:1706.02679

Tunability of Drude Weight

  • f 1D Dirac-Weyl Fermions

$\nu = 1/2$ resonance in chiral fermionic ladders

  • A. Haller, MR, M. Burrello, arXiv:1707.05715

V |"i |#i φ tei φ

2

tei φ

2

Ω Ω U φ = 4kF Ω

|"i |#i

k E

2kF 2kF

h O2,−1 O2,+1 µ(ρ0) jc

Pair Luttinger Liquid in Bosonic Flat Bands

  • C. Cartwright, MR, G. deChiara,

soon on arXiv

x

y

L L

m N

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ −t −t ei 2πφx x y L

FQHE in hard-core bosons: Harper-Hofstadter model via Tensor Networks

  • M. Gerster, et al. (MR)

arXiv:1705.06515

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