Many-body chiral edge currents and sliding phases of atomic spinwaves in momentum-space lattice

Da-Wei Wang; Han Cai; Jianmin Yuan; Lin Li; Weibin Li; Yongqiang Li

REVIEW 2 major objections 5 minor 91 references

Many-body chiral edge currents and sliding phases of atomic spinwaves in momentum-space lattice

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A momentum-space ladder of atomic spinwaves can host anti-chiral edge currents.

desk verdict Solid anti-chiral edge current result, but the sliding-phase section is likely an artifact of the flat-interaction limit; the paper needs a referee to catch it. read the letter →

arxiv 1909.00750 v4 pith:TLID4OU3 submitted 2019-09-02 cond-mat.quant-gas cond-mat.str-elphysics.atom-ph

Yongqiang Li , Han Cai , Da-wei Wang , Lin Li , Jianmin Yuan , Weibin Li This is my paper · ORCID

classification cond-mat.quant-gascond-mat.str-elphysics.atom-ph

keywords momentum-spacelatticeatomicspinwavesRydbergdressingchiraledgecurrentsanti-chiralslidingphasesBose-Hubbardladdersyntheticmagneticfield

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a cold-atom scheme in which collective excitations (spinwaves) of two long-lived hyperfine states form a two-leg ladder in momentum space, with laser-induced complex hoppings acting as a staggered synthetic magnetic field and Rydberg dressing supplying state-dependent long-range interactions. The central claim is that the many-body ground state of this ladder supports an anti-chiral edge-current phase: when intra-leg hopping dominates inter-leg hopping, the currents on the two legs flow in the same direction, something not found in standard real-space ladder systems. The paper further claims that dressing both legs to Rydberg states makes the interaction effectively all-to-all in momentum space, producing massively degenerate sliding insulating phases and, once hopping is added, sliding superfluid and supersolid phases. If these phases survive in experiment, the scheme would give access to many-body chiral transport and exotic correlated phases in a long-coherence-time atomic platform, avoiding the motional heating and short lifetimes that hamper real-space cold-atom topological experiments.

What carries the argument

The load-bearing object is the two-leg momentum-space Bose-Hubbard Hamiltonian $H = H_c + H_d - \sum_{i,\sigma} \mu_\sigma n_{i\sigma}$, built from spinwave creation operators $a^\dagger_i$ (momentum $2i k_c$) and $b^\dagger_i$ (momentum $(2i-1)k_c$). The coupling $H_c$ contains complex nearest-neighbor hoppings $h_o e^{i\varphi}$ along the legs and $h_r$ between the legs; the staggered synthetic flux ($\varphi$ and $\pi-\varphi$ in neighboring plaquettes) is what breaks the usual current pattern. The interaction $H_d$ is the momentum-space image of the Rydberg soft-core potential $V_\sigma(x) = C_\sigma/(r_\sigma^6 + |x|^6)$, whose Fourier transform $\tilde{V}_\sigma(k)$ is sharply peaked at $k=0$ because $r_c \gg \lambda$; that flatness is what turns the interaction into an effectively all-to-all coupling. The paper solves the noninteracting band structure exactly and uses a bosonic dynamical mean-field theory with an Anderson-impurity solver to obtain the ground-state phases and the bond currents $J_{\sigma\sigma'} = -2\,\mathrm{Im}\langle \sigma^\dagger_i \sigma'_{j}\rangle$.

What would settle it

Check whether the momentum-space interaction $\tilde{V}(k)$ is flat over the occupied band: for the proposed $^{87}$Rb parameters ($r_c \approx 4.5$ $\mu$m, $\lambda \approx 785$ nm), compute $\tilde{V}(k)/\tilde{V}(0)$ at the band minima where particles condense; if it deviates from 1 by more than the hopping amplitudes, the sliding phases' degeneracy-energy argument $E = V(0)N(N-1)/2$ fails and the predicted sliding insulator and supersolid would not survive. Alternatively, measure the real-space density distributions in the CSFp phase: the anti-chiral claim predicts the A- and B-leg density peaks separated by about $\pi$ in $2kx$, with both leg currents having the same sign; observing counter-propagating currents would refute it.

Watch

Extended reading notes

Core claim

The paper establishes, through exact solution of the noninteracting ladder and bosonic dynamical mean-field calculations of the interacting model, that a two-leg momentum-space lattice with staggered flux $\varphi$ and $\pi-\varphi$ on adjacent plaquettes has three chiral superfluid ground states. In the CSFm (Meissner-like) and CSFv (vortex-like) phases the two leg currents oppose each other, as in real-space ladders. In the CSFp phase, reached when $h_o/h_r$ is large, the staggered flux makes A-leg and B-leg atoms condense into different band minima, so the leg currents satisfy $J_{AA}\times J_{BB} > 0$: both edges carry current in the same direction. The paper calls this anti-chiral edge current and argues it is a new many-body phase absent from real-space ladder studies. With both legs Rydberg-dressed, the momentum-space interaction is nearly constant over the occupied band, so the ground-state energy depends only on particle numbers per leg, not on where the particles sit; this yields a sliding insulator at zero hopping, a Devil's-staircase filling structure, and sliding superfluid and supersolid phases at finite hopping.

Load-bearing premise

The sliding-phase results rest on the assumption that the Rydberg-dressed interaction is effectively the same for every pair of occupied momentum sites, so that $\tilde{V}_\sigma(k) \approx \tilde{V}_\sigma(0)$ over the occupied band; if the interaction is noticeably momentum-dependent at the experimentally occupied momenta, the massive degeneracy that defines the sliding insulator is lifted and the sliding phases become ordinary correlated phases.

Editorial extensions

If this is right

The anti-chiral CSFp phase should be observable as a many-body ground state with both leg currents pointing in the same direction, accessible through time-of-flight imaging of the spinwave density in real space.
Single-leg Rydberg dressing yields a momentum-space blockade: strong interaction suppresses double occupation in the dressed leg and can amplify anti-chiral currents for flux $\varphi = 5\pi/4$.
Double Rydberg dressing produces a sliding insulator whose many-body ground state is massively degenerate, with a Devil's-staircase filling as the chemical potential changes, and a transition to sliding superfluid and supersolid phases at finite hopping.
In the presence of flux the sliding phases carry nonzero local currents, connecting sliding order to chiral response.
With the proposed $^{87}$Rb parameters, the interaction strength $V \approx 158$ kHz and soft-core radius $r_c \approx 4.5$ $\mu$m place the predicted phases within currently accessible coherent times, with a dressed-state lifetime around 2.9 ms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same staggered-flux mechanism might produce anti-chiral edge currents in real-space synthetic ladders with long-range interactions, not just in momentum-space spinwaves; a direct test would be to look for $J_{AA}\times J_{BB} > 0$ in a real-space zigzag ladder with alternating plaquette fluxes.
The effectively all-to-all momentum-space interaction generated by $r_c \gg \lambda$ could be exploited as a tunable platform for infinite-range boson or spin models, with dressing parameters controlling the range and sign of the coupling.
The predicted sliding phases should leave a distinctive experimental signature: at fixed parameters, repeated preparations would show random, run-dependent density distributions in momentum space, unlike the reproducible distributions of ordinary superfluids.
If the momentum dependence of $\tilde{V}_\sigma(k)$ is not negligible at the occupied sites, the sliding phases should cross over to pinned density-wave or supersolid phases; tuning $r_c/\lambda$ would map this crossover.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper has one solid new result and one unsubstantiated one. The solid part is the prediction of an anti-chiral edge current (CSF_p) in a staggered-flux two-leg momentum-space ladder with single Rydberg dressing. The noninteracting band analysis is clean, and the perturbative expansion in the SM for ho/hr >> 1 shows how asymmetric condensation produces same-direction leg currents. This is genuinely new relative to the real-space ladder literature.

The soft spot is the sliding-phase section. The authors assume a nearly flat Rydberg interaction V~(k) in momentum space, so the interaction energy depends only on leg populations, not on the distribution. In that limit the interacting term commutes with the intra-leg hopping, and the exact ground state for any nonzero hopping is the noninteracting Bloch condensate; the degeneracy is split at first order in h0. The paper instead argues that the relative bandwidth vanishes and that BDMFT finds a sliding insulator. That argument compares hN to VN^2, but the relevant level spacing within the fixed-N manifold is order h. The BDMFT energy histograms in Fig. S4 show many random density configurations at nearly equal energy—evidence that the solver is sampling a flat landscape, not equilibrating. So the sliding insulator/supersolid claim is likely an artifact of the flat-interaction assumption plus a metastable solver.

Other soft spots are smaller: BDMFT is approximate for a z=4 ladder, and no code or unbiased benchmark is provided. But the anti-chiral phase does not rest on those approximations. The citation pattern looks appropriate; the momentum-space lattice and Rydberg dressing references are the right ones.

Verdict: this deserves a serious referee, because the first half is a real contribution and the second half needs to be caught. I would not accept the paper as is. If the sliding-phase claim is removed or reframed as the trivial flat-limit condensate, the paper could be publishable on the strength of the CSF_p phase. Bring it to the reading group if you want to discuss how easy it is to fool BDMFT on a flat energy surface.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes a momentum-space two-leg Bose-Hubbard model for atomic spinwaves, with staggered synthetic flux from laser coupling and state-dependent Rydberg-dressed interactions. For the noninteracting ladder, the authors classify three current-carrying superfluid phases, including a CSF_p phase with co-propagating (anti-chiral) edge currents on the two legs. They then use bosonic dynamical mean-field theory to map the interacting phase diagram for single and double Rydberg dressing, reporting interaction-stabilized chiral currents, a momentum-space excitation blockade, and, in the double-dressing case, sliding insulating and sliding superfluid/supersolid phases. The Supplemental Material provides a perturbative derivation of the anti-chiral current in the h0/hr ≫ 1 limit, exact-diagonalization checks of energy gaps for a single chain, and a detailed experimental proposal.

Significance. If the anti-chiral CSF_p claim holds, it is a genuinely new many-body current phase that has no direct real-space ladder analogue, and the paper offers a concrete experimental route to observe it. The noninteracting band-structure analysis and the perturbative derivation in the Supplemental Material are clean, explicit, and parameter-free. However, the sliding-phase pillar is built directly on the flat-interaction degeneracy; for the model as written, the claimed sliding insulator is not a ground state once intra-leg hopping is switched on. The double-dressing section therefore does not currently support its advertised conclusions, although the anti-chiral current and blockade results remain significant.

major comments (2)

[Fig. 1(c) and SM experimental parameters] The sliding-phase prediction is conditioned on the assumption that \tilde V_σ(k) is constant over the occupied band. The text states that \tilde V_σ(k) decays rapidly because rc≫λ, and the experimental section quotes rc=4.54 μm versus λ=785 nm, making \tilde V(k_c)/\tilde V(0) exponentially small. With any finite momentum dependence the massive degeneracy is lifted, and the phases labelled SI and SSF become ordinary correlated phases; no calculation is presented for a small but nonzero \tilde V(k_c)/\tilde V(0). Because the degeneracy is the entire mechanism for the claimed sliding behavior, the authors must either demonstrate robustness against a finite momentum dependence of the dressed interaction or explicitly restrict the claim to the exact flat limit with h0=0.
[Fig. 2 and SM BDMFT section] The interacting phase diagrams in Fig. 2 and Fig. 4 rely on BDMFT, an approximation that the SM itself notes is exact only in infinite dimensions and uncontrolled for a two-leg ladder with z=4 and long-range interactions. The anti-chiral CSF_p claim has independent support from the exact noninteracting solution and from the perturbative expansion in the SM, but the quantitative interacting phase boundaries, including the discontinuous transition in Fig. 2(c) and the SI-SSF transition in Fig. 4(e), are not benchmarked against exact diagonalization or quantum Monte Carlo for this specific model. A direct ED check on small ladders, at least along one representative cut, is needed before these phase boundaries can be regarded as reliable.

minor comments (5)

[Throughout] There are several typographical errors: 'Interdisciplanery' in the affiliation line, 'superfuid' in the Fig. 4 caption, 'sliding superﬂuity phase' in the text, and 'continues CSFp-CSFm' in the Fig. 2 caption.
[Main text, definition of μσ] The sentence 'with niσ=σ†_iσ_i and μσ to be the atomic density at site (momentum) i and chemical potential in state |σ⟩' is garbled and should be rewritten to state that μσ is the chemical potential for species σ.
[SM Eq. (S9)] Equation (S9) renders as 'G(0)^{-1}(τ−τ′)≡−' followed by a matrix with an equation number inserted mid-expression; the display should be corrected.
[Fig. 2(c)-(e)] The panels showing currents lack a legend identifying J_AA, J_BB, and J_AB; since the sign of J_AA×J_BB is central to the phase classification, the curves should be labelled directly.
[Notation for hopping amplitudes] The main text uses both 'h0' and 'ho' for the intra-leg hopping amplitude; the notation should be harmonized throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: anti-chiral currents are derived from the stated Hamiltonian, and the sliding-phase degeneracy is a transparent consequence of the explicitly assumed flat-interaction limit.

full rationale

The paper's central new result, the anti-chiral edge current (CSF_p), is derived from the noninteracting ladder Hamiltonian H_c and a perturbative expansion in the Supplemental Material (SM section 'CSF_p PHASE WITHOUT TWO-BODY INTERACTIONS'), not from any fitted parameter or self-citation. The many-body BDMFT calculations are used only to show stability of these phases with interactions, and the method is benchmarked against an external quantum Monte Carlo study [68]. The sliding phases are introduced in the section 'Sliding phases when both chains are laser dressed to Rydberg states', where the paper explicitly states that for hr=ho=0 the energy 'depends on the total number of particles but not their distributions in the momentum-space ladder, i.e. changing their locations in the corresponding leg costs no energy.' This is a direct and openly acknowledged consequence of the assumed flat interaction limit, Vtilde_sigma(0) dominating for rc >> lambda; the paper does not conceal this input or claim it was fitted from data. Calling the resulting massively degenerate state a sliding insulator is a naming/interpretation choice, and the finite-hopping SI/SSF boundaries are numerical predictions from the stated model, but they do not reduce by construction to a hidden fit or to an unverified self-citation. The citations to prior momentum-space lattice work [49,50] establish the mapping and the synthetic field, but the paper's own equations and numerical solver generate the phase diagrams. No enumerated circularity pattern (self-definition, fitted input called prediction, load-bearing self-citation, author-imported uniqueness, ansatz smuggled via citation, or renaming of a known result) is exhibited, so the derivation chain is self-contained as presented.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claims depend on several hand-chosen model ingredients rather than fitted data: the flat interaction limit, the hopping amplitudes, the flux, and the low filling factor. The bosonic mapping and BDMFT approximation are additional unproven assumptions. No new physical particles or forces are introduced.

free parameters (6)

Rydberg soft-core range rc = 4.5 to 4.54 micrometers
Sets the flatness of the momentum-space interaction; the sliding phases require rc much larger than the laser wavelength so that all momentum sites couple equally.
Interaction energy scale V = Vb(0) = 158.2 kHz in the example
Used as the energy unit and controls the relative strength of interaction versus hopping; derived from the dressing parameters.
Intra-leg hopping amplitude ho = 0.1 V, 0.2 V, etc.
Varied to produce the CSF_m, CSF_v, and CSF_p phases; the anti-chiral phase requires ho/hr much larger than 1.
Inter-leg hopping amplitude hr = varied from 0 to about 0.3 V
Controls the band minima and the phase transitions between Meissner-like, vortex-like, and anti-chiral phases.
Synthetic flux phi = pi/4, 5pi/4, pi/2, and 0
The staggered flux is essential for the anti-chiral current; the chosen values probe different chiral symmetry sectors.
Filling factor Ntot/Llat = 0.125 for phase diagrams
Low filling supports the bosonic spinwave description and is used in the BDMFT phase diagrams.

assumptions (4)

domain assumption Collective spinwave operators a_l and b_l are bosonic when the excitation number is much smaller than the atom number.
Used to map the many-atom spinwave Hamiltonian to the Bose-Hubbard model; the Supplemental Material states the transformation is valid for many excitations if the excitation number is much less than the atom number.
domain assumption Rydberg dressing produces a soft-core two-body potential V_sigma(x) = C/(r_sigma^6 + |x|^6), and its momentum-space Fourier transform is flat over the occupied low-k states because rc >> lambda.
Justifies V(k) approximately equal to V(0) so the interaction energy depends only on total particle numbers; this is the mechanism behind the sliding phases, stated near Fig. 1(c).
domain assumption Bosonic dynamical mean-field theory with z = 4 is a quantitatively reliable approximation for the ladder ground states.
BDMFT is exact in infinite dimensions and approximate on a finite-coordination ladder; the paper relies on it for the many-body phase diagrams in Figs. 2 and 4.
domain assumption The tight-binding ladder Hamiltonian H_c with complex hoppings exactly describes the laser-induced couplings, including the synthetic magnetic field.
Borrowed from the prior momentum-space lattice construction in Refs. [49,50]; it underlies all band-structure and current calculations.

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Pith. "Pith review of Many-body chiral edge currents and sliding phases of atomic spinwaves in momentum-space lattice." pith.science (2026). https://pith.science/paper/TLID4OU3

@misc{pith2026190900750,
  author       = {Pith},
  title        = {Pith review of: Many-body chiral edge currents and sliding phases of atomic spinwaves in momentum-space lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TLID4OU3}},
  note         = {Machine review of arXiv:1909.00750}
}

read the original abstract

Collective excitations (spinwaves) of long-lived atomic hyperfine states can be synthesized into a Bose-Hubbard model in momentum space. We explore many-body ground states and dynamics of a two-leg momentum-space lattice formed by two coupled hyperfine states. Essential ingredients of this setting are a staggered artificial magnetic field engineered by lasers that couple the spinwave states, and a state-dependent long-range interaction, which is induced by laser-dressing a hyperfine state to a Rydberg state. The Rydberg dressed two-body interaction gives rise to a state-dependent blockade in momentum space, and can amplify staggered flux induced anti-chiral edge currents in the many-body ground state in the presence of magnetic flux. When the Rydberg dressing is applied to both hyperfine states, exotic sliding insulating and superfluid/supersolid phases emerge. Due to the Rydberg dressed long-range interaction, spinwaves slide along a leg of the momentum-space lattice without costing energy. Our study paves a route to the quantum simulation of topological phases and exotic dynamics with interacting spinwaves of atomic hyperfine states in momentum-space lattice.

Figures

Figures reproduced from arXiv: 1909.00750 by the authors.

**Figure 2.** (a) and (b)]. Actually, the two-body interaction breaks the chiral symmetry [CH(φ)C −1 = H(π + φ)]. The broken symmetry is most apparent in the CSFp phase, where the phase region shrinks when φ = π/4 [ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** (a), we illustrate this phase with an example of five bosons along each leg. We attribute this phase as a sliding insulating phase (SI), analogous to long-sought sliding superfluid phase [71–76]. The underlying physics of the emergent SI studied here is a result of momentumspace Rydberg coupling (all sites equally coupled) from the long-range real-space Rydberg interaction. Varying the chemical potential, the groun… view at source ↗

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Works this paper leans on

91 extracted references · 58 canonical work pages

[1]

ocTLVc574NOAe/DoMycp9A9e/3g=

The current ¯JAA≈ 2h0 Llat ∑ j sin(−∆k1)n(a) j ≈ 2h0 Llat ∑ j|∆k1|n(a) j > 0. Similarly, we ﬁnd that ¯JBB > 0. Here ¯JAA > ¯JBB because the state |b⟩ is weakly occupied in the ground state, due to small hr. This explains the results shown in the main text. SLIDING PHASES IN THE STRONGL Y INTERACTING REGIME 2 4 6 8 Llat 1 2 3 4 5 6 7 8 9Energy gap EN-EN-1 ...
[2]

K. V. Klitzing, G. Dorda, and M. Pepper, Physical Re- view Letters 45, 494 (1980)

1980
[3]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)

1982
[4]

M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010)

2010
[5]

Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015)

T. Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015)

2015
[6]

Sinova, S

J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Reviews of Modern Physics 87, 1213 (2015)

2015
[7]

T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, Reviews of Modern Physics 89, 025005 (2017)

2017
[8]

Bloch, J

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

2008

Show all 91 references

[9]

Lewenstein, A

M. Lewenstein, A. Sanpera, and V. Ahuﬁnger, Ultracold Atoms in Optical Lattices: Simulating quantum many- body systems (Oxford University Press, 2012)

2012
[10]

S. L. Bromley, S. Kolkowitz, T. Bothwell, D. Kedar, A. Safavi-Naini, M. L. Wall, C. Salomon, A. M. Rey, and J. Ye, Nature Physics 14, 399 (2018)

2018
[11]

Goldman, J

N. Goldman, J. C. Budich, and P. Zoller, Nature Physics 12, 639 (2016)

2016
[12]

Dalibard, F

J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. ¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011)

2011
[13]

Goldman, G

N. Goldman, G. Juzeli¯ unas, P.¨Ohberg, and I. B. Spiel- 15 man, Reports on Progress in Physics 77, 126401 (2014)

2014
[14]

N. R. Cooper, J. Dalibard, and I. B. Spielman, Rev. Mod. Phys. 91, 015005 (2019)

2019
[15]

Y.-J. Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa, J. V. Porto, and I. B. Spielman, Nature 462, 628 (2009)

2009
[16]

Y. Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa, W. D. Phillips, J. V. Porto, and I. B. Spielman, Nature Physics 7, 531 (2010)

2010
[17]

Mancini, G

M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, and M. Dal- monte, Science 349, 1510 (2015)

2015
[18]

L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, Phys. Rev. Lett. 117, 220401 (2016)

2016
[19]

Atala, M

M. Atala, M. Aidelsburger, M. Lohse, J. Barreiro, B. Paredes, and I. Bloch, Nature Physics 10, 588 (2014)

2014
[20]

B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Science 349, 1514 (2015)

2015
[21]

J. H. Kang, J. H. Han, and Y. Shin, Phys. Rev. Lett. 121, 150403 (2018)

2018
[22]

Aidelsburger, M

M. Aidelsburger, M. Atala, S. Nascimb` ene, S. Trotzky, Y.-A. Chen, and I. Bloch, Physical Review Letters 107, 487 (2011)

2011
[23]

Struck, C

J. Struck, C. ¨Olschl¨ ager, M. Weinberg, P. Hauke, J. Si- monet, A. Eckardt, M. Lewenstein, K. Sengstock, and P. Windpassinger, Phys. Rev. Lett. 108, 225304 (2012)

2012
[24]

Aidelsburger, M

M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 185301 (2013)

2013
[25]

Hirokazu, G

M. Hirokazu, G. A. Siviloglou, C. J. Kennedy, B. William Cody, and K. Wolfgang, Physical Review Letters 111, 185302 (2013)

2013
[26]

Gregor, M

J. Gregor, M. Michael, D. R´ emi, L. Martin, U. Thomas, G. Daniel, and E. Tilman, Nature 515, 237 (2014)

2014
[27]

C. J. Kennedy, W. C. Burton, W. C. Chung, and W. Ket- terle, Nature Physics 11, 1106 (2015)

2015
[28]

Fl¨ aschner, B

N. Fl¨ aschner, B. S. Rem, M. Tarnowski, D. Vogel, D.- S. L¨ uhmann, K. Sengstock, and C. Weitenberg, Science 352, 1091 (2016)

2016
[29]

Asteria, D

L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski, B. S. Rem, N. Fl¨ aschner, K. Sengstock, B. Goldman, and C. Weitenberg, Nature Physics 15, 449 (2019)

2019
[30]

M. L. Wall, A. P. Koller, S. Li, X. Zhang, N. R. Cooper, J. Ye, and A. M. Rey, Physical Review Letters 116, 035301 (2016)

2016
[31]

Zhou, J.-S

X. Zhou, J.-S. Pan, Z.-X. Liu, W. Zhang, W. Yi, G. Chen, and S. Jia, Phys. Rev. Lett. 119, 185701 (2017)

2017
[32]

Kolkowitz, S

S. Kolkowitz, S. L. Bromley, T. Bothwell, M. L. Wall, G. E. Marti, A. P. Koller, X. Zhang, A. M. Rey, and J. Ye, Nature 542, 66 (2016)

2016
[33]

M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, T. Menke, B. Dan, P. M. Preiss, F. Grusdt, A. M. Kaufman, and M. Greiner, Nature 546, 519 (2017)

2017
[34]

P. He, M. A. Perlin, S. R. Muleady, R. J. Lewis-Swan, R. B. Hutson, J. Ye, and A. M. Rey, 1904.07866 (2019)

2019 arXiv
[35]

Saﬀman, T

M. Saﬀman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys. 82, 2313 (2010)

2010
[36]

Labuhn, D

H. Labuhn, D. Barredo, S. Ravets, S. De L´ es´ eleuc, T. Macr` ı, T. Lahaye, and A. Browaeys, Nature 534, 667 (2016)

2016
[37]

Levine, A

H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Phys. Rev. Lett. 121, 123603 (2018)

2018
[38]

Omran, H

A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, et al. , Science 365, 570 (2019)

2019
[39]

de L´ es´ eleuc, V

S. de L´ es´ eleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. B¨ uchler, T. Lahaye, and A. Browaeys, Science 365, 775 (2019)

2019
[40]

A. Celi, B. Vermersch, O. Viyuela, H. Pichler, M. D. Lukin, and P. Zoller, arXiv preprint arXiv:1907.03311 (2019)

2019 arXiv
[41]

Bouchoule and K

I. Bouchoule and K. Mølmer, Phys. Rev. A 65, 041803 (2002)

2002
[42]

Pupillo, A

G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, Phys. Rev. Lett. 104, 223002 (2010)

2010
[43]

Honer, H

J. Honer, H. Weimer, T. Pfau, and H. P. B¨ uchler, Phys. Rev. Lett. 105, 160404 (2010)

2010
[44]

Cinti, P

F. Cinti, P. Jain, M. Boninsegni, A. Micheli, P. Zoller, and G. Pupillo, Phys. Rev. Lett. 105, 135301 (2010)

2010
[45]

W. Li, L. Hamadeh, and I. Lesanovsky, Phys. Rev. A 85, 053615 (2012)

2012
[46]

Henkel, R

N. Henkel, R. Nath, and T. Pohl, Phys. Rev. Lett. 104, 195302 (2010)

2010
[47]

A. D. Bounds, N. C. Jackson, R. K. Hanley, R. Faoro, E. M. Bridge, P. Huillery, and M. P. A. Jones, Phys. Rev. Lett. 120, 183401 (2018)

2018
[48]

Y.-Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, Nature Physics 12, 71 (2016)

2016
[49]

Zeiher, R

J. Zeiher, R. Van Bijnen, P. Schauß, S. Hild, J.-y. Choi, T. Pohl, I. Bloch, and C. Gross, Nature Physics 12, 1095 (2016)

2016
[50]

Wang, R.-B

D.-W. Wang, R.-B. Liu, S.-Y. Zhu, and M. O. Scully, Phys. Rev. Lett. 114, 043602 (2015)

2015
[51]

H. Cai, J. Liu, J. Wu, Y. He, S.-Y. Zhu, J.-X. Zhang, and D.-W. Wang, Phys. Rev. Lett. 122, 023601 (2019)

2019
[52]

We consider the system with lattice sites up toLlat = 256 to verify the ﬁnite-size eﬀects on phase diagrams
[53]

Vasi´ c, A

I. Vasi´ c, A. Petrescu, K. Le Hur, and W. Hofstetter, Phys. Rev. B 91, 094502 (2015)

2015
[54]

Plekhanov, I

K. Plekhanov, I. Vasi´ c, A. Petrescu, R. Nirwan, G. Roux, W. Hofstetter, and K. Le Hur, Phys. Rev. Lett. 120, 157201 (2018)

2018
[55]

Orignac and T

E. Orignac and T. Giamarchi, Phys. Rev. B 64, 144515 (2001)

2001
[56]

Boada, A

O. Boada, A. Celi, J. I. Latorre, and M. Lewenstein, Phys. Rev. Lett. 108, 133001 (2012)

2012
[57]

Petrescu and K

A. Petrescu and K. Le Hur, Phys. Rev. Lett. 111, 150601 (2013)

2013
[58]

A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeli¯ unas, and M. Lewenstein, Phys. Rev. Lett. 112, 043001 (2014)

2014
[59]

Petrescu and K

A. Petrescu and K. Le Hur, Phys. Rev. B 91, 054520 (2015)

2015
[60]

Tokuno and A

A. Tokuno and A. Georges, New Journal of Physics 16, 073005 (2014)

2014
[61]

Piraud, F

M. Piraud, F. Heidrich-Meisner, I. P. McCulloch, S. Greschner, T. Vekua, and U. Schollw¨ ock, Phys. Rev. B 91, 140406 (2015)

2015
[62]

Greschner, M

S. Greschner, M. Piraud, F. Heidrich-Meisner, I. P. Mc- Culloch, U. Schollw¨ ock, and T. Vekua, Phys. Rev. A94, 063628 (2016)

2016
[63]

Anisimovas, M

E. Anisimovas, M. Raˇ ci¯ unas, C. Str¨ ater, A. Eckardt, I. B. Spielman, and G. Juzeli¯ unas, Phys. Rev. A 94, 063632 (2016)

2016
[64]

Calvanese Strinati, E

M. Calvanese Strinati, E. Cornfeld, D. Rossini, S. Bar- 16 barino, M. Dalmonte, R. Fazio, E. Sela, and L. Mazza, Phys. Rev. X 7, 021033 (2017)

2017
[65]

H. M. Price, T. Ozawa, and N. Goldman, Phys. Rev. A 95, 023607 (2017)

2017
[66]

Sundar, B

B. Sundar, B. Gadway, and K. R. Hazzard, Scientiﬁc reports 8, 3422 (2018)

2018
[67]

S. N. L. Barbiero, L. Chomaz and N. Goldman, arXiv preprint arXiv:1907.10555 (2019)

2019 arXiv
[68]

Y.-Q. Li, M. R. Bakhtiari, L. He, and W. Hofstetter, Phys. Rev. B 84, 144411 (2011)

2011
[69]

Anders, E

P. Anders, E. Gull, L. Pollet, M. Troyer, and P. Werner, Phys. Rev. Lett. 105, 096402 (2010)

2010
[70]

Urban, T

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saﬀman, Nat. Phys. 5, 110 (2009)

2009
[71]

Gaetan, Y

A. Gaetan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, Nat. Phys. 5, 115 (2009)

2009
[72]

M. Y. Choi and S. Doniach, Phys. Rev. B31, 4516 (1985)

1985
[73]

Granato and J

E. Granato and J. M. Kosterlitz, Phys. Rev. B 33, 4767 (1986)

1986
[74]

C. S. O’Hern and T. C. Lubensky, Phys. Rev. Lett. 80, 4345 (1998)

1998
[75]

C. S. O’Hern, T. C. Lubensky, and J. Toner, Phys. Rev. Lett. 83, 2745 (1999)

1999
[76]

Pekker, G

D. Pekker, G. Refael, and E. Demler, Phys. Rev. Lett. 105, 085302 (2010)

2010
[77]

L. Niu, S. Jin, X. Chen, X. Li, and X. Zhou, Phys. Rev. Lett. 121 (2018)

2018
[78]

L. Chen, P. Wang, Z. Meng, L. Huang, H. Cai, D.-W. Wang, S.-Y. Zhu, and J. Zhang, Phys. Rev. Lett. 120, 193601 (2018)

2018
[79]

Georges, G

A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen- berg, Rev. Mod. Phys. 68, 13 (1996)

1996
[80]

Byczuk and D

K. Byczuk and D. Vollhardt, Phys. Rev. B 77, 235106 (2008)

2008
[81]

Svidzinsky and J.-T

A. Svidzinsky and J.-T. Chang, Phys. Rev. A 77, 043833 (2008)

2008
[82]

M. O. Scully and M. S. Zubairy, Quantum Optics, 1st ed. (Cambridge University Press, 1997)

1997
[83]

A. J. Daley, arXiv:1405.6694 [cond-mat, physics:quant- ph] (2014)

2014 arXiv
[84]

Gadway, Phys

B. Gadway, Phys. Rev. A 92, 043606 (2015)

2015
[85]

F. A. An, E. J. Meier, and B. Gadway, Science advances 3, e1602685 (2017)

2017
[86]

F. A. An, E. J. Meier, J. Ang’ong’a, and B. Gadway, Phys. Rev. Lett. 120, 040407 (2018)

2018
[87]

F. A. An, E. J. Meier, and B. Gadway, Phys. Rev. X 8, 031045 (2018)

2018
[88]

E. J. Meier, F. A. An, A. Dauphin, M. Maﬀei, P. Massig- nan, T. L. Hughes, and B. Gadway, Science 362, 929 (2018)

2018
[89]

M. O. Scully, E. S. Fry, C. H. R. Ooi, and K. W´ odkiewicz, Phys. Rev. Lett. 96, 010501 (2006)

2006
[90]

Fleischhauer and M

M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000)

2000
[91]

Wang and M

D.-W. Wang and M. O. Scully, Physical Review Letters 113, 083601 (2014)

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

ocTLVc574NOAe/DoMycp9A9e/3g=

The current ¯JAA≈ 2h0 Llat ∑ j sin(−∆k1)n(a) j ≈ 2h0 Llat ∑ j|∆k1|n(a) j > 0. Similarly, we ﬁnd that ¯JBB > 0. Here ¯JAA > ¯JBB because the state |b⟩ is weakly occupied in the ground state, due to small hr. This explains the results shown in the main text. SLIDING PHASES IN THE STRONGL Y INTERACTING REGIME 2 4 6 8 Llat 1 2 3 4 5 6 7 8 9Energy gap EN-EN-1 ...

[2] [2]

K. V. Klitzing, G. Dorda, and M. Pepper, Physical Re- view Letters 45, 494 (1980)

1980

[3] [3]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)

1982

[4] [4]

M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010)

2010

[5] [5]

Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015)

T. Senthil, Annual Review of Condensed Matter Physics 6, 299 (2015)

2015

[6] [6]

Sinova, S

J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Reviews of Modern Physics 87, 1213 (2015)

2015

[7] [7]

T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, Reviews of Modern Physics 89, 025005 (2017)

2017

[8] [8]

Bloch, J

I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008)

2008