Pith. sign in

REVIEW 3 major objections 5 minor 81 references

Topology meets superconductivity in a one-dimensional $t-J$ model of magnetic atoms

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A one-dimensional t-J model of magnetic atoms can realize a topological triplet superconductor.

desk verdict A new lanthanide t-J realization with a mostly convincing phase diagram; the TTS claim needs a zero-field, large-L check and the temperature estimate is off by orders of magnitude. read the letter →

arxiv 2509.03387 v2 pith:Y7IRO2FD submitted 2025-09-03 cond-mat.quant-gas cond-mat.str-elquant-ph

Leonardo Bellinato Giacomelli , Thomas Bland , Louis Lafforgue , Francesca Ferlaino , Manfred J. Mark , Luca Barbiero This is my paper
classification cond-mat.quant-gascond-mat.str-elquant-ph
keywords t-Jmodelmagneticlanthanideatomstopologicaltripletsuperconductorsymmetry-protectedorderbosonizationDMRGopticallatticedipolarinteractions
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 concrete way to realize a strongly interacting fermionic t-J model in a one-dimensional optical lattice of magnetic lanthanide atoms, where hopping, onsite interaction, and spin-exchange coupling can be tuned independently. The central claim is that for attractive onsite interactions combined with a positive spin-flip coupling, the ground state becomes a topological triplet superconductor: symmetry-protected topological order—witnessed by string order, edge magnetization, and even entanglement-spectrum degeneracy—coexists with dominant triplet superconducting correlations. If correct, the setup would give cold-atom experiments access to phases that have so far been out of reach: one-dimensional superconductors, a topological liquid, and a phase where topology and superconductivity are intertwined. The derivation relies on the large total spin of erbium and dysprosium to amplify the spin-flip exchange relative to the Ising term, making the effective spin-1/2 model anisotropic and not limited to strong repulsion.

What carries the argument

The object that carries the argument is the derived Hamiltonian H_t-J: a spin-1/2 fermionic t-J model in which the spin-flip coupling J⊥=(1/4)γV is boosted by the large total angular momentum of erbium or dysprosium, while the Ising coupling Jz remains small, explicitly breaking SU(2) symmetry and allowing doubly occupied sites. The analysis then pairs two tools: a bosonization/Sine-Gordon mapping, which determines when the spin field pins—ϕs=0 giving parity order and ϕs=±√(π/8) giving string order—and when Kc>1 signals superconducting correlations; and DMRG simulations, which compute the spin gap, the string and parity correlators, edge magnetization, and entanglement-spectrum degeneracy to

What would settle it

In a chain of erbium or dysprosium atoms at n̄≈0.5, U/t≈-1.5, and J⊥/t≈1.25, measure the spin-resolved site occupancy; the topological-triplet-superconductor claim is settled if the string correlator does not saturate to a finite value at long distance, if no edge magnetization appears, or if the triplet pairing correlator does not decay more slowly than the singlet, charge-density-wave, and spin-density-wave correlators. Alternatively, a calculation including the neglected Jz term in the same parameter region should preserve the coexisting order; if it destroys it, the approximation is load-b

Watch

Extended reading notes

Core claim

On its own terms, the paper derives a generalized spin-1/2 t-J model for magnetic lanthanides in a one-dimensional optical lattice, with the spin-flip coupling J⊥ enhanced by the large total angular momentum (γ≈100 for erbium and 121 for dysprosium) so that the Ising coupling Jz can be neglected and double occupancy is not suppressed. The central finding is that for attractive onsite interaction U<0 and J⊥>-U/2 at density n̄≈0.5, the ground state simultaneously develops long-range string order, finite edge magnetization, even entanglement-spectrum degeneracy, and dominant triplet-superconducting correlations CTS(r); the paper identifies this coexistence as a topological triplet superconducto

Load-bearing premise

The prediction depends on the spin-selective light-shift technique confining the atoms to the mF=±1/2 pair and on the enhanced spin-flip coupling J⊥ being so much larger than the Ising term Jz that Jz can be neglected; if either condition is not met in the dense one-dimensional lattice, the claimed phase diagram does not describe the realized system.

Editorial extensions

If this is right

  • At n̄=0.5, U/t≈-1.5, and J⊥/t≈1.25, the ground state of H_t-J is a topological triplet superconductor with string order, edge magnetization, even entanglement-spectrum degeneracy, and dominant triplet-pairing correlations.
  • The same model hosts a topological liquid, a local singlet superconductor, an extended singlet superconductor, a Luther-Emery liquid, a Luttinger liquid, and a gapless Luttinger triplet superconductor across its phase diagram.
  • The required temperature scale T/t≲0.2 lies within reach of existing ultracold-atom entropy-control experiments, making the predicted phases realistic to access.
  • Unlike the Kitaev chain, the proposed topological triplet superconductor conserves particle number and has massless edge states, which may make it less demanding to realize and probe.
  • The paper's preparation and single-site spin-and-density detection protocol would allow direct measurement of the string correlator, edge magnetization, and the Luttinger parameter Kc.

Reading between the lines

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

  • A direct numerical test would include the neglected Ising term Jz and longer-range dipolar tails; if the topological-triplet-superconductor region survives small Jz, the phase would be robust to imperfect spin projection and the observable parameter window would widen.
  • The role of double occupancy could be isolated experimentally by comparing a U/t→+∞ realization, where the paper finds no topological phases, with the attractive regime; this would test the claim that the topological triplet superconductor is interaction-induced by finite doublon density.
  • Because the edge states are predicted to be massless, measuring the local excitation spectrum near the edges could distinguish this phase from a Kitaev-like topological superconductor with massive Majorana edge modes.
  • Mapping the same phase boundaries with an independent probe of the spin gap, such as local spin-flip spectroscopy, would provide a finite-temperature check of the T/t≲0.2 requirement.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a cold-atom realization of a spin-1/2 t-J model using ultracold magnetic lanthanide atoms (Er, Dy) in a 1D optical lattice, obtained by projecting the large-F dipolar Hamiltonian onto mF=±1/2 via spin-selective quadratic light shifts. The model has independently tunable hopping t, onsite interaction U, and anisotropic spin-flip coupling J_perp. Using bosonization, the authors map the weak-coupling regime to a sine-Gordon model and predict several phases: Luttinger liquid, Luther-Emery liquid, extended/local singlet superconductors, a topological liquid, a Luttinger triplet superconductor, and a topological triplet superconductor (TTS). DMRG calculations (L=240, bond dimension up to 800) are used to map the phase diagram and to identify the TTS phase for U<0 and J_perp>0 through string order, edge magnetization, entanglement-spectrum degeneracy, and dominant triplet-pair correlations. The paper also provides concrete experimental parameters and a state preparation/detection protocol.

Significance. If the TTS coexistence claim is correct, this is a significant result: it would provide a realistic, tunable cold-atom platform for interaction-induced topological superconductivity, going beyond alkaline-atom t-J models and previous dipolar proposals. The bosonization treatment is standard, the DMRG calculations use a respectable bond dimension and multiple diagnostics, and the Table I parameters plus the proposed detection scheme make the proposal concrete and falsifiable. The main caveat is that the numerical evidence for the central TTS phase is not yet quantitatively conclusive: the topological diagnostics are shown at a single system size with an applied edge field, and the correlation-function dominance is assessed at a single distance rather than through extracted exponents.

major comments (3)
  1. [Methods: Details on the DMRG analysis; Figs. 4(c,d) and 5(c,d)] The SPT identification uses edge magnetization |<S^z_L>| computed in the presence of an applied edge field mu=±0.01. Finite edge magnetization is not by itself a topological diagnostic, because any gapped phase with open boundaries is polarized by a boundary field. The even entanglement-spectrum degeneracy xi=0 is a stronger diagnostic, but it is reported only at L=240 and at this nonzero mu. Please provide a mu→0 extrapolation at fixed L and an L→∞ extrapolation at fixed mu (or at mu=0) for xi in both the TL and TTS regions. Without this, the coexistence claim at the center of the paper is not quantitatively established.
  2. [Results, 'Strongly interacting...' and Methods, 'Decay of correlation functions'; Figs. 3, 4, 5] Phase assignments are based on comparing 'asymptotic' correlator values at a single distance r=162 within a central window of L=240. For gapless power-law correlators, the largest value at r=162 does not determine asymptotic dominance: a correlator with a smaller prefactor but slower decay exponent can dominate at larger r. Please extract decay exponents from log-log fits, or provide finite-size scaling of crossing points, especially for CTS versus CLSS/CSDW in Fig. 5(e) and Fig. 9(b). This is required to support the statement that triplet superconducting correlations are dominant in the TTS phase.
  3. [Figs. 4(b), 5(b) and string order] The long-range order of the string correlator O_S^s(r) is inferred from one value at r=162 at L=240. This should be supported by showing saturation with r and evolution with L, and ideally combined with the mu→0 check. Because the TTS phase is defined by coexistence of this SPT order with dominant triplet pairing, this finite-size evidence is load-bearing rather than merely presentational.
minor comments (5)
  1. [Methods, Eq. (22)] The bosonization expansion appears to have a typo: the second term should involve the left-moving field Psi_Lsigma(x), not Psi_Rsigma(x).
  2. [Eqs. (14) and (15)] The correlation functions are written with a comma inside the expectation value: <O†_TS(i), O_TS(i+r)>. For consistency with Eq. (13), the comma should be removed.
  3. [Fig. 5 caption] The sentence 'In all panels we set J⊥/t = 1.25, density n̄ = 1/2 and chain length L = 240' is repeated twice in the caption.
  4. [References] Reference 70 duplicates reference 61 (same arXiv:2505.17009). One of the two should be removed or the citation should be consolidated.
  5. [Methods, 'Decay of correlation functions'] The phrase 'central r = 162 sites' is ambiguous: it could mean the central 162 sites or a distance r=162 from the center. Please clarify the convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the parameters are independently computed, the phase assignments are not fitted to the target order parameters, and the central TTS claim is cross-checked by DMRG diagnostics that do not reduce to the model inputs.

full rationale

The paper's derivation chain is not circular. The t–J Hamiltonian (Eq. 2) is obtained from the dipolar Hamiltonian (Eq. 1) by a spin-space projection, with the enhancement factor γ computed from angular-momentum algebra (Methods, Eq. 21) and the experimental parameters (t, U, J⊥) obtained from Wannier-function integrals (Eqs. 16–20, Table I), not from the target phases. The bosonization analysis uses standard identities for the parity and string correlators (Eqs. 6–9), and the phase boundaries are determined by the sign of the mass term gs and the Luttinger parameter Kc, which are derived from U and J⊥. The DMRG simulations independently compute spin gaps, correlators, edge magnetization, and entanglement-spectrum degeneracy; the phase labels are not obtained by fitting the same quantities that define the phases. The self-citations (e.g., refs 44, 46, 47, 62) provide prior bosonization results and earlier predictions of similar phases, but the present central claim—coexistence of string order and dominant triplet superconductivity in the U<0, J⊥>0 region—is supported by the numerical DMRG data in Fig. 5, so the self-citations are not load-bearing in the sense of forcing the result. The applied edge field μ=±0.01 and the finite-size central-window analysis are legitimate correctness caveats (boundary artifacts could affect edge-magnetization diagnostics), but they are not circularity: the topological identification is also based on the even entanglement-spectrum degeneracy and on string-order long-range order, which are independent of the edge-field perturbation. Similarly, the reliance on spin-selective light shifts (refs 28, 34) is an experimental applicability assumption, not a circular theoretical step. No fitted input is renamed as a prediction, and no uniqueness claim is imported solely from the authors' prior work.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model rests on these domain assumptions and standard field-theoretic tools. No free parameters are fitted to the target phases; control parameters are scanned numerically. No invented entities are introduced.

assumptions (5)
  • domain assumption Spin-selective quadratic light shifts can isolate the mF = +/-1/2 subspace and suppress all other spin states without relevant losses.
    Used to project the large-spin dipolar Hamiltonian (1) to the spin-1/2 t-J model Eq. (2); cited to refs 28 and 34, but not demonstrated for the specific lattice parameters in Table I.
  • domain assumption The Ising term Jz is negligible compared to J_perp.
    J_perp = (1/4) gamma V with gamma about F(F+1)+1/4 (about 100 to 121), so Jz = V/4 is dropped; this underpins the SU(2)-broken model used in bosonization and DMRG.
  • domain assumption Dipolar spin-spin interactions are well approximated by 1/|i-j|^3 up to third neighbors, with relative error below 4% for Dy and assumed similar for Er.
    Methods section; justifies truncating long-range coupling in DMRG to third nearest neighbor.
  • standard math Standard bosonization dictionary and RG flow for the Sine-Gordon model are valid.
    Used to derive Kc_s, gs, pinning values, and phase boundaries in the low-energy analysis.
  • domain assumption DMRG at L=240, bond dimension 800 approximates the thermodynamic limit, and central r=162 correlations are asymptotic.
    Methods section; not backed by finite-size scaling data.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Topology meets superconductivity in a one-dimensional $t-J$ model of magnetic atoms." pith.science (2026). https://pith.science/paper/Y7IRO2FD

@misc{pith2026250903387,
  author       = {Pith},
  title        = {Pith review of: Topology meets superconductivity in a one-dimensional $t-J$ model of magnetic atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y7IRO2FD}},
  note         = {Machine review of arXiv:2509.03387}
}
abstract

Strongly interacting fermions represent the key constituent of several intriguing phases of matter. However, due to the inherent complexity of these systems, important regimes are still inaccessible. Here, we derive a realistic and flexible setup based on ultracold magnetic lanthanide atoms trapped in a one-dimensional optical lattice. Leveraging their large magnetic moments, we design a fermionic $t-J$ model with independently tunable hopping, spin-spin couplings, and onsite interaction. Through combined analytical and numerical analysis, we uncover a variety of many-body quantum phases$-$including superconducting and topological states. Crucially, in the regime of attractive onsite interaction we reveal that topology and superconductivity coexist, thus giving rise to an exotic state of matter: a topological triplet superconductor. We also outline a practical protocol to prepare and detect all discovered phases using current experimental techniques. Our results establish an alternative and powerful route for a deeper understanding of strongly interacting fermionic quantum matter.

Figures

Figures reproduced from arXiv: 2509.03387 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

81 extracted references · 60 canonical work pages

  1. [1]

    Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994)

    A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994)

    1994

  2. [2]

    Sachdev, Quantum magnetism and criticality, Nature Physics 4, 173 (2008)

    S. Sachdev, Quantum magnetism and criticality, Nature Physics 4, 173 (2008)

    work page 2008

  3. [3]

    Dagotto, Correlated electrons in high-temperature supercon- ductors, Rev

    E. Dagotto, Correlated electrons in high-temperature supercon- ductors, Rev. Mod. Phys. 66, 763 (1994)

    work page 1994

  4. [4]

    P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, Rev. Mod. Phys. 78, 17 (2006)

    work page 2006

  5. [5]

    Leijnse and K

    M. Leijnse and K. Flensberg, Introduction to topological su- perconductivity and Majorana fermions, Semiconductor Science and Technology 27, 124003 (2012)

    work page 2012

  6. [6]

    Sato and S

    M. Sato and S. Fujimoto, Majorana fermions and topology in superconductors, Journal of the Physical Society of Japan 85, 072001 (2016), https://doi.org/10.7566/JPSJ.85.072001

    doi:10.7566/jpsj.85.072001 2016

  7. [7]

    Lewenstein, A

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

    work page 2012

  8. [8]

    Gross and I

    C. Gross and I. Bloch, Quantum simulations with ultra- cold atoms in optical lattices, Science 357, 995 (2017), https://www.science.org/doi/pdf/10.1126/science.aal3837

    doi:10.1126/science.aal3837 2017

Show all 81 references
  1. [9]

    J ¨ordens, N

    R. J ¨ordens, N. Strohmaier, K. G ¨unter, H. Moritz, and T. Esslinger, A Mott insulator of fermionic atoms in an optical lattice, Nature 455, 204 (2008)

    2008

  2. [10]

    D. Tusi, L. Franchi, L. F. Livi, K. Baumann, D. Bene- dicto Orenes, L. Del Re, R. E. Barfknecht, T. W. Zhou, M. In- guscio, G. Cappellini, M. Capone, J. Catani, and L. Fallani, Flavour-selective localization in interacting lattice fermions, Na- ture Physics 18, 1201 (2022)

    2022

  3. [11]

    Mazurenko, C

    A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kan ´asz- Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom Fermi–Hubbard antiferromagnet, Na- ture 545, 462 (2017)

    2017

  4. [12]

    Shao, Y .-X

    H.-J. Shao, Y .-X. Wang, D.-Z. Zhu, Y .-S. Zhu, H.-N. Sun, S.- Y . Chen, C. Zhang, Z.-J. Fan, Y . Deng, X.-C. Yao, Y .-A. Chen, and J.-W. Pan, Observation of the antiferromagnetic phase transi- tion in the fermionic Hubbard model (2024), arXiv:2402.14605 [cond-mat.quant-gas]

    2024 arXiv

  5. [13]

    Lebrat, A

    M. Lebrat, A. Kale, L. H. Kendrick, M. Xu, Y . Gang, A. Nikolaenko, S. Sachdev, and M. Greiner, Ferrimagnetism of ultracold fermions in a multi-band Hubbard system (2024), arXiv:2404.17555 [cond-mat.quant-gas]

    2024 arXiv

  6. [14]

    Sompet, S

    P. Sompet, S. Hirthe, D. Bourgund, T. Chalopin, J. Bibo, J. Koepsell, P. Bojovi´c, R. Verresen, F. Pollmann, G. Salomon, C. Gross, T. A. Hilker, and I. Bloch, Realizing the symmetry- protected Haldane phase in Fermi–Hubbard ladders, Nature606, 484 (2022)

    2022

  7. [15]

    Walter, Z

    A.-S. Walter, Z. Zhu, M. G ¨achter, J. Minguzzi, S. Roschinski, K. Sandholzer, K. Viebahn, and T. Esslinger, Quantization and its breakdown in a Hubbard–Thouless pump, Nature Physics 19, 1471 (2023)

    2023

  8. [16]

    P. T. Brown, D. Mitra, E. Guardado-Sanchez, R. Nourafkan, A. Reymbaut, C.-D. H ´ebert, S. Bergeron, A.-M. S. Tremblay, J. Kokalj, D. A. Huse, P. Schauß, and W. S. Bakr, Bad metallic transport in a cold atom Fermi-Hubbard system, Science 363, 379 (2019), https://www.science.org...

    2019 doi

  9. [17]

    Hartke, B

    T. Hartke, B. Oreg, C. Turnbaugh, N. Jia, and M. Zwier- lein, Direct observation of nonlocal fermion pairing in an attractive Fermi-Hubbard gas, Science 381, 82 (2023), https://www.science.org/doi/pdf/10.1126/science.ade4245

    2023 doi

  10. [18]

    T.-W. Zhou, G. Cappellini, D. Tusi, L. Franchi, J. Parravicini, C. Repellin, S. Greschner, M. In- guscio, T. Giamarchi, M. Filippone, J. Catani, and L. Fallani, Observation of universal hall response in strongly interacting fermions, Science 381, 427 (2023), https://www.scienc...

    2023 doi

  11. [19]

    Bourgund, T

    D. Bourgund, T. Chalopin, P. Bojovi ´c, H. Schl ¨omer, S. Wang, T. Franz, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and T. A. Hilker, Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system, Nature 637, 57 (2025)

    2025

  12. [20]

    Chalopin, P

    T. Chalopin, P. Bojovi ´c, S. Wang, T. Franz, A. Sinha, Z. Wang, D. Bourgund, J. Obermeyer, F. Grusdt, A. Bohrdt, L. Pollet, A. Wietek, A. Georges, T. Hilker, and I. Bloch, Probing the mag- netic origin of the pseudogap using a Fermi-Hubbard quantum simulator (2024), arXiv:241...

    2024

  13. [21]

    Hubbard and B

    J. Hubbard and B. H. Flowers, Electron correlations in narrow energy bands, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238 (1963), https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1963.0204

    1963

  14. [22]

    K. A. Chao, J. Spałek, and A. M. Ole ´s, Canonical perturbation expansion of the Hubbard model, Phys. Rev. B 18, 3453 (1978)

    1978

  15. [23]

    Hirthe, T

    S. Hirthe, T. Chalopin, D. Bourgund, P. Bojovi ´c, A. Bohrdt, E. Demler, F. Grusdt, I. Bloch, and T. A. Hilker, Magnetically mediated hole pairing in fermionic ladders of ultracold atoms, Nature 613, 463 (2023)

    2023

  16. [24]

    Chalopin, P

    T. Chalopin, P. Bojovi ´c, D. Bourgund, S. Wang, T. Franz, I. Bloch, and T. Hilker, Optical superlattice for engineering Hub- bard couplings in quantum simulation, Phys. Rev. Lett. 134, 053402 (2025)

    2025

  17. [25]

    A. V . Gorshkov, S. R. Manmana, G. Chen, J. Ye, E. Demler, M. D. Lukin, and A. M. Rey, Tunable superfluidity and quantum magnetism with ultracold polar molecules, Phys. Rev. Lett. 107, 115301 (2011)

    2011

  18. [26]

    A. N. Carroll, H. Hirzler, C. Miller, D. Wellnitz, S. R. Muleady, J. Lin, K. P. Zamarski, R. R. W. Wang, J. L. Bohn, A. M. 12 Rey, and J. Ye, Observation of generalized t-j spin dynam- ics with tunable dipolar interactions, Science 388, 381 (2025), https://www.science.org/doi/...

    2025 doi

  19. [27]

    Douglas, V

    A. Douglas, V . Kaxiras, L. Su, M. Szurek, V . Singh, O. Markovi ´c, and M. Greiner, Spin squeezing with magnetic dipoles (2024), arXiv:2411.07219 [quant-ph]

    2024 arXiv

  20. [28]

    Claude, L

    F. Claude, L. Lafforgue, J. J. A. Houwman, M. J. Mark, and F. Ferlaino, Optical manipulation of spin states in ultracold mag- netic atoms via an inner-shell Hz transition, Phys. Rev. Res. 6, L042016 (2024)

    2024

  21. [29]

    C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach reso- nances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010)

    2010

  22. [30]

    Baier, D

    S. Baier, D. Petter, J. H. Becher, A. Patscheider, G. Natale, L. Chomaz, M. J. Mark, and F. Ferlaino, Realization of a strongly interacting Fermi gas of dipolar atoms, Phys. Rev. Lett. 121, 093602 (2018)

    2018

  23. [31]

    Gross and W

    C. Gross and W. S. Bakr, Quantum gas microscopy for single atom and spin detection, Nature Physics 17, 1316 (2021)

    2021

  24. [32]

    Laupr ˆetre, A

    T. Laupr ˆetre, A. M. Rey, L. Vernac, and B. Laburthe-Tolra, Prob- ing coherences and itinerant magnetism in a dipolar lattice gas (2025), arXiv:2501.11402 [cond-mat.quant-gas]

    2025 arXiv

  25. [33]

    Esslinger, Fermi-Hubbard physics with atoms in an optical lat- tice, Annual Review of Condensed Matter Physics1, 129 (2010)

    T. Esslinger, Fermi-Hubbard physics with atoms in an optical lat- tice, Annual Review of Condensed Matter Physics1, 129 (2010)

    2010

  26. [34]

    Lecomte, A

    M. Lecomte, A. Journeaux, J. Veschambre, J. Dalibard, and R. Lopes, Production and stabilization of a spin mixture of ul- tracold dipolar bose gases, Phys. Rev. Lett. 134, 013402 (2025)

    2025

  27. [35]

    As a consequence, this results in a boost of J⊥ by approximately two orders of magnitude for both erbium ( γEr = 100 ) and dysprosium ( γDy = 121 ), thereby allowing us to neglect the much smaller Jz term. For a realistic lattice spacing dx = 266 nm30,35–39, it is possible to ...

    2018

  28. [36]

    Patscheider, B

    A. Patscheider, B. Zhu, L. Chomaz, D. Petter, S. Baier, A.- M. Rey, F. Ferlaino, and M. J. Mark, Controlling dipolar ex- change interactions in a dense three-dimensional array of large- spin fermions, Phys. Rev. Res. 2, 023050 (2020)

    2020

  29. [37]

    de Paz, A

    A. de Paz, A. Sharma, A. Chotia, E. Mar ´echal, J. H. Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, and B. Laburthe- Tolra, Nonequilibrium quantum magnetism in a dipolar lattice gas, Phys. Rev. Lett. 111, 185305 (2013)

    2013

  30. [38]

    Baier, M

    S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Extended Bose-Hubbard models with ultracold magnetic atoms, Science 352, 201 (2016), https://www.science.org/doi/pdf/10.1126/science.aac9812

    2016 doi

  31. [39]

    Lepoutre, J

    S. Lepoutre, J. Schachenmayer, L. Gabardos, B. Zhu, B. Naylor, E. Mar´echal, O. Gorceix, A. M. Rey, L. Vernac, and B. Laburthe- Tolra, Out-of-equilibrium quantum magnetism and thermaliza- tion in a spin-3 many-body dipolar lattice system, Nature Com- munications 10, 1714 (2019)

    2019

  32. [40]

    L. Su, A. Douglas, M. Szurek, R. Groth, S. F. Ozturk, A. Krahn, A. H. H´ebert, G. A. Phelps, S. Ebadi, S. Dickerson, F. Ferlaino, O. Markovi´c, and M. Greiner, Dipolar quantum solids emerging in a Hubbard quantum simulator, Nature 622, 724 (2023)

    2023

  33. [41]

    N. D. Mermin and H. Wagner, Absence of ferromagnetism or an- tiferromagnetism in one- or two-dimensional isotropic Heisen- berg models, Phys. Rev. Lett. 17, 1133 (1966)

    1966

  34. [42]

    Giamarchi, Quantum physics in one dimension (Oxford Sci- ence Publications, 2003)

    T. Giamarchi, Quantum physics in one dimension (Oxford Sci- ence Publications, 2003)

    2003

  35. [43]

    A. M. T. Alexander O. Gogolin, Alexander A. Nersesyan, Bosonization and Strongly Correlated Systems (Cambridge Uni- versity Press, 1998)

    1998

  36. [44]

    Nakamura, Tricritical behavior in the extended hubbard chains, Phys

    M. Nakamura, Tricritical behavior in the extended hubbard chains, Phys. Rev. B 61, 16377 (2000)

    2000

  37. [45]

    Barbiero, A

    L. Barbiero, A. Montorsi, and M. Roncaglia, How hidden orders generate gaps in one-dimensional fermionic systems, Phys. Rev. B 88, 035109 (2013)

    2013

  38. [46]

    Baldelli, A

    N. Baldelli, A. Montorsi, S. Juli `a-Farr´e, M. Lewenstein, M. Rizzi, and L. Barbiero, Deconfined quantum critical points in fermionic systems with spin-charge separation (2024), arXiv:2407.04073 [cond-mat.str-el]

    2024 arXiv

  39. [47]

    Montorsi and M

    A. Montorsi and M. Roncaglia, Nonlocal order parameters for the 1D Hubbard model, Phys. Rev. Lett. 109, 236404 (2012)

    2012

  40. [48]

    Fazzini, L

    S. Fazzini, L. Barbiero, and A. Montorsi, Interaction-induced fractionalization and topological superconductivity in the polar molecules anisotropic t − J model, Phys. Rev. Lett.122, 106402 (2019)

    2019

  41. [49]

    F. D. M. Haldane, Nonlinear field theory of large-spin Heisen- berg antiferromagnets: Semiclassically quantized solitons of the one-dimensional easy-axis N´eel state, Phys. Rev. Lett. 50, 1153 (1983)

    1983

  42. [50]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Symme- try protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B 85, 075125 (2012)

    2012

  43. [51]

    Tang and X.-G

    E. Tang and X.-G. Wen, Interacting one-dimensional fermionic symmetry-protected topological phases, Phys. Rev. Lett. 109, 096403 (2012)

    2012

  44. [52]

    Senthil, Symmetry-protected topological phases of quantum matter, Annual Review of Condensed Matter Physics 6, 299 (2015)

    T. Senthil, Symmetry-protected topological phases of quantum matter, Annual Review of Condensed Matter Physics 6, 299 (2015)

    2015

  45. [53]

    Shapourian, K

    H. Shapourian, K. Shiozaki, and S. Ryu, Many-body topological invariants for fermionic symmetry-protected topological phases, Phys. Rev. Lett. 118, 216402 (2017)

    2017

  46. [54]

    Montorsi, F

    A. Montorsi, F. Dolcini, R. C. Iotti, and F. Rossi, Symmetry- protected topological phases of one-dimensional interacting fermions with spin-charge separation, Phys. Rev. B 95, 245108 (2017)

    2017

  47. [55]

    S. R. White, Density matrix formulation for quantum renormal- ization groups, Phys. Rev. Lett. 69, 2863 (1992)

    1992

  48. [56]

    Schollw ¨ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

    U. Schollw ¨ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

    2011

  49. [57]

    Luther and V

    A. Luther and V . J. Emery, Backward scattering in the one- dimensional electron gas, Phys. Rev. Lett. 33, 589 (1974)

    1974

  50. [58]

    Li and F

    H. Li and F. D. M. Haldane, Entanglement spectrum as a gener- alization of entanglement entropy: Identification of topological order in non-abelian fractional quantum Hall effect states, Phys. Rev. Lett. 101, 010504 (2008)

    2008

  51. [59]

    Pollmann, A

    F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entan- glement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010)

    2010

  52. [60]

    A. M. Turner, F. Pollmann, and E. Berg, Topological phases of one-dimensional fermions: An entanglement point of view, Phys. Rev. B 83, 075102 (2011)

    2011

  53. [61]

    de L ´es´eleuc, V

    S. de L ´es´eleuc, V . Lienhard, P. Scholl, D. Barredo, S. We- ber, N. Lang, H. P. B ¨uchler, T. Lahaye, and A. Browaeys, Ob- servation of a symmetry-protected topological phase of inter- acting bosons with Rydberg atoms, Science 365, 775 (2019), https://www.science.org/doi/pd...

    2019 doi

  54. [63]

    Montorsi, S

    A. Montorsi, S. Fazzini, and L. Barbiero, Homogeneous and domain-wall topological Haldane conductors with dressed Ry- dberg atoms, Phys. Rev. A 101, 043618 (2020)

    2020

  55. [64]

    Guardado-Sanchez, A

    E. Guardado-Sanchez, A. Morningstar, B. M. Spar, P. T. Brown, D. A. Huse, and W. S. Bakr, Subdiffusion and heat transport in a tilted two-dimensional Fermi-Hubbard system, Phys. Rev. X10, 011042 (2020)

    2020

  56. [65]

    Weitenberg, M

    C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, Single-spin addressing in an atomic Mott insulator, Nature 471, 319 (2011)

    2011

  57. [66]

    Sohmen, M

    M. Sohmen, M. J. Mark, M. Greiner, and F. Ferlaino, A ship-in- a-bottle quantum gas microscope setup for magnetic mixtures, SciPost Phys. 15, 182 (2023). 13

    2023

  58. [67]

    L. R. B. Picard, M. J. Mark, F. Ferlaino, and R. van Bijnen, Deep learning-assisted classification of site-resolved quantum gas microscope images, Measurement Science and Technology 31, 025201 (2019)

    2019

  59. [68]

    Impertro, J

    A. Impertro, J. F. Wienand, S. H ¨afele, H. von Raven, S. Hubele, T. Klostermann, C. R. Cabrera, I. Bloch, and M. Aidelsburger, An unsupervised deep learning algorithm for single-site recon- struction in quantum gas microscopes, Communications Physics 6, 166 (2023)

    2023

  60. [69]

    Asteria, H

    L. Asteria, H. P. Zahn, M. N. Kosch, K. Sengstock, and C. Weit- enberg, Quantum gas magnifier for sub-lattice-resolved imaging of 3D quantum systems, Nature 599, 571 (2021)

    2021

  61. [70]

    L. Su, A. Douglas, M. Szurek, A. H. H´ebert, A. Krahn, R. Groth, G. A. Phelps, O. Markovi ´c, and M. Greiner, Fast single atom imaging for optical lattice arrays, Nature Communications 16, 1017 (2025)

    2025

  62. [71]

    L. Su, R. Sahay, M. Szurek, A. Douglas, O. Markovic, C. B. Dag, R. Verresen, and M. Greiner, Topological phases, critical- ity, and mixed state order in a Hubbard quantum simulator, arXiv preprint arXiv:2505.17009 (2025)

    2025 arXiv

  63. [72]

    M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Ne- spolo, L. Pollet, I. Bloch, and C. Gross, Spin- and density-resolved microscopy of antiferromagnetic correla- tions in Fermi-Hubbard chains, Science 353, 1257 (2016), https://www.science.org/doi/pdf/10.1126/science.aag1635

    2016 doi

  64. [73]

    M. C. Strinati, F. Gerbier, and L. Mazza, Spin-gap spectroscopy in a bosonic flux ladder, New Journal of Physics 20, 015004 (2018)

    2018

  65. [74]

    Pichler, G

    H. Pichler, G. Zhu, A. Seif, P. Zoller, and M. Hafezi, Mea- surement protocol for the entanglement spectrum of cold atoms, Phys. Rev. X 6, 041033 (2016)

    2016

  66. [75]

    Zheng, C.-M

    B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. K.-L. Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155 (2017), https://www.science.org/doi/pdf/10.1126/science.aam7127

    2017 doi

  67. [76]

    Wietek, Y .-Y

    A. Wietek, Y .-Y . He, S. R. White, A. Georges, and E. M. Stoudenmire, Stripes, antiferromagnetism, and the pseudogap in the doped Hubbard model at finite temperature, Phys. Rev. X11, 031007 (2021)

    2021

  68. [77]

    M. Xu, L. H. Kendrick, A. Kale, Y . Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat, and M. Greiner, A neutral-atom Hub- bard quantum simulator in the cryogenic regime, Nature 642, 909 (2025)

    2025

  69. [78]

    A. Y . Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi 44, 131 (2001)

    2001

  70. [79]

    S ´en´echal, An introduction to bosonization, in Theoret- ical Methods for Strongly Correlated Electrons , edited by D

    D. S ´en´echal, An introduction to bosonization, in Theoret- ical Methods for Strongly Correlated Electrons , edited by D. S ´en´echal, A.-M. Tremblay, and C. Bourbonnais (Springer New York, New York, NY , 2004) pp. 139–186

    2004

  71. [80]

    G. I. Japaridze and E. M ¨uller-Hartmann, Triplet superconductiv- ity in a one-dimensional ferromagnetic t − J model, Phys. Rev. B 61, 9019 (2000)

    2000

  72. [81]

    Dziurzik, G

    C. Dziurzik, G. I. Japaridze, A. Schadschneider, and J. Zittartz, Triplet superconductivity vs. easy-plane ferromagnetism in a 1D itinerant electron system with transverse spin anisotropy, The European Physical Journal B - Condensed Matter and Complex Systems 37, 453 (2004)

    2004

  73. [82]

    Hauschild and F

    J. Hauschild and F. Pollmann, Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy), Sci- Post Phys. Lect. Notes , 5 (2018), code available from https: //github.com/tenpy/tenpy, arXiv:1805.00055

    2018 arXiv

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.