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Multipole phases in a type of spin ladders with local conserved quantities and generalizations

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Exact multipole phases in spin ladders reduce to transverse-field Ising models.

desk verdict Exact mappings of these ladders to Ising/ANNNI models are checkable and correct; the phase diagrams are more conditional than the title suggests, but the central construction deserves a serious referee. read the letter →

arxiv 2507.04811 v4 pith:WCJD6QBD submitted 2025-07-07 cond-mat.str-el

Jianlong Fu This is my paper · ORCID
classification cond-mat.str-el MSC 82B2082B2681V70 PACS 75.10.Jm75.10.Pq
keywords multipolephasesspinladderslocalconservedquantitiesfragmentedHilbertspacetransverse-fieldIsingmodelRydbergatomarraysJordan-Wignertransformationdynamicalstructurefactor
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

The paper claims that certain spin ladders support exact multipole phases, meaning ordinary spin order carried by composite multipole moments with zero net magnetization, and that these phases can be solved exactly. The solubility comes from dimer-local conserved quantities, products $\tau_n = \sigma^z_{2n-1}\sigma^z_{2n}$, which split the Hilbert space into sectors; the degrees of freedom left inside each sector form a first-order spin $S$. On the vertical ladder the Hamiltonian becomes exactly a transverse-field Ising model coupled to static $\tau$ variables, and on the horizontal ladder an ANNNI model, so the phases are labeled by the $\tau$ configuration together with the ordinary ordered or disordered phase of $S$. Adding more conserved quantities produces quadrupole and octopole versions of the same idea. A reader should care because these are exactly realized zero-magnetization ordered phases in a solvable family, with a proposed Rydberg-atom realization and a dynamical-structure-factor signature that could distinguish them.

What carries the argument

The central object is the dimer-local conserved quantity $\tau_n = \sigma^z_{2n-1}\sigma^z_{2n}$, together with a two-step Jordan-Wigner transformation along a zig-zag string. First the original spins become Majorana fermions and the dimer products become static $\mathbb{Z}_2$ variables; then pairing the two sites of each $J$-bond defines a first-order spin $S_n$ whose $x$-component is $\sigma^y_{2n-1}\sigma^y_{2n}$ and whose $z$-component is a string of $\sigma^z$ operators. This machinery turns the vertical dipole Hamiltonian into a transverse-field Ising model coupled to static $\tau$ variables, and the horizontal ladder into a transverse-field ANNNI model; the same ladder-of-spins construction, iterated, carries quadrupole and octopole models down to transverse-field Ising models of second- and higher-order spins.

What would settle it

Exactly diagonalize the vertical ladder for a small system, say $4N$ up to 16 spins, over all $2^{2N}$ dimer-label sectors without assuming translational symmetry, and compare the lowest-energy sector with the two-site-periodic calculation used for the phase diagram; finding a non-periodic or longer-period sector with lower energy in the labeled regions would revise those boundaries. Alternatively, in a Rydberg realization, measure $D^{zz}(q_x,q_y,\omega)$ and check whether the zero lines occur at $q_y=0$ for the dipole phase and $q_y=\pi$ for the charge-pair phase as predicted.

Watch

Extended reading notes

Core claim

The paper constructs exactly solvable spin-ladder Hamiltonians whose ground states are exact multipole phases: conventional spin order carried by composite multipole moments, with zero net magnetization. The construction works because every $J$-bond carries a conserved $\mathbb{Z}_2$ variable $\tau = \sigma^z_i \sigma^z_j$, so the Hilbert space fragments into sectors; inside a sector the residual degree of freedom is a first-order spin $S$ whose Hamiltonian is exactly a transverse-field Ising model (vertical ladder) or an ANNNI model (horizontal ladder). The phases are therefore labeled by the static $\tau$ configuration, which may be dipole, charge-pair, or staggered, together with the ordinary ordered or disordered phase of $S$. The same ladder-of-spins construction produces quadrupole and octopole phases by adding more conserved dimer quantities, and the paper gives a Rydberg-atom realization and a zero-line structure-factor signature for the dipole case.

Load-bearing premise

The load-bearing premise is that the ground state can be restricted to dimer-label configurations repeating every two sites, since a longer-period configuration with lower energy would shift the phase boundaries.

Editorial extensions

If this is right

  • The vertical dipole ladder realizes an exact transverse-field Ising transition for the dipole moment $S$, with a weak zero mode in the ordered phase and zero net magnetization of the original spins.
  • The zero-temperature dynamical structure factor of the vertical ladder is $(1\pm\cos q_y)$ times the Ising-chain structure factor, so dipole and charge-pair phases produce distinct lines of zeros in momentum space.
  • The horizontal ladder maps to an ANNNI model in a transverse field, so its phase diagram contains the anti-phase and the Majumdar-Ghosh point in the appropriate parameter limit.
  • Adding quartic plaquette terms promotes the first-order spin to a general XYZ model in a transverse field, and adding more dimer conserved quantities yields quadrupole and octopole phases described by second- and higher-order spins.
  • The proposed Rydberg setup, with an electric field tuned so the exchange coupling vanishes on horizontal bonds and a microwave field canceling the effective longitudinal field, realizes the vertical dipole Hamiltonian to nearest-neighbor order.

Reading between the lines

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

  • The paper leaves implicit that a cold-atom emulator of the vertical ladder could directly measure Ising-critical behavior, such as the correlation-length exponent or entanglement scaling, inside a zero-magnetization multipole phase; the paper does not compute these quantities.
  • The phase diagrams in the paper assume the ground state has a two-site-periodic pattern of the conserved dimer labels; a longer-period or non-periodic dimer pattern with lower energy would change the displayed phase boundaries, and checking this numerically is a natural extension.
  • A natural next step is to extend the electric-field Rydberg design to the quadrupole model, since the required quartic plaquette terms involve four-spin couplings not directly present in the nearest-neighbor Rydberg Hamiltonian considered in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces a family of spin ladder models, called multipole models, in which dimer-local conserved quantities (products of two Z2 spin operators) fragment the Hilbert space into sectors described by a 'higher-order spin'. For the dipole models, the author derives exact mappings: the vertical ladder (Eq. 1) maps to free fermions coupled to static Z2 variables and then to a transverse-field Ising model of first-order spins (Eq. 4), and the horizontal ladder (Eq. 6) maps to an ANNNI model in a transverse field (Eq. 7). The quadrupole model (Eq. 8) is shown to map to a transverse-field Ising model of second-order spins (Eq. 9). Phase diagrams are presented for the vertical and horizontal ladders and for a quadrupole model, together with a proposal for realizing the vertical-ladder dipole model with electric-field-controlled Rydberg atom arrays and a prediction of zero lines in the dynamical structure factor that distinguish dipole and charge-pair phases.

Significance. If the results hold, the paper provides a family of exactly solvable parent Hamiltonians exhibiting multipole order with zero net magnetization, thereby connecting local conserved quantities to multipole phases and giving concrete experimental signatures (Rydberg realization and DSF zero lines). The exact mappings (Eqs. 4, 7, 9) are elegant and internally consistent, and the identification of phase labels with values of the conserved quantities plus the higher-order spin phases is conceptually appealing. The DSF factorization into (1 +/- cos q_y) times an Ising-chain structure factor is a clean, falsifiable prediction. However, the quantitative phase diagrams rely on an unproven restriction to translationally invariant configurations of the conserved quantities, and the quadrupole phase diagram is computed for a modified Hamiltonian, so the full ground-state phase structure of the announced models is not yet established.

major comments (2)
  1. [Supplemental Materials, after Eq. (S11) and Eq. (S13)] The sentence 'Only phases which preserve the translational symmetry are considered' restricts the vertical-ladder phase diagram (Fig. 2) to tau configurations with period 1 or 2. The fermionic Hamiltonian (S11)-(S13) has couplings K + tilde-K tau_n tau_{n+1} that depend on the tau configuration; a period-4, period-6, or aperiodic tau configuration with lower fermionic ground-state energy would change the phase boundaries and the extent of the 'staggered' phase. Because the paper claims exact multipole phases as ground states of the parent Hamiltonian, the comparison over all tau sectors is load-bearing. The 16-spin open-boundary ED in Fig. S2 is not decisive for long-period tau order. I ask the author to either (i) prove that the ground-state tau configuration is translationally invariant (e.g., via a reflection-positivity or Perron-Frobenius argument on the effective fermionic Hamiltonian), (ii) extend the calculation to larger unit cells (period 4, 6, ...) over the parameter range of Fig. 2 and show that the phase boundaries do not change, or (iii) explicitly state in the main text that Fig. 2 is a restricted variational phase diagram and correspondingly soften the claims about the 'staggered' phase.
  2. [Supplemental Materials, Eq. (S25) and the preceding paragraph] The phase diagram in Fig. S3 is computed for a modified Hamiltonian in which extra ZZ couplings on horizontal bonds are added and tilde-J^z is set equal to J^z, whereas the quadrupole model H^(q) announced in Eq. (8) of the main text contains only J^z sigma^z_{4n-2} sigma^z_{4n-1} + tilde-J^z sigma^z_{4n-3} sigma^z_{4n}. As the SM states, the original model has degeneracies (quadrupole vs. CP-dipole and dipole-CP vs. CP-CP) that the added terms lift. Consequently, Fig. S3 is not the phase diagram of the model (8) as written in the main text. The author should either present the phase diagram of the original Hamiltonian (for instance, by adding an infinitesimal symmetry-breaking field and identifying the selection rule), or clearly state that the phase diagram applies to the modified model and explain the physical motivation for the added couplings.
minor comments (5)
  1. [Rydberg simulation section] The text says 'we thus obtained the dipole model' but should read 'we thus obtain the dipole model'. More substantively, the proposal tunes V(R,theta)=0 for horizontal bonds, which sets the XY coupling to zero, but the Rydberg Hamiltonian (SM Eq. S31) still has an Ising coupling J_z on those bonds (SM Eq. S32); the author should explain how this J_z is matched to the desired K and tilde-K couplings in Eq. (1).
  2. [Supplemental Materials, Eqs. (S40)-(S41)] In the sentence 'For the dipole phase of the horizontal ladder, S_{x,1,y} ≡ -S_{x,0,y}', the spin component superscript z is missing; it should read S^z_{x,1,y} ≡ -S^z_{x,0,y} for consistency with Eq. (S39).
  3. [Figures 2 and 3] The captions do not explain the color/pattern scheme used in the lower phase diagrams, nor the correspondence between the upper illustrative configurations and the phase labels. Adding a legend or explicit description would improve readability.
  4. [Discussion of zero modes] The statement 'Weak zero mode exists in Dipole-o phase [42]' relies on the author's prior work; a brief derivation or an explicit expression for the zero-mode operator in the notation of this paper would make the paper more self-contained.
  5. [Abstract] The sentence 'The dipole models can in principle be realized in experiments, we propose such realization by electric field controlled Rydberg atom arrays' contains a comma splice; it should be split into two sentences or joined with a semicolon.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the ladder models are solved by exact Jordan-Wigner mappings to independent Ising/ANNNI results, with only a minor non-load-bearing self-citation.

full rationale

The central derivations are self-contained exact mappings. Eq. (4) is obtained from Eq. (1) by two Jordan-Wigner transformations, and the phase diagram in Fig. 2 is computed by comparing fermionic ground-state energies (Supplemental Eqs. S13-S15) rather than imposed by the phase labels. Similarly, Eq. (7) is mapped to the ANNNI model and its phase diagram is obtained by 16-site exact diagonalization. The tau-sector labels (dipole, charge-pair, staggered) are definitions, but the ground-state selection among sectors is a computed energy competition, so the existence claim is not simply the input restated. The only self-citation that carries a physical assertion is Ref. [42] for the weak zero mode in the Dipole-o phase; that is a peripheral property and the central existence and phase-diagram results do not reduce to it. The Supplemental explicitly restricts to translationally symmetric tau configurations ('Only phases which preserve the translational symmetry are considered'), which is a completeness caveat for the infinite-system phase diagram but not a circular step, because the exact multipole phases and their Ising/ANNNI characterization are established independently of that search restriction. No fitted parameter is relabeled as a prediction, and the DSF zero-line signatures follow directly from the phase definitions but are stated as experimental discriminators, not as evidence for the definitions.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new fundamental entities are introduced; the 'higher-order spin' is an exact composite variable defined by Jordan-Wigner transformations, not a freely postulated particle or force. The main ledger items are the hand-chosen Hamiltonian parameters, the translational-symmetry restriction on tau configurations, the adopted ANNNI phase diagram, and the explicit idealizations in the Rydberg proposal.

free parameters (3)
  • Vertical ladder coupling choice = Jx=Jy=1, K~=0.8
    Chosen for the phase diagram so the transverse term vanishes in the charge-pair sector; not fitted to external data.
  • Horizontal ladder coupling choice = Jx=Jy=-1, K=-4
    Chosen to hit the Gamma/J1 = 0.5 (dipole) and 0 (charge-pair) slices of the ANNNI phase diagram; not fitted.
  • Quadrupole coupling choice = Jp^x=Jp^y=1, Jz=0.1, K~=0.8
    Chosen to lift degeneracies and obtain the quadrupole phase diagram; not fitted.
assumptions (4)
  • standard math Jordan-Wigner and Majorana fermionizations faithfully represent the spin Hilbert space.
    Both vertical and horizontal solutions rely on these fermionizations; this is standard but unproved within the paper.
  • ad hoc to paper Only translationally invariant tau configurations need to be compared for the vertical-ladder ground state.
    Invoked in Supplemental Materials: 'Only phases which preserve the translational symmetry are considered'; non-uniform configurations with larger unit cells are not checked.
  • domain assumption The adopted schematic ANNNI phase diagram is correct for the horizontal ladder parameter regime.
    Fig. 3 upper panel is a 'schematic (simplified) phase diagram' adopted from Refs. [44,52-54]; the lower panel uses it to label phases from a 16-spin exact diagonalization.
  • domain assumption For the Rydberg proposal, interactions beyond nearest neighbors are negligible and the effective longitudinal field can be compensated by microwaves.
    The author states 'we have neglected the atomic interaction beyond nearest neighbors' and assumes microwave compensation; this is an idealization for the proposal.

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Pith. "Pith review of Multipole phases in a type of spin ladders with local conserved quantities and generalizations." pith.science (2026). https://pith.science/paper/WCJD6QBD

@misc{pith2026250704811,
  author       = {Pith},
  title        = {Pith review of: Multipole phases in a type of spin ladders with local conserved quantities and generalizations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCJD6QBD}},
  note         = {Machine review of arXiv:2507.04811}
}
read the original abstract

We study spin ladder models with exact multipole phases, which are traditional spin phases formed by multipole moments. These phases feature non-trivial order with zero magnetization. The multipole models have dimer local conserved quantities that are Ising terms of spin. The Hilbert spaces are locally fragmented into independent sectors described effectively by ``higher-order spin". For dipole models, we consider two ladder geometries with quadratic spin couplings and work out the phase diagrams. Higher-order multipole models are obtained by introducing more dimer conserved quantities. The phases are characterized by the values of the local conserved quantities and the traditional spin phases of the higher-order spin. The dipole models can in principle be realized in experiments, we propose such realization by electric field controlled Rydberg atom arrays.

Figures

Figures reproduced from arXiv: 2507.04811 by the authors.

Figure 1
Figure 1. FIG. 1: Two ladder geometries for dipole models with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Phases of the vertical ladder. Upper: [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: The quadrupole ladder with zeroth-order spin [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figures from the paper (2 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Phases of the horizontal ladder. Upper: [PITH_FULL_IMAGE:figures/full_fig_p004_3.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The simulation of the dipole model on vertical [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Works this paper leans on

77 extracted references · 59 canonical work pages

  1. [1]

    charge-pair-antiferromagnetic (CP-a)

    TheZ 2 conserved quantities onJ-bonds areτ n = σz 2n−1σz 2n =−iη α 2n−1ηβ 2n. With these the Hamiltonian is given by [43] H(d) v = 2NX n=1 h (Jy −J xτn)iηβ 2n−1ηα 2n +J zτn i + 2N−1X n=1 K+ ˜Kτ nτn+1 (−i)ηα 2nηβ 2n+1, (3) which is a free fermion model coupled toτvariables. We pair up everyJ-bond with two sites (2n−1,2n) and label it with numbern, then int...

    work page

  2. [2]

    Wilczek, Phys

    F. Wilczek, Phys. Rev. Lett.48, 1144 (1982)

    work page 1982

  3. [3]

    Kitaev, Annals of Physics321, 2 (2006), january Spe- cial Issue

    A. Kitaev, Annals of Physics321, 2 (2006), january Spe- cial Issue

    2006

  4. [4]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys.80, 1083 (2008)

    2008

  5. [5]

    Hotta, Phys

    C. Hotta, Phys. Rev. B82, 241104 (2010)

    work page 2010

  6. [6]

    Naka and S

    M. Naka and S. Ishihara, Journal of the Physical Society of Japan79, 063707 (2010), https://doi.org/10.1143/JPSJ.79.063707

    work page doi:10.1143/jpsj.79.063707 2010

  7. [7]

    Naka and S

    M. Naka and S. Ishihara, Sci Rep6, 20781 (2016)

    work page 2016

  8. [8]

    Naka and S

    M. Naka and S. Ishihara, Phys. Rev. B93, 195114 (2016)

    work page 2016

Show all 77 references
  1. [9]

    M. G. Yamada and Y. Tada, Phys. Rev. Res.2, 043077 (2020)

    2020

  2. [10]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 031042 (2022)

    2022

  3. [11]

    ˇSmejkal, J

    L. ˇSmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022)

    2022

  4. [12]

    Bhowal and N

    S. Bhowal and N. A. Spaldin, Phys. Rev. X14, 011019 (2024)

    2024

  5. [13]

    P. A. McClarty and J. G. Rau, Phys. Rev. Lett.132, 176702 (2024)

    2024

  6. [14]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Phys. Rev. X10, 011047 (2020)

    2020

  7. [15]

    Khemani, M

    V. Khemani, M. Hermele, and R. Nandkishore, Phys. Rev. B101, 174204 (2020)

    2020

  8. [16]

    Gaiotto, A

    D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, J. High Energ. Phys.02, 172 (2015)

    2015

  9. [17]

    McGreevy, Annual Review of Condensed Matter Physics14, 57 (2023)

    J. McGreevy, Annual Review of Condensed Matter Physics14, 57 (2023)

    2023

  10. [18]

    Yao and S

    H. Yao and S. A. Kivelson, Phys. Rev. Lett.99, 247203 (2007)

    2007

  11. [19]

    Fu, Phys

    J. Fu, Phys. Rev. B100, 195131 (2019)

    2019

  12. [20]

    DeGottardi, D

    W. DeGottardi, D. Sen, and S. Vishveshwara, New Jour- nal of Physics13, 065028 (2011)

    2011

  13. [21]

    F. L. Pedrocchi, S. Chesi, S. Gangadharaiah, and D. Loss, Phys. Rev. B86, 205412 (2012)

    2012

  14. [22]

    Wu, Physics Letters A376, 3530 (2012)

    N. Wu, Physics Letters A376, 3530 (2012)

    2012

  15. [23]

    Saffman, T

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

    2010

  16. [24]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Nature551, 579 (2017)

    2017

  17. [25]

    Zeiher, R

    J. Zeiher, R. Van Bijnen, P. Schauß, S. Hild, J.-y. Choi, T. Pohl, I. Bloch, and C. Gross, Nature Physics12, 1095 (2016)

    2016

  18. [26]

    de L´ es´ eleuc, S

    S. de L´ es´ eleuc, S. Weber, V. Lienhard, D. Barredo, H. P. B¨ uchler, T. Lahaye, and A. Browaeys, Phys. Rev. Lett. 120, 113602 (2018)

    2018

  19. [27]

    Browaeys and T

    A. Browaeys and T. Lahaye, Nature Physics16, 132 (2020)

    2020

  20. [28]

    T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Corti˜ nas, T. Cantat-Moltrecht, F. Assemat, I. Dotsenko, S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, and M. Brune, Phys. Rev. X8, 011032 (2018)

    2018

  21. [29]

    A. W. Glaetzle, M. Dalmonte, R. Nath, C. Gross, I. Bloch, and P. Zoller, Phys. Rev. Lett.114, 173002 (2015)

    2015

  22. [30]

    Chen, B.-Z

    Y.-H. Chen, B.-Z. Wang, T.-F. J. Poon, X.-C. Zhou, Z.- X. Liu, and X.-J. Liu, Phys. Rev. Res.6, L042054 (2024)

    2024

  23. [31]

    Yang, B.-Z

    T.-H. Yang, B.-Z. Wang, X.-C. Zhou, and X.-J. Liu, Phys. Rev. A106, L021101 (2022)

    2022

  24. [32]

    Tsitsishvili, T

    M. Tsitsishvili, T. Chanda, M. Votto, P. Fromholz, M. Dalmonte, and A. Nersesyan, Phys. Rev. B105, 155159 (2022)

    2022

  25. [33]

    Fromholz, M

    P. Fromholz, M. Tsitsishvili, M. Votto, M. Dalmonte, A. Nersesyan, and T. Chanda, Phys. Rev. B106, 155411 (2022)

    2022

  26. [34]

    Eck and P

    L. Eck and P. Fendley, Phys. Rev. B108, 125135 (2023)

    2023

  27. [35]

    S.-A. Liao, J. Zhang, and L.-P. Yang, Phys. Rev. B111, 165154 (2025)

    2025

  28. [36]

    Sarkar, M

    M. Sarkar, M. Pal, A. Sen, and K. Sengupta, SciPost Phys.14, 004 (2023)

    2023

  29. [37]

    Zhang, S

    J. Zhang, S. H. Cant´ u, F. Liu, A. Bylinskii, B. Braver- man, F. Huber, J. Amato-Grill, A. Lukin, N. Gemelke, A. Keesling, et al., Nature Communications16, 712 (2025)

    2025

  30. [38]

    T. J. Carroll, K. Claringbould, A. Goodsell, M. J. Lim, and M. W. Noel, Phys. Rev. Lett.93, 153001 (2004)

    2004

  31. [39]

    L. F. Gon¸ calves and L. G. Marcassa, Phys. Rev. A94, 043424 (2016)

    2016

  32. [40]

    Comparat and P

    D. Comparat and P. Pillet, J. Opt. Soc. Am. B27, A208 (2010)

    2010

  33. [41]

    M. T. Eiles, H. Lee, J. P´ erez-R ´ ıos, and C. H. Greene, Phys. Rev. A95, 052708 (2017)

    2017

  34. [42]

    Jahangiri, J

    A. Jahangiri, J. P. Shaffer, L. F. Gon¸ calves, and L. G. Marcassa, Journal of Physics B: Atomic, Molecular and Optical Physics53, 014001 (2019)

    2019

  35. [43]

    Fu, Phys

    J. Fu, Phys. Rev. B106, L161412 (2022)

    2022

  36. [44]

    See Supplemental Materials

  37. [45]

    Suzuki, J

    S. Suzuki, J. Inoue, and B. K. Chakrabarti, Quantum Ising Phases and Transitions in Transverse Ising Models, Lecture Notes in Physics (Springer Berlin, 2013)

    2013

  38. [46]

    Greiter, V

    M. Greiter, V. Schnells, and R. Thomale, Annals of Physics351, 1026 (2014)

    2014

  39. [47]

    Y. Niu, S. B. Chung, C.-H. Hsu, I. Mandal, S. Raghu, and S. Chakravarty, Phys. Rev. B85, 035110 (2012)

    2012

  40. [48]

    Fu, Annals of Physics432, 168564 (2021)

    J. Fu, Annals of Physics432, 168564 (2021)

    2021

  41. [49]

    Selke, Physics Reports170, 213 (1988)

    W. Selke, Physics Reports170, 213 (1988)

    1988

  42. [50]

    M. E. Fisher and W. Selke, Phys. Rev. Lett.44, 1502 (1980)

    1980

  43. [51]

    Hornreich, R

    R. Hornreich, R. Liebmann, H. Schuster, and W. Selke, Z. Physik B - Condensed Matter35, 91–97 (1979)

    1979

  44. [52]

    Sen and B

    P. Sen and B. K. Chakrabarti, Phys. Rev. B40, 760 (1989)

    1989

  45. [53]

    Arizmendi, A

    C. Arizmendi, A. Rizzo, L. Epele, and C. Garc ´ ıa Canal, Z. Physik B - Condensed Matter83, 273–276 (1991)

    1991

  46. [54]

    Sen and B

    P. Sen and B. K. Chakrabarti, Phys. Rev. B43, 13559 (1991)

    1991

  47. [55]

    P. Sen, S. Chakraborty, S. Dasgupta, and B. Chakrabarti, Z. Physik B - Condensed Matter 88, 333–338 (1992). 7

    1992

  48. [56]

    Majumdar and D

    C. Majumdar and D. Ghosh, J. Math. Phys.10, 1388–1398 (1969)

    1969

  49. [57]

    Majumdar and D

    C. Majumdar and D. Ghosh, J. Math. Phys.10, 1399–1402 (1969)

    1969

  50. [58]

    Schwettmann, J

    A. Schwettmann, J. Crawford, K. R. Overstreet, and J. P. Shaffer, Phys. Rev. A74, 020701(R) (2006)

    2006

  51. [59]

    Whitlock, A

    S. Whitlock, A. W. Glaetzle, and P. Hannaford, Journal of Physics B: Atomic, Molecular and Optical Physics50, 074001 (2017)

    2017

  52. [60]

    R. v. Bijnen, Quantum engineering with ultracold atoms, PhD thesis (Technische Universiteit Eindhoven, 2013)

    2013

  53. [61]

    Y. Jiao, J. Bai, R. Song, S. Bao, J. Zhao, and S. Jia, Frontiers in PhysicsV olume 10 - 2022(2022), 10.3389/fphy.2022.892542

    2022

  54. [62]

    Jordan and E

    P. Jordan and E. Wigner, Zeitschrift f¨ ur Physik47, 631 (1928)

    1928

  55. [63]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961)

    1961

  56. [64]

    Feng, G.-M

    X.-Y. Feng, G.-M. Zhang, and T. Xiang, Phys. Rev. Lett.98, 087204 (2007)

    2007

  57. [65]

    Wang, Y.-H

    Y.-Y. Wang, Y.-H. Shi, Z.-H. Sun, C.-T. Chen, Z.-A. Wang, K. Zhao, H.-T. Liu, W.-G. Ma, Z. Wang, H. Li, J.-C. Zhang, Y. Liu, C.-L. Deng, T.-M. Li, Y. He, Z.- H. Liu, Z.-Y. Peng, X. Song, G. Xue, H. Yu, K. Huang, Z. Xiang, D. Zheng, K. Xu, and H. Fan, PRX Quantum 6, 010325 (2025)

    2025

  58. [66]

    Marshall and R

    W. Marshall and R. D. Lowde, Reports on Progress in Physics31, 705 (1968)

    1968

  59. [67]

    S. M. Girvin and K. Yang, Modern Condensed Matter Physics (Cambridge Univer- sity Press, 2019)

    2019

  60. [68]

    J. Fu, J. G. Rau, M. J. P. Gingras, and N. B. Perkins, Phys. Rev. B96, 035136 (2017)

    2017

  61. [69]

    Ko, Z.-X

    W.-H. Ko, Z.-X. Liu, T.-K. Ng, and P. A. Lee, Phys. Rev. B81, 024414 (2010)

    2010

  62. [70]

    Derzhko and T

    O. Derzhko and T. Krokhmalskii, Phys. Rev. B56, 11659 (1997)

    1997

  63. [71]

    J.-S. Caux, F. H. L. Essler, and U. L¨ ow, Phys. Rev. B 68, 134431 (2003)

    2003

  64. [72]

    A. J. A. James, W. D. Goetze, and F. H. L. Essler, Phys. Rev. B79, 214408 (2009)

    2009

  65. [73]

    Jia and S

    X. Jia and S. Chakravarty, Phys. Rev. B74, 172414 (2006)

    2006

  66. [74]

    R. G. Pereira, J. Sirker, J.-S. Caux, R. Hagemans, J. M. Maillet, S. R. White, and I. Affleck, Journal of Statis- tical Mechanics: Theory and Experiment2007, P08022 (2007)

    2007

  67. [75]

    Haga and S.-i

    N. Haga and S.-i. Suga, Phys. Rev. B66, 132415 (2002)

    2002

  68. [76]

    staggered

    A. P. Young, Phys. Rev. B56, 11691 (1997). 1 Supplemental Materials INTERACTING FERMIONIC VERSION OF THE DIPOLE MODELS The dipole models on the ladder geometry with Hamiltonians H(d) σ = X J-bonds Jxσx i σx j +J yσy i σy j +J zσz i σz j + X K-bonds Kijσz i σz j .(S1) are equiv...

    1997

  69. [77]

    (S39) For thedipole phaseof the horizontal ladder,S x,1,y ≡ −Sx,0,y for all unit cells

    The dynamical structural factor (S35) can be written as Dzz (qx, qy, ω) = 1 2N X x,x′ X y,y ′ Z dteiωteiqy(y−y ′)eiqx(x−x′) ⟨Sz x,0,y(t)Sz x′,0,y′(0)⟩ +⟨S z x,0,y(t)Sz x′,1,y′(0)⟩e−iqx +⟨S z x,1,y(t)Sz x′,0,y′(0)⟩eiqx +⟨S z x,1,y(t)Sz x′,1,y′(0)⟩ . (S39) For thedipole phaseof ...

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