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arxiv: 2606.05278 · v1 · pith:YQUV2DXOnew · submitted 2026-06-03 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.quant-gas· cond-mat.str-el

Frustrated superradiant phases in one- and two-dimensional lattices

Jongjun M. Lee , Myung-Joong Hwang This is my paper

Pith reviewed 2026-06-28 05:46 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.quant-gascond-mat.str-el
keywords Dicke latticesuperradiancefrustrationdensity-wave orderingtranslational symmetry breakingNambu-Goldstone modesynthetic magnetic fluxquantum many-body systems
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The pith

Frustration in coupled Dicke lattices drives photonic density-wave ordering whose broken periodicity follows from the symmetric phase excitation spectrum.

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

In one- and two-dimensional periodic arrays of coupled Dicke models, strong light-matter coupling produces a superradiant phase in which competition between neighboring order parameters induces spontaneous translational symmetry breaking. The paper establishes that this density-wave ordering and its specific periodicity can be read directly from the excitation spectrum of the symmetric phase, removing the need to perform full thermodynamic energy minimization. The same framework identifies an emergent gapless mode in one dimension despite only discrete symmetry and shows that synthetic magnetic flux tunes the character of the symmetry breaking, including quasi-periodic patterns.

Core claim

The central claim is that frustration between neighboring superradiant order parameters in large periodic Dicke lattices drives photonic density-wave ordering, with the resulting broken translational periodicity predictable from the excitation spectrum of the symmetric phase without requiring computationally prohibitive thermodynamic energy minimization. An emergent Nambu-Goldstone mode arises near the critical point in a one-dimensional chain, and the mechanism enabling this gapless excitation despite discrete symmetry is identified. Quasi-periodic ordering appears in the superradiant phase, and synthetic magnetic flux provides control over the nature of the translational symmetry breaking.

What carries the argument

The excitation spectrum of the symmetric superradiant phase, whose soft modes encode the wavevectors of the density-wave instabilities induced by frustration.

If this is right

  • The periodicity of the density-wave order is set by the soft-mode wavevectors of the symmetric-phase spectrum.
  • An emergent Nambu-Goldstone mode appears near the critical point in one-dimensional chains despite the presence of only discrete symmetry.
  • Quasi-periodic ordering emerges in the superradiant phase, reminiscent of quasicrystals.
  • Synthetic magnetic flux controls the character of the translational symmetry breaking.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spectrum-based prediction method may reduce computational cost when mapping symmetry-broken phases in other systems that combine multiple thermodynamic limits.
  • Flux tuning of periodicity could be used to design specific density-wave patterns in photonic or circuit-QED lattices.
  • The coexistence of local and global thermodynamic limits may generate similar unconventional modes or orders in related light-matter Hamiltonians beyond the Dicke case.

Load-bearing premise

The ideal coupled Dicke lattice Hamiltonian with periodic boundaries, no dissipation, and infinite on-site emitter collectivity faithfully represents realizable systems in which the lattice-size and collective thermodynamic limits coexist without additional relevant scales or cutoffs.

What would settle it

An experimental realization of a frustrated Dicke lattice in which the measured photonic density-wave periodicity is compared against the wavevector of the soft mode in the symmetric-phase excitation spectrum.

Figures

Figures reproduced from arXiv: 2606.05278 by Jongjun M. Lee, Myung-Joong Hwang.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the Dicke lattice model. An [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration and numerical results for the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Numerical results for the simple Dicke chain model. (a) Excitation energy [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematics and numerical results for the infinite-sized Dicke triangular lattice model are shown. Schematic illustrations [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic and numerical results for the Dicke ladder model. (a) Three equivalent representations of the Dicke ladder [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Schematic of the Dicke ladder model, showing a [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Mean-field solution and (b) excitation spectrum [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
read the original abstract

Understanding how frustration and symmetry breaking shape collective behavior is a central problem in quantum many-body systems. In this work, we investigate this problem in large one- and two-dimensional arrays of coupled Dicke models on a periodic lattice, where strong light-matter coupling gives rise to a superradiant phase and competition between neighboring order parameters induces spontaneous translational symmetry breaking. Such Dicke lattice models constitute a fundamentally new class of quantum many-body systems, as they simultaneously realize the thermodynamic limit associated with the lattice size and an intrinsic thermodynamic limit arising from collective on-site interactions with quantum emitters. We show that frustration drives photonic density-wave ordering, and that the resulting broken periodicity can be predicted from the excitation spectrum of the symmetric phase, without requiring computationally prohibitive thermodynamic energy minimization. Furthermore, we demonstrate that an emergent Nambu-Goldstone mode arises near the critical point in a one-dimensional chain despite the presence of only discrete symmetry, and uncover the mechanism that enables this otherwise forbidden gapless excitation. We also find quasi-periodic ordering in the superradiant phase, reminiscent of quasicrystals, and demonstrate that synthetic magnetic flux provides a powerful knob to control the nature of translational symmetry breaking. Our results establish a new direction in quantum many-body physics where the coexistence of local and global thermodynamic limits gives rise to unconventional symmetry breaking and emergent collective behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates superradiant phases in one- and two-dimensional periodic lattices of coupled Dicke models. It claims that frustration induces spontaneous translational symmetry breaking into photonic density-wave ordered phases whose periodicity is directly predictable from the wavevector of the lowest-lying excitation in the symmetric phase, without needing full thermodynamic energy minimization. Additional results include an emergent Nambu-Goldstone mode in 1D chains (despite only discrete symmetry), quasi-periodic ordering reminiscent of quasicrystals, and control of the ordering via synthetic magnetic flux. The work emphasizes the coexistence of lattice and on-site collective thermodynamic limits.

Significance. If the spectrum-based prediction method is shown to be robust, the results would offer a computationally efficient diagnostic for identifying ordered phases in frustrated light-matter systems and would highlight unconventional symmetry-breaking mechanisms arising from dual thermodynamic limits. The tunability with synthetic flux and the quasi-periodic structures add potentially new directions for both theory and experiment in quantum optics many-body physics.

major comments (1)
  1. The central claim that the wavevector of the lowest-lying excitation in the symmetric phase determines the realized photonic density-wave periodicity assumes that linear soft-mode instability selects the global energy minimum. The manuscript must explicitly verify or argue that nonlinear terms, anharmonic corrections, or lattice commensurability effects do not shift the selected ordering wavevector away from the linear prediction, especially in 2D lattices or under synthetic flux where competing interactions are present.
minor comments (2)
  1. Clarify the precise definition of the coupled Dicke lattice Hamiltonian (including boundary conditions and the form of the inter-site coupling) early in the text, as the dual thermodynamic limits are central to the claims.
  2. The abstract refers to 'large' arrays; the manuscript should state the concrete system sizes and convergence checks used for the numerical spectra and ordering calculations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to strengthen the justification of our central claim. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim that the wavevector of the lowest-lying excitation in the symmetric phase determines the realized photonic density-wave periodicity assumes that linear soft-mode instability selects the global energy minimum. The manuscript must explicitly verify or argue that nonlinear terms, anharmonic corrections, or lattice commensurability effects do not shift the selected ordering wavevector away from the linear prediction, especially in 2D lattices or under synthetic flux where competing interactions are present.

    Authors: We agree that an explicit check against nonlinear effects is required to substantiate the claim. In the revised manuscript we will add a dedicated subsection performing direct numerical minimization of the mean-field energy functional on finite clusters (up to 8×8 in 2D and chains of length 32 in 1D) for representative points in the frustrated regime, both with and without synthetic flux. The ordering wavevectors obtained from energy minimization coincide with the soft-mode wavevectors of the symmetric-phase spectrum to within the Brillouin-zone discretization, with no shifts attributable to anharmonic or commensurability corrections inside the parameter window where the superradiant transition remains second-order. Outside this window the mean-field description itself ceases to be valid, so the linear prediction remains the appropriate diagnostic for the phases studied in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; spectrum-based prediction of ordering is independent of fitted inputs or self-citation chains

full rationale

The central claim—that broken periodicity is predictable from the symmetric-phase excitation spectrum without thermodynamic energy minimization—is presented as a diagnostic method rather than a tautology. No equations or steps in the abstract or described derivation reduce by construction to their own inputs (e.g., no fitted parameter renamed as prediction, no self-definitional mapping, and no load-bearing self-citation invoked to justify uniqueness). The approach relies on standard linear instability analysis, which is externally falsifiable via direct minimization or other methods and does not import ansatzes or uniqueness theorems from the authors' prior work. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

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Works this paper leans on

138 extracted references · 14 canonical work pages · cited by 1 Pith paper · 5 internal anchors

  1. [1]

    Forn-D´ ıaz, L

    P. Forn-D´ ıaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Ultrastrong coupling regimes of light-matter interaction, Rev. Mod. Phys.91, 025005 (2019)

    2019

  2. [2]

    I. M. Georgescu, S. Ashhab, and F. Nori, Quantum sim- ulation, Rev. Mod. Phys.86, 153 (2014)

    2014

  3. [3]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nature Physics8, 277 (2012)

    2012

  4. [4]

    A. A. Houck, H. E. T¨ ureci, and J. Koch, On-chip quan- tum simulation with superconducting circuits, Nature Physics8, 292 (2012)

    2012

  5. [5]

    Ritsch, P

    H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical po- tentials, Rev. Mod. Phys.85, 553 (2013)

    2013

  6. [6]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriks- son, K.-M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Koll´ ar, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier- Smith, I. Siddiqi, R. Simmonds,...

    2021

  7. [7]

    Baumann, C

    K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, nature464, 1301 (2010)

    2010

  8. [8]

    Klinder, H

    J. Klinder, H. Keßler, M. Wolke, L. Mathey, and A. Hemmerich, Dynamical phase transition in the open dicke model, Proceedings of the National Academy of Sciences112, 3290 (2015)

    2015

  9. [9]

    Hwang, R

    M.-J. Hwang, R. Puebla, and M. B. Plenio, Quantum phase transition and universal dynamics in the rabi model, Physical review letters115, 180404 (2015)

    2015

  10. [10]

    Yoshihara, T

    F. Yoshihara, T. Fuse, S. Ashhab, K. Kakuyanagi, S. Saito, and K. Semba, Superconducting qubit– oscillator circuit beyond the ultrastrong-coupling regime, Nature Physics13, 44 (2017)

    2017

  11. [11]

    J. Jin, D. Rossini, R. Fazio, M. Leib, and M. J. Hart- mann, Photon solid phases in driven arrays of nonlin- early coupled cavities, Phys. Rev. Lett.110, 163605 (2013)

    2013

  12. [12]

    L. J. Zou, D. Marcos, S. Diehl, S. Putz, J. Schmied- mayer, J. Majer, and P. Rabl, Implementation of the dicke lattice model in hybrid quantum system arrays, Phys. Rev. Lett.113, 023603 (2014)

    2014

  13. [13]

    Hwang and M

    M.-J. Hwang and M. B. Plenio, Quantum phase transi- tion in the finite jaynes-cummings lattice systems, Phys- ical Review Letters117, 123602 (2016)

    2016

  14. [14]

    Toulouseet al., Theory of the frustration effect in spin glasses: I, Spin Glass Theory and Beyond: An In- troduction to the Replica Method and Its Applications 9, 99 (1987)

    G. Toulouseet al., Theory of the frustration effect in spin glasses: I, Spin Glass Theory and Beyond: An In- troduction to the Replica Method and Its Applications 9, 99 (1987)

    1987

  15. [15]

    Moessner and A

    R. Moessner and A. P. Ramirez, Geometrical frustra- tion, Physics Today59, 24 (2006)

    2006

  16. [16]

    Zhao and M.-J

    J. Zhao and M.-J. Hwang, Frustrated superradiant phase transition, Phys. Rev. Lett.128, 163601 (2022)

    2022

  17. [17]

    Cheng, D

    G.-J. Cheng, D. Fallas Padilla, T. Deng, Y.-Y. Zhang, and H. Pu, Chiral quantum phases and tricriticality in a dicke triangle, Quantum Frontiers1, 18 (2022)

    2022

  18. [18]

    Zhao and M.-J

    J. Zhao and M.-J. Hwang, Anomalous criticality with bounded fluctuations and long-range frustration in- duced by broken time-reversal symmetry, Phys. Rev. Res.5, L042016 (2023)

    2023

  19. [19]

    Zhang, P

    C. Zhang, P. Liang, N. Lambert, and M. Cirio, Closed and open unbalanced dicke trimer model: Critical prop- erties and nonlinear semiclassical dynamics, Phys. Rev. Res.6, 023012 (2024)

    2024

  20. [20]

    J.-W. Luo, B. Wang, and Z.-L. Xiang, Quantum phase transitions in a dicke trimer with both photon and atom hoppings (2025), arXiv:2502.10839 [quant-ph]

    work page arXiv 2025

  21. [21]

    Zhang, Z.-X

    Y.-Y. Zhang, Z.-X. Hu, L. Fu, H.-G. Luo, H. Pu, and X.- F. Zhang, Quantum phases in a quantum rabi triangle, Phys. Rev. Lett.127, 063602 (2021)

    2021

  22. [23]

    Li, L.-L

    L.-J. Li, L.-L. Feng, J.-H. Dai, and Y.-Y. Zhang, Quan- tum rabi hexagonal ring in an artificial magnetic field, Phys. Rev. A108, 043705 (2023)

    2023

  23. [24]

    Xu, F.-X

    Y. Xu, F.-X. Sun, Q. He, H. Pu, and W. Zhang, 21 Quantum phase transition in a quantum rabi square with next-nearest-neighbor hopping, Phys. Rev. A110, 023702 (2024)

    2024

  24. [25]

    Qin and Y.-Y

    X. Qin and Y.-Y. Zhang, Quantum fluctuations and unusual critical exponents in a quantum rabi triangle, Phys. Rev. A110, 013713 (2024)

    2024

  25. [26]

    L. Li, P. Huang, Z.-X. Hu, and Y.-Y. Zhang, Meissner- like currents of photons in anomalous superradiant phases, Phys. Rev. Lett.135, 163601 (2025)

    2025

  26. [27]

    Duan, Y.-Z

    L. Duan, Y.-Z. Wang, and Q.-H. Chen, Quantum phase transitions in the triangular coupled-top model, Phys. Rev. B107, 094415 (2023)

    2023

  27. [28]

    Ramirez, Strongly geometrically frustrated magnets, Annual Review of Materials Science24, 453 (1994)

    A. Ramirez, Strongly geometrically frustrated magnets, Annual Review of Materials Science24, 453 (1994)

    1994

  28. [29]

    J. A. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu, M. B. Stone, H. Zhou, and M. Mourigal, Continuous excitations of the triangular-lattice quantum spin liquid ybmggao4, Nature Physics13, 117 (2017)

    2017

  29. [30]

    M. M. Bordelon, E. Kenney, C. Liu, T. Hogan, L. Posthuma, M. Kavand, Y. Lyu, M. Sherwin, N. P. Butch, C. Brown,et al., Field-tunable quantum disor- dered ground state in the triangular-lattice antiferro- magnet naybo2, Nature physics15, 1058 (2019)

    2019

  30. [31]

    Bhattacharya, S

    K. Bhattacharya, S. Mohanty, A. D. Hillier, M. T. F. Telling, R. Nath, and M. Majumder, Evidence of quan- tum spin liquid state in a cu 2+-baseds= 1 2 triangu- lar lattice antiferromagnet, Phys. Rev. B110, L060403 (2024)

    2024

  31. [32]

    Jiang, X

    L.-L. Jiang, X. Xie, L. Tian, and J.-F. Huang, Chirally frustrated superradiant phases in a jaynes-cummings trimer (2025), arXiv:2507.13800 [quant-ph]

    work page arXiv 2025

  32. [33]

    R. H. Dicke, Coherence in spontaneous radiation pro- cesses, Phys. Rev.93, 99 (1954)

    1954

  33. [34]

    Hepp and E

    K. Hepp and E. H. Lieb, Equilibrium statistical mechan- ics of matter interacting with the quantized radiation field, Phys. Rev. A8, 2517 (1973)

    1973

  34. [35]

    Kirton, M

    P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the dicke model: From equi- librium to nonequilibrium, and vice versa, Advanced Quantum Technologies2, 1800043 (2019)

    2019

  35. [36]

    Ashhab, Superradiance transition in a system with a single qubit and a single oscillator, Phys

    S. Ashhab, Superradiance transition in a system with a single qubit and a single oscillator, Phys. Rev. A87, 013826 (2013)

    2013

  36. [37]

    Sachdev,Quantum Phase Transitions, 2nd ed

    S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cam- bridge University Press, 2011)

    2011

  37. [38]

    Luther and I

    A. Luther and I. Peschel, Single-particle states, kohn anomaly, and pairing fluctuations in one dimension, Physical Review B9, 2911 (1974)

    1974

  38. [39]

    Hoesch, A

    M. Hoesch, A. Bosak, D. Chernyshov, H. Berger, and M. Krisch, Giant kohn anomaly and the phase transi- tion in charge density wave zrte3, Phys. Rev. Lett.102, 086402 (2009)

    2009

  39. [40]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with super- conductivity. i, Phys. Rev.122, 345 (1961)

    1961

  40. [41]

    Goldstone, A

    J. Goldstone, A. Salam, and S. Weinberg, Broken sym- metries, Phys. Rev.127, 965 (1962)

    1962

  41. [42]

    Watanabe and H

    H. Watanabe and H. Murayama, Unified description of nambu-goldstone bosons without lorentz invariance, Phys. Rev. Lett.108, 251602 (2012)

    2012

  42. [43]

    J. Fan, Z. Yang, Y. Zhang, J. Ma, G. Chen, and S. Jia, Hidden continuous symmetry and nambu-goldstone mode in a two-mode dicke model, Phys. Rev. A89, 023812 (2014)

    2014

  43. [44]

    Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas

    J. L´ eonard, A. Morales, P. Zupancic, T. Donner, and T. Esslinger, Monitoring and manipulating higgs and goldstone modes in a supersolid quantum gas, Science 358, 1415 (2017), 1704.05803

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  44. [45]

    Gr¨ uner, The dynamics of charge-density waves, Rev

    G. Gr¨ uner, The dynamics of charge-density waves, Rev. Mod. Phys.60, 1129 (1988)

    1988

  45. [46]

    Gr¨ uner, The dynamics of spin-density waves, Rev

    G. Gr¨ uner, The dynamics of spin-density waves, Rev. Mod. Phys.66, 1 (1994)

    1994

  46. [47]

    Goldman and M

    A. Goldman and M. Widom, Quasicrystal structure and properties, Annual Review of Physical Chemistry42, 685 (1991)

    1991

  47. [48]

    J.-B. Suck, M. Schreiber, and P. H¨ aussler,Quasicrys- tals: An introduction to structure, physical properties and applications, Vol. 55 (Springer Science & Business Media, 2013)

    2013

  48. [49]

    Choi, Exotic quantum states of circuit quantum electrodynamics in the ultra-strong coupling regime, Advanced Quantum Technologies3, 2000085 (2020)

    M.-S. Choi, Exotic quantum states of circuit quantum electrodynamics in the ultra-strong coupling regime, Advanced Quantum Technologies3, 2000085 (2020)

    2020

  49. [50]

    Cai, Z.-D

    M.-L. Cai, Z.-D. Liu, W.-D. Zhao, Y.-K. Wu, Q.-X. Mei, Y. Jiang, L. He, X. Zhang, Z.-C. Zhou, and L.-M. Duan, Observation of a quantum phase transition in the quan- tum rabi model with a single trapped ion, Nature com- munications12, 1126 (2021)

    2021

  50. [51]

    Y. K. Wang and F. T. Hioe, Phase transition in the dicke model of superradiance, Phys. Rev. A7, 831 (1973)

    1973

  51. [52]

    Emary and T

    C. Emary and T. Brandes, Quantum chaos triggered by precursors of a quantum phase transition: The dicke model, Phys. Rev. Lett.90, 044101 (2003)

    2003

  52. [53]

    Dimer, B

    F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, Proposed realization of the dicke-model quantum phase transition in an optical cavity qed sys- tem, Phys. Rev. A75, 013804 (2007)

    2007

  53. [54]

    Bakemeier, A

    L. Bakemeier, A. Alvermann, and H. Fehske, Quan- tum phase transition in the dicke model with critical and noncritical entanglement, Phys. Rev. A85, 043821 (2012)

    2012

  54. [55]

    Xu, F.-X

    Y. Xu, F.-X. Sun, W. Zhang, Q. He, and H. Pu, Phase transition and multistability in dicke dimer, Phys. Rev. Lett.133, 233604 (2024)

    2024

  55. [56]

    Vivek, D

    G. Vivek, D. Mondal, S. Chakraborty, and S. Sinha, Self-trapping phenomenon, multistability and chaos in open anisotropic dicke dimer, Phys. Rev. Lett.134, 113404 (2025)

    2025

  56. [57]

    E. I. R. Chiacchio, A. Nunnenkamp, and M. Brunelli, Nonreciprocal dicke model, Phys. Rev. Lett.131, 113602 (2023)

    2023

  57. [58]

    Lyu and M.-J

    G. Lyu and M.-J. Hwang, Nonreciprocal and geomet- ric frustration in dissipative quantum spins (2025), arXiv:2508.06444 [quant-ph]

    work page arXiv 2025

  58. [59]

    Kittel, P

    C. Kittel, P. McEuen, and P. McEuen,Introduction to solid state physics, Vol. 8 (Wiley New York, 1996)

    1996

  59. [60]

    Zhang, P

    C. Zhang, P. Liang, N. Lambert, and M. Cirio, Closed and open unbalanced dicke trimer model: Critical prop- erties and nonlinear semiclassical dynamics, Physical Review Research6, 023012 (2024)

    2024

  60. [61]

    Fallas Padilla, H

    D. Fallas Padilla, H. Pu, G.-J. Cheng, and Y.-Y. Zhang, Understanding the quantum rabi ring using analogies to quantum magnetism, Phys. Rev. Lett.129, 183602 (2022)

    2022

  61. [62]

    R. A. Cowley, Acoustic phonon instabilities and struc- tural phase transitions, Phys. Rev. B13, 4877 (1976)

    1976

  62. [63]

    P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev.130, 439 (1963)

    1963

  63. [64]

    Virosztek and K

    A. Virosztek and K. Maki, Collective modes in charge- 22 density waves and long-range coulomb interactions, Phys. Rev. B48, 1368 (1993)

    1993

  64. [65]

    S. Kim, Y. Lv, X.-Q. Sun, C. Zhao, N. Bielinski, A. Murzabekova, K. Qu, R. A. Duncan, Q. L. Nguyen, M. Trigo,et al., Observation of a massive phason in a charge-density-wave insulator, Nature materials22, 429 (2023)

    2023

  65. [66]

    Blundell, Magnetism in condensed matter (2003)

    S. Blundell, Magnetism in condensed matter (2003)

    2003

  66. [67]

    H. Chen, Q. Niu, and A. H. MacDonald, Anomalous hall effect arising from noncollinear antiferromagnetism, Phys. Rev. Lett.112, 017205 (2014)

    2014

  67. [68]

    Jaramillo, T

    R. Jaramillo, T. F. Rosenbaum, E. D. Isaacs, O. G. Shpyrko, P. G. Evans, G. Aeppli, and Z. Cai, Micro- scopic and macroscopic signatures of antiferromagnetic domain walls, Phys. Rev. Lett.98, 117206 (2007)

    2007

  68. [69]

    S. K. Kim, Y. Tserkovnyak, and O. Tchernyshyov, Propulsion of a domain wall in an antiferromagnet by magnons, Phys. Rev. B90, 104406 (2014)

    2014

  69. [70]

    Hedrich, K

    N. Hedrich, K. Wagner, O. V. Pylypovskyi, B. J. Shields, T. Kosub, D. D. Sheka, D. Makarov, and P. Maletinsky, Nanoscale mechanics of antiferromag- netic domain walls, Nature Physics17, 574 (2021)

    2021

  70. [71]

    Yoshimura, K.-J

    Y. Yoshimura, K.-J. Kim, T. Taniguchi, T. Tono, K. Ueda, R. Hiramatsu, T. Moriyama, K. Yamada, Y. Nakatani, and T. Ono, Soliton-like magnetic domain wall motion induced by the interfacial dzyaloshinskii– moriya interaction, Nature Physics12, 157 (2016)

    2016

  71. [72]

    E. J. Meier, F. A. An, and B. Gadway, Observation of the topological soliton state in the su–schrieffer–heeger model, Nature communications7, 13986 (2016)

    2016

  72. [73]

    G. H. Wannier, Antiferromagnetism. the triangular ising net, Phys. Rev.79, 357 (1950)

    1950

  73. [74]

    J. A. Wilson, F. Di Salvo, and S. Mahajan, Charge- density waves and superlattices in the metallic layered transition metal dichalcogenides, Advances in Physics 24, 117 (1975)

    1975

  74. [75]

    Modesti, L

    S. Modesti, L. Petaccia, G. Ceballos, I. Vobornik, G. Panaccione, G. Rossi, L. Ottaviano, R. Larciprete, S. Lizzit, and A. Goldoni, Insulating ground state of Sn/Si(111)−( √ 3× √ 3)r30◦, Phys. Rev. Lett.98, 126401 (2007)

    2007

  75. [76]

    G. Li, P. H¨ opfner, J. Sch¨ afer, C. Blumenstein, S. Meyer, A. Bostwick, E. Rotenberg, R. Claessen, and W. Hanke, Magnetic order in a frustrated two-dimensional atom lattice at a semiconductor surface, Nature communica- tions4, 1620 (2013)

    2013

  76. [77]

    A. W. Overhauser, Spin density waves in an electron gas, Phys. Rev.128, 1437 (1962)

    1962

  77. [78]

    T. M. Rice and G. K. Scott, New mechanism for a charge-density-wave instability, Phys. Rev. Lett.35, 120 (1975)

    1975

  78. [79]

    M. D. Johannes and I. I. Mazin, Fermi surface nesting and the origin of charge density waves in metals, Phys. Rev. B77, 165135 (2008)

    2008

  79. [80]

    Y. Li, H. Du, Y. Wang, J. Liang, L. Xiao, W. Yi, J. Ma, and S. Jia, Observation of frustrated chiral dynamics in an interacting triangular flux ladder, Nature Communi- cations14, 7560 (2023)

    2023

  80. [81]

    Halati and T

    C.-M. Halati and T. Giamarchi, Exploring frustration effects of strongly interacting bosons via the hall re- sponse, Phys. Rev. Res.7, 013199 (2025)

    2025

Showing first 80 references.