pith. sign in

arxiv: 2604.11880 · v1 · submitted 2026-04-13 · ❄️ cond-mat.str-el

Classification and correlation signatures of chiral spin liquids on the pyrochlore lattice

Chunxiao Liu , Leon Balents , Yasir Iqbal This is my paper

Pith reviewed 2026-05-10 15:54 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords chiral spin liquidspyrochlore latticespin structure factorpinch pointsU(1) gauge fieldsvariational Monte CarloGutzwiller projectionprojective symmetry group
0
0 comments X

The pith

Chiral spin liquids on the pyrochlore lattice fall into distinct classes distinguished by the strength of emergent U(1) gauge fields, shown in the geometry and contrast of pinch-point singularities.

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

The paper classifies chiral U(1) and Z2 quantum spin liquids on the pyrochlore lattice using fermionic parton constructions and a projective symmetry group framework to characterize their flux structures and spinon spectra. It then employs Gutzwiller-projected wave functions with variational Monte Carlo to compute equal-time spin structure factors and extract correlation diagnostics. The central result is that distinct chiral flux sectors, even when close in energy, display markedly different degrees of emergent U(1) gauge-field dominance versus matter-field contributions, visible as variations in pinch-point geometry and contrast. A sympathetic reader would care because these diagnostics identify which states could be stabilized as proximate phases in extended pyrochlore Hamiltonians, even if they are not the ground state of the nearest-neighbor Heisenberg model.

Core claim

We present a systematic classification and variational study of chiral quantum spin liquids on the pyrochlore lattice based on fermionic parton constructions. Focusing on chiral U(1) and Z2 spin-liquid Ansätze, we characterize their symmetry properties, flux structures, and low-energy spinon spectra within a projective symmetry group framework, and incorporate gauge fluctuations through Gutzwiller-projected wave functions studied by variational Monte Carlo. From the equal-time spin structure factor, we develop correlation-based diagnostics that distinguish gauge-dominated Coulomb phases from states with substantial matter-field and short-range contributions. Distinct chiral flux sectors,虽虽虽虽

What carries the argument

Pinch-point singularities in the equal-time spin structure factor, whose geometry and contrast serve as diagnostics for the degree of emergent U(1) gauge-field dominance in each chiral flux sector.

Load-bearing premise

The Gutzwiller-projected fermionic wave functions and variational Monte Carlo sampling accurately capture the low-energy physics and correlation signatures of the target chiral spin-liquid states.

What would settle it

Unbiased simulations such as quantum Monte Carlo or tensor-network methods on the same models that produce spin structure factors lacking the predicted differences in pinch-point contrast and geometry for the corresponding flux sectors would falsify the distinction in gauge dominance.

Figures

Figures reproduced from arXiv: 2604.11880 by Chunxiao Liu, Leon Balents, Yasir Iqbal.

Figure 1
Figure 1. Figure 1: FIG. 1. Pyrochlore lattice. The sites occupy the vertices of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Brillouin zones for the (i) 0-flux ( [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Nearest-neighbor hopping patterns for the 12 U(1) spin-singlet [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Spinon band structures for the 12 nearest-neighbor spin-singlet U(1) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Equal-time spin structure factors of the Gutzwiller-projected nearest-neighbor U(1) chiral spin-liquid [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. One-dimensional cuts of the equal-time spin structure factors for the eight [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
read the original abstract

We present a systematic classification and variational study of chiral quantum spin liquids on the pyrochlore lattice based on fermionic parton constructions. Focusing on chiral $\mathrm{U(1)}$ and $\mathbb{Z}_2$ spin-liquid Ans\"atze, we characterize their symmetry properties, flux structures, and low-energy spinon spectra within a projective symmetry group framework, and incorporate gauge fluctuations through Gutzwiller-projected wave functions studied by variational Monte Carlo. From the equal-time spin structure factor, we develop correlation-based diagnostics that distinguish gauge-dominated Coulomb phases from states with substantial matter-field and short-range contributions. Distinct chiral flux sectors, though close in energy, exhibit markedly different degrees of emergent $\mathrm{U(1)}$ gauge-field dominance, reflected in the geometry and contrast of pinch-point singularities. Although these states are not competitive ground states of the nearest-neighbor Heisenberg model, they define a physically meaningful family of proximate chiral phases relevant to extended pyrochlore Hamiltonians.

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

2 major / 2 minor

Summary. The paper classifies chiral U(1) and Z2 quantum spin liquids on the pyrochlore lattice via fermionic parton constructions in the projective symmetry group (PSG) framework. It characterizes symmetry properties, flux structures, and spinon spectra for various chiral flux sectors, then incorporates gauge fluctuations by constructing Gutzwiller-projected wave functions and evaluating them with variational Monte Carlo (VMC). From the resulting equal-time spin structure factors, the authors introduce correlation diagnostics based on the geometry and contrast of pinch-point singularities to distinguish gauge-dominated Coulomb phases from those with substantial matter-field or short-range contributions. They report that distinct chiral flux sectors, although close in variational energy, display markedly different degrees of emergent U(1) gauge-field dominance and note that these states are not competitive ground states of the nearest-neighbor Heisenberg model but may be relevant for extended pyrochlore Hamiltonians.

Significance. If the VMC-based diagnostics reliably separate the flux sectors by emergent gauge dominance, the work supplies a concrete, correlation-function-based toolkit for identifying and distinguishing proximate chiral spin-liquid phases on the pyrochlore lattice. The systematic PSG classification and the explicit construction of projected wave functions constitute a useful reference for future studies of extended models or material candidates.

major comments (2)
  1. [VMC results and correlation diagnostics (inferred from abstract and § on numerical evaluation)] The central claim that distinct chiral flux sectors exhibit markedly different degrees of emergent U(1) gauge-field dominance rests on the geometry and contrast of pinch-point singularities in the equal-time spin structure factor. However, the manuscript provides no quantitative measures (e.g., contrast ratios, integrated intensities, or finite-size scaling) nor error bars from the VMC sampling, rendering the reported differences difficult to assess for robustness. This is load-bearing because the diagnostics are the primary evidence distinguishing the sectors.
  2. [Gutzwiller projection and VMC methodology] The assumption that Gutzwiller projection and VMC sampling faithfully preserve the mean-field flux sectors' low-energy gauge physics without substantial contamination from matter fields or finite-size effects is not verified against known limits. No comparisons to the unprojected mean-field structure factors, to exact diagonalization on small clusters, or to established U(1) spin-liquid benchmarks are reported, which directly affects the reliability of the pinch-point proxy for gauge dominance.
minor comments (2)
  1. [Classification section] Notation for the PSG labels and flux sectors could be made more uniform across the classification tables and the numerical sections to aid readability.
  2. [Introduction or discussion] The abstract states that the states are 'not competitive ground states' of the nearest-neighbor Heisenberg model; a brief quantitative comparison of variational energies to the known Heisenberg ground-state energy per site would strengthen this statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the work's potential utility. We address each major comment below and describe the revisions we will implement to strengthen the presentation.

read point-by-point responses

  1. Referee: [VMC results and correlation diagnostics (inferred from abstract and § on numerical evaluation)] The central claim that distinct chiral flux sectors exhibit markedly different degrees of emergent U(1) gauge-field dominance rests on the geometry and contrast of pinch-point singularities in the equal-time spin structure factor. However, the manuscript provides no quantitative measures (e.g., contrast ratios, integrated intensities, or finite-size scaling) nor error bars from the VMC sampling, rendering the reported differences difficult to assess for robustness. This is load-bearing because the diagnostics are the primary evidence distinguishing the sectors.

    Authors: We agree that quantitative measures are needed to allow readers to assess the robustness of the differences in pinch-point features. In the revised manuscript we will add explicit contrast ratios, integrated intensities around the pinch points, and statistical error bars obtained from the VMC sampling. We will also include finite-size scaling analysis for the key diagnostics where the computational cost permits. revision: yes

  2. Referee: [Gutzwiller projection and VMC methodology] The assumption that Gutzwiller projection and VMC sampling faithfully preserve the mean-field flux sectors' low-energy gauge physics without substantial contamination from matter fields or finite-size effects is not verified against known limits. No comparisons to the unprojected mean-field structure factors, to exact diagonalization on small clusters, or to established U(1) spin-liquid benchmarks are reported, which directly affects the reliability of the pinch-point proxy for gauge dominance.

    Authors: We acknowledge that direct validation against known limits would increase in the methodology. In the revised version we will add comparisons between the Gutzwiller-projected and unprojected mean-field spin structure factors, results from exact diagonalization on small clusters, and benchmarks against established U(1) spin-liquid states on the pyrochlore lattice to demonstrate that the essential gauge physics is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: classification and pinch-point diagnostics are independent of input Ansätze

full rationale

The paper classifies chiral U(1) and Z2 spin-liquid Ansätze on the pyrochlore lattice via projective symmetry group analysis of fermionic parton constructions, then evaluates equal-time spin structure factors from Gutzwiller-projected wave functions using variational Monte Carlo. The correlation-based diagnostics for U(1) gauge dominance (geometry and contrast of pinch points) are extracted from these computed structure factors rather than being tautological with the mean-field flux sectors. No step reduces by construction to the inputs; the numerical results supply independent content. Any self-citations are not load-bearing for the central claims, and the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard domain assumptions of the parton construction and projective symmetry group framework; no new free parameters or invented entities are introduced beyond those conventional in the literature.

axioms (2)
  • domain assumption Fermionic parton representation faithfully encodes the spin-1/2 Hilbert space when projected
    Invoked throughout the classification and variational study as the basis for constructing the Ansätze.
  • domain assumption Projective symmetry group analysis correctly labels distinct spin-liquid phases by their flux patterns
    Used to organize the chiral U(1) and Z2 states.

pith-pipeline@v0.9.0 · 5467 in / 1406 out tokens · 38898 ms · 2026-05-10T15:54:06.886215+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

88 extracted references · 88 canonical work pages

  1. [1]

    Due to three-fold rotation symmetries2, all four tri- angle faces of a tetrahedron must have the same flux (ϕ for up tetrahedron andϕ for down tetra- hedron). Given that the sum of four triangle fluxes must be multiples of 2π, the triangle flux can only beϕ , = 0, πorϕ , =±π/2 (ϕ andϕ are inde- pendent at this point), corresponding to Φ , =∀ in the former...

    work page

  2. [2]

    down” triangle faces, which is opposite to that for the “up

    We see from Table I that Φ =π/2 breaks time-reversal symmetry, and can only appear in the classes (IT, S, @T) and (I, ST, @T). Specifi- cally, Φ = Φ =±π/2 can only appear in class (IT, S, @T), while Φ =−Φ =±π/2 can only ap- pear in class (I, ST, @T). 3.ϕ ,ϕ / , andϕ are not all independent; they are constrained by ϕ =ϕ +ϕ −ϕ ,(29) here the minus sign in f...

    work page

  3. [3]

    PT theorem

    We can then enumerate all the possible values forϕ , , to obtain the complete set of NN spin- singlet states: (ϕ , ϕ , ϕ ) = (ϕ 1,0,0), (ϕ 1,0, π), (ϕ1, π 2 , π 2 ), (ϕ1, π 2 ,− π 2 ) withϕ 1 = 0,π, or π 2 . The NN spin-singlet states resulting from these general rules are given in Table VI. The U(1) PSG classification correctly produces these states, as ...

    work page

  4. [4]

    [40] and further analyzed in Ref

    Monopole flux state and nodal line spin liquids The (0, π/2,0) state corresponds to the monopole flux state introduced in Ref. [40] and further analyzed in Ref. [27]. In this configuration all triangular plaquettes carry the same chiral fluxπ/2, while the hexagon flux vanishes. As a result, each tetrahedron encloses a net 2πgauge flux, which can be interp...

    work page

  5. [5]

    [41], up to unitary transformations (see Appendix A)

    Staggered flux state and gappedU(1)spin liquids The (π, π/2,0) state defined here can be identified with the staggered flux (SF) state introduced in Ref. [41], up to unitary transformations (see Appendix A). The feature of this state is that the triangular plaquettes on up and down tetrahedra carry opposite chiral fluxes±π/2, while the hexagon flux vanish...

    work page

  6. [6]

    Energetics We now turn to the variational Monte Carlo (VMC) evaluation of the ground-state energies of the eight U(1) chiral spin-liquidAns¨ atzeintroduced above. The result- ing energies, summarized in Table VIII, are systemati- cally higher than those obtained from state-of-the-art nu- merical approaches such as exact diagonalization, tensor- network me...

    work page

  7. [7]

    form factor

    Equal-time spin structure factors The equal-time spin structure factor S(q) = 1 N X i,j e−iq·(ri−rj)⟨ˆSi · ˆSj⟩(36) provides a principal probe of magnetic correlations in frustrated quantum magnets and plays a particularly im- portant role in diagnosing Coulomb phases on the py- rochlore lattice. In contrast to magnetically ordered states, which exhibit B...

    work page 2048

  8. [8]

    Hermele, M

    M. Hermele, M. P. A. Fisher, and L. Balents, Py- rochlore Photons: TheU(1) spin Liquid in aS= 1 2 Three-Dimensional Frustrated Magnet, Phys. Rev. B69, 064404 (2004)

    work page 2004

  9. [9]

    C. L. Henley, Ordering due to Disorder in a Frus- trated Vector Antiferromagnet, Phys. Rev. Lett.62, 2056 (1989)

    work page 2056

  10. [10]

    J. N. Reimers, A. J. Berlinsky, and A.-C. Shi, Mean-Field Approach to Magnetic Ordering in Highly Frustrated Py- rochlores, Phys. Rev. B43, 865 (1991)

    work page 1991

  11. [11]

    Canals and C

    B. Canals and C. Lacroix, Pyrochlore Antiferromagnet: A Three-Dimensional Quantum Spin Liquid, Phys. Rev. Lett.80, 2933 (1998)

    work page 1998

  12. [12]

    Canals and C

    B. Canals and C. Lacroix, Quantum Spin Liquid: The Heisenberg Antiferromagnet on the Three-Dimensional Pyrochlore Lattice, Phys. Rev. B61, 1149 (2000)

    work page 2000

  13. [13]

    H. Yan, O. Benton, L. D. C. Jaubert, and N. Shannon, Rank–2u(1) spin liquid on the breathing pyrochlore lat- tice, Phys. Rev. Lett.124, 127203 (2020)

    work page 2020

  14. [14]

    Lozano-G´ omez, O

    D. Lozano-G´ omez, O. Benton, M. J. P. Gingras, and H. Yan, An Atlas of Classical Pyrochlore Spin Liquids (2025), arXiv:2411.03547 [cond-mat.str-el]

    work page arXiv 2025

  15. [15]

    Kalmeyer and R

    V. Kalmeyer and R. B. Laughlin, Equivalence of the resonating-valence-bond and fractional quantum Hall states, Phys. Rev. Lett.59, 2095 (1987)

    work page 2095

  16. [16]

    X. G. Wen, F. Wilczek, and A. Zee, Chiral spin states and superconductivity, Phys. Rev. B39, 11413 (1989)

    work page 1989

  17. [17]

    X. G. Wen, Mean-field theory of spin-liquid states with finite energy gap and topological orders, Phys. Rev. B 44, 2664 (1991)

    work page 1991

  18. [18]

    R. B. Laughlin and Z. Zou, Properties of the chiral-spin- liquid state, Phys. Rev. B41, 664 (1990)

    work page 1990

  19. [19]

    Tikofsky, S

    A. Tikofsky, S. Libby, and R. Laughlin, Gauge theory of the three-dimensional chiral spin liquid, Nucl. Phys. B 413, 579 (1994)

    work page 1994

  20. [20]

    Si and Y

    T. Si and Y. Yu, Anyonic loops in three-dimensional spin liquid and chiral spin liquid, Nucl. Phys. B803, 428 (2008)

    work page 2008

  21. [21]

    Wu and Y

    X. Wu and Y. Zhang, Two-dimension to three-dimension transition of chiral spin liquid and fractional quantum Hall phases (2023), arXiv:2307.06366 [cond-mat.str-el]

    work page arXiv 2023

  22. [22]

    M. F. Lapa and C. L. Henley, Ground States of the Clas- sical Antiferromagnet on the Pyrochlore Lattice (2012), arXiv:1210.6810 [cond-mat.str-el]

    work page arXiv 2012

  23. [23]

    Domenge, P

    J.-C. Domenge, P. Sindzingre, C. Lhuillier, and L. Pierre, Twelve Sublattice Ordered Phase in theJ 1-J2 Model on the Kagom´ e Lattice, Phys. Rev. B72, 024433 (2005)

    work page 2005

  24. [24]

    Domenge, C

    J.-C. Domenge, C. Lhuillier, L. Messio, L. Pierre, and P. Viot, Chirality andZ 2 vortices in a Heisenberg spin model on the kagome lattice, Phys. Rev. B77, 172413 (2008)

    work page 2008

  25. [25]

    Messio, C

    L. Messio, C. Lhuillier, and G. Misguich, Lattice symme- tries and regular magnetic orders in classical frustrated antiferromagnets, Phys. Rev. B83, 184401 (2011)

    work page 2011

  26. [26]

    Iqbal, T

    Y. Iqbal, T. M¨ uller, K. Riedl, J. Reuther, S. Rachel, R. Valent´ ı, M. J. P. Gingras, R. Thomale, and H. O. Jeschke, Signatures of a Gearwheel Quantum Spin Liq- uid in a Spin- 1 2 Pyrochlore Molybdate Heisenberg Anti- ferromagnet, Phys. Rev. Mater.1, 071201 (2017)

    work page 2017

  27. [27]

    S.-S. Gong, W. Zhu, L. Balents, and D. N. Sheng, Global phase diagram of competing ordered and quantum spin- liquid phases on the kagome lattice, Phys. Rev. B91, 075112 (2015)

    work page 2015

  28. [28]

    W.-J. Hu, W. Zhu, Y. Zhang, S. Gong, F. Becca, and D. N. Sheng, Variational Monte Carlo study of a chi- ral spin liquid in the extended Heisenberg model on the kagome lattice, Phys. Rev. B91, 041124 (2015)

    work page 2015

  29. [29]

    Wietek and A

    A. Wietek and A. M. L¨ auchli, Chiral spin liquid and quantum criticality in extendedS= 1 2 Heisenberg mod- els on the triangular lattice, Phys. Rev. B95, 035141 (2017)

    work page 2017

  30. [30]

    Hickey, L

    C. Hickey, L. Cincio, Z. Papi´ c, and A. Paramekanti, Emergence of chiral spin liquids via quantum melting of noncoplanar magnetic orders, Phys. Rev. B96, 115115 (2017)

    work page 2017

  31. [31]

    Cookmeyer, J

    T. Cookmeyer, J. Motruk, and J. E. Moore, Four-Spin Terms and the Origin of the Chiral Spin Liquid in Mott Insulators on the Triangular Lattice, Phys. Rev. Lett. 127, 087201 (2021)

    work page 2021

  32. [32]

    Y. Liu, W. Hou, X. Han, and J. Zang, Three-Dimensional Dynamics of a Magnetic Hopfion Driven by Spin Transfer Torque, Phys. Rev. Lett.124, 127204 (2020)

    work page 2020

  33. [33]

    C. Naya, D. Schubring, M. Shifman, and Z. Wang, Skyrmions and hopfions in three-dimensional frustrated magnets, Phys. Rev. B106, 094434 (2022)

    work page 2022

  34. [34]

    C. Liu, G. B. Hal´ asz, and L. Balents, Symmetric U(1) andZ 2 spin liquids on the pyrochlore lattice, Phys. Rev. B104, 054401 (2021)

    work page 2021

  35. [35]

    K. C. Sahu and S. Sanyal, U(1) quantum spin liquids in dipolar-octupolar pyrochlore magnets: A fermionic par- ton approach, Phys. Rev. B110, 115144 (2024)

    work page 2024

  36. [36]

    Schneider, J

    B. Schneider, J. C. Halimeh, and M. Punk, Projective symmetry group classification of chiralZ 2 spin liquids on the pyrochlore lattice: Application to the spin- 1 2 XXZ Heisenberg model, Phys. Rev. B105, 125122 (2022)

    work page 2022

  37. [37]

    Iqbal, T

    Y. Iqbal, T. M¨ uller, P. Ghosh, M. J. P. Gingras, H. O. Jeschke, S. Rachel, J. Reuther, and R. Thomale, Quan- tum and Classical Phases of the Pyrochlore Heisenberg Model with Competing Interactions, Phys. Rev. X9, 011005 (2019)

    work page 2019

  38. [38]

    Hagym´ asi, R

    I. Hagym´ asi, R. Sch¨ afer, R. Moessner, and D. J. Luitz, Possible Inversion Symmetry Breaking in theS= 1/2 Pyrochlore Heisenberg Magnet, Phys. Rev. Lett.126, 117204 (2021)

    work page 2021

  39. [39]

    Astrakhantsev, T

    N. Astrakhantsev, T. Westerhout, A. Tiwari, K. Choo, A. Chen, M. H. Fischer, G. Carleo, and T. Neu- pert, Broken-Symmetry Ground States of the Heisen- berg Model on the Pyrochlore Lattice, Phys. Rev. X11, 041021 (2021)

    work page 2021

  40. [40]

    Hering, V

    M. Hering, V. Noculak, F. Ferrari, Y. Iqbal, and J. Reuther, Dimerization tendencies of the pyrochlore Heisenberg antiferromagnet: A functional renormaliza- tion group perspective, Phys. Rev. B105, 054426 (2022)

    work page 2022

  41. [41]

    Sch¨ afer, B

    R. Sch¨ afer, B. Placke, O. Benton, and R. Moessner, Abundance of Hard-Hexagon Crystals in the Quan- tum Pyrochlore Antiferromagnet, Phys. Rev. Lett.131, 096702 (2023)

    work page 2023

  42. [42]

    Pohle, Y

    R. Pohle, Y. Yamaji, and M. Imada, Ground state of theS= 1/2 pyrochlore Heisenberg antiferromagnet: A quantum spin liquid emergent from dimensional reduc- 27 tion (2023), arXiv:2311.11561 [cond-mat.str-el]

    work page arXiv 2023

  43. [43]

    Cheng and T

    R. Cheng and T. Li, Closely competing valence bond crystal orders in the ground state of the spin- 1 2 anti- ferromagnetic Heisenberg model on the pyrochlore lat- tice: a large scale unrestricted variational study (2025), arXiv:2509.13746 [cond-mat.str-el]

    work page arXiv 2025

  44. [44]

    C. Liu, G. B. Hal´ asz, and L. Balents, Competing orders in pyrochlore magnets from aZ 2 spin liquid perspective, Phys. Rev. B100, 075125 (2019)

    work page 2019

  45. [45]

    D. B. Litvin, Magnetic group tables, International Union of Crystallography10, 9780955360220001 (2013)

    work page 2013

  46. [46]

    Bieri, L

    S. Bieri, L. Messio, B. Bernu, and C. Lhuillier, Gapless chiral spin liquid in a kagome Heisenberg model, Phys. Rev. B92, 060407 (2015)

    work page 2015

  47. [47]

    F. J. Burnell, S. Chakravarty, and S. L. Sondhi, Monopole Flux State on the Pyrochlore Lattice, Phys. Rev. B79, 144432 (2009)

    work page 2009

  48. [48]

    J. H. Kim and J. H. Han, Chiral Spin States in the Pyrochlore Heisenberg Magnet: Fermionic Mean-Field Theory and Variational Monte Carlo Calculations, Phys. Rev. B78, 180410 (2008)

    work page 2008

  49. [49]

    Bieri, C

    S. Bieri, C. Lhuillier, and L. Messio, Projective Sym- metry Group Classification of Chiral Spin Liquids, Phys. Rev. B93, 094437 (2016)

    work page 2016

  50. [50]

    Becca and S

    F. Becca and S. Sorella,Quantum Monte Carlo Ap- proaches for Correlated Systems(Cambridge University Press, Cambridge, England, 2017)

    work page 2017

  51. [51]

    V. R. Chandra and J. Sahoo, Spin- 1 2 Heisenberg antifer- romagnet on the pyrochlore lattice: An exact diagonal- ization study, Phys. Rev. B97, 144407 (2018)

    work page 2018

  52. [52]

    M¨ uller, A

    P. M¨ uller, A. Lohmann, J. Richter, and O. Derzhko, Thermodynamics of the pyrochlore-lattice quantum Heisenberg antiferromagnet, Phys. Rev. B100, 024424 (2019)

    work page 2019

  53. [53]

    Sch¨ afer, I

    R. Sch¨ afer, I. Hagym´ asi, R. Moessner, and D. J. Luitz, PyrochloreS= 1 2 Heisenberg antiferromagnet at finite temperature, Phys. Rev. B102, 054408 (2020)

    work page 2020

  54. [54]

    Derzhko, T

    O. Derzhko, T. Hutak, T. Krokhmalskii, J. Schnack, and J. Richter, Adapting Planck’s route to investigate the thermodynamics of the spin-half pyrochlore Heisenberg antiferromagnet, Phys. Rev. B101, 174426 (2020)

    work page 2020

  55. [55]

    S. V. Isakov, K. Gregor, R. Moessner, and S. L. Sondhi, Dipolar Spin Correlations in Classical Pyrochlore Mag- nets, Phys. Rev. Lett.93, 167204 (2004)

    work page 2004

  56. [56]

    C. L. Henley, The Coulomb Phase in Frustrated Systems, Annu. Rev. Condens. Matter Phys.1, 179 (2010)

    work page 2010

  57. [57]

    Castelnovo, R

    C. Castelnovo, R. Moessner, and S. L. Sondhi, Magnetic monopoles in spin ice, Nature (London)451, 42 (2008)

    work page 2008

  58. [58]

    Savary and L

    L. Savary and L. Balents, Quantum spin liquids: a re- view, Rep. Prog. Phys.80, 016502 (2017)

    work page 2017

  59. [59]

    Fennell, S

    T. Fennell, S. T. Bramwell, D. F. McMorrow, P. Manuel, and A. R. Wildes, Pinch Points and Kasteleyn Transi- tions in Kagome Ice, Nat. Phys.3, 566 (2007)

    work page 2007

  60. [60]

    Benton, O

    O. Benton, O. Sikora, and N. Shannon, Seeing the Light: Experimental Signatures of Emergent Electromagnetism in a Quantum Spin Ice, Phys. Rev. B86, 075154 (2012)

    work page 2012

  61. [61]

    Savary and L

    L. Savary and L. Balents, Coulombic Quantum Liquids in Spin-1/2 Pyrochlores, Phys. Rev. Lett.108, 037202 (2012)

    work page 2012

  62. [62]

    Wen, Quantum Orders and Symmetric Spin Liq- uids, Phys

    X.-G. Wen, Quantum Orders and Symmetric Spin Liq- uids, Phys. Rev. B65, 165113 (2002)

    work page 2002

  63. [63]

    Sachdev and N

    S. Sachdev and N. Read, Large N Expansion for Frus- trated and Doped Quantum Antiferromagnets, Int. J. Mod. Phys. B05, 219 (1991)

    work page 1991

  64. [64]

    X. G. Wen, Topological Orders in Rigid States, Int. J. Mod. Phys. B04, 239 (1990)

    work page 1990

  65. [65]

    S. H. Lee, C. Broholm, W. Ratcliff, G. Gasparovic, Q. Huang, T. H. Kim, and S. W. Cheong, Emergent ex- citations in a geometrically frustrated magnet, Nature (London)418, 856 (2002)

    work page 2002

  66. [66]

    Benton, L

    O. Benton, L. D. C. Jaubert, H. Yan, and N. Shannon, A Spin-Liquid with Pinch-Line Singularities on the Py- rochlore Lattice, Nat. Commun.7, 11572 (2016)

    work page 2016

  67. [67]

    Niggemann, Y

    N. Niggemann, Y. Iqbal, and J. Reuther, Quantum Ef- fects on Unconventional Pinch Point Singularities, Phys. Rev. Lett.130, 196601 (2023)

    work page 2023

  68. [68]

    Gresista, D

    L. Gresista, D. Lozano-G´ omez, M. Vojta, S. Trebst, and Y. Iqbal, Quantum effects on pyrochlore higher-rank U(1) spin liquids: Pinch-line singularities, spin nematics, and connections to oxide materials, Phys. Rev. Res.7, 033109 (2025)

    work page 2025

  69. [69]

    Okubo, T

    T. Okubo, T. H. Nguyen, and H. Kawamura, Cubic and Noncubic Multiple-qStates in the Heisenberg Antifer- romagnet on the Pyrochlore Lattice, Phys. Rev. B84, 144432 (2011)

    work page 2011

  70. [70]

    Kr´ anitz and K

    P. Kr´ anitz and K. Penc, Rigorous Anderson-type lower bounds on the ground-state energy of the py- rochlore Heisenberg antiferromagnet, Z. Naturforsch. A doi:10.1515/zna-2026-0009

    work page doi:10.1515/zna-2026-0009 2026

  71. [71]

    O. I. Motrunich, Variational study of triangular lattice spin-1/2 model with ring exchanges and spin-liquid state, Phys. Rev. B72, 045105 (2005)

    work page 2005

  72. [72]

    Balents, Spin liquids in frustrated magnets, Na- ture (London)464, 199 (2010)

    L. Balents, Spin liquids in frustrated magnets, Na- ture (London)464, 199 (2010)

    work page 2010

  73. [73]

    Noculak, D

    V. Noculak, D. Lozano-G´ omez, J. Oitmaa, R. R. P. Singh, Y. Iqbal, M. J. P. Gingras, and J. Reuther, Classi- cal and quantum phases of the pyrochloreS= 1 2 magnet with Heisenberg and Dzyaloshinskii-Moriya interactions, Phys. Rev. B107, 214414 (2023)

    work page 2023

  74. [74]

    Gresista, D

    L. Gresista, D. Lozano-G´ omez, M. Vojta, S. Trebst, and Y. Iqbal, Quantum Coulomb Liquids of Different Rank in the Breathing Pyrochlore Antiferromagnet (2026), arXiv:2602.15662 [cond-mat.str-el]

    work page arXiv 2026

  75. [75]

    Lozano-G´ omez, Y

    D. Lozano-G´ omez, Y. Iqbal, and M. Vojta, A classi- cal chiral spin liquid from chiral interactions on the py- rochlore lattice, Nat. Commun.15, 10162 (2024)

    work page 2024

  76. [76]

    T. Meng, T. Neupert, M. Greiter, and R. Thomale, Coupled-wire construction of chiral spin liquids, Phys. Rev. B91, 241106 (2015)

    work page 2015

  77. [77]

    R. G. Pereira and S. Bieri, Gapless chiral spin liquid from coupled chains on the kagome lattice, SciPost Phys.4, 004 (2018)

    work page 2018

  78. [78]

    C. L. Kane, R. Mukhopadhyay, and T. C. Lubensky, Fractional Quantum Hall Effect in an Array of Quantum Wires, Phys. Rev. Lett.88, 036401 (2002)

    work page 2002

  79. [79]

    J. C. Y. Teo and C. L. Kane, From Luttinger liquid to non-Abelian quantum Hall states, Phys. Rev. B89, 085101 (2014)

    work page 2014

  80. [80]

    Fradkin and S

    E. Fradkin and S. H. Shenker, Phase diagrams of lattice gauge theories with Higgs fields, Phys. Rev. D19, 3682 (1979)

    work page 1979

Showing first 80 references.