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arxiv: 2606.23234 · v1 · pith:FBD73SWXnew · submitted 2026-06-22 · ❄️ cond-mat.str-el · quant-ph

Wess-Zumino terms in 0+1 SU(N) superspin systems

J. S. Morales , M. N. Kiselev This is my paper

Pith reviewed 2026-06-26 06:42 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords Wess-Zumino termsSU(N) superspincoherent statesCP^{N-1}Berry phaseHeisenberg modelsspin-orbital systems
0
0 comments X

The pith

The SU(N) superspin coherent-state path integral is constructed on CP^{N-1} phase space with explicit local Wess-Zumino terms for SU(3) and SU(4).

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

The paper introduces Wess-Zumino terms in systems with SU(N) symmetry by beginning with the SU(2) spin coherent-state path integral in which the Berry phase appears as a WZ term that encodes the symplectic structure of the Bloch sphere. It then extends the construction to higher N by identifying the phase space with CP^{N-1} and deriving explicit local WZ terms for SU(3) and SU(4). A sympathetic reader would care because these geometric terms enter the dynamics of SU(N) Heisenberg models, spin-orbital systems, and multipolar exchange interactions, offering a route to incorporate topological effects into condensed-matter Hamiltonians. The notes also relate the terms to integral cohomology classes and to Berry curvature as the first Chern class of the canonical U(1) bundle.

Core claim

The central claim is that the 0+1-dimensional SU(N) superspin coherent-state path integral can be built by direct analogy with the SU(2) case, with the phase space identified as CP^{N-1}, yielding explicit local Wess-Zumino terms for SU(3) and SU(4) that encode the symplectic structure without additional global topological obstructions.

What carries the argument

The SU(N) superspin coherent-state path integral with phase space identified as CP^{N-1}.

If this is right

  • Geometric terms of this form enter the dynamics of adiabatic processes and geometric quantum noise in magnetic quantum dots.
  • SU(N) Heisenberg models and multipolar exchange interactions can incorporate these local WZ terms directly into their path-integral descriptions.
  • Higher-spin multipolar orders in spin-orbital and spin-pseudospin systems become accessible through the same coherent-state construction.
  • Appendices supply algebraic dictionaries that map the abstract superspin generators to concrete physical embeddings and SU(4) parametrizations.

Where Pith is reading between the lines

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

  • The same local-term construction could be used to simplify Monte Carlo sampling of path integrals for SU(N) models in one dimension.
  • If the terms remain local for general N, they may allow systematic comparison of topological contributions across different symmetry groups in cold-atom realizations.
  • The explicit SU(3) and SU(4) forms could be inserted into effective models of multipolar magnets to predict measurable shifts in level splittings under slow driving.

Load-bearing premise

The SU(N) superspin coherent-state path integral can be constructed by direct analogy with the SU(2) case, with the phase space identified as CP^{N-1} and local WZ terms obtainable without additional global topological obstructions or inconsistencies for N greater than 2.

What would settle it

An explicit calculation that shows the proposed local WZ term for SU(3) produces a Berry curvature that fails to match the first Chern class of the U(1) bundle over CP^2, or that leads to inconsistent quantization in a simple closed path, would falsify the construction.

Figures

Figures reproduced from arXiv: 2606.23234 by J. S. Morales, M. N. Kiselev.

Figure 1
Figure 1. Figure 1: Possible energy configurations (assuming [PITH_FULL_IMAGE:figures/full_fig_p042_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Adiabatic evolution of instantaneous groundstate. [PITH_FULL_IMAGE:figures/full_fig_p049_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Forward and backward insertions of the an observable operator [PITH_FULL_IMAGE:figures/full_fig_p050_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic structure of the Keldysh technique [PITH_FULL_IMAGE:figures/full_fig_p051_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Groundstate configurations of a N/2 − N/2 SU(N) single color (nc = 1) Heisenberg model. (a) Peierls phase/ Bond states, where χ1 ̸= 0, χ2,3,4 = 0, and Flux phase, where χ1 = χ2,3,4. (b) Alternative degenerate groundstate configurations giving rise to disconnected dimerized plaquettes. Now, even though Affleck and Marston first proposed these VBS bond states, their work focused on the single-color case nc =… view at source ↗
Figure 6
Figure 6. Figure 6: Possible metastable configurations of the [PITH_FULL_IMAGE:figures/full_fig_p107_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Uniform on-site interaction strength U. Here, orbitals α = 1, 2 are diagram￾matically represented by pz , px atomic levels. 109 [PITH_FULL_IMAGE:figures/full_fig_p109_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Electron orbital filling in neutral degenerate (left) and ionic split (right) [PITH_FULL_IMAGE:figures/full_fig_p110_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Scaling of the normalization coefficients [PITH_FULL_IMAGE:figures/full_fig_p157_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Four-terminal X-junction with channels (1, 2) and (3, 4) forming two wires. The left and right rephasings then acquire a simple meaning. They are just changes of phase convention for outgoing and incoming channel amplitudes. If S −→ RLSRR , then each matrix element transforms as Si j −→ e iθ L i Si j e iθ R j , |Si j| 2 −→ |Si j| 2 . (769) Therefore, any quantity depending only on scattering probabilities… view at source ↗
read the original abstract

These notes present a self-contained introduction to Wess-Zumino (WZ) terms in quantum systems with $SU(N)$ symmetry, emphasizing the interplay between geometry, topology, and condensed-matter applications. We begin with the $SU(2)$ spin coherent-state path integral, where the Berry phase appears as a WZ term encoding the symplectic structure of the Bloch sphere. This example is then used to introduce the geometric origin of topological terms, their relation to integral cohomology classes, and the role of Berry curvature as the first Chern class of the canonical $U(1)$ bundle. We next discuss physical realizations in which such geometric terms affect dynamics, including adiabatic Berry phases and geometric quantum noise in magnetic quantum dots. A substantial part of the notes is devoted to the condensed-matter motivation for higher $SU(N)$ symmetries, covering $SU(N)$ Heisenberg models, $SU(4)$ spin-orbital and spin-pseudospin systems, multipolar exchange interactions, and higher-spin multipolar orders. Finally, we develop the 0+1-dimensional $SU(N)$ superspin coherent-state construction, identify the phase space with $CP^{N-1}$, and derive explicit local WZ terms for $SU(3)$ and $SU(4)$. The appendices provide algebraic dictionaries connecting the abstract superspin language with concrete physical embeddings, including multipolar generator bases and several useful $SU(4)$ parametrizations.

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

0 major / 2 minor

Summary. The manuscript presents pedagogical notes on Wess-Zumino terms in 0+1-dimensional SU(N) superspin systems. It begins with the SU(2) spin coherent-state path integral and Berry phase as a WZ term on the Bloch sphere, explains the geometric and topological origins including the relation to integral cohomology and the first Chern class, discusses physical realizations such as adiabatic phases and geometric noise, motivates SU(N) symmetries via Heisenberg models, spin-orbital systems and multipolar interactions, and derives the SU(N) superspin coherent-state construction identifying the phase space with CP^{N-1} together with explicit local WZ terms for SU(3) and SU(4). Appendices supply algebraic dictionaries to physical embeddings.

Significance. If the derivations hold, the notes provide a consolidated pedagogical resource linking the standard geometric construction (U(1) bundle over CP^{N-1} with Fubini-Study connection) to condensed-matter applications in higher-SU(N) models. The explicit local WZ terms for SU(3) and SU(4) and the appendices connecting abstract superspin language to multipolar generators constitute a practical strength for researchers working on SU(N) Heisenberg or multipolar systems.

minor comments (2)
  1. [SU(N) construction section] The abstract states that explicit local WZ terms are derived for SU(3) and SU(4), but the main text would benefit from a brief statement (e.g., near the end of the SU(N) construction section) confirming that the expressions reduce to the known SU(2) Berry phase when N=2.
  2. [Appendices] Several appendices are referenced; adding a short table of contents or explicit cross-references in the main text (e.g., 'see Appendix A for the generator basis') would improve navigability for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of the manuscript, as well as the recommendation to accept. The report correctly identifies the pedagogical focus on the geometric construction of Wess-Zumino terms for SU(N) superspins via CP^{N-1} and the appendices linking to physical multipolar embeddings.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a self-contained pedagogical derivation that starts from the established SU(2) Berry phase on the Bloch sphere, identifies the phase space with the standard CP^{N-1} manifold, and obtains local WZ terms via the Fubini-Study connection and first Chern class. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz imported from the authors' prior work; the construction is the standard geometric one for coherent-state path integrals and is externally verifiable in the literature on symplectic geometry and U(1) bundles over projective spaces.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the explicit assumptions stated there. The work relies on standard differential geometry and bundle theory rather than new postulates.

axioms (2)
  • domain assumption The phase space of the SU(N) superspin is CP^{N-1}
    Directly stated in the abstract as the identification used for the construction.
  • domain assumption Local WZ terms exist and can be derived explicitly for SU(3) and SU(4) from the coherent-state path integral
    The central derivation claimed in the abstract.

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Works this paper leans on

108 extracted references · 94 canonical work pages · 4 internal anchors

  1. [1]

    Austin , title =

    S. Blundell,Magnetism in Condensed Matter, Oxford University Press, ISBN 9780198505921, doi:10.1093/oso/9780198505921.001.0001 (2001). 177

    work page doi:10.1093/oso/9780198505921.001.0001 2001

  2. [2]

    Monaco and C

    D. Monaco and C. Tauber,Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess-Zumino, and Fu-Kane-Mele, Lett. Math. Phys.107(7), 1315 (2017), doi:10.1007/s11005-017-0946-y

    work page doi:10.1007/s11005-017-0946-y 2017

  3. [3]

    I. J. R. Aitchison,Berry Phases, Magnetic Monopoles and Wess-Zumino Terms or How the Skyrmion Got Its Spin, Acta Phys. Polon. B18, 207 (1987)

    1987

  4. [4]

    Wilczek and A

    F . Wilczek and A. Shapere,Geometric Phases in Physics, Advanced series in mathematical physics. World Scientific, ISBN 9789971506216 (1989)

    1989

  5. [5]

    Zhang and C

    H. Zhang and C. D. Batista,Classical spin dynamics based on SU(n)coherent states, Phys. Rev. B104, 104409 (2021), doi:10.1103/PhysRevB.104.104409

    work page doi:10.1103/physrevb.104.104409 2021

  6. [6]

    A. M. Perelomov,Generalized Coherent States and Their Applications, Texts and Mono- graphs in Physics. Springer-Verlag, Berlin, Heidelberg, New York, ISBN 978-3-540- 15912-4, doi:10.1007/978-3-642-61629-7 (1986)

    work page doi:10.1007/978-3-642-61629-7 1986

  7. [7]

    Zhang, D

    W .-M. Zhang, D. H. Feng and R. Gilmore,Coherent states: Theory and some applications, Rev. Mod. Phys.62, 867 (1990), doi:10.1103/RevModPhys.62.867

    work page doi:10.1103/revmodphys.62.867 1990

  8. [8]

    Altland and B

    A. Altland and B. D. Simons,Condensed Matter Field Theory, Cambridge University Press, 2 edn. (2010)

    2010

  9. [9]

    A. G. Abanov and A. Abanov,Berry phase for a ferromagnet with fractional spin, Phys. Rev. B65, 184407 (2002), doi:10.1103/PhysRevB.65.184407. [10]M. Nakahara,Geometry, Topology and Physics, Taylor and Francis Group (2003)

    work page doi:10.1103/physrevb.65.184407 2002

  10. [10]

    Atiyah,Geometry of yang-mills fields, In G

    F . Atiyah,Geometry of yang-mills fields, In G. Dell’Antonio, S. Doplicher and G. Jona- Lasinio, eds.,Mathematical Problems in Theoretical Physics: International Conference Held in Rome, June 6–15, 1977, pp. 216–221. Springer Berlin Heidelberg, Berlin, Heidelberg, doi:10.1007/3-540-08853-9_18 (1978)

    work page doi:10.1007/3-540-08853-9_18 1977

  11. [11]

    Mombelli,Homotopy theory and characteristic classes, InETH Zürich - Proseminar in Theoretical Physics(2018)

    S. Mombelli,Homotopy theory and characteristic classes, InETH Zürich - Proseminar in Theoretical Physics(2018)

    2018

  12. [12]

    R. A. Bertlmann,Anomalies in Quantum Field Theory, Oxford University Press, ISBN 9780198507628 (2000). [14]E. Fradkin,Quantum Field Theory: An Integrated Approach, Princeton University Press, ISBN 9780691189550 (2021)

    2000

  13. [13]

    Bogoliubov and D

    N. Bogoliubov and D. Shirkov,Introduction to the Theory of Quantized Fields, Interscience monographs in physics and astronomy , v.3. Interscience Publishers (1959)

    1959

  14. [14]

    Scharf,Finite Quantum Electrodynamics: The Causal Approach, Third Edition, Dover Books on Physics

    G. Scharf,Finite Quantum Electrodynamics: The Causal Approach, Third Edition, Dover Books on Physics. Dover Publications, ISBN 9780486492735 (2014)

    2014

  15. [15]

    Shnirman, Y

    A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev and A. Alt- land,Geometric quantum noise of spin, Phys. Rev. Lett.114, 176806 (2015), doi:10.1103/PhysRevLett.114.176806

    work page doi:10.1103/physrevlett.114.176806 2015

  16. [16]

    Michler, ed.,Single Semiconductor Quantum Dots, NanoScience and Technology

    P . Michler, ed.,Single Semiconductor Quantum Dots, NanoScience and Technology. Springer, ISBN 978-3-540-87445-4, doi:10.1007/978-3-540-87446-1 (2009)

    work page doi:10.1007/978-3-540-87446-1 2009

  17. [17]

    C. W . J. Beenakker,Random-matrix theory of quantum transport, Reviews of Modern Physics69(3), 731 (1997), doi:10.1103/RevModPhys.69.731. 178

    work page doi:10.1103/revmodphys.69.731 1997

  18. [18]

    Aleiner, P

    I. Aleiner, P . Brouwer and L. Glazman,Quantum effects in coulomb blockade, Physics Reports358(5–6), 309–440 (2002), doi:10.1016/s0370-1573(01)00063-1

    work page doi:10.1016/s0370-1573(01)00063-1 2002

  19. [19]

    A. Saha, Y. Gefen, I. S. Burmistrov, A. Shnirman and A. Altland,A quantum dot close to stoner instability: The role of the berry phase, Annals of Physics327(10), 2543 (2012), doi:10.1016/j.aop.2012.07.013

    work page doi:10.1016/j.aop.2012.07.013 2012

  20. [20]

    Kamenev,Field Theory of Non-Equilibrium Systems, Cambridge Modern Physics

    A. Kamenev,Field Theory of Non-Equilibrium Systems, Cambridge Modern Physics. Cambridge University Press, ISBN 9780521760829, doi:10.1017/CBO9781139003667 (2011)

    work page doi:10.1017/cbo9781139003667 2011

  21. [21]

    A. V . Gorshkov, M. Hermele, V . Gurarie, C. Xu, P . S. Julienne, J. Ye, P . Zoller, E. Demler, M. D. Lukin and A. M. Rey,Two-orbital su(n) magnetism with ultracold alkaline-earth atoms, Nature Physics6, 289 (2010), doi:10.1038/nphys1535

    work page doi:10.1038/nphys1535 2010

  22. [22]

    M. A. Cazalilla and A. M. Rey ,Ultracold fermi gases with emergent su(n) symmetry, Reports on Progress in Physics77(12), 124401 (2014), doi:10.1088/0034-4885/77/12/124401

    work page doi:10.1088/0034-4885/77/12/124401 2014

  23. [23]

    Zhang, M

    X. Zhang, M. Bishof, S. L. Bromley , C. V . Kraus, M. S. Safronova, P . Zoller, A. M. Rey and J. Ye,Spectroscopic observation of su(<i>n</i>)-symmetric interactions in sr orbital magnetism, Science345(6203), 1467 (2014), doi:10.1126/science.1254978, https: //www.science.org/doi/pdf/10.1126/science.1254978

    work page doi:10.1126/science.1254978 2014

  24. [24]

    F . D. M. Haldane,O(3) nonlinear sigma model and the topological distinction between integer- and half-integer-spin antiferromagnets in two dimensions, Physical Review Letters 61(8), 1029 (1988), doi:10.1103/PhysRevLett.61.1029

    work page doi:10.1103/physrevlett.61.1029 1988

  25. [25]

    Read and S

    N. Read and S. Sachdev,Some features of the phase diagram of the square lattice su(n) anti- ferromagnet, Nuclear Physics B316(3), 609 (1989), doi:10.1016/0550-3213(89)90061- 8

    work page doi:10.1016/0550-3213(89)90061- 1989

  26. [26]

    D. P . Arovas and A. Auerbach,Functional integral theories of low-dimensional quantum heisenberg models, Phys. Rev. B38, 316 (1988), doi:10.1103/PhysRevB.38.316

    work page doi:10.1103/physrevb.38.316 1988

  27. [27]

    Auerbach and D

    A. Auerbach and D. P . Arovas,Spin dynamics in the square-lattice antiferromagnet, Phys. Rev. Lett.61, 617 (1988), doi:10.1103/PhysRevLett.61.617

    work page doi:10.1103/physrevlett.61.617 1988

  28. [28]

    Harada, N

    K. Harada, N. Kawashima and M. Troyer,Néel and spin-peierls ground states of two- dimensional SU(n)quantum antiferromagnets, Phys. Rev. Lett.90, 117203 (2003), doi:10.1103/PhysRevLett.90.117203

    work page doi:10.1103/physrevlett.90.117203 2003

  29. [29]

    K. S. D. Beach, F . Alet, M. Mambrini and S. Capponi, SU(n)heisenberg model on the square lattice: A continuous- n quantum monte carlo study, Phys. Rev. B80, 184401 (2009), doi:10.1103/PhysRevB.80.184401

    work page doi:10.1103/physrevb.80.184401 2009

  30. [30]

    Senthil, A

    T . Senthil, A. Vishwanath, L. Balents, S. Sachdev and M. P . A. Fisher,Deconfined quantum critical points, Science303(5663), 1490 (2004), doi:10.1126/science.1091806, https: //www.science.org/doi/pdf/10.1126/science.1091806

    work page doi:10.1126/science.1091806 2004

  31. [31]

    Senthil,Deconfined quantum critical points: A review, In50 Years of the Renormalization Group, chap

    T . Senthil,Deconfined quantum critical points: A review, In50 Years of the Renormalization Group, chap. 14, pp. 169–195. World Scientific, doi:10.1142/9789811282386_0014 (2022)

    work page doi:10.1142/9789811282386_0014 2022

  32. [32]

    A. W . Sandvik,Evidence for deconfined quantum criticality in a two-dimensional heisenberg model with four-spin interactions, Phys. Rev. Lett.98, 227202 (2007), doi:10.1103/PhysRevLett.98.227202. 179

    work page doi:10.1103/physrevlett.98.227202 2007

  33. [33]

    R. K. Kaul and A. W . Sandvik,Lattice model for the SU(n)néel to valence-bond solid quantum phase transition at large n, Phys. Rev. Lett.108, 137201 (2012), doi:10.1103/PhysRevLett.108.137201

    work page doi:10.1103/physrevlett.108.137201 2012

  34. [34]

    Ghaemi and T

    P . Ghaemi and T . Senthil,Néel order , quantum spin liquids, and quantum criticality in two dimensions, Phys. Rev. B73, 054415 (2006), doi:10.1103/PhysRevB.73.054415

    work page doi:10.1103/physrevb.73.054415 2006

  35. [35]

    Vishwanath, L

    A. Vishwanath, L. Balents and T . Senthil,Quantum criticality and deconfinement in phase transitions between valence bond solids, Phys. Rev. B69, 224416 (2004), doi:10.1103/PhysRevB.69.224416

    work page doi:10.1103/physrevb.69.224416 2004

  36. [36]

    Fradkin, D

    E. Fradkin, D. A. Huse, R. Moessner, V . Oganesyan and S. L. Sondhi,Bipartite rokhsar–kivelson points and cantor deconfinement, Phys. Rev. B69, 224415 (2004), doi:10.1103/PhysRevB.69.224415

    work page doi:10.1103/physrevb.69.224415 2004

  37. [37]

    Grover and T

    T . Grover and T . Senthil,Topological spin hall states, charged skyrmions, and superconductivity in two dimensions, Phys. Rev. Lett.100, 156804 (2008), doi:10.1103/PhysRevLett.100.156804

    work page doi:10.1103/physrevlett.100.156804 2008

  38. [38]

    Affleck and J

    I. Affleck and J. B. Marston,Large-N limit of the Heisenberg-Hubbard model: Im- plications for high-T c superconductors, Physical Review B37(7), 3774 (1988), doi:10.1103/PhysRevB.37.3774

    work page doi:10.1103/physrevb.37.3774 1988

  39. [39]

    K. I. Kugel and D. I. Khomskii,Crystal-structure and magnetic properties of substances with orbital degeneracy, Soviet Physics JETP37, 725 (1973), Zh. Eksp. Teor. Fiz. 64, 1429 (1973)

    1973

  40. [40]

    Inagaki,Theory of magnetic structure in orbitally degenerate systems, Journal of the Physical Society of Japan39(3), 596 (1975), doi:10.1143/JPSJ.39.596

    Y. Inagaki,Theory of magnetic structure in orbitally degenerate systems, Journal of the Physical Society of Japan39(3), 596 (1975), doi:10.1143/JPSJ.39.596

    work page doi:10.1143/jpsj.39.596 1975

  41. [41]

    Y. Q. Li, M. Ma, D. N. Shi and F . C. Zhang,Su(4) theory for spin systems with orbital degeneracy, Phys. Rev. Lett.81, 3527 (1998), doi:10.1103/PhysRevLett.81.3527

    work page doi:10.1103/physrevlett.81.3527 1998

  42. [42]

    Azaria, A

    P . Azaria, A. O. Gogolin, P . Lecheminant and A. A. Nersesyan,One-dimensional su(4) spin-orbital model: A low-energy effective theory, Phys. Rev. Lett.83, 624 (1999), doi:10.1103/PhysRevLett.83.624

    work page doi:10.1103/physrevlett.83.624 1999

  43. [43]

    Frischmuth, F

    B. Frischmuth, F . Mila and M. Troyer,Thermodynamics of the one-dimensional su(4) symmetric spin-orbital model, Phys. Rev. Lett.82, 835 (1999), doi:10.1103/PhysRevLett.82.835

    work page doi:10.1103/physrevlett.82.835 1999

  44. [44]

    Corboz, A

    P . Corboz, A. M. Läuchli, K. Penc, M. Troyer and F . Mila,Simultaneous dimerization and su(4) symmetry breaking of 4-color fermions on the square lattice, Phys. Rev. Lett.107, 215301 (2011), doi:10.1103/PhysRevLett.107.215301

    work page doi:10.1103/physrevlett.107.215301 2011

  45. [45]

    Corboz, M

    P . Corboz, M. Lajkó, A. M. Läuchli, K. Penc and F . Mila,Spin-orbital quantum liquid on the honeycomb lattice, Phys. Rev. X2, 041013 (2012), doi:10.1103/PhysRevX.2.041013

    work page doi:10.1103/physrevx.2.041013 2012

  46. [46]

    Tokura \ and\ author N

    Y . Tokura and N. Nagaosa,Orbital physics in transition-metal oxides, Science288(5465), 462 (2000), doi:10.1126/science.288.5465.462, https://www.science.org/doi/pdf/10. 1126/science.288.5465.462

    work page doi:10.1126/science.288.5465.462 2000

  47. [47]

    Khaliullin,Orbital order and fluctuations in mott insulators, Progress of Theoretical Physics Supplement160, 155 (2005), doi:10.1143/PTPS.160.155, https://academic

    G. Khaliullin,Orbital order and fluctuations in mott insulators, Progress of Theoretical Physics Supplement160, 155 (2005), doi:10.1143/PTPS.160.155, https://academic. oup.com/ptps/article-pdf/doi/10.1143/PTPS.160.155/5162453/160-155.pdf. 180

    work page doi:10.1143/ptps.160.155 2005

  48. [48]

    M. G. Yamada, M. Oshikawa and G. Jackeli,Emergent SU(4)symmetry in α−zrcl3 and crystalline spin-orbital liquids, Phys. Rev. Lett.121, 097201 (2018), doi:10.1103/PhysRevLett.121.097201

    work page doi:10.1103/physrevlett.121.097201 2018

  49. [49]

    M. G. Yamada, M. Oshikawa and G. Jackeli,Su(4)-symmetric quantum spin-orbital liquids on various lattices, Phys. Rev. B104, 224436 (2021), doi:10.1103/PhysRevB.104.224436

    work page doi:10.1103/physrevb.104.224436 2021

  50. [50]

    W . M. H. Natori, R. Nutakki, R. G. Pereira and E. C. Andrade,Su(4) heisenberg model on the honeycomb lattice with exchange-frustrated perturbations: Implications for twistronics and mott insulators, Phys. Rev. B100, 205131 (2019), doi:10.1103/PhysRevB.100.205131

    work page doi:10.1103/physrevb.100.205131 2019

  51. [51]

    Lajkó and K

    M. Lajkó and K. Penc,Tetramerization in a su(4) heisenberg model on the honeycomb lattice, Phys. Rev. B87, 224428 (2013), doi:10.1103/PhysRevB.87.224428

    work page doi:10.1103/physrevb.87.224428 2013

  52. [52]

    H.-K. Jin, W . M. H. Natori and J. Knolle,Twisting the dirac cones of the su(4) spin-orbital liquid on the honeycomb lattice, Phys. Rev. B107, L180401 (2023), doi:10.1103/PhysRevB.107.L180401

    work page doi:10.1103/physrevb.107.l180401 2023

  53. [53]

    Vörös and K

    D. Vörös and K. Penc,Dynamical structure factor of the su(4) algebraic spin liquid on the honeycomb lattice, Phys. Rev. B108, 214407 (2023), doi:10.1103/PhysRevB.108.214407

    work page doi:10.1103/physrevb.108.214407 2023

  54. [54]

    Jin, R.-Y

    H.-K. Jin, R.-Y. Sun, H.-H. Tu and Y. Zhou,Unveiling a critical stripy state in the triangular-lattice su(4) spin-orbital model, Science Bulletin67(9), 918 (2022), doi:https://doi.org/10.1016/j.scib.2022.03.004

    work page doi:10.1016/j.scib.2022.03.004 2022

  55. [55]

    Keselman, L

    A. Keselman, L. Savary and L. Balents,Dimer description of the SU(4) antiferromagnet on the triangular lattice, SciPost Phys.8, 076 (2020), doi:10.21468/SciPostPhys.8.5.076

    work page doi:10.21468/scipostphys.8.5.076 2020

  56. [56]

    Zhang, H.-K

    C. Zhang, H.-K. Jin and Y. Zhou,Variational monte carlo approach to the su(4) spin-orbital model on the triangular lattice, Phys. Rev. B109, 125103 (2024), doi:10.1103/PhysRevB.109.125103

    work page doi:10.1103/physrevb.109.125103 2024

  57. [57]

    L. C. Biedenharn and J. D. Louck,Angular Momentum in Quantum Physics: Theory and Application, Encyclopedia of Mathematics and its Applications. Cambridge University Press (1984)

    1984

  58. [58]

    Balla,The Equation of Motion Method for Spin Systems with Multipolar Hamiltonians, Master’s thesis, Budapest University of Technology and Economics, Budapest, Hungary (2014)

    P . Balla,The Equation of Motion Method for Spin Systems with Multipolar Hamiltonians, Master’s thesis, Budapest University of Technology and Economics, Budapest, Hungary (2014)

    2014

  59. [59]

    Lindgård,Tables of products of tensor operators and stevens operators, Tech

    P .-A. Lindgård,Tables of products of tensor operators and stevens operators, Tech. Rep. Risoe-R-313, Risø National Laboratory , Roskilde, Denmark (1974)

    1974

  60. [60]

    Jensen and A

    J. Jensen and A. R. Mackintosh,Rare Earth Magnetism: Structures and Excitations, Clarendon Press, Oxford, Updated versions available through 2024 (1991)

    2024

  61. [61]

    Suzuki, H

    M.-T . Suzuki, H. Ikeda and P . M. Oppeneer,First-principles theory of magnetic multipoles in condensed matter systems, Journal of the Physical Society of Japan87(4), 041008 (2018), doi:10.7566/JPSJ.87.041008, https://doi.org/10.7566/JPSJ.87.041008

    work page doi:10.7566/jpsj.87.041008 2018

  62. [62]

    T . A. Tóth,Quadrupolar Ordering in Two-Dimensional Spin-One Systems, Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, doi:10.5075/epfl- thesis-5037, Thèse n° 5037 (2011). 181

    work page doi:10.5075/epfl- 2011

  63. [63]

    Kuramoto, H

    Y. Kuramoto, H. Kusunose and A. Kiss,Multipole orders and fluctuations in strongly correlated electron systems, Journal of the Physical Society of Japan78(7), 072001 (2009), doi:10.1143/JPSJ.78.072001, https://doi.org/10.1143/JPSJ.78.072001

    work page doi:10.1143/jpsj.78.072001 2009

  64. [64]

    Mila and F .-C

    F . Mila and F .-C. Zhang,On the origin of biquadratic exchange in spin 1 chains, The European Physical Journal B - Condensed Matter and Complex Systems16(1), 7 (2000), doi:10.1007/s100510070242

    work page doi:10.1007/s100510070242 2000

  65. [65]

    S.-T . Pi, R. Nanguneri and S. Savrasov,Calculation of multipolar exchange in- teractions in spin-orbital coupled systems, Phys. Rev. Lett.112, 077203 (2014), doi:10.1103/PhysRevLett.112.077203

    work page doi:10.1103/physrevlett.112.077203 2014

  66. [66]

    Hoffmann and S

    M. Hoffmann and S. Blügel,Systematic derivation of realistic spin models for beyond- heisenberg solids, Phys. Rev. B101, 024418 (2020), doi:10.1103/PhysRevB.101.024418

    work page doi:10.1103/physrevb.101.024418 2020

  67. [67]

    R. Soni, N. Kaushal, C. ¸ Sen, F . A. Reboredo, A. Moreo and E. Dagotto,Estimation of biquadratic and bicubic heisenberg effective couplings from multiorbital hubbard models, New Journal of Physics24(7), 073014 (2022), doi:10.1088/1367-2630/ac7b9c

    work page doi:10.1088/1367-2630/ac7b9c 2022

  68. [68]

    On analysis of steady flows of fluids with shear- dependent viscosity based on the Lipschitz truncation method

    K. Penc and A. M. Läuchli,Spin nematic phases in quantum spin systems, In C. Lacroix, P . Mendels and F . Mila, eds.,Introduction to Frustrated Magnetism: Materials, Experiments, Theory, pp. 331–362. Springer Berlin Heidelberg, Berlin, Heidelberg, doi:10.1007/978- 3-642-10589-0_13 (2011)

    work page doi:10.1007/978- 2011

  69. [69]

    Affleck, T

    I. Affleck, T . Kennedy, E. H. Lieb and H. Tasaki,Rigorous results on valence- bond ground states in antiferromagnets, Phys. Rev. Lett.59, 799 (1987), doi:10.1103/PhysRevLett.59.799

    work page doi:10.1103/physrevlett.59.799 1987

  70. [70]

    Tasaki,Physics and Mathematics of Quantum Many-Body Systems, Graduate Texts in Physics

    H. Tasaki,Physics and Mathematics of Quantum Many-Body Systems, Graduate Texts in Physics. Springer, Cham, ISBN 978-3-030-41264-7, doi:10.1007/978-3-030-41265-4 (2020)

    work page doi:10.1007/978-3-030-41265-4 2020

  71. [71]

    Pollmann, E

    F . Pollmann, E. Berg, A. M. Turner and M. Oshikawa,Symmetry protection of topological phases in one-dimensional quantum spin systems, Phys. Rev. B85, 075125 (2012), doi:10.1103/PhysRevB.85.075125

    work page doi:10.1103/physrevb.85.075125 2012

  72. [72]

    Läuchli, G

    A. Läuchli, G. Schmid and S. Trebst,Spin nematics correlations in bilinear-biquadratic s=1spin chains, Phys. Rev. B74, 144426 (2006), doi:10.1103/PhysRevB.74.144426

    work page doi:10.1103/physrevb.74.144426 2006

  73. [73]

    De Chiara, M

    G. De Chiara, M. Lewenstein and A. Sanpera,Bilinear-biquadratic spin-1 chain undergoing quadratic zeeman effect, Phys. Rev. B84, 054451 (2011), doi:10.1103/PhysRevB.84.054451

    work page doi:10.1103/physrevb.84.054451 2011

  74. [74]

    Niesen and P

    I. Niesen and P . Corboz,A tensor network study of the complete ground state phase diagram of the spin-1 bilinear-biquadratic Heisenberg model on the square lattice, SciPost Phys.3, 030 (2017), doi:10.21468/SciPostPhys.3.4.030

    work page doi:10.21468/scipostphys.3.4.030 2017

  75. [75]

    Kartsev, M

    A. Kartsev, M. Augustin, R. F . L. Evans, K. S. Novoselov and E. J. G. Santos,Biquadratic exchange interactions in two-dimensional magnets, npj Computational Materials6(1), 150 (2020), doi:10.1038/s41524-020-00416-1

    work page doi:10.1038/s41524-020-00416-1 2020

  76. [76]

    K. Ren, M. Wu, S.-S. Gong, D.-X. Yao and H.-Q. Wu,Haldane phases and phase diagrams of the s = 3 2 and s =1bilinear-biquadratic heisenberg model on the orthogonal dimer chain, Phys. Rev. B108, 245104 (2023), doi:10.1103/PhysRevB.108.245104. 182

    work page doi:10.1103/physrevb.108.245104 2023

  77. [77]

    Lian, S.-L

    L. Lian, S.-L. Yu, Z.-Y. Dong and J.-X. Li,Phase diagram and spin dynamics of the spin- 3 2 bilinear-biquadratic-bicubic heisenberg model on the square lattice, Phys. Rev. B pp. – (2026), doi:10.1103/ccyk-yzqm

    work page doi:10.1103/ccyk-yzqm 2026

  78. [78]

    E. M. Stoudenmire, S. Trebst and L. Balents,Quadrupolar correlations and spin freez- ing in s =1triangular lattice antiferromagnets, Phys. Rev. B79, 214436 (2009), doi:10.1103/PhysRevB.79.214436

    work page doi:10.1103/physrevb.79.214436 2009

  79. [79]

    Pohle, N

    R. Pohle, N. Shannon and Y. Motome,Spin nematics meet spin liquids: Exotic quantum phases in the spin-1 bilinear-biquadratic model with kitaev interactions, Phys. Rev. B107, L140403 (2023), doi:10.1103/PhysRevB.107.L140403

    work page doi:10.1103/physrevb.107.l140403 2023

  80. [80]

    Iwahara, Z

    N. Iwahara, Z. Huang, I. Neefjes and L. F . Chibotaru,Multipolar exchange interac- tion and complex order in insulating lanthanides, Phys. Rev. B105, 144401 (2022), doi:10.1103/PhysRevB.105.144401

    work page doi:10.1103/physrevb.105.144401 2022

Showing first 80 references.