Gapped topological spin-orbital liquid on the honeycomb lattice

Masahiko G. Yamada

REVIEW 3 major objections 5 minor 33 references

Gapped topological spin-orbital liquid on the honeycomb lattice

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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

Pith's one-line read The ground state of the SU(4) Heisenberg model on the honeycomb lattice is a gapped Z4 spin-orbital liquid, the paper argues, not the gapless Dirac liquid proposed earlier.

desk verdict Serious large-scale DMRG study that likely rules out the gapless Dirac spin liquid as the 2D ground state, but the Z4 topological label rests on an underdetermined three-point fit and should be treated as provisional. read the letter →

arxiv 2601.06549 v4 pith:73BCA6B6 submitted 2026-01-10 cond-mat.str-el

Masahiko G. Yamada This is my paper

classification cond-mat.str-el

keywords SU(4)Heisenbergmodelhoneycomblatticespin-orbitalliquidZ4topologicalorderentanglemententropydensitymatrixrenormalizationgroupquantumspinDirac

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Using density matrix renormalization group simulations with full SU(4) symmetry on honeycomb cylinders, the paper argues that the SU(4) Heisenberg model has a gapped spin-orbital liquid ground state with Z4 topological order in the two-dimensional limit. The main evidence is a topological entanglement entropy close to ln 4, the absence of any SU(4) or lattice symmetry breaking, and a ground-state energy per site of -0.9210(6), clearly lower than the competing pi-flux variational state. If correct, this settles a long-standing debate about the model and provides a two-dimensional quantum magnet with exact continuous symmetry that hosts topological order, making it a concrete target for cold-atom simulators and spin-orbital materials. The paper also identifies a gapless critical phase on narrow cylinders with central charge about 3, interpreting it as a quasi-one-dimensional remnant of the unstable Dirac spin-orbital liquid rather than the thermodynamic state.

What carries the argument

The load-bearing objects are the SU(4)-symmetric Heisenberg Hamiltonian built from nearest-neighbor swap operators P_ij = (2 S_i·S_j + 1/2)(2 T_i·T_j + 1/2), the full non-Abelian SU(4) implementation of DMRG in which the wavefunction is kept as sparse matrices labeled by Young-tableau irreps, and the finite-size scaling relation S_EE = a + b Ly + c cos(pi Ly/2). The constant a in this relation defines the topological entanglement entropy gamma_top = -a; its near-ln(4) value is the primary signature identifying Z4 order. The paper's phase separation by cylinder width is equally central: Ly=4 is a rung singlet, Ly=6 is gapless with central charge about 3, and Ly>=8 is gapped, which justifies u

What would settle it

Extend the cylinder simulations to Ly=14, 16, and 18 and check whether the entanglement entropy still follows a + b Ly + c cos(pi Ly/2) with the same a; alternatively, perform exact diagonalization on a torus with enough sites to resolve the ground-state degeneracy. If gamma_top moves away from ln 4 or the degeneracy is not 16, the Z4 identification would be ruled out.

Watch

Extended reading notes

Core claim

The central claim is that the SU(4) Heisenberg model on the honeycomb lattice, H = sum over nearest-neighbor bonds of the swap operator P_ij, is gapped and topologically ordered in the two-dimensional limit, with Z4 topological order. The paper reaches this through finite-size scaling of cylinder DMRG data: for circumferences Ly = 8, 10, and 12, the entanglement entropy follows S_EE = a + b Ly + c cos(pi Ly/2), and the constant a gives gamma_top = -a = 1.33(3), close to ln 4, the value expected for a Z4 liquid. No Anderson tower of states appears in the entanglement spectrum, ruling out spontaneous SU(4) breaking, and bond-operator fluctuations rule out translation-symmetry breaking. The the

Load-bearing premise

The topological-entanglement-entropy value comes from fitting exactly three cylinder widths (Ly=8, 10, 12) to S = a + b Ly + c cos(pi Ly/2), so with three points and three parameters the fit is exact and the resulting gamma_top = 1.33(3) depends entirely on that oscillatory ansatz being the correct form for larger Ly; if the mod-4 oscillation is wrong or the gapped phase does not persist beyond Ly=12, the Z4 conclusion collapses.

Editorial extensions

If this is right

If correct, the SU(4) honeycomb Heisenberg model becomes a concrete two-dimensional quantum magnet with exact continuous symmetry that realizes topological order, giving cold-atom emulators a specific ground state to target.
The inferred Z4 topological order implies a 16-fold ground-state degeneracy on a torus, a sharp signature that can be checked by exact diagonalization on finite clusters.
Because the phase is gapped and shows no symmetry breaking, a generalized Lieb-Schultz-Mattis theorem forces the ground state to be degenerate and topologically ordered, so the two observational facts (gap and disorder) already imply the topological nature.
The thermodynamic-limit energy lower than the pi-flux variational state rules out the previously proposed Dirac spin-orbital liquid as the true ground state in two dimensions, redirecting theoretical attention to gapped ansatze.
The gapless Ly=6 cylinder, with central charge close to 3, provides a controlled setting in which a Dirac spin-orbital liquid can be studied in quasi-one-dimensional form before it becomes unstable in two dimensions.

Reading between the lines

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

A directly testable extension would be to simulate Ly=14 and Ly=16 cylinders; if the entanglement entropy continues to follow a + b Ly + c cos(pi Ly/2) with the same a, the gamma_top about ln 4 result gains real support, while any drift would weaken the Z4 conclusion.
The paper leaves open the full anyon content of the suspected Z4 phase; a concrete next step is a cylinder transfer-matrix or infinite-DMRG computation of the modular matrices to verify the fractional statistics.
If the Z4 order is real, doping the model may produce exotic pairing, including the charge-4e superconductivity the author mentions; computing pairing correlations on a doped cylinder would be a natural numerical probe.
The same SU(N) symmetry techniques could be applied to SU(N) honeycomb models with N>4, where analogous gapped topological liquids may appear, but the finite-size separation by Ly would need to be redone before drawing conclusions.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a serious numerical attack on a long-standing problem, and it likely kills the gapless Dirac spin-orbital liquid as the 2D ground state. The more specific claim of Z4 topological order, however, rests on an underdetermined three-point fit; treat the Z4 label as provisional.

What is new and good: the DMRG campaign is genuinely large — full SU(4) symmetry, up to 12,800 multiplets, bond dimensions exceeding a million U(1) states. The 2D energy estimate −0.9210(6) sits well below the previous VMC value −0.894, which is meaningful. The identification of the Ly=6 cylinder as a gapless critical point with c=2.90(11), consistent with an SU(4)1 WZW theory, is a nice piece of finite-size analysis and gives a coherent picture: Dirac spin liquid as a critical remnant, gapped phase at larger widths. The paper is also honest about its caveats.

The soft spots, in order. First, the topological entanglement entropy: Eq. (6) has three parameters and is fit to three widths (Ly=8,10,12), so the fit is exact only because it is underdetermined. γtop=1.33(3) is not a measured value with that error bar; it is the intercept of an assumed ansatz. If the subleading correction is not the cosine form, or if the gapped regime does not persist, γtop could shift by several tenths. A Ly=14 or 16 point would settle this. Second, the conversion of 'gapped plus no symmetry breaking' into 'topological order' uses an LSM-type theorem from the author's own prior papers. That is acceptable if the theorem is solid, but it is load-bearing and not re-derived here. Third, the algorithm details are deferred to a future publication and no code is released; that limits reproducibility, though the main qualitative results are probably robust.

None of this is fatal to the paper's main thrust. The energy comparison and symmetry-breaking checks are solid, and a gapped spin-orbital liquid is the most plausible reading of the data. What is not established is the Z4 quantum number and the 16-fold degeneracy. The paper deserves a serious referee, and I would push for more cylinder widths and a systematic TEE analysis before accepting the topological label. I'd bring it to a reading group; people working on SU(N) magnets or spin liquids will want to see it.

Referee Report

3 major / 5 minor

Summary. The manuscript reports large-scale DMRG simulations of the SU(4) Heisenberg model on the honeycomb lattice, exploiting full SU(4) symmetry to reach bond dimensions up to 12,800 multiplets. The authors claim that the ground state is a gapped Z4 spin-orbital liquid, based on three pieces of evidence: (i) an extrapolated 2D energy E2D/N = -0.9210(6) below the π-flux VMC value -0.894, (ii) the absence of SU(4) and translation symmetry breaking as inferred from entanglement spectra and bond-operator fluctuations, and (iii) a topological entanglement entropy γtop = 1.33(3) close to ln(4). They also identify a gapless critical state at cylinder width Ly=6, which they interpret as a quasi-1D remnant of the unstable Dirac spin-orbital liquid. The central claim is that the thermodynamic-limit ground state is a gapped topological spin-orbital liquid with 16-fold degeneracy on a torus.

Significance. If the conclusion is correct, it resolves a long-standing open problem in SU(4) quantum magnetism and provides a concrete microscopic model of a gapped topological spin-orbital liquid, with relevance to cold-atom and material platforms. The paper makes a technical contribution by implementing SU(4)-symmetric DMRG at unprecedented bond dimensions, and the energy comparison with the previous VMC ansatz is a valuable step. However, the central claim currently rests on finite-size extrapolations that are underdetermined: the topological entanglement entropy comes from an exact three-parameter fit to three data points, and the energy extrapolation is similarly fragile. The significance is therefore conditional on additional numerical evidence that would make these extrapolations robust.

major comments (3)

[2D limit, Eq. (6)] The topological entanglement entropy is extracted from SEE at Ly=8,10,12 using SEE = a + bLy + c cos(πLy/2). With exactly three data points and three parameters, the fit is an interpolation, so γtop = -a has no statistical meaning; the quoted error 1.33(3) must originate from the Lx extrapolation, but this propagation is not described. The cosine term models a mod-4 oscillation inferred from the single Ly=10 point; any other subleading correction (e.g., 1/Ly, or a different oscillation period) would shift γtop by O(0.1-0.5). Since the Z4 topological-order claim rests on this value, the authors should add Ly=14 (and preferably Ly=16) data and test multiple fitting forms to demonstrate robustness.
[2D limit, Eq. (5) and Fig. 4(a)] The 2D energy E2D/N = -0.9210(6) is obtained from E/N = qL_y^{-p} + r with p=3.235. The manuscript does not specify which Ly values enter the fit. If only Ly=8,10,12 are used, the fit is exact (3 parameters, 3 points) and the error 0.0006 is not an extrapolation uncertainty. If Ly=4 and Ly=6 are included, the fit mixes the rung-singlet and gapless phases, which would bias the result. Because the energy comparison with VMC (-0.894) is used to rule out the Dirac spin liquid, this scaling must be made robust (e.g., by adding Ly=14,16 or by fixing p from a theoretical argument).
[Methods and Supplemental Material] The custom SU(4) DMRG algorithm, including the gauge fixing of subduction coefficients, the matrix-form wavefunction storage, and the eigenstate prediction, is described only as 'details to be discussed in the future publication.' The correctness of the central numerical result depends on this implementation, and earlier DMRG studies are criticized for limited bond dimension. To allow independent verification, the authors should provide a complete algorithmic description or release the code; deferring these details is insufficient for a paper whose main evidence is a large-scale numerical simulation.

minor comments (5)

[2D limit, Eq. (6)] The phrase 'An exact fit is achieved by γtop=1.33(3)' is misleading because the fit is an interpolation. Also, the origin of the error (3) should be explicitly stated (e.g., uncertainty from Lx extrapolation propagated through the fit).
[Fig. 3 and Sec. Simulations] The statement that bond-operator fluctuations 'decay when Ly gets larger' lacks quantitative support. A plot of the maximum fluctuation versus Ly, or a fitted decay length, would strengthen the absence-of-tetramerization claim.
[SM Sec. III] The central charge estimate c=2.90(11) for Ly=6 is based on a fit that the authors themselves call 'not very accurate'; the error bar and the limited bond dimension should be clearly visible in the figure, and the conclusion of gaplessness should be justified independently (e.g., by the nearly power-law correlation decay shown in Fig. S1(a)).
[Sec. 2D limit, LSM theorem] The inference 'gapped + no symmetry breaking ⇒ topological order' is based on the LSM-AYOJ theorem. The paper should state the precise assumptions of the theorem (e.g., projective representation per unit cell) and verify that they hold for the SU(4) honeycomb model, rather than only citing Refs. [10,11,18,19].
[Introduction and Methods] The text contains repeated phrasing ('compelling numerical evidence' appears twice) and some typos. Additionally, 'M=220 is sufficient until m=12800 and Ly=12' would be more convincing with a convergence plot of energy and entanglement entropy versus M.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central evidence is independent numerical data; self-citations are to established theorems and prior labels, not to fitted conclusions.

full rationale

The paper's central derivation chain is not circular. The gapped phase on Ly=8,10,12 is established by entanglement-entropy plateaus, exponential spin correlations, and the absence of SU(4) and translation symmetry breaking, all of which are direct DMRG outputs. The application of the Lieb-Schultz-Mattis-type theorem is a citation to a proven external result (with classical proofs in Refs [18,19] and SU(4) extensions in Refs [10,11]); citing a theorem does not reduce the argument to its own inputs. The 2D energy E2D/N=-0.9210(6) is compared against the external VMC value -0.894, an independent benchmark. The topological entanglement entropy γtop=1.33(3) is extracted by fitting SEE=a+bLy+c cos(pi Ly/2) to exactly three cylinder widths (Ly=8,10,12). This is an underdetermined finite-size fit and a real robustness concern: with three points and three parameters the fit is exact, so the extracted intercept is not a statistically protected prediction. However, this is a data-fitting/extrapolation limitation, not a circular reduction: γtop is not defined in terms of the Z4 conclusion, and the entropy values are independent raw data. The 'Z4' label is inherited from the author's prior work Ref [8], but it is presented as a characterization ('presumably', 'most consistent with') rather than derived by a chain that re-uses the conclusion. No equation is shown to be equivalent to its own input, and no fitted parameter is renamed as an independent prediction. Therefore no step satisfies the hard rule for circularity.

Assumptions & free parameters 7 free parameters · 6 assumptions · 1 invented entities

The central claim depends on several inputs from prior literature and on numerical convergence assumptions. The model itself has no free physical parameters (the exchange constant sets the energy scale), but the analysis introduces fit parameters (p, a, b, c, r), a hand-chosen irrep truncation αmax=9, and relies on the author's own published Z4 spin-orbital-liquid and LSM-type theorem papers. No machine-checked proofs or shipped code support the numerical claims.

free parameters (7)

a (entropy fit intercept) = -1.33(3)
Used to extract γtop=-a from SEE fit to Ly=8,10,12 with oscillatory term; this is the numerical basis for the Z4 topological order claim.
b (area-law coefficient) = not given
Fitted linear term in SEE vs Ly in Eq. (6).
c (oscillatory amplitude) = not given
Ad hoc oscillatory term added to account for the 'mod 4' effect; with only three data points the fit is exact.
p (energy finite-size exponent) = 3.235
Power-law exponent in E/N = q Ly^{-p} + r used to obtain the 2D energy; fitted to Ly=4,8,10,12 with mod-4 scatter.
r (2D energy per site) = -0.9210(6)
Extrapolated 2D ground-state energy; central benchmark against VMC -0.894.
αmax (irrep width truncation) = 9 (M=220 irreps)
DMRG truncation cutoff; authors state M=220 is sufficient until m=12800 but no systematic M-scan is shown.
m (bond dimension) = 12800 (Ly=8,10,12); 3200 (Ly=6)
DMRG bond dimension; results extrapolated in truncation error with an ad hoc error bar of 1/5 of the extrapolation difference.

assumptions (6)

domain assumption Lieb-Schultz-Mattis-Affleck-Yamada-Oshikawa-Jackeli theorem (Refs [10,11,18,19]): a gapped SU(4)-symmetric spin-orbital model without symmetry breaking has degenerate ground states and hence topological order.
Invoked in the 'absence of symmetry breaking automatically means degenerate' sentence in the main text (just before Fig. 4) to convert numerical absence of order into a topological phase.
domain assumption The π-flux Dirac spin-orbital liquid state from Corboz et al. (Ref [2]) is the only competing variational state; a lower energy rules it out.
Used to argue that E2D=-0.9210 below -0.894 'strongly disfavors' the Dirac liquid; assumes the VMC energy is a valid upper bound for the same thermodynamic limit.
domain assumption Calvera-Wang scenario that the Dirac spin-orbital liquid is unstable in 2D and appears as a proximate critical state (Ref [3]).
Used to interpret the Ly=6 gapless critical state as a quasi-1D remnant of the Dirac liquid rather than evidence against a gapped 2D phase.
ad hoc to paper DMRG truncation with αmax=9 (M=220 irreps) is sufficient to converge the quantities of interest for the reported cylinders.
Asserted in Methods ('We have checked that M=220 is sufficient...') but no systematic convergence data appear in the main text or Supplemental; this assumption underlies all reported extrapolations.
domain assumption The entanglement spectrum being 'random' (no Anderson tower of states) confirms the absence of SU(4) symmetry breaking.
The tower-of-states criterion is used as a diagnostic; absence of a tower is interpreted as absence of spontaneous symmetry breaking, an inference rather than a proof.
ad hoc to paper The modified SU(N) DMRG algorithm, including the gauge fixing and eigenstate prediction, is implemented correctly even though details are deferred to future publications.
The paper relies on a different gauge and a sparse-matrix trick; correctness is assumed, but the reader cannot verify the implementation without the deferred details.

invented entities (1)

Z4 topological spin-orbital liquid (gapped Z4 spin liquid ground state) independent evidence
purpose: Explains the numerical ground state of the SU(4) honeycomb Heisenberg model: gapped, no symmetry breaking, finite topological entropy, 16-fold degeneracy.
The paper provides a falsifiable handle: the predicted topological entanglement entropy γtop≈ln(4)≈1.386, which future independent simulations or quantum-simulation measurements could confirm or refute. However, the quoted value 1.33(3) is itself the output of a fit, so the handle is only as strong as the fit.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Gapped topological spin-orbital liquid on the honeycomb lattice." pith.science (2026). https://pith.science/paper/73BCA6B6

@misc{pith2026260106549,
  author       = {Pith},
  title        = {Pith review of: Gapped topological spin-orbital liquid on the honeycomb lattice},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73BCA6B6}},
  note         = {Machine review of arXiv:2601.06549}
}

abstract

We perform large-scale density matrix renormalization group simulations of the $\mathrm{SU}(4)$ Heisenberg model on the honeycomb lattice to address the long-standing question of its ground state in an unbiased and quantitatively controlled manner. We find reliable numerical evidence that the ground state is a gapped spin-orbital liquid, presumably with a $Z_4$ topological order, characterized by a finite topological entanglement entropy close to $\ln(4)$, the absence of both $\mathrm{SU}(4)$ and lattice symmetry breaking, and a variationally optimized ground-state energy well below the previously proposed $\pi$-flux variational state. By exploiting full $\mathrm{SU}(4)$ symmetry and keeping up to 12,800 $\mathrm{SU}(4)$ multiplets, corresponding to more than one million $\mathrm{U}(1)$ states, we achieve unprecedented accuracy for two-dimensional $\mathrm{SU}(4)$ quantum magnets. Finite-size scaling of energies and entanglement entropies supports a robust gapped phase in the two-dimensional limit, while a gapless critical state on narrow cylinders is identified as a proximate remnant of a Dirac spin-orbital liquid. Our results find the $\mathrm{SU}(4)$ honeycomb Heisenberg model a realization of a gapped topological spin-orbital liquid and provide convincing numerical evidence for topological order in a highly symmetric two-dimensional quantum magnet.

Figures

Figures reproduced from arXiv: 2601.06549 by the authors.

**Figure 2.** FIG. 2. (a) Entanglement spectrum of the SU(3) Heisenberg [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. (a) Finite size scaling of energy about [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

33 extracted references · 2 linked inside Pith

[1]

Savary and L

L. Savary and L. Balents, Quantum spin liquids: a re- view, Rep. Prog. Phys. 80, 016502 (2016)

2016
[2]

Corboz, M

P. Corboz, M. Lajk´ o, A. M. L¨ auchli, K. Penc, and F. Mila, Spin-Orbital Quantum Liquid on the Honey- comb Lattice, Phys. Rev. X 2, 041013 (2012)

2012
[4]

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

1992
[5]

I. P. McCulloch and M. Gul´ acsi, The non-Abelian density matrix renormalization group algorithm, Europhys. Lett. 57, 852 (2002)

2002
[6]

Nataf and F

P. Nataf and F. Mila, Density matrix renormalization group simulations of SU(N) Heisenberg chains using stan- dard Young tableaus: Fundamental representation and comparison with a ﬁnite-size Bethe ansatz, Phys. Rev. B 97, 134420 (2018)

2018
[7]

Chen, J.-W

J.-Y. Chen, J.-W. Li, P. Nataf, S. Capponi, M. Mambrini, K. Totsuka, H.-H. Tu, A. Weichselbaum, J. von Delft, and D. Poilblanc, Abelian su( n)1 chiral spin liquids on the square lattice, Phys. Rev. B 104, 235104 (2021)

2021
[8]

M. G. Yamada and S. Fujimoto, Thermodynamic signa- ture of SU(4) spin-orbital liquid and symmetry fraction- alization from Lieb-Schultz-Mattis theorem, Phys. Rev. B 105, L201115 (2022)

2022
[9]

M. A. Cazalilla and A. M. Rey, Ultracold Fermi gases with emergent SU(N) symmetry, Rep. Prog. Phys. 77, 124401 (2014). 5

2014

Show all 33 references

[10]

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)

2018
[11]

M. G. Yamada, M. Oshikawa, and G. Jackeli, SU(4)- symmetric quantum spin-orbital liquids on various lat- tices, Phys. Rev. B 104, 224436 (2021)

2021
[13]

[3, 6, 12, 23–28]

See Supplemental Material at [URL will be inserted by publisher] for more details, which includes Refs. [3, 6, 12, 23–28]
[14]

Sharma and G

S. Sharma and G. K.-L. Chan, Spin-adapted density ma- trix renormalization group algorithms for quantum chem- istry, J. Chem. Phys. 136, 124121 (2012)

2012
[15]

S.-S. Gong, D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Phase diagram of the spin- 1 2 J1-J2 Heisenberg model on a honeycomb lattice, Phys. Rev. B 88, 165138 (2013)

2013
[16]

M. A. Metlitski and T. Grover, Entanglement Entropy of Systems with Spontaneously Broken Continuous Sym- metry (2015), arXiv:1112.5166 [cond-mat.str-el]

2015 arXiv
[17]

Bauer, P

B. Bauer, P. Corboz, A. M. L¨ auchli, L. Messio, K. Penc, M. Troyer, and F. Mila, Three-sublattice order in the SU(3) Heisenberg model on the square and triangular lattice, Phys. Rev. B 85, 125116 (2012)

2012
[18]

E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys. 16, 407 (1961)

1961
[19]

Aﬄeck and E

I. Aﬄeck and E. H. Lieb, A proof of part of Haldane’s conjecture on spin chains, Lett. Math. Phys. 12, 57 (1986)

1986
[20]

H.-K. Jin, W. M. H. Natori, and J. Knolle, Twisting the Dirac cones of the SU(4) spin-orbital liquid on the hon- eycomb lattice, Phys. Rev. B 107, L180401 (2023)

2023
[21]

Anderson, Resonating valence bonds: A new kind of insulator?, Mater

P. Anderson, Resonating valence bonds: A new kind of insulator?, Mater. Res. Bull. 8, 153 (1973)

1973
[22]

P. W. Anderson, The Resonating Valence Bond State in La 2CuO4 and Superconductivity, Science 235, 1196 (1987)

1987
[24]

H. S. Wilf, A uniﬁed setting for sequencing, ranking, and selection algorithms for combinatorial objects, Adv. Math. 24, 281 (1977)

1977
[25]

A. G. Rao and N. R. Shankar, Graphical Approach to Index and Retrieve Standard Young Tableaux, Int. J. Comput. Appl. 111, 10 (2015)

2015
[26]

Weichselbaum, QSpace - An open-source tensor li- brary for Abelian and non-Abelian symmetries, SciPost Phys

A. Weichselbaum, QSpace - An open-source tensor li- brary for Abelian and non-Abelian symmetries, SciPost Phys. Codebases , 40 (2024)

2024
[28]

Topological Z4 spin-orbital liquid on the honeycomb lattice

T. Ziman and H. J. Schulz, Are antiferromagnetic spin chains representations of the higher Wess-Zumino- Witten models?, Phys. Rev. Lett. 59, 140 (1987). Supplemental Material for “Topological Z4 spin-orbital liquid on the honeycomb lattice” Masahiko G. Yamada 1 1Department of ...

1987
[29]

Nataf and F

P. Nataf and F. Mila, Density matrix renormalization group sim ulations of SU(N) Heisenberg chains using standard Young tableaus: Fundamental representation and comparison with a ﬁn ite-size Bethe ansatz, Phys. Rev. B 97, 134420 (2018)

2018
[30]

https://github.com/simple-dmrg/simple-dmrg
[31]

J. Ping, F. Wang, and J. Chen, Group Representation Theory For Physicists (2nd Edition) (World Scientiﬁc Publishing Company, Singapore, 2002)

2002
[32]

H. S. Wilf, A uniﬁed setting for sequencing, ranking, and sel ection algorithms for combinatorial objects, Adv. Math. 24, 281 (1977)

1977
[33]

A. G. Rao and N. R. Shankar, Graphical Approach to Index and Re trieve Standard Young Tableaux, Int. J. Comput. Appl. 111, 10 (2015)

2015
[34]

Weichselbaum, QSpace - An open-source tensor library for A belian and non-Abelian symmetries, SciPost Phys

A. Weichselbaum, QSpace - An open-source tensor library for A belian and non-Abelian symmetries, SciPost Phys. Codebases , 40 (2024)

2024
[35]

Calabrese and J

P. Calabrese and J. Cardy, Entanglement entropy and conformal ﬁeld theory, J. Phys. A: Math. Theor. 42, 504005 (2009)

2009
[36]

Ziman and H

T. Ziman and H. J. Schulz, Are antiferromagnetic spin chains re presentations of the higher Wess-Zumino-Witten models?, Phys. Rev. Lett. 59, 140 (1987)

1987
[37]

Calvera and C

V. Calvera and C. Wang, Theory of Dirac Spin-Orbital Liquids: m onopoles, anomalies, and applications to SU (4) honeycomb models (2021), arXiv:2103.13405 [cond-mat.str-el]

2021 arXiv

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Savary and L

L. Savary and L. Balents, Quantum spin liquids: a re- view, Rep. Prog. Phys. 80, 016502 (2016)

2016

[2] [2]

Corboz, M

P. Corboz, M. Lajk´ o, A. M. L¨ auchli, K. Penc, and F. Mila, Spin-Orbital Quantum Liquid on the Honey- comb Lattice, Phys. Rev. X 2, 041013 (2012)

2012

[3] [4]

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

1992

[4] [5]

I. P. McCulloch and M. Gul´ acsi, The non-Abelian density matrix renormalization group algorithm, Europhys. Lett. 57, 852 (2002)

2002

[5] [6]

Nataf and F

P. Nataf and F. Mila, Density matrix renormalization group simulations of SU(N) Heisenberg chains using stan- dard Young tableaus: Fundamental representation and comparison with a ﬁnite-size Bethe ansatz, Phys. Rev. B 97, 134420 (2018)

2018

[6] [7]

Chen, J.-W

J.-Y. Chen, J.-W. Li, P. Nataf, S. Capponi, M. Mambrini, K. Totsuka, H.-H. Tu, A. Weichselbaum, J. von Delft, and D. Poilblanc, Abelian su( n)1 chiral spin liquids on the square lattice, Phys. Rev. B 104, 235104 (2021)

2021

[7] [8]

M. G. Yamada and S. Fujimoto, Thermodynamic signa- ture of SU(4) spin-orbital liquid and symmetry fraction- alization from Lieb-Schultz-Mattis theorem, Phys. Rev. B 105, L201115 (2022)

2022

[8] [9]

M. A. Cazalilla and A. M. Rey, Ultracold Fermi gases with emergent SU(N) symmetry, Rep. Prog. Phys. 77, 124401 (2014). 5

2014