Vacancy induced expansion of spin-liquid regime in J1-J2 Heisenberg model

Anish Koley; Sanjeev Kumar; Soumyaranjan Dash

REVIEW 3 major objections 5 minor 45 references

Vacancy induced expansion of spin-liquid regime in J1-J2 Heisenberg model

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Mobile vacancies widen the spin-liquid regime of the square-lattice J1–J2 Heisenberg antiferromagnet.

desk verdict A novel semi-classical Monte Carlo with dynamic singlet dimers gives a qualitatively reasonable no-vacancy J1-J2 phase diagram and a plausible vacancy broadening, but the raw structure-factor diagnostics cannot yet separate the effect from trivial dilution. read the letter →

arxiv 2507.20561 v1 pith:KFEB3Q5H submitted 2025-07-28 cond-mat.str-el

Soumyaranjan Dash , Anish Koley , Sanjeev Kumar This is my paper · ORCID

classification cond-mat.str-el

keywords spinliquidJ1-J2HeisenbergmodelvacancydopingsemiclassicalMonteCarlosingletdimersfrustratedmagnetismsquarelattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a semiclassical Monte Carlo method for frustrated quantum antiferromagnets in which singlet dimers are treated as classical degrees of freedom that can form, break, and rotate. Using it for the spin-1/2 J1–J2 Heisenberg model on the square lattice, the paper recovers the known qualitative physics: Néel order for J2/J1 below 0.5, stripe order above, and an intermediate nonmagnetic spin-liquid-like regime near J2/J1 = 0.5. It then adds spin vacancies and finds that the low-temperature nonmagnetic window widens monotonically as the vacancy concentration increases. The authors argue this vacancy-induced broadening is an entropy effect and propose mobile vacancies as a route to stabilize spin-liquid ground states. The value of the claim, if right, is experimental: doping or optically realizing mobile vacancies could turn a magnetically ordered material into a spin-liquid candidate.

What carries the argument

The load-bearing object is the semi-classical Hamiltonian Hsc = J1∑⟨ij⟩∈U SiSj + J2∑⟨⟨ij⟩⟩∈U SiSj + Eb(J1)∑⟨ij⟩∈C bij + Eb(J2)∑⟨⟨ij⟩⟩∈C bij, where lattice sites are dynamically partitioned into uncorrelated classical spins (U) and correlated singlet sites (C), and bij are binary bond variables indicating whether a singlet occupies a bond. The dimer energy Eb(Jn) = −(3Jn/4)(1 − e−Jn/T)/(1 + 3e−Jn/T) is the thermal expectation value of an isolated pair of spins, and Monte Carlo moves create, annihilate, or rotate dimers while enforcing one-dimer-per-site occupancy. This machinery carries the argument because it lets singlet formation compete with classical magnetic order through both energy and entropy, and it handles vacancies simply by setting the spin magnitude to vi/2 with vi ∈ {0,1} and forbidding dimers on vacant bonds.

What would settle it

Compute the ground-state phase diagram of the spin-1/2 J1–J2 model with one or two vacancies on finite clusters using an unbiased method that does not rely on the isolated-dimer energy, such as exact diagonalization. If the nonmagnetic window does not broaden monotonically with vacancy density—or if vacancies pin Néel order—the central claim fails.

Watch

Extended reading notes

Core claim

The central discovery, stated on the paper's terms, is that the spin-liquid regime of the square-lattice spin-1/2 J1–J2 Heisenberg antiferromagnet expands when a fraction of sites are empty. In the semiclassical simulation, the region with no conventional magnetic order at low temperature grows monotonically with vacancy concentration ρ = 0, 0.1, 0.2, 0.3, 0.4 (Fig. 5), pushing the Néel and stripe boundaries apart. In the spin-liquid regime vacancies stay mobile and disordered, whereas in the Néel phase they cluster and in the stripe phase they order in a staggered pattern; the extra entropy available to the disordered state is what stabilizes it. The paper also shows that the same semiclassical method, without vacancies, reproduces the qualitative phase diagram of the clean model, including the order-by-disorder narrowing of the all-dimer ground state at finite temperature. This is proposed as a new route—tuning J2/J1 by pressure in doped Mott insulators or using optical lattices with mobile spin vacancies—toward realizing spin-liquid ground states.

Load-bearing premise

The load-bearing premise is that a correlated singlet pair can be replaced by an independent classical dimer whose energy is the isolated-pair value Eb(Jn); if singlet–neighbour couplings are not negligible, the vacancy-induced broadening could be an artifact of this replacement.

Editorial extensions

If this is right

If the claim is right, doping square-lattice J1–J2 antiferromagnets with mobile vacancies is a practical knob for expanding the parameter window in which a spin-liquid ground state appears.
The monotonic widening with ρ means higher vacancy content should suppress magnetic order over a broader J2/J1 range, a trend that can be tested in doped Mott insulators under pressure.
Distinct vacancy patterns—clustering in the Néel phase, staggered ordering in the stripe phase, and persistent disorder in the spin-liquid phase—give observable real-space signatures of which phase is realized.
The semiclassical method itself is a general tool: it applies to short-range frustrated spin models with disorder where quantum Monte Carlo suffers from the sign problem.

Reading between the lines

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

An extension the paper leaves implicit is that quenched (immobile) vacancies should not produce the same broadening, since the entropy gain requires vacancy mobility; comparing mobile versus pinned vacancy distributions would isolate the proposed mechanism.
Because Eb(Jn) is computed for an isolated pair, the predicted width of the spin-liquid window may be quantitatively too generous; benchmarking against exact small-cluster calculations with one or two vacancies would calibrate how much of the broadening is real.
The mechanism suggests a link to randomness-induced spin-liquid work: positional entropy from mobile vacancies may act alongside random bond disorder, so materials with both quenched disorder and mobile carriers might show the effect at lower ρ.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

You should know this paper for its method, not for its central claim. The authors put classical spins and dynamic singlet dimers into one Monte Carlo scheme, with dimer energies taken from the isolated two-spin problem. It has no fitted parameters, reproduces the expected Néel and stripe phases without vacancies, and produces a nonmagnetic intermediate window that they call a spin-liquid. That benchmark is a real plus: the method is simple, transparent, and qualitatively tracks known physics at zero doping. With vacancies, they see that the nonmagnetic window widens as vacancy density grows. This is a new observation within this model, and the vacancy clustering and ordering patterns are interesting side findings.

The soft spots are serious, though. The dimer energy ignores any interaction between a singlet and the surrounding classical moments; the singlet is just taken out of the spin lattice. That mapping is uncontrolled and could easily bias the phase boundaries. Second, the label "spin liquid" is too strong for an all-dimer state in a classical effective model. Without a quantum treatment, it is at best a nonmagnetic valence-bond-like regime, not a demonstrated spin liquid. Third, and most important for the vacancy claim, the phase boundaries are read from raw S(pi,pi) and S(pi,0)/S(0,pi). A random dilution of any antiferromagnet reduces raw structure-factor values roughly as (1-rho)^2 and lowers the apparent ordering temperature, so the broadening in Figure 5 may be mostly a detection artifact. The entropy argument in the text is plausible but the data as presented cannot separate it from trivial dilution. The authors need to normalize per occupied site, show error bars, and compare against a control where the dimer mechanism is disabled or vacancies are pinned.

That said, the paper is honest about its semi-classical limits and does not claim to settle the J1-J2 debate. The method is worth having in the toolbox, and the vacancy question is worth pursuing with better diagnostics. I would send it to a referee, but with the expectation of major revision: the dilution artifact must be addressed, and the language about spin liquids needs to be tamed.

Referee Report

3 major / 5 minor

Summary. The paper proposes a semiclassical Monte Carlo approach for the spin-1/2 J1-J2 Heisenberg model on the square lattice with spin vacancies. In this approach, uncorrelated spins are treated as classical vectors, while correlated spin pairs are replaced by binary dimer variables with a temperature-dependent energy taken from the exact finite-temperature internal energy of an isolated singlet pair (Eq. (3)). The authors first show that, without vacancies, the method qualitatively reproduces the known phase diagram of the J1-J2 model: Neel order for small J2/J1, stripe order for large J2/J1, and an intermediate all-dimer nonmagnetic region. They then introduce mobile vacancies and report, in Figs. 4 and 5, that this intermediate nonmagnetic region broadens monotonically with vacancy concentration rho, and they attribute this effect to an enhanced entropy of the dimerized state in the presence of mobile vacancies. The paper concludes that doping with mobile vacancies may be a route to stabilize spin-liquid ground states in square-lattice antiferromagnets.

Significance. If the central claim were quantitatively established, the paper would offer a new and experimentally relevant route to stabilizing spin-liquid behavior in frustrated square-lattice magnets, and the semiclassical method could be a useful low-cost tool for disordered frustrated spin systems. The manuscript includes concrete Monte Carlo data, reproducible algorithmic details in the Supplemental Material, and a no-vacancy test that matches qualitative expectations. However, the central claim is currently supported by raw spin structure factors that are trivially suppressed by dilution, by a spin-liquid identification that is largely an assumption built into the effective model, and by simulations at vacancy concentrations approaching the percolation threshold. These issues make the central physical conclusion, as stated, not yet convincing, although they are addressable with additional analysis and more careful framing.

major comments (3)

[Effect of spin vacancies; Eq. (S1)] The raw spin structure factor S(q), defined in Eq. (S1), is not normalized by the number of occupied sites. For a perfectly ordered Neel state with randomly distributed vacancies, S(pi,pi) is proportional to (1-rho)^2, so the monotonic decrease of S(pi,pi) with rho shown in Fig. 3(a) is exactly the behavior expected from dilution alone, independent of any spin-liquid or dimer-entropy mechanism. The phase boundaries in Fig. 5 are therefore potentially tracking the weakening of conventional order by dilution rather than an expansion of a genuinely nonmagnetic regime. To support the paper's claim, the authors should present a per-occupied-site normalized order parameter, such as S(q)/(1-rho)^2 or the squared staggered magnetization per occupied site, and should compare with a control simulation in which the dimer channel is disabled or the dimer energy is artificially set to zero, so that the proposed vacancy-entropy mechanism can be isolated.
[Model Hamiltonian and the Monte Carlo approach; Eqs. (2)-(3)] The intermediate nonmagnetic state in this model is a classical all-dimer state in which the binary variables b_ij are static and have no quantum resonance or kinetic energy. Identifying this state with a quantum spin liquid is an assumption built into the effective model, not a result derived from it; the expanded region in Fig. 5 is, strictly speaking, an expansion of a classical dimerized valence-bond-solid-like phase. The paper should either provide a diagnostic that distinguishes a spin liquid from a valence-bond solid within the model, such as the dimer structure factor or the absence of long-range dimer order, or should explicitly label the regime as a 'nonmagnetic/dimerized regime' rather than a 'spin-liquid regime'.
[Figs. 4 and 5; vacancy concentration] The simulations include rho = 0.4, which is very close to the site-percolation threshold for vacancies on the square lattice (pc ~ 0.4073). At these densities, long-range Neel and stripe order disappear for purely geometric reasons, independent of any spin-liquid physics. The monotonic broadening of the nonmagnetic region with rho in Fig. 5 may therefore be, at least in part, a percolation artifact. The authors should either restrict the analysis to rho well below pc (for example, rho <= 0.2), or explicitly demonstrate that the broadening persists when compared with a nonmagnetic model without dimer degrees of freedom, and they should discuss the role of percolation at the concentrations studied.

minor comments (5)

[Model Hamiltonian and the Monte Carlo approach] The local constraint that each site can participate in at most one dimer is stated only in the Supplemental Material; it should be stated in the main text immediately after Eq. (2), since it is essential to the model.
[Supplemental Material D; main text phase diagram construction] The procedure for locating phase boundaries is described inconsistently: the main text refers to inflection points in the spin structure factor and bright spots in the specific-heat map, while the Supplemental states that boundaries are estimated from specific-heat peaks. Please unify the criterion and state it clearly in the main text.
[Fig. 3 caption] The caption for panel (b) lists 'S(0,pi)/S(pi,0)' while the text describes S(0,pi) and S(pi,0) as separate open and filled symbols; the caption should be corrected to describe both components.
[Supplemental Material A; equilibrium of vacancies] The vacancy-exchange moves used in the Monte Carlo updates mean that the system is annealed over vacancy positions, corresponding to mobile vacancies in equilibrium rather than quenched disorder. The text should clarify that the results describe annealed disorder and that the vacancy positions are part of the thermal ensemble.
[Fig. 2(d) and related text] The term 'order-by-thermal-disorder' used to explain the narrowing of the all-dimer state is not defined; a brief explanatory sentence would help readers understand the mechanism.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the semiclassical Hamiltonian is an explicit ansatz benchmarked against external methods, and the vacancy-induced broadening is a parameter-free Monte Carlo result.

full rationale

I walked the derivation chain and found no step in which a prediction reduces by construction to its input. The semiclassical Hamiltonian Hsc in Eq. (2) is introduced explicitly as an ansatz: sites are partitioned into classical U sites and correlated C sites, with C sites carrying no local moment and b_ij representing singlet bonds. The bond energy Eb(J_n) in Eq. (3) is the exact thermal energy of an isolated antiferromagnetic pair, not a parameter fitted to the phase diagram or to the vacancy data. The existence of an intermediate nonmagnetic all-dimer state is indeed a consequence of this ansatz, but the paper does not present that existence as a first-principles derivation; it states that the approach 'effectively capture[s]' a known feature of the quantum problem and explicitly disclaims settling the SL debate: 'Clearly, our approximation is not tailored to settle the ongoing debate regarding the range of stability of the SL phase.' The central new claim, vacancy-induced broadening of the nonmagnetic regime, is obtained from Monte Carlo simulations with no fitted constants; the dynamics of dimer formation, vacancy mobility, and entropy are simulated rather than inserted as the final phase diagram. The only self-citation, Ref. [33] for a similar semiclassical approach in Kondo lattices, is methodological provenance and is not load-bearing; no uniqueness theorem or ansatz is imported through it. The concern that raw S(q) is diluted by the (1-rho)^2 normalization because S(q) is defined with total N and S_i = v_i/2 is a legitimate correctness risk, but the phase boundaries in the paper are estimated from specific-heat peaks (Supplement D), and the dilution scaling is not equated with the broadening claim by any equation in the paper. I therefore find no circular step under the stated evidentiary standard.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central claim rests on no fitted numerical parameters: J1, J2, rho, and T are inputs, and Eb is the exact thermal energy of an isolated spin pair. What the claim rests on instead are modeling assumptions about how quantum singlets are represented, how they interact with the surrounding classical spins, and what the nonmagnetic dimer state means physically. The mobile-vacancy assumption is explicitly acknowledged by the authors as the main experimental challenge.

assumptions (5)

ad hoc to paper Correlated spin pairs are replaced by classical dimer bonds with energy Eb(Jn) from Eq. (3), independent of the neighboring spin configuration.
This is the defining ansatz of Hsc in Eq. (2). It is exact only for an isolated dimer and neglects interactions between a singlet and the surrounding classical moments; the paper offers no controlled expansion or benchmark for the vacancy case.
ad hoc to paper The all-dimer nonmagnetic state of the effective model is identified as a spin-liquid regime.
Phase diagrams in Figs. 2(d), 4(d), and 5 label the no-order region 'SL' based only on spin structure factor and dimer fraction; no quantum diagnostic such as entanglement entropy or fractionalized excitations is computed.
domain assumption The classical partition function over spins and dimer variables, sampled by Metropolis Monte Carlo, represents the quantum partition function of H in Eq. (1).
Standard classical statistical mechanics is applied to Hsc; the mapping from the quantum Hamiltonian to this classical effective model is assumed rather than derived.
domain assumption Vacancies are mobile, implemented through spin-vacancy exchange moves.
The experimental route in the discussion requires mobile vacancies; in real site-diluted magnets vacancies are typically quenched. The authors explicitly acknowledge this as the main challenge, yet all broadening results are computed with mobile vacancies.
domain assumption Finite-size lattices with L=40 and 60 and periodic boundary conditions approximate the thermodynamic limit.
Standard assumption in Monte Carlo studies; the paper does not show a systematic finite-size scaling analysis to confirm that the reported phase boundaries are converged.

invented entities (1)

Classical singlet dimer variable bij
purpose: To describe quantum singlet formation as an independent binary degree of freedom in the effective Hamiltonian.
The dimer variables are introduced in Eq. (2) as an effective construct. The energy function Eb is exact for an isolated pair, but the independent treatment of dimers has no falsifiable handle outside the model; the qualitative nonmagnetic phase is governed by these degrees of freedom.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Vacancy induced expansion of spin-liquid regime in J1-J2 Heisenberg model." pith.science (2026). https://pith.science/paper/KFEB3Q5H

@misc{pith2026250720561,
  author       = {Pith},
  title        = {Pith review of: Vacancy induced expansion of spin-liquid regime in J1-J2 Heisenberg model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFEB3Q5H}},
  note         = {Machine review of arXiv:2507.20561}
}

read the original abstract

We study the model for spin-1/2 J1-J2 Heisenberg antiferromagnets on a square lattice in the presence of spin vacancies. In order to overcome the methodological challenges associated with analyzing models with magnetic frustration and inhomogeneities, we introduce a new semi-classical approach in which singlet dimers are treated as effective classical degrees of freedom. The energetic and entropic aspects of the dimer formation are included via a classical Monte Carlo scheme that allows for the dynamical conversion of spin pairs into dimers and vice versa. We show that our semi-classical approach recovers the qualitative physics of the J1-J2 model in the absence of vacancies. The vacancies lead to a broadening of the spin-liquid regime between the N\'eel and the stripe antiferromagnetic phases. This suggests a possible new route to discover spin-liquid ground states by tuning the J2/J1 ratio in doped square lattice antiferromagnets.

Figures

Figures reproduced from arXiv: 2507.20561 by the authors.

**Figure 2.** (Color online) Real-space patterns of spins (red: [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** (Color online) Real-space patterns of spins (red: [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 35 canonical work pages

[1]

Broholm, R

C. Broholm, R. J. Cava, S. Kivelson, D. Nocera, M. Nor- man, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020)

2020
[2]

Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017)

2017
[3]

Savary and L

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

2016
[4]

Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

L. Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

2010
[5]

Yu and Y.-J

J.-F. Yu and Y.-J. Kao, Spin-1/2 j1-j2 heisenberg antifer- romagnet on a square lattice: A plaquette renormalized tensor network study, Phys. Rev. B 85, 094407 (2012)

work page 2012
[6]

Jiang, H

H.-C. Jiang, H. Yao, and L. Balents, Spin liquid ground state of the spin-1/2 square j1-j2 heisenberg model, Phys. Rev. B 86, 024424 (2012)

work page 2012
[7]

Morita, R

S. Morita, R. Kaneko, and M. Imada, Quantum spin liquid in spin 1/2 j1–j2 heisenberg model on square lattice: Many-variable variational monte carlo study combined with quantum-number pro- jections, https://doi.org/10.7566/JPSJ.84.024720 84, 10.7566/JPSJ.84.024720 (2015)

doi:10.7566/jpsj.84.024720 2015
[8]

Richter and J

J. Richter and J. Schulenburg, The spin-1/2 j1–j2 heisen- berg antiferromagnet on the square lattice: Exact diago- nalization for n=40 spins, The European Physical Jour- nal B 2009 73:1 73, 117 (2009)

work page 2009

Show all 45 references

[9]

Richter, R

J. Richter, R. Zinke, and D. J. Farnell, The spin-1/2 square-lattice j1-j2 model: the spin-gap issue, The Eu- ropean Physical Journal B 2015 88:1 88, 1 (2015)

2015
[10]

Qian and M

X. Qian and M. Qin, Absence of spin liquid phase in the j1-j2 heisenberg model on the square lattice, Phys. Rev. B 109, L161103 (2024)

2024
[11]

W.-Y. Liu, S. Dong, C. Wang, Y. Han, H. An, G.-C. Guo, and L. He, Gapless spin liquid ground state of the spin- 1 2 J1−J2 heisenberg model on square lattices, Phys. Rev. B 98, 241109 (2018)

2018
[12]

W.-J. Hu, F. Becca, A. Parola, and S. Sorella, Direct evidence for a gapless Z2 spin liquid by frustrating n´ eel antiferromagnetism, Phys. Rev. B 88, 060402 (2013)

2013
[13]

T. Li, F. Becca, W. Hu, and S. Sorella, Gapped spin- liquid phase in theJ1−J2 heisenberg model by a bosonic resonating valence-bond ansatz, Phys. Rev. B 86, 075111 (2012)

2012
[14]

Wang and A

L. Wang and A. W. Sandvik, Critical level crossings and gapless spin liquid in the square-lattice spin-1 /2 J1−J2 heisenberg antiferromagnet, Phys. Rev. Lett.121, 107202 (2018)

2018
[15]

Ferrari and F

F. Ferrari and F. Becca, Gapless spin liquid and valence- bond solid in the J1 J2 heisenberg model on the square lattice: Insights from singlet and triplet excitations, Phys. Rev. B 102, 014417 (2020)

2020
[16]

S.-S. Gong, W. Zhu, D. N. Sheng, O. I. Motrunich, and M. P. A. Fisher, Plaquette ordered phase and quantum phase diagram in the spin- 1 2 J1−J2 square heisenberg model, Phys. Rev. Lett. 113, 027201 (2014)

2014
[17]

Mezzacapo, Ground-state phase diagram of the quan- tum J1−J2 model on the square lattice, Phys

F. Mezzacapo, Ground-state phase diagram of the quan- tum J1−J2 model on the square lattice, Phys. Rev. B 86, 045115 (2012)

2012
[18]

O. P. Sushkov, J. Oitmaa, and Z. Weihong, Quantum phase transitions in the two-dimensional J1−j2 model, Phys. Rev. B 63, 104420 (2001)

2001
[19]

Coldea, S

R. Coldea, S. M. Hayden, G. Aeppli, T. G. Perring, C. D. Frost, T. E. Mason, S.-W. Cheong, and Z. Fisk, Spin waves and electronic interactions in la2cuo4, Phys. Rev. Lett. 86, 5377 (2001)

2001
[20]

Carretta, N

P. Carretta, N. Papinutto, C. B. Azzoni, M. C. Moz- zati, E. Pavarini, S. Gonthier, and P. Millet, Frustration- driven structural distortion in vomoo4, Phys. Rev. B 66, 094420 (2002)

2002
[21]

Melzi, P

R. Melzi, P. Carretta, A. Lascialfari, M. Mambrini, M. Troyer, P. Millet, and F. Mila, li2VO(si,ge )O4, a pro- totype of a two-dimensional frustrated quantum heisen- berg antiferromagnet, Phys. Rev. Lett. 85, 1318 (2000)

2000
[22]

Bombardi, J

A. Bombardi, J. Rodriguez-Carvajal, S. Di Matteo, F. de Bergevin, L. Paolasini, P. Carretta, P. Millet, and R. Caciuffo, Direct determination of the magnetic ground state in the square lattice s = 1/2 antiferromag- net li2vosio4, Phys. Rev. Lett. 93, 027202 (2004)

2004
[23]

Pavarini, S

E. Pavarini, S. C. Tarantino, T. B. Ballaran, M. Zema, P. Ghigna, and P. Carretta, Effect of high pressure on competing exchange couplings in li2vosio4, Phys. Rev. B 77, 014425 (2008)

2008
[24]

Uematsu and H

K. Uematsu and H. Kawamura, Randomness-induced quantum spin liquid behavior in the s=1/2 random j1-j2 heisenberg antiferromagnet on the square lattice, Phys. Rev. B 98, 134427 (2018)

2018
[25]

Savary and L

L. Savary and L. Balents, Quantum spin liquids: a re- view, Reports on Progress in Physics 80, 016502 (2016)

2016
[26]

Yamaguchi, M

H. Yamaguchi, M. Okada, Y. Kono, S. Kittaka, T. Sakakibara, T. Okabe, Y. Iwasaki, and Y. Hosokoshi, Randomness-induced quantum spin liquid on honeycomb lattice, Scientific Reports 2017 7:1 7, 1 (2017)

2017
[27]

A. C. Potter, M. Barkeshli, J. McGreevy, and T. Senthil, Quantum spin liquids and the metal-insulator transition in doped semiconductors, Phys. Rev. Lett. 109, 077205 (2012)

2012
[28]

M. P. Gelfand, Series investigations of magnetically disor- dered ground states in two-dimensional frustrated quan- tum antiferromagnets, Phys. Rev. B 42, 8206 (1990)

1990
[29]

Capriotti and S

L. Capriotti and S. Sorella, Spontaneous plaquette dimer- ization in the j1−− j2 heisenberg model, Phys. Rev. Lett. 84, 3173 (2000)

2000
[30]

Darradi, O

R. Darradi, O. Derzhko, R. Zinke, J. Schulenburg, S. E. Kr¨ uger, and J. Richter, Ground state phases of the spin- 1/2 J1−−J2 heisenberg antiferromagnet on the square lattice: A high-order coupled cluster treatment, Phys. Rev. B 78, 214415 (2008)

2008
[31]

Sirker, Z

J. Sirker, Z. Weihong, O. P. Sushkov, and J. Oitmaa, j1−j2 model: First-order phase transition versus decon- finement of spinons, Phys. Rev. B 73, 184420 (2006)

2006
[32]

Ren, N.-H

Y.-Z. Ren, N.-H. Tong, and X.-C. Xie, Cluster mean- field theory study of j1- j2 heisenberg model on a square lattice, Journal of Physics: Condensed Matter 26, 115601 (2014)

2014
[33]

Dash and S

S. Dash and S. Kumar, Semiclassical approach to par- tial magnetic order in kondo lattices, Phys. Rev. B 111, 174443 (2025)

2025
[34]

Schick, T

R. Schick, T. Ziman, and M. E. Zhitomirsky, Quantum versus thermal fluctuations in the fcc antiferromagnet: Alternative routes to order by disorder, Phys. Rev. B 102, 220405 (2020)

2020
[35]

Glittum, A

C. Glittum, A. ˇStrkalj, D. Prabhakaran, P. A. Goddard, C. D. Batista, and C. Castelnovo, A resonant valence 9 bond spin liquid in the dilute limit of doped frustrated mott insulators, Nature Physics , 1 (2025)

2025
[36]

Badola, D

S. Badola, D. Negi, A. Joshi, A. Ali, R. Shankar Singh, and S. Saha, Emergence of a spin-liquid-like phase in the quantum spin ladder compound ba2cuteo6 with chemical substitution, Phys. Rev. B 109, L100405 (2024)

2024
[37]

Duncan, F

C. Duncan, F. Bellotti, P. ¨Ohberg, N. Zinner, and M. Va- liente, Mobile spin impurity in an optical lattice, New Journal of Physics 19, 075001 (2017)

2017
[38]

W. Yue, Q. Wei, S. Kais, B. Friedrich, and D. Her- schbach, Realization of heisenberg models of spin systems with polar molecules in pendular states, Phys. Chem. Chem. Phys. 24, 25270 (2022)

2022
[39]

S. Yoon, W. Lee, S. Lee, J. Park, C. H. Lee, Y. S. Choi, S.-H. Do, W.-J. Choi, W.-T. Chen, F. Chou, D. I. Gorbunov, Y. Oshima, A. Ali, Y. Singh, A. Berlie, I. Watanabe, and K.-Y. Choi, Quantum dis- ordered state in the J1−J2 square-lattice antiferromag- net sr 2Cu(te0.95w0.05)...

2021
[40]

Mustonen, S

O. Mustonen, S. Vasala, E. Sadrollahi, K. Schmidt, C. Baines, H. Walker, I. Terasaki, F. Litterst, E. Baggio- Saitovitch, and M. Karppinen, Spin-liquid-like state in a spin-1/2 square-lattice antiferromagnet perovskite in- duced by d 10–d 0 cation mixing, Nature communications...

2018
[41]

Watanabe, N

M. Watanabe, N. Kurita, H. Tanaka, W. Ueno, K. Mat- sui, and T. Goto, Valence-bond-glass state with a singlet gap in the spin- 1 2 square-lattice random J1−J2 heisen- berg antiferromagnet sr 2cute1−xwxo6, Phys. Rev. B 98, 054422 (2018)

2018
[42]

Mustonen, S

O. Mustonen, S. Vasala, K. P. Schmidt, E. Sadrollahi, H. C. Walker, I. Terasaki, F. J. Litterst, E. Baggio- Saitovitch, and M. Karppinen, Tuning the s = 1 /2 square-lattice antiferromagnet Sr 2Cu(Te1−xwx)o6 from n´ eel order to quantum disorder to columnar order, Phys. Rev. B ...

2018
[43]

Skolimowski, Y

J. Skolimowski, Y. Gerasimenko, and R. ˇZitko, Mottness collapse without metallization in the domain wall of the triangular-lattice mott insulator 1 t− tas2, Phys. Rev. Lett. 122, 036802 (2019)

2019
[44]

C. M. Yim, G.-R. Siemann, S. Stavri´ c, S. Khim, I. Benediˇ ciˇ c, P. A. Murgatroyd, T. Antonelli, M. D. Wat- son, A. P. Mackenzie, S. Picozzi, et al., Avoided metallic- ity in a hole-doped mott insulator on a triangular lattice, Nature Communications 15, 8098 (2024)

2024
[45]

Yee and L

C.-H. Yee and L. Balents, Phase separation in doped mott insulators, Phys. Rev. X 5, 021007 (2015)

2015

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Broholm, R

C. Broholm, R. J. Cava, S. Kivelson, D. Nocera, M. Nor- man, and T. Senthil, Quantum spin liquids, Science 367, eaay0668 (2020)

2020

[2] [2]

Y. Zhou, K. Kanoda, and T.-K. Ng, Quantum spin liquid states, Rev. Mod. Phys. 89, 025003 (2017)

2017

[3] [3]

Savary and L

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

2016

[4] [4]

Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

L. Balents, Spin liquids in frustrated magnets, Nature 464, 199 (2010)

2010

[5] [5]

Yu and Y.-J

J.-F. Yu and Y.-J. Kao, Spin-1/2 j1-j2 heisenberg antifer- romagnet on a square lattice: A plaquette renormalized tensor network study, Phys. Rev. B 85, 094407 (2012)

work page 2012

[6] [6]

Jiang, H

H.-C. Jiang, H. Yao, and L. Balents, Spin liquid ground state of the spin-1/2 square j1-j2 heisenberg model, Phys. Rev. B 86, 024424 (2012)

work page 2012

[7] [7]

Morita, R

S. Morita, R. Kaneko, and M. Imada, Quantum spin liquid in spin 1/2 j1–j2 heisenberg model on square lattice: Many-variable variational monte carlo study combined with quantum-number pro- jections, https://doi.org/10.7566/JPSJ.84.024720 84, 10.7566/JPSJ.84.024720 (2015)

doi:10.7566/jpsj.84.024720 2015

[8] [8]

Richter and J

J. Richter and J. Schulenburg, The spin-1/2 j1–j2 heisen- berg antiferromagnet on the square lattice: Exact diago- nalization for n=40 spins, The European Physical Jour- nal B 2009 73:1 73, 117 (2009)

work page 2009