The Ising phase in the J1-J2 Heisenberg Model

A. Parola; V. Lante

REVIEW

The Ising phase in the J1-J2 Heisenberg Model

V. Lante, A. Parola

Strongly Correlated Electrons · Submitted 23 Feb 2006 · 25 references

Pith's verdict: A genuinely new NLSM derivation for the collinear phase, but the zero-temperature nematic phase rests on two small clusters.

T0 · AI review

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arXiv IDcond-mat/0602544 (version 1)

Pith IDQVDKIMNG, Pith's permanent record of this paper and its review

CitationA. Parola, V. Lante. "The Ising phase in the J1-J2 Heisenberg Model." arXiv:cond-mat/0602544 (2006). https://pith.science/paper/QVDKIMNG

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Claim paper

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Editor's note

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You should read this paper for the field theory, not for the phase diagram. The authors derive, from the microscopic J1-J2 Hamiltonian, a 2+1-dimensional NLSM in terms of two Néel fields n+ and n-, with an O(3)×Z2 invariant action. This looks new: CCL constructed their effective action from spin-wave perturbation theory and only addressed finite T. The present derivation by coherent-state path integral, keeping 1/S and derivative terms, reproduces the known spin-wave dispersions, and after specializing to weak fluctuations it reduces to the CCL action. That is a real contribution and deserves careful study.

The numerical part is weaker. All the evidence for a zero-temperature valence bond nematic phase comes from 4x4 and 6x6 Lanczos runs. The s-wave and d-wave singlets become quasi-degenerate and cross on the 6x6 cluster, and the combinations (|s> +- |d>)/√2 give a large expectation value of the bond anisotropy operator. But a level crossing at one cluster size and a local order parameter in a symmetry-restored finite system are exactly what finite-size precursors of a nearby transition look like. There is no scaling of the s-d gap in L, no correlation length for the orientational order, no tunneling splitting. The 'remarkable size independence' of S(k) concerns short-range correlations and cannot certify long-range order. So αc≈0.62 is essentially an estimate from two points.

Two more soft spots, though neither is fatal. First, the Berry phase is set to zero under the assumption of no hedgehog singularities; the paper flags this in footnote 12 but does not resolve it. If hedgehogs are present, the extra Berry term can alter the symmetry argument. Second, the effective Lagrangian is obtained after 'tedious but straightforward calculations' that are not shown; a referee would need that algebra to check the result.

To the paper's credit, the conclusions are appropriately hedged: they say the scenario is 'suggested' and 'may be' realized, they explicitly cannot distinguish second-order from weakly first-order, and they call for Monte Carlo simulations. The title and abstract are more assertive than the actual evidence, but the body is honest.

My take: the field-theory derivation is worth engaging with and likely correct; the existence of a nematic phase remains unproven. I would cite this for the NLSM derivation, not for the phase diagram.

Referee report

Significance

If the central claim is correct, the paper would provide an early field-theoretic and numerical case for a valence bond nematic ground state in one of the most studied frustrated spin models, which would be a meaningful contribution. The paper has two clear strengths: first, it presents a first-principles derivation of a 2+1-dimensional NLSM tailored to the collinear phase of the J1-J2 model, and checks the effective action against the known long-wavelength spin-wave dispersion, which is a nontrivial consistency test; second, the authors are honest about limitations, explicitly calling for Monte Carlo simulations of the NLSM and admitting that their analysis cannot distinguish second-order from weakly first-order transitions. The significance is, however, limited by the fact that the affirmative claim of a thermodynamic VBN phase rests on Lanczos data on only two small clusters with no finite-size scaling analysis, so the more novel part of the paper is, at this stage, suggestive rather than established.

Major objections (4)

[Sec. III, Figs. 2–4] The central numerical claim that a valence bond nematic phase exists near α ≈ 0.6 rests on Lanczos results for only two cluster sizes, L = 4 and L = 6. The paper does not provide a scaling analysis of the s-wave/d-wave singlet gap, the tunneling splitting between the two symmetry-broken combinations, the correlation length of the orientational order parameter O_r, or the order parameter itself as a function of L. The level crossing at a single cluster size and the large expectation value of O_r in a symmetry-restored finite system are precisely what one expects as finite-size precursors of a nearby transition or crossover, so they cannot by themselves certify a thermodynamic VBN phase. The 'remarkable size independence' of S(k) concerns short-range correlations and does not imply long-range orientational order. This is a load-bearing gap, because the existence claim in the abstract and conclusions is carried by these data.
[Sec. II, Eq. (13)] The derivation of the effective Lagrangian is not shown: after introducing the constrained Gaussian integration and the modified matrix A and array B in Eq. (11), the paper jumps to Eq. (13) with the statement 'after tedious but straightforward calculations'. The final Lagrangian is the basis for the symmetry argument that an O(3)-symmetric, Z2-broken state may occur, so the omitted algebra is not merely a presentation detail. The coefficients in LS, LA, and L×, in particular the factors involving D, D*, and σ, should be derived in an appendix or a supplementary file; without this, the reader cannot audit the calculation or check whether the claimed O(3) × Z2 structure and the saddle-point dispersion in Eq. (14) follow from the same truncation.
[Sec. II, footnote 12 and discussion of Berry phase] The Berry phase is set to zero in the effective action under the assumption that the Néel fields have no hedgehog singularities in 2+1-dimensional spacetime. The paper itself notes in footnote 12 that if such singularities are allowed, a Berry term may contribute. This caveat is directly relevant to the zero-temperature argument, because topological defects can affect the O(3) × Z2 symmetry analysis that motivates the Ising phase. The manuscript neither rules out the relevant hedgehog configurations in the frustrated collinear regime nor explains why they are harmless for the proposed ground state. This should be addressed explicitly before the symmetry-based prediction is presented as robust.
[Sec. III, statement of αc ≈ 0.62] The numerical value αc ≈ 0.62 is presented as the location of the zero-temperature transition to the collinear phase, but no extrapolation to the thermodynamic limit is given. The orderings of low-lying states on 4×4 and 6×6 clusters are used to exclude plaquette and columnar VBC states, but on such small clusters the same level ordering may also be compatible with short-range columnar tendencies or with a finite-size artifact of the open boundary conditions of the Hilbert space sectors. The paper should either soften the claim to a qualitative crossover estimate or provide a systematic finite-size analysis, for example by computing the relevant gaps and the orientational correlation function on several larger clusters or on the same clusters with twisted boundary conditions.

Minor objections (5)

[Sec. II, Eq. (9)–(12)] The notation for the constrained Gaussian integration is dense and would benefit from a short explanatory paragraph: in particular, the meaning of the indices in A_ik and the projection operators in Eq. (11) is not spelled out, and the reader has to infer the range of the Greek and Latin indices from the text.
[Sec. III, Fig. 3] The caption of Fig. 3 does not state how the probabilities P_x and P_y are normalized or whether the order parameter values are computed on the state (|s>+|d>)/√2 or (|s>−|d>)/√2; a sentence clarifying this would improve reproducibility.
[Sec. III, fitting function for S(k)] The parameterized form used to fit S(k) is described only in reference [23] as a square-root ratio of two first-order polynomials in γ(k) and γ′(k); the actual coefficients and fitting procedure are not given, which makes the comparison in Fig. 4 difficult to assess quantitatively.
[Sec. II, Eq. (13)] The symbol σ is used both for the Ising order parameter n+ · n− and, in the preceding notation, as a Greek spin index; this double use is confusing and should be disambiguated by renaming one of the two.
[References] Reference [4] cites a preprint (cond-mat/0504306) without indicating whether it was subsequently published; the authors should provide the published version if one exists, or mark the reference clearly as a preprint.

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The paper

The paper claims

The square-lattice J1-J2 Heisenberg model has a zero-temperature valence bond nematic phase near J2/J1 ≈ 0.6.

The reading

This paper attempts to establish that the strongly frustrated square-lattice Heisenberg antiferromagnet with couplings $J_1$ (nearest neighbor) and $J_2$ (next-nearest neighbor) has a zero-temperature phase, near $J_2/J_1 \approx 0.6$, in which the spins are not magnetically ordered but the lattice's 90-degree rotation symmetry is broken. In that phase, which the authors identify as a valence bond nematic, spin singlet bonds point preferentially along one lattice direction while the full $O(3)$ spin-rotation symmetry of the Hamiltonian is preserved. The case rests on two legs: a new effective nonlinear $\sigma$ model (a low-energy field theory for the staggered magnetization directions) derived from the microscopic Hamiltonian, whose $O(3)\times Z_2$ symmetry permits such an intermediate Ising phase, and exact diagonalizations on $4\times4$ and $6\times6$ clusters showing a quasi-degenerate $s$-wave/$d$-wave singlet pair, a large orientational order parameter, and a transition signature at $\alpha_c \approx 0.62$. If the claim is right, the $J_1$-$J_2$ model is a clean example of a quantum phase that breaks only lattice rotation symmetry, separating the Néel phase from collinear magnetic order.

What carries the argument

The load-bearing object is the effective low-energy theory: a $O(3)\times Z_2$ nonlinear $\sigma$ model in $2+1$ dimensions written in terms of two coupled Néel fields $\hat{n}_+$ and $\hat{n}_-$, one for each sublattice of the square lattice. The action contains gradient terms for each field, a coupling term $\mathcal{L}_A$ that is not invariant under $\pi/2$ rotations, and the relative-angle combination $\sigma=\hat{n}_+\cdot\hat{n}_-$, the 'Ising order parameter'. Gaussian integration over the transverse fluctuations, under the constraints $\hat{n}_\pm\cdot L_\pm=0$, produces the effective Lagrangian; its saddle-point spectrum matches the known spin-wave dispersions, and its symmetry structure allows a phase with $\langle\hat{n}_\pm\rangle=0$ but $\langle\sigma\rangle\neq0$, exactly the valence bond nematic. The role of this machinery is to show why a rotationally broken, spin-symmetric phase is a natural intermediate state between Néel and collinear order.

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the $J_1$-$J_2$ Heisenberg model on the square lattice, for frustration ratios $\alpha=J_2/J_1$ above the classical threshold $1/2$, passes from the collinear antiferromagnet to a non-magnetic state that breaks the $\pi/2$ rotational symmetry of the lattice while keeping the full $SU(2)$ spin symmetry. The authors derive an effective $2+1$-dimensional action with two staggered fields, one per sublattice, and show that its symmetry group is $O(3)\times Z_2$, with the relative orientation $\sigma = \hat{n}_+\cdot\hat{n}_-$ acting as an Ising order parameter. Exact diagonalizations on $4\times4$ and $6\times6$ clusters show that the lowest $s$-wave and $d$-wave singlets become quasi-degenerate for $0.6\lesssim\alpha\lesssim0.7$, the rotational order parameter is large and anisotropic, and the magnetic structure factor is size independent until a peak develops at momentum $(\pi,0)$ for $\alpha\gtrsim0.62$. The authors conclude that a zero-temperature transition at $\alpha_c\sim0.62$ separates the collinear phase from a gapped valence bond nematic, and they leave open whether the transition is second order or weakly first order.

Load-bearing premise

The derivation assumes the staggered magnetization field is smooth everywhere in $2+1$-dimensional space-time, so the Berry phase terms vanish from the effective action; if point-like topological defects (hedgehogs) are allowed, those terms can reappear and the symmetry argument for the Ising phase may fail.

Editorial extensions

If this is right

A zero-temperature phase boundary at $\alpha_c\approx0.62$ separates the collinear antiferromagnet from a gapped, $SU(2)$-symmetric state with broken $\pi/2$ lattice rotation symmetry.
In the quasi-degenerate region $0.6\lesssim\alpha\lesssim0.7$, the fully symmetric spin liquid and the plaquette valence bond crystal are incompatible with the observed low-energy level ordering.
The size independence of the magnetic structure factor means short-range spin correlations from small clusters can be compared directly with neutron scattering data on frustrated two-dimensional antiferromagnets.
At finite temperature, the same $O(3)\times Z_2$ effective action is consistent with a two-dimensional Ising transition associated with the order parameter $\sigma$, connecting to the earlier finite-temperature scenario.
The transition into the Ising/nematic phase could be second order, tied to the vanishing of the sublattice staggered magnetization, whereas a direct collinear-to-valence-bond-crystal transition would be first order.

Reading between the lines

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

Beyond the paper: the same effective action suggests that other frustrated two-sublattice antiferromagnets in two dimensions should have a similar rotationally ordered, spin-symmetric intermediate phase; a testable signature would be an anisotropic spin structure factor in the nematic region.
Beyond the paper: the smooth-Néel-field assumption can be probed by restoring the Berry phase term in a numerical simulation of the effective action; if hedgehog configurations remove the Ising phase, the predicted nematic state is an artifact of the derivative expansion.
Beyond the paper: finite-size scaling of the rotational susceptibility $\chi_O$ on clusters larger than $6\times6$ would sharpen the location of $\alpha_c$, since the quasi-degeneracy of the $s$ and $d$ singlets implies a diverging susceptibility in the thermodynamic limit.
Beyond the paper: translational invariance of the bond order parameter would distinguish the nematic from a valence bond crystal; checking $\langle O_r\rangle$ independence of $r$ on larger clusters is a direct way to disambiguate the two states.

Original abstract

The two dimensional Heisenberg antiferromagnet on the square lattice with nearest (J1) and next-nearest (J2) neighbor couplings is investigated in the strong frustration regime (J2/J1>1/2). A new effective field theory describing the long wavelength physics of the model is derived from the quantum hamiltonian. The structure of the resulting non linear sigma model allows to recover the known spin wave results in the collinear regime, supports the presence of an Ising phase transition at finite temperature and suggests the possible occurrence of a non-magnetic ground state breaking rotational symmetry. By means of Lanczos diagonalizations we investigate the spin system at T=0, focusing our attention on the region where the collinear order parameter is strongly suppressed by quantum fluctuations and a transition to a non-magnetic state occurs. Correlation functions display a remarkable size independence and allow to identify the transition between the magnetic and non-magnetic region of the phase diagram. The numerical results support the presence of a non-magnetic phase with orientational ordering.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,051 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.40 · compares with prior results 0.63 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The two dimensional Heisenberg antiferromagnet on the square lattice with nearest (J1) and next-nearest (J2) neighbor couplings is investigated in the strong frustration regime (J2/J1>1/2). A new effective field theory describing the long wavelength physics of the model is derived from the quantum hamiltonian. The structure of the resulting non linear sigma model allows to recover the known spin wave results in the collinear regime, supports the presence of an Ising phase transition at finite temperature and suggests the possible occurrence of a non-magnetic ground state breaking rotational symmetry. By means of Lanczos diagonalizations we investigate the spin system at T=0, focusing our attention on the region where the collinear order parameter is strongly suppressed by quantum fluctuations and a transition to a non-magnetic state occurs. Correlation functions display a remarkable size independence and allow to identify the transition between the magnetic and non-magnetic region of the phase diagram. The numerical results support the presence of a non-magnetic phase with orientational ordering.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The two dimensional Heisenberg antiferromagnet on the square lattice with nearest (J1) and next-nearest (J2) neighbor couplings is investigated in the strong frustration regime (J2/J1>1/2).	background 0.77	0.73	0.03	asserted 0.85
s1	A new effective field theory describing the long wavelength physics of the model is derived from the quantum hamiltonian.	method 0.55	0.95	0.05	asserted 1.00
s2	The structure of the resulting non linear sigma model allows to recover the known spin wave results in the collinear regime, supports the presence of an Ising phase transition at finite temperature and suggests the possible occurrence of a non-magnetic ground state breaking rotational symmetry.	interpretation 0.56	0.79	0.34	hedged possibility 0.86
s3	By means of Lanczos diagonalizations we investigate the spin system at T=0, focusing our attention on the region where the collinear order parameter is strongly suppressed by quantum fluctuations and a transition to a non-magnetic state occurs.	method 0.79	0.97	0.07	asserted 0.89
s4	Correlation functions display a remarkable size independence and allow to identify the transition between the magnetic and non-magnetic region of the phase diagram.	result 0.97	0.94	0.12	asserted 1.00
s5	The numerical results support the presence of a non-magnetic phase with orientational ordering.	result 0.73	0.91	0.58	asserted 0.80

What kind of paper the classifier reports this is: simulation study 0.51 · evidence from simulation or computation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 2.** FIG. 2: Low energy states referenced to the GS as a function

**Figure 4.** The close similarity between the spin correlations

Showing 3 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0602544 by the authors.

References

Works cited (25)

[1]

See for instance: Misguich and C. Lhuillier in ``Frustrated spin systems'', H. T. Diep editor, World-Scientific (2005) and cond-mat/0310405 and Refs. therein

View on Pith Pith review arXiv 2005
[2]

R. Melzi et al. , 85 , 1318 (2000); 64 , 024409 (2001)

View on Pith 2000
[3]

P. Chandra and B. Doucot, 38 , R9335 (1988)

View on Pith 1988
[4]

S. Moukouri, cond-mat/0504306 (2005)

View on Pith Pith review arXiv 2005
[5]

F. D. M. Haldane, 50 , 1153 (1983)

View on Pith 1983
[6]

P. Chandra, P. Coleman, A. I. Larkin, 64 , 88 (1990)

View on Pith 1990
[7]

R.R.P. Singh, W. Zheng, J. Oitmaa, O.P. Sushkov, C.J. Hamer, 91 , 017201 (2003)

View on Pith 2003
[8]

C. Weber et al. , 91 , 177202 (2003)

View on Pith 2003
[9]

L. Capriotti , A. Fubini, T. Roscilde, V. Tognetti, 92 , 157202 (2004)

View on Pith 2004
[10]

S. Chakravarty, B. I. Halperin and D. R. Nelson, 39 2344 (1989)

View on Pith 1989
[11]

K. Takano, Y. Kito, Y. \=Ono and K. Sano, 91 , 197202 (2003)

View on Pith 2003
[12]

This is true within the O(3) NLSM approach where it is assumed that the N\'eel field is well defined at all points in (2+1) dimensional space-time. However if this assumption is relaxed to allow ``hedgehog'' (spatial) singularities of the N\'eel field then a dependence on the quantized value of the spin S appears and the Berry term may contribute to the B...

View on Pith
[13]

F. D. M. Haldane, 61 , 1029 (1988)

View on Pith 1988
[14]

A. Moreo, E. Dagotto, T. Jolicoeur and J. Riera, 42 , 6283 (1990)

View on Pith 1990
[15]

N. D. Mermin, H. Wagner 17 , 1133 (1966)

View on Pith 1966
[16]

D. Poilblanc, E. Gagliano, S. Bacci and E. Dagotto, 43 , 10970 (1991)

View on Pith 1991
[17]

T. Einarsson and H. J. Schulz, 51 , R6151 (1994); H. J. Schulz and T. A. L. Ziman, J. Phys. I 6 , 675 (1996)

View on Pith 1994
[18]

M. E. Zhitomirsky, K. Ueda, 54 , 9007 (1996)

View on Pith 1996
[19]

O. P. Sushkov, J.Oitmaa and Z. Weihong, 63 , 104420 (2001)

View on Pith 2001
[20]

R. R. P. Singh, Z. Weihong, C. J. Hamer, J. Oitmaa, 60 , 7278 (1999)

View on Pith 1999
[21]

These two conditions together imply that the susceptibility associated to the order parameter O diverges

View on Pith
[22]

L. Capriotti, F. Becca, A. Parola and S. Sorella, 87 , 097201 (2001)

View on Pith 2001
[23]

The fitting function is the square root of the ratio between two first order polynomials in ( k )= k_x + k_y and '( k )= k_x k_y

View on Pith
[24]

F. Figueirido et al. 41 , 4619 (1990)

View on Pith 1990
[25]

V. N. Kotov, J. Oitmaa, O. P. Sushkov and Z. Weihong, 60 , 14613 (1999)

View on Pith 1999

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

See for instance: Misguich and C. Lhuillier in ``Frustrated spin systems'', H. T. Diep editor, World-Scientific (2005) and cond-mat/0310405 and Refs. therein

View on Pith Pith review arXiv 2005

[2] [2]

R. Melzi et al. , 85 , 1318 (2000); 64 , 024409 (2001)

View on Pith 2000

[3] [3]

P. Chandra and B. Doucot, 38 , R9335 (1988)

View on Pith 1988

[4] [4]

S. Moukouri, cond-mat/0504306 (2005)

View on Pith Pith review arXiv 2005

[5] [5]

F. D. M. Haldane, 50 , 1153 (1983)

View on Pith 1983

[6] [6]

P. Chandra, P. Coleman, A. I. Larkin, 64 , 88 (1990)

View on Pith 1990

[7] [7]

R.R.P. Singh, W. Zheng, J. Oitmaa, O.P. Sushkov, C.J. Hamer, 91 , 017201 (2003)

View on Pith 2003

[8] [8]

C. Weber et al. , 91 , 177202 (2003)

View on Pith 2003

[9] [9]

L. Capriotti , A. Fubini, T. Roscilde, V. Tognetti, 92 , 157202 (2004)

View on Pith 2004

[10] [10]

S. Chakravarty, B. I. Halperin and D. R. Nelson, 39 2344 (1989)

View on Pith 1989

[11] [11]

K. Takano, Y. Kito, Y. \=Ono and K. Sano, 91 , 197202 (2003)

View on Pith 2003

[12] [12]

This is true within the O(3) NLSM approach where it is assumed that the N\'eel field is well defined at all points in (2+1) dimensional space-time. However if this assumption is relaxed to allow ``hedgehog'' (spatial) singularities of the N\'eel field then a dependence on the quantized value of the spin S appears and the Berry term may contribute to the B...

View on Pith

[13] [13]

F. D. M. Haldane, 61 , 1029 (1988)

View on Pith 1988

[14] [14]

A. Moreo, E. Dagotto, T. Jolicoeur and J. Riera, 42 , 6283 (1990)

View on Pith 1990

[15] [15]

N. D. Mermin, H. Wagner 17 , 1133 (1966)

View on Pith 1966

[16] [16]

D. Poilblanc, E. Gagliano, S. Bacci and E. Dagotto, 43 , 10970 (1991)

View on Pith 1991

[17] [17]

T. Einarsson and H. J. Schulz, 51 , R6151 (1994); H. J. Schulz and T. A. L. Ziman, J. Phys. I 6 , 675 (1996)

View on Pith 1994

[18] [18]

M. E. Zhitomirsky, K. Ueda, 54 , 9007 (1996)

View on Pith 1996

[19] [19]

O. P. Sushkov, J.Oitmaa and Z. Weihong, 63 , 104420 (2001)

View on Pith 2001

[20] [20]

R. R. P. Singh, Z. Weihong, C. J. Hamer, J. Oitmaa, 60 , 7278 (1999)

View on Pith 1999

[21] [21]

These two conditions together imply that the susceptibility associated to the order parameter O diverges

View on Pith

[22] [22]

L. Capriotti, F. Becca, A. Parola and S. Sorella, 87 , 097201 (2001)

View on Pith 2001

[23] [23]

The fitting function is the square root of the ratio between two first order polynomials in ( k )= k_x + k_y and '( k )= k_x k_y

View on Pith

[24] [24]

F. Figueirido et al. 41 , 4619 (1990)

View on Pith 1990

[25] [25]

V. N. Kotov, J. Oitmaa, O. P. Sushkov and Z. Weihong, 60 , 14613 (1999)

View on Pith 1999