# Critical parameters and universal amplitude ratios

of two-dimensional spin- Ising models

using high- and low-temperature expansions

###### Abstract

For the study of Ising models of general spin on the square lattice, we have combined our recently extended high-temperature expansions with the low-temperature expansions derived some time ago by Enting, Guttmann and Jensen. We have computed for the first time various critical parameters and improved the estimates of others. Moreover the properties of hyperscaling and of universality (spin independence) of exponents and of various dimensionless amplitude combinations have been verified accurately. Assuming the validity of the lattice-lattice scaling, from our estimates of critical amplitudes for the square lattice we have also obtained estimates of the corresponding amplitudes for the spin Ising model on the triangular, honeycomb, and kagomé lattices.

###### pacs:

PACS numbers: 05.50+q, 11.15.Ha, 64.60.Cn, 75.10.Hk^{†}

^{†}preprint: Bicocca-FT-02-14 July 2002

## I Introduction

The properties of the spin two-dimensional Ising model with nearest-neighbor interactions in zero magnetic field, have been extensively explored in the last six decades. Much more modest efforts have been devoted to the study of the simplest generalizations of the model to spin . The main reason is probably that these models are not known to be solvable or, at least, to have any simple duality property which can help to extend the small body of information coming from numerical methods of limited accuracy such as stochastic simulations, series expansions or transfer-matrix calculations.

The first important result from a comparative study of Ising models for different values of the spin came from pioneering work by Domb and Sykes dosy . They analysed the high-temperature(HT) expansion of the susceptibility through in the three-dimensional case and conjectured that the value of the critical exponent is independent of the spin magnitude. This was the first step towards the modern formulation of the critical-universality hypothesis. Similar analyses were soon repeated by other authors using both HTmoore ; camp and low-temperature (LT) expansionsfoxgu for two-dimensional systems. Unfortunately, the series derived in those years were rather short and therefore the results of the analyses could not reach a sufficient accuracy or were inconclusive. It was only in 1980 that Nickelnick21 ; nr90 finally extended through the HT series in two dimensions on the square (sq) lattice and in three dimensions on the body-centered cubic lattice. The expansions of and of the second moment of the spin-spin correlation function were then published only for . More recently also the LT expansions on the sq lattice for were considerably extended by Enting, Jensen and Guttmannejg . We have summarized in Table 1 and Table 2 the state of HTcamp ; bakin and LTejg expansions before our work.

Observable | Lattice | Order | Ref. |
---|---|---|---|

, | sq | 21 | nr90 |

sq | 10 | bakin | |

, | tr | 10 | camp |

tr | 10 | bakin |

Observable | Lattice | Order | Ref. |
---|---|---|---|

, , | sq | 113 | ejg |

, , | sq | 100 | ejg |

, , | sq | 119 | ejg |

, , | sq | 126 | ejg |

, , | sq | 154 | ejg |

In spite of the very large number of LT expansion coefficients now available, the analysis of the series remains arduous due to occurrenceejg of numerous unphysical singularities in the complex temperature planefisherzeroes which are closer to the origin than the physical singularity and whose structure becomes increasingly complicated with . As a consequence, the LT study of Ref.ejg has been an alarming lesson on the subtleties in the analysis of slowly convergent series more than a source of accurate estimates of the critical parameters of the models.

Many intriguing indications and conjectures about the structure of these unphysical singularities also came in the same period from work by Matveev and Shrockms who examined the spin models on various two-dimensional lattices using transfer-matrix methods.

Here we discuss some results of an analysis of HT series for the sq lattice recently extendedbcesse25 by linked-cluster expansion techniques.

For the nearest-neighbor correlation function , for , and our series reach order , while for the second field derivative of the susceptibility they extend through . In order to make alternative analyses possible, our vast collection of series data both for two- and three-dimensional lattices was made easily accessiblebcol on-line for . It should be noted that HT and LT expansions as extensive as those obtained by Nickel et al. in Ref.nick21 and more recently in Ref.orri (only for ) in the very special case of the (partially solvable) two-dimensional Ising model seem presently beyond reach for .

The HT series show somewhat simpler and faster convergence properties than the LT series, because the behavior of the coefficients is dominated by the physical singularity. Although, even in this case, these favorable properties slightly deteriorate for , we can hope to determine basic HT critical parameters with a reasonable accuracy for various values of . Moreover it is also worthwhile to reconsider the LT expansions of Ref.ejg for the sq lattice, because by relying on the results of our HT analysis, we can improve some estimates of the LT critical parameters and thus obtain new determinations of universal combinations aha of LT and HT amplitudes. No theoretical surprises are expected from this analysis, however we believe it is still useful to improve the rather modest numerical precision presently available even for basic critical parameters like the critical temperatures, to determine various critical amplitudes for which no estimates are yet known and to use our results to test with higher accuracy the validity of hyperscaling and of universality with respect to the magnitude of the spin.

Almost all the computational effort in extending series for the two-dimensional Ising model for has been devoted to square-lattice series. However by making use of the theory of lattice-lattice scaling, as developed by Betts et al.bgj and extended by Gaunt and Guttmanngg , using our estimates of the critical amplitudes on the square lattice, we are able to calculate the corresponding amplitudes on other two-dimensional lattices to precisely the same precision as they are known for the square lattice.

## Ii The spin- Ising models

The spin- Ising models with nearest-neighbor interaction are defined by the Hamiltonian:

(1) |

where is the exchange coupling, and with a classical spin variable at the lattice site , taking the values . The sum runs over all nearest-neighbor pairs of sites. We shall restrict ourselves to the square lattice and consider expansions either in the usual HT variable and in the natural LT variable . Here is the temperature, the Boltzmann constant, and will be called “inverse temperature” for brevity. In the critical region we shall also refer to the standard reduced-temperature variable .

In the HT phase, the basic observables are the connected -spin correlation functions. Our seriesbcol cover quantities related to the two-spin correlation functions and to the four-spin correlation functions .

In the LT phase the symmetry is broken and the -spin correlations are non trivial also for odd . In particular, we shall reconsider the LT expansions of the magnetization, the susceptibility and the specific heat derived for in Ref.ejg .

The spontaneous magnetization is defined by

(2) |

The internal energy per spin is given in terms of the nearest-neighbor correlation function by

(3) |

where is a nearest-neighbor lattice vector and is the lattice coordination number.

The specific heat is the temperature-derivative of the internal energy at fixed zero external field

(4) |

In terms of , the zero-field reduced susceptibility,

(5) |

and of , the second moment of the correlation function,

(6) |

the “second-moment correlation length” is defined by

(7) |

The second field-derivative of the susceptibility is defined by

(8) |

## Iii Definitions of critical parameters

In terms of the asymptotic behavior of these observables, we can now define the critical parameters, amplitudes and exponents that we are going to estimate using HT and LT series.

The spontaneous magnetization has the asymptotic behavior

(9) |

as .

The asymptotic behaviour of the susceptibility as , is expected to be

(10) |

The correlation length

(11) |

the specific heat

(12) |

and the second field-derivative of the susceptibility

(13) |

have analogous asymptotic behaviours.

Different (universal) critical exponents and different (non-universal) critical amplitudes , , , , , etc. are associated with the various observables. We have reported in such detail our definitions of the critical amplitudes, because they differ significantly from those of other authors and it is necessary to use a consistent normalisation convention when comparing models expected to belong to the same universality class. Let us notice in particular that the estimates reported in the tables of Ref.ejg for the critical amplitudes of the susceptibility are related to ours by the factor (-ln. A similar remark applies to the specific heat amplitudes for which the conversion factor is (ln. Finally, the magnetization amplitudes of Ref.ejg should be multiplied by the factor -ln to agree with ours. Of course, the amplitudes of the conformal field theory considered in the study of Ref.delf are not comparable to our series quantities.

As indicated in eqs.(9) - (13), for a given spin , all asymptotic forms are moreover expectedweg to contain leading non-analytic confluent corrections characterized by the same exponent . Higher order corrections are also expected to contain logarithmicweg factors. If universality holds, all exponents have to be -independent.

The presence and the value of the confluent exponent has been discussedahafi ; case ; blonij ; barfish ; adler several times. From RG calculationszinn , both in the -expansion and in the fixed-dimension approach, it was conjectured that for the universality class of the two-dimensional Ising model. Aharony and Fisher and later Blöte and den Nijs arguedahafi ; blonij that for and indeed no such correction was revealed by the later very accurate studywu ; orri of the critical asymptotic expansion for . However, in the absence of more general results, the reliable assessment of the subleading asymptotic critical behaviour remained an open problem when .

## Iv Estimates of universal amplitude combinations

In terms of , and , a “hyper-universal” combination of critical amplitudes denoted by and usually called the “dimensionless renormalized coupling constant”, can be defined by

(14) |

Here the normalisation factor is chosen in order to match the usual field theoretic definitionzinn of and denotes the volume per lattice site, measured in units of the square of a lattice constant. For all lattices one has , with the lattice constant. For the triangular lattice, for the honeycomb lattice and for the kagomé lattice

We have also studied the hyper-universal combination usually denoted as

(15) |

and the Watson combinationwats

(16) |

The other frequently considered universal combination

(17) |

is not independent of the previous ones, since .

All of these quantities are accurately known in the case. As indicated in Ref.wu ; aha , it is known that ln, , and ln. In Refs.saso ; case , we find the very accurate estimates and .

Therefore we can conclude , , and .

We have also considered the ratio . In Ref.wu , for , this ratio was computed with arbitrary precision to be .

Finally, we have estimated the ratio for various values of This ratio equals unity for by self-duality. This was arguedande in greater generality for the -state () Potts model on the square lattice, which, for , reduces to the Ising model.

In what follows, we determine the values of these universal amplitude combinations for The preliminary part of our series analysis is aimed at estimating the critical temperatures using the expansions of for . We employed a variety of methods: Zinn-Justin improved-ratio formulazinn81 , Padé approximants (PA) and inhomogeneous differential approximants (DA)gutda . The best results with DA’s were obtained from approximants such that the polynomial coefficient of the highest derivative is even. (As a consequence, the approximants always contain an additional anti-ferromagnetic singularity at , beside the one at ). Similarly to the LT analysis, but to a much smaller extent, the accuracy of our results tends to deteriorate with increasing . In spite of this, our final HT estimates of the critical points, reported in Table 3, show significant improvement in apparent accuracy and sizable discrepancies from the previous LT determinationsejg .

S=1/2 | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | S= | |
---|---|---|---|---|---|---|---|

0.44068679… | 0.590473(5) | 0.684255(6) | 0.748562(8) | 0.79541(1) | 0.83106(2) | 1.09315(2) | |

ejg | 0.44068679… | 0.5904727(9) | 0.684338(46) | 0.7487(14) | 0.8025(35) | 0.839(10) | |

burk | 0.441 | 0.592 | 0.687 | 0.752 | 0.800 | 0.836 | |

burk | 0.458 | 0.610 | 0.704 | 0.770 | 0.818 | 0.855 | |

bln | 0.5904727(10) | ||||||

lipo | 0.590471 | ||||||

yuri | 0.590076… |

For general values of , less accurate estimates of have been obtained in Ref.burk from the ten term susceptibility series of Ref.camp and from a renormalization group method. More recently other estimatesmonr were obtained by a generalized cluster method. To our knowledge other accurate determinations of the critical points are available only for . They have been obtained either by analysingadler the 21 term HT series of Ref.nr90 for the susceptibility or by various transfer-matrix methodsbln ; yuri ; lipo . Some of these results have also been cited in Table 3.

We have then turned to the critical exponents , and and have evaluated them from the log-derivatives of the appropriate HT expansions by first order DA’s biased with our HT estimates of the critical temperatures. This computation shows that the relative variation of the exponents is smaller than , in the worst case, for varying between and . We report these results in Table 4 without further details and simply conclude that universality and hyperscaling appear to be well supported for the leading critical exponents.

S=1/2 | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | S= | |
---|---|---|---|---|---|---|---|

1.75 | 1.7502(4) | 1.7500(4) | 1.7496(5) | 1.7500(5) | 1.7501(5) | 1.7494(8) | |

1.0 | 0.9999(6) | 0.9996(8) | 0.9994(8) | 0.9994(8) | 0.9994(8) | 0.9994(8) | |

5.5 | 5.498(4) | 5.497(5) | 5.497(5) | 5.497(5) | 5.497(5) | 5.497(5) |

It is perhaps also worth noticing that assuming the universality of we can bias and therefore refine the determination of . This procedure does not change the central values of the critical points with respect to the unbiased one, but reduces the error bars.

On the other hand, the estimate of the exponent of the leading singular confluent corrections to scaling in the various observables remains quite elusive. Performing either a Baker-Hunterhb or a Zinn-Justinzinn81 analysis, we can conclude that, at the level of accuracy made possible by the present extension of the HT series, the amplitudes of these corrections are very small, (or perhaps vanishing) for all values of . We should mention that a similar conclusion was suggested for in Ref.blonij , while the opposite conclusion was advocated in Ref.adler .

Once we have estimated the critical temperatures and verified the universality of the leading exponents, we can proceed with the analysis simply assuming that, for all values of , these exponents take exactly the values expected for the universality class of the two-dimensional spin-1/2 Ising model and using them along with our estimated critical temperatures to bias the evaluation of the HT and LT critical amplitudes defined by eqs.(9) - (13).

Our estimates of the critical amplitudes are reported in Table 5.

S=1/2(ex.) | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | S= | |
---|---|---|---|---|---|---|---|

0.5514(2) | 0.4307(2) | 0.3755(2) | 0.3441(2) | 0.3254(3) | 0.2351(2) | ||

0.736(10) | 0.854(4) | 0.917(4) | 0.956(4) | 0.983(4) | 1.054(6) | ||

0.567068 | 0.4640(2) | 0.4309(2) | 0.4159(2) | 0.4082(2) | 0.4033(2) | 0.3900(2) | |

4.379095(8) | 0.9630(4) | 0.5073(3) | 0.3591(3) | 0.2902(3) | 0.2533(3) | 0.1239(3) | |

0.01462(3) | 0.0114(3) | 0.0102(6) | 0.0090(8) | 0.0055(30) | |||

ejg | 0.01462(2) | 0.0109(29) | 0.0094(10) | 0.0096(33) | |||

1.131(4) | 1.076(5) | 1.041(5) | 1.016(5) | 1.001(5) | |||

ejg | 1.1313(2) | 1.077(9) | 1.042(16) | 1.030(19) | 1.016(26) | ||

0.738(6) | 0.855(10) | 0.915(10) | 1.1(2) | 1.1(2) | |||

ejg | 0.73(2) | 0.77(6) | 0.86(8) | 0.87(9) |

We have employed quasi-diagonal non-defective PA’s or DA’s for extrapolating to the effective amplitudes of the susceptibility and of the derivative of the specific heat, from the HT and the LT side of the critical points. We have similarly studied the effective amplitudes of the correlation-length (available only in the HT region) and of the magnetization. For proper comparison, in the same Table we have also cited the LT estimates of the critical amplitudes for the spontaneous magnetization, the specific heat and the susceptibility previously obtained in Ref.ejg . These quantities have been multiplied by the above indicated conversion factors to agree with our normalisation conventions. The uncertainties we have reported, which allow for the observed spreads in the approximant values, provide a subjective assessment of residual trends in the sequence of estimates and for the (unbiased) uncertainties of the critical points. The HT amplitudes can be determined with a relative accuracy ranging from in the case of the susceptibility, to in the case of the specific heat. The LT amplitudes are subject to larger relative uncertainties, increasing with , and reaching up to for . In some cases, in order to improve the accuracy of the estimates of the LT amplitudes for , we have based our extrapolations only on the data for . This unconventional but reasonable procedure reduces the sensitivity of the approximants to the unphysical nearby singularities. Unfortunately, even this prescription fails to work satisfactorily for .

Using only the HT series, we can evaluate , either directly, in terms of the amplitudes reported in Table 5, or by extrapolating to the critical points via DA’s the HT expansion of the inverse effective coupling . A third approach consists in studying the residua at of the series with coefficients , where are the HT coefficients of and are the coefficients of the quantity . In Table 5 we have reported the results of the latter procedure since it yields estimates with smaller spreads.

Several other estimatesbb ; lai ; bc96 ; kim ; case ; balo obtained by a variety of methods are also available in the literature.

Using also the LT series, we have evaluated, directly in terms of the single amplitudes, the other mentioned universal combinations, for a range of values of . We have reported in Table 6, our series estimates of all these quantities for . In conclusion, whenever only HT amplitudes are involved, our estimates, within a precision up to , are independent of , in full agreement with universality. On the other hand, our reanalysis of the LT series has been only partially successful: whenever LT amplitudes also enter into the combinations, universality appears to be fairly well respected for , but the uncertainties grow notably larger for larger values of the spin.

S=1/2(ex.) | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | ||
---|---|---|---|---|---|---|---|

37.71(9) | 38(1). | 37(2). | 38(3). | 59(32). | |||

1.0 | 0.997(21) | 0.999(16) | 1.0(1) | 0.87(16) | 0.89(17) | ||

1.754364(2) | 1.753(2) | 1.753(2) | 1.752(3) | 1.752(3) | 1.752(3) | 1.752(3) | |

0.398(3) | 0.398(1) | 0.398(1) | 0.399(1) | 0.400(2) | 0.400(2) | ||

0.317(5) | 0.318(2) | 0.318(2) | 0.319(3) | 0.319(5) | |||

ejg | 0.317(4) | 0.317(7) | 0.317(11) | 0.31(1) | 0.31(1) |

In the next section we describe the theory of lattice-lattice scaling, and show how it can be used to extend our estimates of the critical amplitudes from the square lattice to other two-dimensional lattices.

## V Lattice-lattice scaling

The theory of lattice-lattice scaling was developed by Betts, Guttmann and Joycebgj in the early ’70s. It explains how amplitudes change within a given universality class, as one moves from one lattice to another. It can also be viewed as a generalisation of the law of corresponding states. In this section we give a terse development of the theory, and apply it to the problem at hand.

In order to review the general ideas let us first consider the Weiss theory or mean field theory of a magnetic system. The equation of state is well known to be

Here and are the reduced temperature, magnetic field and magnetization, respectively.

Then the law of corresponding states says that the equation of state is the same for all lattices. That is,

where and denote two lattices. That is to say, the lattice dependence is entirely contained in the critical temperature

A more complex model is the three-dimensional spherical model, for which the critical equation of state is:

Here both and the amplitude are lattice dependent. Thus

We see that we must scale the field variable, so that but that there is no need to scale the reduced temperature.

Let us now consider the case of the (zero-field) spin Ising model on the triangular () and hexagonal () lattices. The star-triangle relationons ; fis allows us to relate the free-energy, susceptibility and spontaneous magnetization between the lattices:

where and Here we see that the reduced temperature needs to be re-scaled for the free-energy to be universal. This is not restricted to the triangular-honeycomb pair, but in that case it is easy to be explicit.

All these examples can be encapsulated in the following expression for the singular part of the free-energy:

where the reduced temperature and field are scaled by

and

The singular part of the free-energy, is then a universal (lattice independent) function for a given model.

Equivalently, by differentiation we obtain

and

where and are universal functions for a given model.

Writing it follows that

Using this result, and the exact scaling parameters and given below, it is a trivial matter to calculate the magnetization amplitudes for the other lattices we consider (triangular, hexagonal and kagomé ()), taking as input the square lattice amplitudes given in Table V. These amplitude estimates are given in Tables VII, VIII and IX.

Writing it follows that

Similarly, it is a trivial matter to calculate the susceptibility amplitudes for the other lattices we consider, taking as input the square lattice amplitudes given in Table V.

Further differentiation gives the corresponding relationship for higher field derivatives, and we readily obtain

for the high-temperature field derivatives, (where only the even-order derivatives are non-zero). The corresponding result for low-temperature field derivatives is

Taking temperature derivatives, one readily establishes that the specific heat amplitudes satisfy

As for the two-dimensional Ising model, this simplifies to

We have similarly calculated the specific-heat amplitudes for the other lattices we consider, taking as input the square lattice amplitudes given in Table VI. These are also given in Tables VII, VIII, and IX.

S=1/2(ex.) | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | S= | |
---|---|---|---|---|---|---|---|

0.5294(2) | 0.4135(2) | 0.3605(2) | 0.3304(2) | 0.3124(3) | 0.2257(2) | ||

0.743(10) | 0.862(4) | 0.925(4) | 0.965(4) | 0.992(4) | 1.064(6) | ||

0.525315 | 0.4298(2) | 0.3992(2) | 0.3853(2) | 0.3781(2) | 0.3736(2) | 0.3613(2) | |

4.000248(8) | 0.8797(4) | 0.4634(3) | 0.3280(3) | 0.2651(3) | 0.2314(3) | 0.1132(3) | |

0.01404(3) | 0.0109(3) | 0.0098(6) | 0.0086(8) | 0.0053(30) | |||

1.113(4) | 1.059(5) | 1.025(5) | 1.000(5) | 0.985(5) | |||

0.745(6) | 0.863(10) | 0.923(10) | 1.1(2) | 1.1(2) |

S=1/2(ex.) | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | S= | |
---|---|---|---|---|---|---|---|

0.5994(2) | 0.4682(2) | 0.4082(2) | 0.3741(2) | 0.3537(3) | 0.2556(2) | ||

0.712(10) | 0.826(4) | 0.887(4) | 0.924(4) | 0.950(4) | 1.019(6) | ||

0.657331 | 0.5379(2) | 0.4995(2) | 0.4821(2) | 0.4732(2) | 0.4675(2) | 0.4521(2) | |

5.352965(8) | 1.1772(4) | 0.6201(3) | 0.4390(3) | 0.3547(3) | 0.3096(3) | 0.1515(3) | |

0.01589(3) | 0.0124(3) | 0.0111(6) | 0.0098(8) | 0.0060(30) | |||

1.159(4) | 1.103(5) | 1.067(5) | 1.042(5) | 1.026(5) | |||

0.714(6) | 0.827(10) | 0.885(10) | 1.1(2) | 1.1(2) |

S=1/2(ex.) | S=1 | S=3/2 | S=2 | S=5/2 | S=3 | S= | |
---|---|---|---|---|---|---|---|

0.5832(2) | 0.4556(2) | 0.3972(2) | 0.3640(2) | 0.3442(3) | 0.2487(2) | ||

0.714(10) | 0.829(4) | 0.890(4) | 0.928(4) | 0.954(4) | 1.023(6) | ||

0.618474 | 0.5061(2) | 0.4700(2) | 0.4536(2) | 0.4452(2) | 0.4399(2) | 0.4254(2) | |

5.046953(8) | 1.1099(4) | 0.5847(3) | 0.4139(3) | 0.3345(3) | 0.2919(3) | 0.1428(3) | |

0.01546(3) | 0.0121(3) | 0.0108(6) | 0.0095(8) | 0.0058(30) | |||

1.146(4) | 1.090(5) | 1.055(5) | 1.030(5) | 1.014(5) | |||

0.716(6) | 0.830(10) | 0.888(10) | 1.1(2) | 1.1(2) |

Finally, the correlation function amplitudes were calculated exactly from various star-triangle transformations by Thompson and Guttmann TG75 for the true correlation length. It follows from universality that these same transformations should hold also for the second-moment correlation length amplitudes. We show below that this is equivalent to the universality of which is a conclusion of this work.

The universality of taken together with the above lattice-lattice scaling relation for the specific heat amplitude implies the lattice-lattice scaling relation

where is the area per site, and and for the square, triangular, honeycomb and kagomé lattices respectively. This is equivalent to the explicit amplitude scaling reported in TG75 , and confirms the expectation that the true correlation length amplitude and the second-moment correlation length amplitude scale similarly.

For the 2d Ising model we can calculate the scaling parameters and exactly from known spin-1/2 spontaneous magnetization and specific heat amplitudes. The critical points are also exactly known. These are given in the table below:

Tr. | 1 | 1 | |

Sq. | |||

Ho. | 2 | ||

Ka. |

In addition to the above results, one can also derive scaling relations for the amplitudes of sub-dominant singularities. For example, consider the susceptibility of the 2d Ising model on lattice X. Writing

from lattice-lattice scaling we can derive the following amplitude relations:

The second expression is false for the kagomé lattice. It is corrected by the theory of extended lattice-lattice scaling developed by Gaunt and Guttmanngg in 1978. In that theory, a third scaling parameter needs to be introduced.

It is widely accepted, and in complete agreement with the results of the first part of this paper, that the spin- Ising model is in the same universality class as the spin- Ising model. This is the only assumption we require in order to apply the theory of lattice-lattice scaling to the square lattice data given in the previous section. The only subtlety is whether “universality” really extends to lattice-lattice universality. While this assumption seems natural, we did attempt to verify it by estimating amplitudes for other lattices from the rather short data available in camp . The longest effective series is the triangular lattice series. We found the high-temperature susceptibility amplitude, as estimated by Padé approximants, to be from this series, in complete agreement with the more precise value found by lattice-lattice scaling, and reported in Table VII.

Note too that there is no loss of accuracy, as the conversions from lattice to lattice are exact. For example, based on the recent estimate of the leading susceptibility amplitude of the spin square lattice Ising model given inorri , application of lattice-lattice scaling gives the corresponding amplitude on the triangular lattice to the same precision, viz:

Similarly accurate results for other lattices can be readily written down, as can equally accurate sub-dominant amplitudes.

###### Acknowledgements.

This work was completed before the untimely death of Marco Comi, our lifelong friend and coauthor. To his dear memory we dedicate this paper. This work has been partially supported by the Ministry of Education, University and Research (PB, MC), and the Australian Research Council (AJG).## References

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