# Results: Critical Exponents

The *xy* linear sizes *L* = 20, 24, 30,..., 80 have been used in our simulations. Films of thickness from *N _{z} =* 3 up to 160 have been used to evaluate corrections to scaling.

In practice, we use first the standard MC simulations to localize for each size the transition temperatures *T _{0}^{E}(L)* for specific heat and T

_{0}"'(L) for susceptibility. The equilibrating time is from 2 x 10

^{s }to 4 x 10

^{5}MC steps/spin and the averaging time is from 5 x 10

^{s }to 10

^{6}MC steps/spin. Next, we make histograms at eight different temperatures

*Tj(L)*around the transition temperatures

*T*with 2 x 10

_{0}^{E}’"'(L)^{6}MC steps/spin, after discarding 10

^{6}MC steps/spin for equilibrating. Finally, we make again histograms at 8 different temperatures around the new transition temperatures

*T*with 2 x 10

^{E,m}(L)^{6}and 4 x 10

^{6}MC steps/spin for equilibrating and averaging time, respectively. Such an iteration procedure gives extremely good results for systems studied so far. The errors shown in the following have been estimated using statistical errors, which are very small thanks to our multiple-histogram procedure, and fitting errors given by fitting software.

We note that only *v* is directly calculated from MC data. Exponent *у* obtained from /^{max} and *v* suffers little errors (systematic errors and errors from r). Other exponents are obtained by MC data and several-step fitting. For example, to obtain *a* we have to fit C"’^{ax }of Eq. 16.12 by choosing *C _{0}, C* and by using the value of

*v.*So, in practice, in most cases, one calculates

*a*or

*p*from MC data and uses the Rushbrooke scaling law to calculate the remaining exponent.

Now, similar to the discussion given in Section 16.2.2, if we work at a distance away from *T _{C}[L =* oo,

*N*we should observe "effective critical exponents." This is the case because in the finite-size analysis using the multiple-histogram technique, we measure the maxima of

_{z})*V*and

_{n}, C_{v}*x*which occur at different temperatures for a finite

*L.*These temperatures, though close to, are not

*T*= oo,

_{C}(L*N*

_{z}). To give a precision on this point, we show the values of these maxima and the corresponding temperatures for

*N*in Table 16.1. For the value of

_{z}= 1*T*= oo,

_{C}[L*N*= 7), see Table 16.2.

_{z}Given this fact, we emphasize that calculations using Eqs.

(16.11)-(16.16) will give effective critical exponents except of course for the case *N _{z}* = 1 where the results correspond to real 2D critical exponents.

We show now the results obtained by MC simulations with the Hamiltonian (16.1). We have tested that all exponents do not change in the finite size scaling with *L >* 30. So most of results are shown for L > 30 except for u where the lowest sizes *L =* 20, 24 can be used without modifying its value.

Let us show in Fig. 16.1 the layer magnetizations and their corresponding susceptibilities of the first three layers, in the case where *J _{s} =* 1. It is interesting to note that the magnetization of the surface layer is smaller than the magnetizations of the interior layers, as it has been shown theoretically by the Green’s function method a long time ago [75,78]. The surface spins have smaller local field due to the lack of neighbors, so thermal fluctuations will reduce more easily the surface magnetization with respect to the interior ones. The susceptibilities have their peaks at the same temperature, indicating a single transition.

Table 16.1 Maxima and temperatures at the maxima of *V _{n}*

*(n*= 1, 2),

*C*and x for various Lwith

_{v}*N*= 7

_{?}
30 |
2.21115658 |
25.29532589 |
164.05948154 |
275.71581036 |
4.19027500 |
4.22755000 |
4.24277500 |
4.24900000 |

40 |
2.36517434 |
41.20958927 |
219.30094769 |
368.72462473 |
4.19305000 |
4.21895000 |
4.23025000 |
4.23500000 |

50 |
2.50496719 |
60.82008190 |
275.66203381 |
463.17327477 |
4.19275000 |
4.21340000 |
4.22210000 |
4.22600000 |

60 |
2.59177903 |
82.96529587 |
329.65536262 |
554.47606570 |
4.19270000 |
4.20940000 |
4.21710000 |
4.22045000 |

70 |
2.70129995 |
109.00528127 |
387.47245040 |
651.24905512 |
4.19250000 |
4.20640000 |
4.21260000 |
4.21530000 |

80 |
2.76931676 |
138.78113065 |
443.00488386 |
743.61068938 |
4.19220000 |
4.20410000 |
4.20965000 |
4.21205000 |

Table 16.2 Critical exponents, effective dimension and critical temperature at the infinite *xy* limit

1 |
0.9990 ±0.0028 |
1.7520 ±0.0062 |
0.00199 ± 0.00279 |
0.1266 ±0.0049 |
2.0000 ±0.0028 |
2.2699 ±0.0005 |

3 |
0.9922 ±0.0019 |
1.7377 ±0.0035 |
0.00222 ± 0.00192 |
0.1452 ±0.0040 |
2.0135 ±0.0019 |
3.6365 ±0.0024 |

5 |
0.9876 ±0.0023 |
1.7230 ±0.0069 |
0.00222 ± 0.00234 |
0.1639 ±0.0051 |
2.0230 ±0.0023 |
4.0234 ±0.0028 |

7 |
0.9828 ±0.0024 |
1.7042 ±0.0087 |
0.00223 ± 0.00238 |
0.1798 ±0.0069 |
2.0328 ±0.0024 |
4.1939 ±0.0032 |

9 |
0.9780 ±0.0016 |
1.6736 ±0.0084 |
0.00224 ± 0.00161 |
0.1904 ±0.0071 |
2.0426 ±0.0016 |
4.2859 ±0.0022 |

11 |
0.9733 ±0.0025 |
1.6354 ±0.0083 |
0.00224 ± 0.00256 |
0.1995 ±0.0088 |
2.0526 ±0.0026 |
4.3418 ±0.0032 |

13 |
0.9692 ±0.0026 |
1.6122 ±0.0102 |
0.00226 ± 0.00268 |
0.2059 ±0.0092 |
2.0613 ±0.0027 |
4.3792 ±0.0034 |

Figure 16.1 Layer magnetizations (a) and layer susceptibilities (b) versus *T* with *N _{z} =* 5 and

*L =*24.

Figure 16.2 shows total magnetization of the film and the total susceptibility. This indicates clearly that there is only one peak as said above.

## Finite-Size Scaling

Let us show some results obtained from multiple histograms described above. Figure 16.3 shows the susceptibility and the first derivative *V* versus *T* around their maxima for several sizes.

Figure 16.2 Total magnetization (a) and total susceptibility (b) versus *T *with *N _{z}* = 5 and

*L =*24.

We show in Fig. 16.4 the maximum of the first derivative of In *M *with respect to *p* = (/свГ)^{-1} versus *L* in the In — In scale for several film thicknesses up to *N _{z} =* 13. If we use Eq. (16.11) to fit these lines, i.e., without correction to scaling, we obtain 1/v from the slopes of the remarkably straight lines. These values are indicated on the figure. In order to see the deviation from the 2D exponent, we plot in Fig. 16.5

*v*as a function of thickness

*N*We observe here a small but systematic deviation of

_{z}.*v*from its 2D value

*(v*

_{2}d = 1) with increasing thickness. To show the precision of our method, we give here the results of

*N*= 1. For

_{z}*N*= 1, we have l/i> = 1.0010 ± 0.0028

_{z}Figure 16.3 (a) Susceptibility and (b) K_{1(} as functions of *T* for several *L*

with *N _{z} =* 11, obtained by multiple histogram technique.

Figure 16.4 Maximum of the first derivative of In *M* versus *L* in the In — In scale. The slopes are indicated on the figure.

Figure 16.5 Effective exponent i> versus *N _{z}.*

Figure 16.6 Maximum of susceptibility versus *L* in the In — In scale. The slopes are indicated on the figure.

which yields *v* = 0.9990 ± 0.0031 and *y/v =* 1.7537 ± 0.0034 (see Figs. 16.6 and 16.7 below) yielding *у* = 1.7520 ± 0.0062. These results are in excellent agreement with the exact results *v _{2D}* = 1 and

*y*= 1.75. The very high precision of our method is thus verified in the rather modest range of the system sizes

_{2D}*L*= 20 — 80 used in the present work. Note that the result of Ref. [309] gave

*v =*0.96 ± 0.05 for

*N*= 1 which is veiy far from the exact value.

_{z}The deviation of *v* from the 2D value when *N _{z}* increases is due, as discussed earlier, to the crossover to 3D (t is not zero). Other exponents will suffer the same deviations as seen below.

We show in Fig. 16.6 the maximum of the susceptibility versus *L* in the In — In scale for film thicknesses up to *N _{z} =* 13. We have used only results of

*L*> 30. Including

*L =*20 and 24 will result, unlike the case of

*v,*in a decrease of

*у*of about one percent for

*N*From the slopes of these straight lines, we obtain the values of effective

_{z}> 7.*y/v.*Using the values of v obtained above, we deduce the values of

*у*which are plotted in Fig. 16.7 as a function of thickness

*N*Unlike the case of

_{z}.*v,*we observe here a stronger deviation of

*у*from its 2D value (1.75) with increasing thickness. This finding is somewhat interesting: The magnitude of the deviation from the 2D

Figure 16.7 Effective exponent *у* versus *N _{z}.*

Figure 16.8 InfC™^{3}* — C_{0}) versus In *L*for *N _{z}* = 1, 3, 5,..., 13. The slope gives

*a/v*(see Eq. 16.12) indicated on the figure.

value may be different from one critical exponent to another: ~ 3% for *v* and ~ 8% for *у* when *N _{z}* goes from 1 to 13. We will see below that

*p*varies even more strongly.

We show now in Fig. 16.8 the maximum of C™^{ax} versus Lfor *N _{z} = *1, 3, 5,..., 13. Note that for each

*N*we had to look for

_{z}*C*and

_{0}, C*a/v*which give the best fit with data of C™

^{ax}. Due to the fact that there are several parameters which can induce a wrong combination of them, we impose that

*a*should satisfy the condition 0 <

*a <*0.11 where the lower limit of

*a*corresponds to the value of 2D case and the upper limit to the 3D case. In doing so, we get very good results shown in Fig. 16.8. From these ratios of

*a/v*we deduce» for each

*N*The values of

_{z}.*a*are shown in Table 16.2 for several

*N*

_{z}.It is interesting to note that the effective exponents obtained above give rise to the effective dimension of thin film. This is conceptually not rigorous but this is what observed in experiments. Replacing the effective values of» obtained above in d_{e}ff = (2—*a)/v *we obtain *d _{e}ff* shown in Fig. 16.9.

We note that dgff is very close to 2. It varies from 2 to ~ 2.061 for *N _{z}* going from 1 to 13. The 2D character is thus dominant even with larger

*N*This supports the idea that the finite correlation in the

_{z}.*z*

Figure 16.9 Effective dimension of thin film obtained by using effective exponents, as a function of thickness.

direction, though qualitatively causing a deviation, cannot strongly alter the 2D critical behavior. This point is interesting because, as said earlier, some thermodynamic properties may show already their 3D values at a thickness of about a few dozens of layers, but not the critical behavior. To show an example of this, let us plot in Fig. 16.10 the transition temperature at *L* = oo for several *N _{z}, *using Eq. 16.16 for each given

*N*As seen,

_{z}.*T*oc) reaches already ~ 4.379 at

_{c}(*N*13 while its value at 3D is 4.51 [70, 112]. A rough extrapolation shows that the 3D values is attained for

_{z}=*N*~ 25 while the critical exponents at this thickness are far away from the 3D ones.

_{z}Let us show the prediction of Capehart and Fisher [53] on the critical temperature as a function of *N _{z}.* Defining the critical-point shift as

they showed that

where *v* = 0.6289 (3D value). Using *T _{C}(3D) =* 4.51, we fit the above formula with

*T*= oo,

_{C}(L*N*taken from Table 16.2, we obtain

_{z}~)Figure 16.10 Critical temperature at infinite *L* as a function of the film thickness. Points are MC results, continuous line is the prediction of Capehart and Fisher, Eq. (16.22). The agreement is excellent.

*a* = -1.37572 and *b* = -1.92629. The MC results and the fitted curve are shown in Fig. 16.10. Note that if we do not use the correction factor [1 + *a/N _{z}],* the fit is not good for small

*N*The prediction of Capehart and Fisher is thus very well verified.

_{z}.We give here the precise values of *T _{C}[L* = oc,

*N*

_{z}) for each thickness. For

*N*= 1, we have

_{z}*T*= oo,

_{C}(L*N*= 1) = 2.2699 ± 0.0005. Note that the exact value of

_{z}*T*oo) is 2.26919 by solving the equation sinh

_{c}(^{2}(2/

*/T*1. Again here, the excellent agreement of our result shows the efficiency of the multiple histogram technique as applied in the present paper. The values of

_{c}) =*T*oc) for other

_{C}[L =*N*are summarized in Table 16.2.

_{z}Calculating now M(L) at these values of *T _{C}[L* = oo,

*N*and using Eq. 16.15, we obtain

_{z})*p/v*for each

*N*For

_{z}.*N*= 1, we have

_{z}*p/v*= 0.1268 ± 0.0022 which yields

*p*= 0.1266 ± 0.0049 which is in excellent agreement with the exact result 0.125. Note that if we calculate

*p*from

*a + 2p + у*= 2, then

*p*= (2 — 1.75198 - 0.00199)/2 = 0.12302 ± 0.0035 which is in good agreement with the direct calculation within errors. We show in Fig. 16.11 the values of

*p*obtained by direct calculation using Eq. 16.15. Note that the deviation of

*p*from the 2D value when

*N*varies from 1 to 13 is

_{z}Figure 16.11 Effective exponent *fi,* obtained by using Eq. 16.15, versus the film thickness.

due to the crossover effect discussed in Section 16.2.3. It represents about 60%. Remember that the 3D value of *p* is 0.3258 ± 0.0044 [112].

Finally, for convenience, let us summarize our results in Table

16.2 for *N _{z}* = 1, 3,..., 13. Except for

*N*= 1, all other cases are effective exponents discussed above. Due to the smallness of

_{z}*a,*its value is shown with 5 decimals without rounding.

## Larger Sizes and Correction to Scaling

We consider here the effects of larger *L* and of the correction to scaling. For the effect of larger L, we will extend our size up to *L* = 160, for just the case *N _{z}* = 3.

The results indicate that larger *L* does not change the results shown above. Figure 16.12a displays the maximum of *V* as a function of *L* up to 160. Using Eq. (16.11), i.e., without correction to scaling, we obtain 1/v = 1.009 ± 0.001 which is to be compared to *1/v* = 1.008 ± 0.002 using Lup to 80. The change is, therefore, insignificant because it is at the third decimal i.e., at the error level. The same is observed for y/v as shown in Fig. 16.12b: *Y/v* =

Figure 16.12 (a) *V"' ^{ax}* and (b) vs

*L*up to 160 with

*N*3.

_{z}=1.752 ± 0.002 using *L* up to 160 instead of *y/v* = 1.751 ± 0.002 using Lup to 80.

Now, let us allow for correction to scaling, i.e., we use Eq.(16.17) instead of Eq. (16.13) for fitting. We obtain the following values: *y/v* = 1.751 ± 0.002, *B _{x}* = 0.05676,

*B*= 1.57554,

_{2}*w*= 3.26618 if we use

*L =*70 to 160 (figure not shown). The value of

*y/v*in the case of no scaling correction is 1.752 ±0.002. Therefore, we can conclude that this correction is insignificant. The large value of

*u>*explains the smallness of the correction.

Figure 16.13 Maximum of susceptibility versus *L* in the in - In scale for *N _{z}* = 5 (a) without PBC in z direction (b) with PBC in z direction. The points of these cases cannot be distinguished in the figure scale. The slopes are indicated on the figure. See text for comments.

## Role of Boundary Conditions

To close this section, let us touch upon the question: Does the absence of PBC in the *z* direction cause the deviation of the critical exponents? The answer is no: We have calculated *v* and *у* for *N _{z}* = 5 in both cases, with and without PBC in the

*z*direction. The results show no significant difference between the two cases as seen in Figs. 16.14 and 16.13. We have found the same thing with

*N*= 11 (not shown). So, we conclude that the fixed thickness will result in the deviation of the critical exponents, not from the absence of the PBC. This is somewhat surprising since we may think, incorrectly, that the PBC should mimic the infinite dimension so that we should obtain the 3D behavior when applying the PBC. As will be seen below, the 3D behavior is recovered only when the finite size scaling is applied in the

_{z}*z*direction at the same time in the

*xy*plane. To show this, we plot in Figs. 16.15 and 16.16 the results for the 3D case. Even with our modest sizes (up to

*L*=

*N*21, since it is not our purpose to treat the 3D case here), we obtain

_{z}=*v =*0.613 ± 0.005 and

*у*= 1.250 ± 0.005 very close to their 3D best known values

444

*Thin Films and Criticality*

Figure 16.14 Maximum of the first derivative of In *M* versus Lin the In — In scale for *N _{z}* = 5 (a) without PBC in z direction (b) with PBC in z direction. The slopes are indicated on the figure. See text for comments.

Figure 16.15 Maximum of the first derivative of In *M* versus Lin the In — In scale for 3D case.

Figure 16.16 Maximum of susceptibility versus Lin the In - In scale for 3D case.

usd = 0.6302 ± 0.0001 from Ref. [70] and *v _{3D} =* 0.6289 ± 0.0008 and узо = 1.2390 ± 0.0025 obtained by using 24 <

*L <*96 given in Ref. [112].