n a0(n) ∆ a1(n) ∆ a2(n) ∆
10 0.4779764889 0.8·10−9 0.3218551631505 1.0·10−12 0.24324853242244 1.1·10−14 20 0.5892154049 3.0·10−9 0.4316043027242 1.9·10−12 0.35065320435835 2.6·10−14 30 0.6553795277 6.2·10−9 0.4974300032432 9.9·10−12 0.41592189386854 8.8·10−14 40 0.7026231983 9.9·10−9 0.544542664951 3.0·10−11 0.46281476449290 3.4·10−13 50 0.739392926 1.3·10−8 0.581247646267 6.9·10−11 0.4994098682251 1.1·10−12 60 0.769499767 1.6·10−8 0.61131758758 1.3·10−10 0.5294166892935 3.0·10−12
Table 1: Results of extrapolations to infiniteN (and corresponding absolute error bounds ∆(n, l)) forl= 0,1,2. Ranges ofN used to extrapolate: 3000≤N≤5000 forl= 0, 1500≤N ≤3000 for l= 1, and 1000≤N ≤2000 forl= 2. For a fixed set of values ofN at which full calculations are made, the error decreases with increasing l. Although smaller N-values are used for l= 1,2, the estimates on the errors in these cases are smaller than those forl= 0.
The computation of Sl(n, N) with increased precision inMathematicais only possible ifN is not too large. The limitation is either the length of time the computation would take or the available amount of memory. For small l, the extrapolations to infinite N were all performed with M achineP recision. At smaller values of N, results obtained with M achineP recision and results computed with increased precision did not differ significantly (between N = 600 and N = 900, the relative error is below 10−14 for l = 0). Therefore, extrapolations obtained with M achineP recision are reliable within the estimated error bounds, which are of the order 10−8.
For l ≥ 3, we have carried out full computations at only two large N-values. Based on these two numbers we build various estimates to ensure that even if the correction for large N goes only as N12l, rather than N2l+21 , the large-N limit is still recovered with sufficiently high precision.
15.2.2 The infinite sum over l
Having taken the infinite-N limit for finite values of l, we now turn to performing the infinite sum overl. For everyn, we can compute, as described above, the value ofSl(n, N =
∞). We do this for l = 0,1,2, . . . , lmax and then use the leading term in Eq. (15.12) to estimate the remainder of the sum, stemming from contributions starting atl=lmax+ 1 and all the way to l =∞. This procedure can be further improved by performing some calculations at a few selected very large values of l > lmax and looking at the difference between the leading asymptotic form and the numerical result. In this way we get an assessment for the subleading term in Eq. (15.12). With this method, we convince ourselves that the values oflmaxwe use in conjunction with the asymptotic result provide an absolute accuracy on the final numbers of approximately 10−8 (we have to increase lmax with nin order to get a similar absolute accuracy for alln).
15.3 Asymptotics at large R
15.3.1 Fit results for subleading coefficients We end up with a set of numbers for
S(R= (n+ 1/2)a)≡ lim
N→∞S(n, N) = lim
N→∞
∞
X
l=0
(2l+ 1)Sl(n, N) (15.14) for (R/a)2up to about 3.7·103. The results forS(R) vary from order one to a few hundreds and are accurate to about 10−8, i.e., to at least eight digits (see table 2).
n S(n,∞) ∆n 5 8.882458402 3.6·10−9 10 32.509818844 1.4·10−8 15 70.911615387 4.4·10−9 20 124.086187971 5.4·10−9 25 192.033006873 6.1·10−9 30 274.751830288 7.8·10−9 35 372.242527052 8.9·10−9 40 484.505017950 1.1·10−8 45 611.539251453 1.2·10−8 50 753.345192156 1.4·10−8 55 909.922814676 1.5·10−8 60 1081.272100199 1.7·10−8
Table 2: Some numerical results for limN→∞S(n, N) together with the corresponding estimate for the total absolute error ∆n.
500 1000 1500 2000 2500 3000 3500 HRaL2 200
400 600 800 1000
SHRL
Figure 43: Plot ofS(R) as a function of (R/a)2, the red line is obtained from a fit through the last 10 points.
Figure 43 shows a plot ofS(R) as a function of (R/a)2, confirming the area law found in Ref. [71]. The red line through the data points (obtained from a fit through the last 10 points, to the right of the vertical dashed line) is given by
Slin(R) = 0.295406 (R/a)2. (15.15) Srednicki quotes a slope of 0.30, so we confirm the two digits he has found.
Next, we fit the data points to the functional form
Slog(R) =s(R/a)2+c0log(R/a)2+d . (15.16) A least square fit over the last 16 data points, 45≤n≤60, results in
s= 0.295431, c0≡c/2 =−0.005545, d=−0.03537. (15.17) Note the change in sby 2.5·10−5.
194 15.3 Asymptotics at largeR
To estimate the quality of our fit, we compute χ2 =
60
X
i=45
[S(R= (i+ 1/2)a)−Slog(R= (i+ 1/2)a)]2
∆2i ≈2.93, (15.18)
where ∆i denotes the estimate for the error bound for the numerical value of S(R = (i+ 1/2)a) (these estimates are all of the order of 10−8, cf. table 2). Dividing by the number of degrees of freedomNd.o.f.= 16−3 = 13, we obtain
χ2
Nd.o.f. ≈0.23, (15.19)
which indicates that the error estimates ∆i might even be a little bit too conservative.
When we change the range of the data points used in the fit, the result forsdoes not change to the given precision, variations inc0are of the order 10−5, and variations indare of the order 10−4. The fit results can be confirmed within this accuracy by fitting even further subleading terms (with two more subleading terms a fit in the same range of R leads to coefficientss= 0.295431,c0 =−0.0055549,d=−0.03529). Therefore, we expect that our numerical result for the coefficient of the logarithmic term is correct within an accuracy of 0.2 percent.
Figure 44 shows a plot of the difference between the two fits, Slog(R)−Slin(R), as a function of (R/a)2 together with the corresponding data points. Figure 45 shows a similar plot, but this time the linear term Slin, corr(R) = 0.295431(R/a)2 obtained from the fit result (15.17) is subtracted. Both plots show that the numerical results are well described by the functional form (15.16) with coefficients (15.17) in the entire range ofR (only the last 16 points are used for the fit).
500 1000 1500 2000 2500 3000 3500 HRaL2
-0.06 -0.04 -0.02 0.02 SHRL-SlinHRL
Figure 44: Plot of Slog(R)−Slin(R) = 2.5·10−5(R/a)2−0.005545 log(R/a)2−0.03537 as a function of (R/a)2 (solid red curve) and numerically computed data points S(R)−Slin(R) = S(R)−0.295406(R/a)2 (blue dots). Slog(R) is obtained from a fit over the last 16 data points (to the right of the vertical dashed line). Error bounds are of the order 10−8and are not visible in the plot.
15.3.2 Discretized derivative
Based on the results presented above, we have good reason to believe thatS(R) is indeed given by the functional form of Eq. (15.16), up to terms that vanish in the limit R→ ∞.
0 500 1000 1500 2000 2500 3000 3500 HRaL2 -0.080
-0.075 -0.070 -0.065 -0.060 SHRL-Slin,corrHRL
Figure 45: Plot of Slog(R)−0.295431(R/a)2 =−0.005545 log(R/a)2−0.03537 as a function of (R/a)2(solid red curve) and numerically computed data pointsS(R)−0.295431(R/a)2(blue dots).
Slog(R) is obtained from a fit over the last 16 data points (to the right of the vertical dashed line).
Error bounds are of the order 10−8and are not visible in the plot.
This means that the coefficientc0 of the logarithmic term could also be obtained by taking the third derivative w.r.t. R,
∂(R/a)3 S(R) = 4c0a3
R3 +. . . (15.20)
and consequently
c0 = lim
R/a→∞
R3
4a3∂(R/a)3 S(R). (15.21)
Therefore, we now use our numerical data to compute the discretized version of the third derivative
∆3RS(R) = 1
8(S(R+ 3a)−3S(R+a) + 3S(R−a)−S(R−3a)) . (15.22) Then, an estimate for the coefficient of the logarithmic term is obtained from
R3
4a3∆3RS(R) (15.23)
for largeR/a.
Figure 46 shows a plot of 4aR33∆3RS(R) as a function of R/a. The data points seem to be quite stable on a horizontal line, indicating again that the error bounds might be too pessimistic (the error bars in the plot increase with increasing R/a due to the multiplication with (R/a)3). The occurrence of a plateau confirms that there is indeed a subleading logarithmic correction to the area law S(R) ∝ (R/a)2. The numerical value for the coefficient of the logarithm found in the previous section is confirmed and is in agreement with the prediction of−1/180 within an error of 0.2 percent (the red horizontal line in Fig. 46 shows this predicted value).