The surprising convergence of the Monte Carlo renormalization group for the d=3 Ising Model

TL;DR

This paper introduces a simplified Monte Carlo renormalization group method with a tunable parameter, achieving high-accuracy critical property calculations for the 3D Ising model by optimizing convergence through parameter tuning.

Contribution

It presents a novel, adjustable block-spin transformation that enhances convergence and accuracy in Monte Carlo renormalization group calculations for the 3D Ising model.

Findings

Improved convergence with a tunable block-spin parameter.

High-accuracy critical property estimates achieved.

Parameter tuning varies for different exponents.

Abstract

We present a surprisingly simple approach to high-accuracy calculations of critical properties of the three-dimensional Ising model. The method uses a modified block-spin transformation with a tunable parameter to improve convergence in Monte Carlo renormalization group. The block-spin parameter must be tuned differently for different exponents to produce optimal convergence.

Tables

Table 1. Table 1: Data for the y T ​ 1 subscript 𝑦 𝑇 1 y_{T1} simulations.

	${256}^3$	${128}^3$	${64}^3$	${32}^3$	${16}^3$
${\#}_{\text{sites}}/N^3$	$2.2\times {10}^5$	$2.2\times {10}^5$	$2.2\times {10}^5$	$4\times {10}^7$	$4\times {10}^8$
${\#}_{\text{Wolff}}/N^3$	22	91	420	$3.2\times {10}^4$	$1.2\times {10}^6$
$\Delta {\#}_{\text{Wolff}}$	164	87	50	26	15
Cluster	$1.0\times {10}^5$	$2.4\times {10}^4$	$5.2\times {10}^3$	$1.2\times {10}^3$	$2.7\times {10}^2$

Table 2. Table 2: The eigenvalue exponent y T ​ 1 subscript 𝑦 𝑇 1 y_{T1} for majority transformation which is equivalent to using a large w 𝑤 w in Eq. ( 6 ).

$n$	$N_e$	${256}^3$	${128}^3$	${64}^3$	${32}^3$
1	10	1.4189(7)	1.4202(5)	1.4203(2)	1.4206(1)
	20	1.4230(7)	1.4241(5)	1.4240(2)	1.4241(1)
	30	1.4224(7)	1.4243(5)	1.4237(2)	1.4238(1)
2	10	1.5093(6)	1.5106(3)	1.5108(1)	1.5120(1)
	20	1.5076(6)	1.5088(3)	1.5086(1)	1.5084(1)
	30	1.5072(6)	1.5084(3)	1.5082(1)	1.5076(1)
3	10	1.5521(5)	1.5534(5)	1.5544(2)
	20	1.5508(5)	1.5515(5)	1.5507(2)
	30	1.5504(5)	1.5512(5)	1.5501(2)
4	10	1.5733(7)	1.5745(5)
	20	1.5721(7)	1.5711(5)
	30	1.5718(7)	1.5702(5)
5	10	1.5825(11)
	20	1.5797(11)
	30	1.5788(11)

Table 3. Table 3: The eigenvalue exponent y T ​ 1 subscript 𝑦 𝑇 1 y_{T1} . The parameter w = 0.4314 𝑤 0.4314 w=0.4314 .

$n$	$N_e$	${256}^3$	${128}^3$	${64}^3$	${32}^3$	${16}^3$
1	10	1.5870(6)	1.5864(2)	1.5865(1)	1.5872(1)	1.58835(4)
	20	1.5923(7)	1.5914(3)	1.5916(1)	1.5920(1)	1.59266(5)
	30	1.5930(8)	1.5924(3)	1.5922(2)	1.5927(2)	1.59308(5)
2	10	1.5908(5)	1.5903(2)	1.5907(1)	1.5920(1)
	20	1.5917(5)	1.5912(2)	1.5915(1)	1.5923(1)
	30	1.5919(5)	1.5914(2)	1.5916(1)	1.5921(1)
3	10	1.5910(5)	1.5904(2)	1.5922(1)
	20	1.5912(5)	1.5905(2)	1.5920(1)
	30	1.5912(5)	1.5905(2)	1.5918(1)
4	10	1.5906(6)	1.5918(3)
	20	1.5906(6)	1.5914(3)
	30	1.5906(6)	1.5912(3)
5	10	1.5911(6)
	20	1.5909(6)
	30	1.5908(7)

Table 4. Table 4: The eigenvalue exponent y H ​ 1 subscript 𝑦 𝐻 1 y_{H1} . The parameter w = 0.555 𝑤 0.555 w=0.555 .

$n$	$N_o$	${256}^3$	${128}^3$	${64}^3$	${32}^3$	${16}^3$
1	5	2.50830(8)	2.50831(3)	2.50829(1)	2.50828(1)	2.50823(1)
	10	2.50853(14)	2.50869(4)	2.50860(2)	2.50860(2)	2.50856(1)
	15	2.50844(15)	2.50871(5)	2.50859(2)	2.50861(2)	2.50859(1)
	20	2.50843(15)	2.50871(5)	2.50860(2)	2.50862(2)	2.50860(1)
2	5	2.48503(2)	2.48504(1)	2.48503(1)	2.48503(2)	2.48481(1)
	10	2.48507(3)	2.48506(1)	2.48505(1)	2.48506(2)	2.48498(1)
	15	2.48508(3)	2.48507(1)	2.48506(1)	2.48507(2)	2.48493(2)
	20	2.48508(3)	2.48507(1)	2.48506(1)	2.48507(2)	2.48494(2)
3	5	2.48285(3)	2.48287(2)	2.48286(1)	2.48267(4)
	10	2.48284(3)	2.48287(2)	2.48288(1)	2.48279(4)
	15	2.48285(3)	2.48288(2)	2.48288(1)	2.48274(4)
	20	2.48285(3)	2.48288(2)	2.48288(1)	2.48274(4)
4	5	2.48300(5)	2.48293(4)	2.48279(3)
	10	2.48299(5)	2.48295(4)	2.48290(3)
	15	2.48300(5)	2.48295(4)	2.48283(3)
	20	2.48300(5)	2.48294(4)	2.48282(3)
5	5	2.48271(12)	2.48250(9)
	10	2.48272(12)	2.48259(9)
	15	2.48273(12)	2.48251(9)
	20	2.48274(12)	2.48251(9)
6	5	2.48223(23)
	10	2.48234(24)
	15	2.48234(25)
	20	2.48231(25)

Table 5. Table 5: The eigenvalue exponent y T ​ 2 subscript 𝑦 𝑇 2 y_{T2} . The parameter w = 0.4314 𝑤 0.4314 w=0.4314 , the tuned parameter for y T ​ 1 subscript 𝑦 𝑇 1 y_{T1} . The simulations were the same as for y T ​ 1 subscript 𝑦 𝑇 1 y_{T1} , which are given in Table 3 .

$n$	$N_e$	${256}^3$	${128}^3$	${64}^3$	${32}^3$	${16}^3$
1	10	-0.60(2)	-0.548(6)	-0.545(2)	-0.546(3)	-0.5300(10)
	20	-0.70(2)	-0.622(9)	-0.616(3)	-0.622(4)	-0.6081(12)
	30	-0.67(2)	-0.644(12)	-0.626(3)	-0.638(4)	-0.6186(12)
2	10	-0.64(2)	-0.608(5)	-0.611(2)	-0.607(2)
	20	-0.70(2)	-0.664(7)	-0.668(2)	-0.668(3)
	30	-0.71(2)	-0.698(10)	-0.691(3)	-0.688(4)
3	10	-0.66(2)	-0.635(5)	-0.634(2)
	20	-0.71(2)	-0.691(8)	-0.696(3)
	30	-0.72(2)	-0.722(10)	-0.725(4)
4	10	-0.67(1)	-0.658(6)
	20	-0.74(2)	-0.726(9)
	30	-0.73(2)	-0.753(10)
5	10	-0.73(2)
	20	-0.77(2)
	30	-0.76(2)

Table 6. Table 6: The eigenvalue exponent y H ​ 2 subscript 𝑦 𝐻 2 y_{H2} . The parameter w = 0.555 𝑤 0.555 w=0.555 , the tuned parameter for y H ​ 1 subscript 𝑦 𝐻 1 y_{H1} . The simulations were the same as for y H ​ 1 subscript 𝑦 𝐻 1 y_{H1} , which are given in Table 4 .

$n$	$N_o$	${256}^3$	${128}^3$	${64}^3$	${32}^3$	${16}^3$
1	5	0.232(4)	0.236(1)	0.2409(6)	0.2469(7)	0.2561(2)
	10	0.287(5)	0.287(2)	0.2888(7)	0.2910(8)	0.2935(3)
	15	0.287(6)	0.290(2)	0.2911(8)	0.2931(9)	0.2957(3)
	20	0.287(6)	0.297(2)	0.2972(8)	0.2992(9)	0.3012(3)
2	5	0.306(3)	0.307(1)	0.3152(4)	0.3284(5)	0.3405(2)
	10	0.362(4)	0.360(2)	0.3635(5)	0.3689(6)	0.3677(2)
	15	0.366(4)	0.365(2)	0.3680(5)	0.3728(6)	0.3744(2)
	20	0.371(5)	0.372(2)	0.3735(5)	0.3777(6)	0.3770(2)
3	5	0.318(3)	0.331(1)	0.3455(5)	0.3586(5)
	10	0.379(3)	0.384(1)	0.3900(6)	0.3871(6)
	15	0.384(3)	0.390(1)	0.3949(6)	0.3949(6)
	20	0.392(3)	0.396(2)	0.4000(6)	0.3976(7)
4	5	0.335(3)	0.353(1)	0.3638(4)
	10	0.390(3)	0.400(1)	0.3935(4)
	15	0.396(3)	0.406(1)	0.4024(4)
	20	0.402(3)	0.411(1)	0.4053(5)
5	5	0.349(3)	0.366(1)
	10	0.397(3)	0.397(1)
	15	0.403(3)	0.407(1)
	20	0.410(3)	0.410(1)
6	5	0.364(3)
	10	0.395(3)
	15	0.406(3)
	20	0.409(3)

Table 7. Table 7: Five correlation functions obtained from the same simulations as in Table 3 and 5 , with the parameter w = 0.4314 𝑤 0.4314 w=0.4314 . (a=000-100; b=000-110; c=000-111; d=000-100-010-110 and e=000-100-010-001.)

$L^3$	$\Sigma$	${256}^3$	${128}^3$	${64}^3$	${32}^3$	${16}^3$
${256}^3$	a	0.330491(1)
	b	0.208951(1)
	c	0.163751(2)
	d	0.175667(1)
	e	0.115223(1)
${128}^3$	a	0.277505(3)	0.330980(2)
	b	0.193674(4)	0.209600(3)
	c	0.156671(4)	0.164487(4)
	d	0.123980(2)	0.176051(2)
	e	0.096526(2)	0.115679(2)
${64}^3$	a	0.264324(7)	0.278867(6)	0.332281(2)
	b	0.189019(8)	0.195345(8)	0.211325(3)
	c	0.154033(9)	0.158523(8)	0.166440(3)
	d	0.113330(5)	0.124977(4)	0.177075(2)
	e	0.091201(5)	0.097588(4)	0.116894(2)
${32}^3$	a	0.261975(16)	0.267975(16)	0.282485(5)	0.335734(4)
	b	0.189760(18)	0.193421(18)	0.199785(5)	0.215903(5)
	c	0.155992(20)	0.158896(20)	0.16350(6)	0.171623(5)
	d	0.111433(12)	0.115920(11)	0.127626(4)	0.179794(3)
	e	0.090799(11)	0.093920(12)	0.100414(4)	0.120119(3)
${16}^3$	a	0.268326(40)	0.271726(37)	0.277684(11)	0.292087(9)	0.344889(3)
	b	0.199029(42)	0.201440(44)	0.205131(12)	0.211566(11)	0.228043(3)
	c	0.166844(46)	0.168865(48)	0.171831(13)	0.176526(12)	0.185370(3)
	d	0.115740(29)	0.118288(26)	0.122836(8)	0.134672(7)	0.187019(2)
	e	0.095977(28)	0.097952(28)	0.101185(8)	0.107933(7)	0.128690(2)
$8^3$	a	0.291884(85)	0.294200(87)	0.297557(25)	0.303419(22)	0.317542(6)
	b	0.228122(96)	0.229982(95)	0.232453(29)	0.236208(25)	0.242812(7)
	c	0.199280(102)	0.200925(102)	0.203053(32)	0.206160(26)	0.211216(7)
	d	0.132318(70)	0.134094(64)	0.136752(19)	0.141473(17)	0.153630(5)
	e	0.113640(65)	0.115173(63)	0.117304(18)	0.120794(17)	0.128180(5)
$4^3$	a	0.358390(186)	0.360473(181)	0.362650(60)	0.365979(44)	0.371601(13)
	b	0.307968(197)	0.309833(214)	0.311695(66)	0.314279(50)	0.318110(14)
	c	0.286644(206)	0.288429(226)	0.290147(68)	0.292477(54)	0.295778(14)
	d	0.183398(168)	0.185148(152)	0.187090(51)	0.190036(37)	0.195074(11)
	e	0.167453(156)	0.169101(150)	0.170828(51)	0.173341(38)	0.177337(10)

Table 8. Table 8: Estimates of the critical exponents and the eigenvalue exponents from several sources. Values that are marked with an asterisk are calculated to be consistent with the published exponents in the same source. For the last three columns, y T ​ 1 subscript 𝑦 𝑇 1 y_{T1} and y H ​ 1 subscript 𝑦 𝐻 1 y_{H1} are obtained for the corresponding values of ν 𝜈 \nu and η 𝜂 \eta .

	This work	Ref.Blöte et al. (1996)	Ref.Hasenbusch (2010)	Ref.Guida and Zinn-Justin (1998)	Ref.Guida and Zinn-Justin (1998)
	MCRG	MCRG	MC	$ϵ$-expansion	$d=3$
$y_{T1}$	1.591(1)	1.585(3)	1.5872(3)*	1.590(63)*	1.5862(33)*
$y_{H1}$	2.4829(2)	2.481(1)	2.4819(1)*	2.4820(25)*	2.4833(13)*
$\nu$	0.6285(4)*	0.6309(2)*	0.63002(10)	0.6290(25)	0.6304(13)
$\eta$	0.0342(4)*	0.038(2)*	0.03627(10)	0.0360(50)	0.0335(25)
$\beta$	0.3250(2)*	0.3274(9)*	0.32645(10)*	0.3257(26)	0.3258(14)
$\gamma$	1.2356(8)*	1.2378(27)*	1.2372(4)*	1.2355(50)	1.2396(13)
$y_{T2}$	-0.75(5)		-0.832(6)*	-0.814(19)*	-0.799(10)*
$\omega$	0.47(3)*		0.524(4)	0.512(12)	.504(6)

Equations

$$H=K\sum _{⟨j,k⟩}{\sigma }_j{\sigma }_k,$$

$$T_{\alpha ,\beta }^{(n+1,n)}=\frac{\partial K_{\alpha }^{(n+1)}}{\partial K_{\beta }^{(n)}}$$

$$\frac{\partial ⟨S_{\gamma }^{(n+1)}⟩}{\partial K_{\beta }^{(n)}}=\sum _{\alpha }\frac{\partial ⟨S_{\gamma }^{(n+1)}⟩}{\partial K_{\alpha }^{(n+1)}}\frac{\partial K_{\alpha }^{(n+1)}}{\partial K_{\beta }^{(n)}}$$

$$\frac{\partial ⟨S_{\gamma }^{(n+1)}⟩}{\partial K_{\beta }^{(n)}}=⟨S_{\gamma }^{(n+1)}{S}_{\beta }^{(n)}⟩-⟨S_{\gamma }^{(n+1)}⟩⟨S_{\beta }^{(n)}⟩,$$

$$\frac{\partial ⟨S_{\gamma }^{(n+1)}⟩}{\partial K_{\alpha }^{(n+1)}}=⟨S_{\gamma }^{(n+1)}{S}_{\alpha }^{(n+1)}⟩-⟨S_{\gamma }^{(n+1)}⟩⟨S_{\alpha }^{(n+1)}⟩.$$

$$P({\sigma }_{\ell }^{\prime })=\frac{\exp (w{\sigma }_{\ell }^{\prime }{\sum }_{j\in \ell }{\sigma }_j)}{\exp (w{\sum }_{j\in \ell }{\sigma }_j)+\exp (-w{\sum }_{j\in \ell }{\sigma }_j)}$$

Full text

Critical exponents; Monte Carlo renormalization group; optimized convergence

I Introduction

The Monte Carlo renormalization group (MCRG) method is a systematic procedure for computing critical properties of lattice spin modelsMa (1976); Swendsen (1979). It has been shown to be both flexible and effective in the calculation of critical exponents, critical temperatures, and renormalized couplings constantsSwendsen (1984a, b); Brandt and Ron (2001); Ron and Swendsen (2001, 2002); Ron et al. (2002, 2005). A particularly interesting application of MCRG is the three-dimensional Ising modelBlöte and Swendsen (1979); Pawley et al. (1984). This model has proven to be one of the most difficult to obtain accurate estimates for, because the approach to the fixed point is so slow. Attempts have been made to bring the fixed point closer to the nearest-neighbor modelSwendsen (1984c), but these have been controversialFisher and Randeria (1986), and have not resulted in improved results.

The most encouraging result has been that of Blöte et al.Blöte et al. (1996) who used a three-parameter approximation to the fixed point, along with a modified majority rule for the RG transformation. We have discovered a particularly simple modification of this calculation, which simulates the nearest-neighbor critical point and optimizes the RG transformation for the even and odd exponents separately.

In the following sections, we recall the MCRG method, illustrate the slow convergence with the majority rule, and introduce a tunable RG transformation in section IVBlöte et al. (1996). The improved convergence of the tuned RG transformation is then demonstrated in sections V through IX. Finally, we present our conclusions and discuss future work.

II MCRG computations

We consider the three-dimensional Ising model on a simple cubic lattice, of size $N\times N\times N$. The Hamiltonian is given by

[TABLE]

where $\sigma_{j}=\pm 1$, and the sum is over all nearest-neighbor pairs. The dimensionless coupling constant $K$ includes the inverse temperature $\beta=1/k_{B}T$, so as to make the Boltzmann factor $e^{H}$.

We used Wolff algorithmWolff (1989), to simulate the model at an inverse temperature of $K_{c}=0.2216544$Talapov and Blöte (1996). The renormalized configurations were obtained from these sets. For each configuration, the lattice was divided up into cubes, each containing eight sites, so that the scaling factor $b=2$. We will denote this block of spins, as well as the renormalized spins associated with them by $\ell$. A value of plus or minus one was assigned to each renormalized spin to represent the original spins in each cube. We used the ran2 random number generator from Numerical RecipesPress et al. (1999). The lengths of the simulations we used are given in Table 1.

The renormalized configurations can be described by the set of (unknown) renormalized coupling constants, $K_{\alpha}^{(n)}$. The subscript $\alpha$ denotes the type of coupling (nearest-neighbor, next-nearest-neighbor, four-spin, etc.). The nearest-neighbor coupling constant $K$ defined earlier, is also denoted by $K_{nn}^{(0)}$. All other coupling constants at level $n=0$ vanish.

To determine the critical exponents, we then wish to calculate the matrix of derivatives of the couplings at level $n+1$ with respect to the couplings at level $n$.

[TABLE]

This matrix of derivatives is then given by the solution of the equation

[TABLE]

where

[TABLE]

and

[TABLE]

For our calculations we have included $N_{e}=30$ even and $N_{o}=20$ odd interactions. We have followedBaillie et al. (1992), who calculated all $53$ even and $46$ odd interactions that fit in either a $3\times 3$ square or a $2\times 2\times 2$ cube of spins, and used their $34$ first even operators (excluding the $25^{th}$,$29^{th}$,$31^{st}$ and the $32^{nd}$) and their first $20$ odd operations. The eigenvalues of the $T$-matrix in Eq. (2) are found separately for the even and odd operators. The critical exponents are then obtained from the usual equations.

III The slow convergence of the majority rule

The usual majority rule performs very well for the two-dimensional Ising model, which converges to the fixed point values of the exponents by the second iteration of the RG transformationSwendsen (1979). For the three-dimensional Ising model, however, convergence is very slow, as shown in Table 2. Reading across at each level $n$ of RG iterations, the values of the approximations for $y_{T1}$ are quite consistent. There is no problem with finite-size effects on the RG trajectories, indicating that the range of the renormalized Hamiltonian is limited. However, even after five iterations of the renormalization group, the value of $y_{T1}$ does not seem to have converged. Anticipating our final result of $y_{T1}\approx 1.591$, Table 2 is very far from convergence.

IV Tunable block-spin transformation

Instead of using the usual majority rule, the renormalized spin was assigned a value according to the following probabilityBlöte et al. (1996).

[TABLE]

For $w=\infty$, this becomes identical to the majority rule.

A special feature of the present calculation is that the RG transformation in Eq. (6) was optimized separately for the even and odd operators. The determination of the optimal value of the parameter $w$ was also done much more carefully than in earlier work. The value of $w$ was adjusted so that the largest eigenvalue (for the even and odd operators separately) was nearly constant for $n>1$.

V The largest even eigenvalue exponent, $y_{T1}$

The results for the largest even eigenvalue are given in Table 3. By contrast to the slow convergence seen in Table 2, the convergence is striking. For $n=1$, the majority rule has $y_{T1}\approx 1.422$, changes to $y_{T1}\approx 1.507$ for $n=2$, and increases to $y_{T1}\approx 1.579$ for $n=5$. The tuned RG transformation in Table 3 starts with $y_{T1}\approx 1.593$ for $n=1$, moves to $y_{T1}\approx 1.592$ for $n=2$, and stays at $y_{T1}\approx 1.591$ for $n=3,4,\textrm{and }5$. Our best estimate for the largest even eigenvalue is $y_{T1}=1.591(1)$.

VI The largest odd eigenvalue exponent, $y_{H1}$

Table 4 shows the convergence of $y_{H1}$ for the tuned renormalization group. Reading along the rows, we see that there is virtually no effect of the size of the lattice on the estimated values of $y_{H1}$. Only for a renormalized lattice of $4\times 4\times 4$ can a decrease in the value of $y_{H1}$ of about $0.00014$ be seen. Neither is there a noticeable dependence on the number of operators for a given number of RG iterations $n$.

The first iteration of the renormalization group ($n=1$) gives an estimate of about $y_{H1}\approx 2.5086$. For $n=2$, it has dropped slightly to $y_{H1}\approx 2.48507$, and for $n=3,4,\textrm{and }5$, it is $y_{H1}\approx 2.4829$. Our best estimate is $y_{H1}=2.4829(2)$.

VII The second-largest even eigenvalue exponent, $y_{T2}$

Table 5 shows that estimates for the second even eigenvalue as a function of the number of RG iterations, $n$, and the size of the renormalized lattices. This eigenvalue is negative (“irrelevant”), and controls the leading corrections to scaling.

The second largest eigenvalues are naturally not as accurately determined as the largest. We need about 20 operators to see the asymptotic behavior. There is a slight trend for the values of $y_{T2}$ to increase in magnitude with an increasing number of RG iterations, suggesting that the asymptotic eigenvalue exponent is actually larger. Note that similar slow convergence for the calculation of $y_{T2}$ was already observed and reported by Baillie et al.Baillie et al. (1992). Perhaps we can estimate $y_{T2}=-0.75(5)$ from the tables, but that might be overly optimistic.

The correction-to-scaling exponent is given by the ratio of $\omega=-y_{T2}/y_{T1}$, so that we would estimate $\omega=0.75/1.591=0.47(3)$.

VIII The second-largest odd eigenvalue exponent, $y_{H2}$

The second-largest odd eigenvalue exponent, like its counterpart in the two-dimensional Ising model, is positive (“relevant”). It has smaller statistical errors than its even counterpart, but it also shows a slightly slower convergence. As shown in Table 6, the first iteration of the RG transformation ($n=1$) is $0.287(6)$, and therefore quite far from the best estimate. By $n=3$ and greater, the value of the second odd eigenvalues has converged to about $y_{H2}=0.403(4)$.

IX Convergence of the correlation functions

Note that Blöte et al. were able to achieve improved convergence with $w=0.4$ and a different Hamiltonian, which appeared to be closer to the fixed pointBlöte et al. (1996). Our results show that the improved convergence came primarily from the choice of RG transformation. Indeed, there is no evidence that the tuned RG transformation brings the renormalized Hamiltonians closer to the fixed point. Apparently, the renormalization trajectory is such that even though it passes through points a significant distance from the fixed point, the convergence of $y_{T1}$ is very good.

The fact that the RG trajectory itself does not converge rapidly, can be seen in Table 7. This table is organized differently than the tables for the eigenvalue exponents. Rows correspond to equally-sized renormalized lattices, with the size indicated by the first column. The entries are the values of the corresponding correlation functions, divided by the size of the lattice, to facilitate comparisons. It can be seen that although the correlation functions are closer to each other when they correspond to more renormalization iterations, they have not converged for the RG iterations down to $4^{3}$. A comparison with the entries on the diagonal in Table 3 shows a relatively weak size effect for the eigenvalue exponent values.

X Summary and future work

The results of our computations and a comparison with other works are shown in Table 8. The agreement between the various methods is generally good, although some differences exist. Since we don’t have estimates of the systematic errors in our results, we can’t really say what the source of the differences are.

The most reliable of the estimates shown in Table 8 are those of HasenbuschHasenbusch (2010). This was a very careful Monte Carlo finite-size study that included many effects of corrections to scaling to provide limits on the systematic errors.

The largest discrepancies are between our estimate of $\omega$ and the estimates of HasenbuschHasenbusch (2010) and Guida and Zinn-JustinGuida and Zinn-Justin (1998). Our value, $\omega=0.47(3)$ is substantially lower than the others. This could be partly due to the large fluctuations, but it could be that the estimates for $y_{T2}$ are not yet converged. Table 5 could be easily viewed as indicating that the absolute values of the estimates of $y_{T2}$ are still increasing with the iterations of the renormalization group. One possibility for improving the convergence is to optimize the RG parameter $w$ separately for the eigenvalue exponent $y_{T2}$. We are currently exploring this possibility.

The most obvious source of systematic error in our calculation of the largest eigenvalue exponents is the uncertainty of the value of the critical coupling that was used in the Monte Carlo simulation. The usual way to estimate the critical coupling from MCRG is to use the convergence of the renormalized correlation functions. Unfortunately, it is clear from Table 7 that the convergence of the correlation functions for the tuned RG transformation is not sufficient for the purpose. It might be possible to do the calculation with the assistance of an extrapolation of the values, but it seems more promising to use the convergence of the eigenvalues. Preliminary calculations are very encouraging, but more work is needed.