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Estimating bulk values based on finite size scaling
I | LII[I I W,-~"| ~ | ll'b,~]I L l L~] Nuclear Physics B (Proc. Suppl.) 34 (1994) 702-704 North-Holland
PROCEEDINGS SUPPLEMENTS
Estimating bulk values based on finite size scaling Jae-Kwon Kim a • a Physics Department, University of Arizona Tucson, AZ 85721, USA We present a new method of evaluating the bulk values (thermodynamic values) of various physical quantities based on a formula of finite size scaling. This method enables one to extract accurate bulk values (BV) without using the large size of the lattice required in the traditional direct Monte Cazlo measurement. We illustrate our
method taking as example the 2 dimensional (2D) Ising model, and present BV of the correlation length (~¢¢) for the 2D 0(3) model up to f~ = 2.4.
1. I N T R O D U C T I O N
The theory of finite size scaling (FSS)[1] combined with Monte Carlo (MC) simulations has been increasingly used to extrapolate the information available from the finite system to the thermodynamic limit. The standard application of the FSS makes use of two basic formulae of the theory applicable to the power-law criticalbehavior (PL); one is Fisher's round formula[i] to extract a critical point and v, and the other is the FSS formula at a critical point to evaluate the ratio of other critical index to u. The standard usage of FSS is, however, limited to the system exhibiting a PL critical behavior; moreover, it is not possible to extract BV of any physical quantities. We present here a simple and powerful technique for MC simulations by uncovering a new aspect of FSS. 2. T H E O R Y
AND METHOD
For a physical quantity P having a PL critical behavior, P~(t) ,,, t-P, the fundamental formula of the FSS[1] is
PL(t) ,,, LP/"Y(s(t)),
(1)
which implies both the round formula and the FSS formula at a criticality. Here s(t) is a scaling variable defined as
s(t) - Ll~oo(t ) = (Po~(t)ILpI")-"I p. *The author would llke to thank the organizin~ committee of the Lattice'93 for financial support.
Eq. 1 states that PL(t)/LPl v is a function of pcc(t)/Le/V only. As a consequence, we obtain
PL(t) = Poo(t)/e(s(t))
(2)
One notices from Eq. 2 that x(t) =_ ~L(t)/L is just a function of s(t) and vice versa, so instead of s(t) in Eq. 2 we write
PL(t) = Poo(t)qp(x(t)),
(3)
where qp is another scaling function. The significance of Eq. 3 is that qe(z) does not explicitly depend on the temperature, so that once qp(x) is known for a certain point of temperature one can interpolate its value at any other temperature. We take as example the 2D Ising model for an illustration. (See Ref. [2] for details.). At fl = 0.425, we measure ~L and XL varying L from L=20 until the BVs are obtained. Thus, q¢(x) and qx(x) are available for some discrete values of x. At fl = 0.430, we choose a suitably small L and measure XL and ~L on this lattice. The value of x thus obtained at this j3 should be in its range at fl = 0.425, so that one can interpolate qp(x). Once qp(x) and PL are given, P ~ is directly calculated from Eq. 3. In this way, ~ and X~ were estimated up to fl = 0.438 (Table 1). The ~ thus estimated are compared with the exact values ( ~ in Table 1) , showing excellent agreement. For a check of the data of X, we fit our X~ to yield 7 = 1.77(3) and fie = 0.4407(1). It is worth stressing that for this model L/~¢~ >__6 is required to get BVs using the traditional men-
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00379-A
J-K. Kim /Estimating bulk values based on finite size scaling surement, so at fl = 0.438, for example, at least L=570 would be required. We used the rational function interpolation scheme (the subroutine RATINT in Ref. [3]) with the degree of the interpolation always four. In principle, the error involved in an interpolation should be explicitly estimated, but the subroutine we used does not give a rigorous error estimate. Now, it is time to question (i) whether Eq. 2 holds for the non-PL critical behavior as well, and (ii) when the correction to Eq. 2 would be significant. Concerning (i), for any multiplicative renormalizable quantity P, Br@zin and ZinnJustin[4] derived Eq. 2 in the 2D O(N) non-linear sigma models (N > 3); This might be regarded as a generalization of Liischer's pioneering work[5], where f~(s) was explicitly obtained. Moreover, for the 2D XY model and the 2D standard 0(3) model there are excellent numerical evidences[6] for Eq. 3. It thus appears that FSS is indeed of the kinematical nature, so that the validity of Eq. 2 does not depend on the Hamiltonian of the system (accordingly, on the critical behavior of the system, at least in dimensions less than the upper marginal dimensions. Concerning(ii), Eq. 2 does not hold when L is sufficiently small. One can easily check that when L < 6, qp(x) is not even a single-valued function for most models such as the 2D and 3D Ising models; for L > 10, however, Eq. 2 starts to be accurate within the numerical error of the very precise measurements. How about the correction to Eq. 2 due to other scaling variables from the irrelevant operators? We believe that this is an artifact of the real space renormalization group equation treatment of the FSS. In particular, we numerically checked this point for the 2D four state Potts model (where the leading PL has a multiplicative logarithmic correction[7]), and observed that the correction due to another scaling variable is indeed negligible[6]. 3. R E S U L T F O R T H E 2D 0 ( 3 ) M O D E L
We applied this method to the 2D standard, nearest neighbor, 0(3) non-linear sigma model. First, we measured ~L(fl) at fl = 1.5, 1.7, 1.8, 1.9, and 2.0 varying L from 20 up to L=200 (for
Table 1 in the 2D Ising model. L Z fL 40 0.430 18.97(7) 60 0.432 25.10(9) 60 0.434 29.62(12) 80 0.436 40.92(21) 150 0.438 73.91(40)
703
23.28(16) 28.80(28) 37.27(15) 53.40(62) 92.85(50)
23.215... 28.60... 37.209... 53.16... 92.87_
/3 = 1.9). Evaluating qf(x) was facilitated by the use of the BVs available at these values of /3[8]. In the range of x thus calculated, Eq. 3 is verified by the observation that q~(x) at different values of fl tend to overlap each other (Figure 1). In spite of some fluctuations depending on the points of interpolation (in the region where qp(x) is steep one needs highly accurate value of x for a reliable interpolation), the BVs obtained by this method agree well with those from the direct measurements in this range of ft. Although our data for fl > 2.2 are more or less preliminary, we present our resultup to fl = 2.4 without providing all the detailed numerics here (Table 2). (A detailed paper will be published elsewhere.) For 2.1 < fl < 2.4, our ~oo(fl) are compared with the analytical prediction, that is, for the 2D O(N) model =
+
+...],
(4)
where C~ = (e/8)l/(N-2)2-5/2e-*/[2(N-2)]r(1 + 1 / ( N - 2))[9] and a: x -0.0915 for N=3 from the 3-loop perturbation calculation[10]. In this range, our values deviate from Eq. 4 by about 11,~ 14 percent (Table 2), which certainly improve the previous report of 20 percent deviation at ~ = 2.05[8]. To check asymptotic scaling we introduce ~ - ~oo/[ e2~(1 + al/fl)/fl ], and the results are: 6~ x 103=1.642(31), 1.715(16), 1.729(41), 1.769(38), and 1.766(28)for ~=2.0, 2.1, 2.2, 2.3, and 2.4 respectively. Asymptotic scaling seems to set in around fl = 2.3, but more data at fl > 2.4 are needed to clarify the issue.
J..-K. Kim /Estimating bulk values based on finite size scaling
704
Table 2 ~oo in the 2D 0(3) model. ~ represents the BVs directly measured for 1.7 < /~ < 2.0, while it represents the analytical prediction for 2.1
80 100 90 120 150 60 100 200 60 60 140 60 120
29.19(5) 31.50(9) 41.37(9) 49.22(14) 54.83(14) 35.03(5) 53.14(12) 86.48(18) 38.29(6) 40.88(6) 93.10(9) 45.61(8) 90.50(7)
34.3(1) 34.4(3) 64.2(4) 65.0(3) 64.6(5) 121.2(4) 120.9(9) 121.7(14) 227.9(17) 423.9(28) 759(10) 1395(18) 2505(27)
1.0
x ~.1.s
%
0 ~b- 1.7
t. p - l . a a p-1.9 o 13-2.o
Q ®
A ~--2.2 v ~-2.4
[D
34.4(1)
(D
64.5(5) 0.5 ®
122.7(11) 224.3(42) 485.5 872.3 1566.9 2819.5
\ 0'%.o
o:s
1.o
Figure 1. q~(x) versus x for the 2D 0(3) model.
4. C O N C L U S I O N A N D D I S C U S S I O N S The method described above can certainly be a way to overcome the difficulties in large scMe simulations. Finally a few comments are in order: * In general, Eq. 2 is violated in the upper marginal dimensions[11]; however, it is only weakly violated. (See, for example, Ref. [12] for the glueball mass in 4D QCD.) • qp(X) characterizes a universality class.
* For the 2D O(N) model (N _> 3) where the critical behavior is of the form P~(/~) ,,~ exp(pfl)/fl d (v and v' for the ~) as fl ~ ~ , Eq. 2 predicts pip' = v/u ~ [6]; this contradicts the analytical prediction based on the perturbation theory[13]. REFERENCES
1. M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28, (1972) 1516 2. J.-K. Kim, Controlling finite size effects in Monte Carlo simulations, Univ. of Arizona Preprint, AZPH-TH/93-36 (1993)
3. W.tt. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, (Cambridge University Press, 1986) 4. E. Br6zin and J. Zinn-Justin, Nucl. Phys. B 257 (1985) 867 5. M. Liischer, Phys. Lett. B118 (1982) 391; Nucl. Phys. B 219 (1983) 233 6. J.-K. Kim, Finite size scaling for general critical behavior, Univ. of Arizona Preprint, AZPIt-TH/93-38 (1993) 7. J.L. Cardy, M. Nauenberg and D.J. Sealpino, Phys. Rev. B 22 (1980) 2560 8. J. Apostolakis, C.F. Baillie, and J. Fox, Phys. Rev. D 43 (1991) 2687 9. P. Hasenfratz, M. Maggiore and F. Niedermayer, Phys. Lett. B 245 (1990) 522; P. Hasenfratz and F. Niedermayer, Phys. Lett. B 245 (1990) 529 10. M. Falcioni and A. Treves, Phys. Lett. B 159 (1985) 140 11. E. Br6zin, J. de Phys. 43 (1982) 15 12. M. Liiseher, Com. Math. Phys. 104 (1986) 177; Com. Math. Phys. 105 (1986) 153; Nucl. Phys. B 354 (1991) 531 13. E. Br6zin and J. Zinn-Justin, Phys. Rev. B 14 (1976) 3110
PROCEEDINGS SUPPLEMENTS
Estimating bulk values based on finite size scaling Jae-Kwon Kim a • a Physics Department, University of Arizona Tucson, AZ 85721, USA We present a new method of evaluating the bulk values (thermodynamic values) of various physical quantities based on a formula of finite size scaling. This method enables one to extract accurate bulk values (BV) without using the large size of the lattice required in the traditional direct Monte Cazlo measurement. We illustrate our
method taking as example the 2 dimensional (2D) Ising model, and present BV of the correlation length (~¢¢) for the 2D 0(3) model up to f~ = 2.4.
1. I N T R O D U C T I O N
The theory of finite size scaling (FSS)[1] combined with Monte Carlo (MC) simulations has been increasingly used to extrapolate the information available from the finite system to the thermodynamic limit. The standard application of the FSS makes use of two basic formulae of the theory applicable to the power-law criticalbehavior (PL); one is Fisher's round formula[i] to extract a critical point and v, and the other is the FSS formula at a critical point to evaluate the ratio of other critical index to u. The standard usage of FSS is, however, limited to the system exhibiting a PL critical behavior; moreover, it is not possible to extract BV of any physical quantities. We present here a simple and powerful technique for MC simulations by uncovering a new aspect of FSS. 2. T H E O R Y
AND METHOD
For a physical quantity P having a PL critical behavior, P~(t) ,,, t-P, the fundamental formula of the FSS[1] is
PL(t) ,,, LP/"Y(s(t)),
(1)
which implies both the round formula and the FSS formula at a criticality. Here s(t) is a scaling variable defined as
s(t) - Ll~oo(t ) = (Po~(t)ILpI")-"I p. *The author would llke to thank the organizin~ committee of the Lattice'93 for financial support.
Eq. 1 states that PL(t)/LPl v is a function of pcc(t)/Le/V only. As a consequence, we obtain
PL(t) = Poo(t)/e(s(t))
(2)
One notices from Eq. 2 that x(t) =_ ~L(t)/L is just a function of s(t) and vice versa, so instead of s(t) in Eq. 2 we write
PL(t) = Poo(t)qp(x(t)),
(3)
where qp is another scaling function. The significance of Eq. 3 is that qe(z) does not explicitly depend on the temperature, so that once qp(x) is known for a certain point of temperature one can interpolate its value at any other temperature. We take as example the 2D Ising model for an illustration. (See Ref. [2] for details.). At fl = 0.425, we measure ~L and XL varying L from L=20 until the BVs are obtained. Thus, q¢(x) and qx(x) are available for some discrete values of x. At fl = 0.430, we choose a suitably small L and measure XL and ~L on this lattice. The value of x thus obtained at this j3 should be in its range at fl = 0.425, so that one can interpolate qp(x). Once qp(x) and PL are given, P ~ is directly calculated from Eq. 3. In this way, ~ and X~ were estimated up to fl = 0.438 (Table 1). The ~ thus estimated are compared with the exact values ( ~ in Table 1) , showing excellent agreement. For a check of the data of X, we fit our X~ to yield 7 = 1.77(3) and fie = 0.4407(1). It is worth stressing that for this model L/~¢~ >__6 is required to get BVs using the traditional men-
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00379-A
J-K. Kim /Estimating bulk values based on finite size scaling surement, so at fl = 0.438, for example, at least L=570 would be required. We used the rational function interpolation scheme (the subroutine RATINT in Ref. [3]) with the degree of the interpolation always four. In principle, the error involved in an interpolation should be explicitly estimated, but the subroutine we used does not give a rigorous error estimate. Now, it is time to question (i) whether Eq. 2 holds for the non-PL critical behavior as well, and (ii) when the correction to Eq. 2 would be significant. Concerning (i), for any multiplicative renormalizable quantity P, Br@zin and ZinnJustin[4] derived Eq. 2 in the 2D O(N) non-linear sigma models (N > 3); This might be regarded as a generalization of Liischer's pioneering work[5], where f~(s) was explicitly obtained. Moreover, for the 2D XY model and the 2D standard 0(3) model there are excellent numerical evidences[6] for Eq. 3. It thus appears that FSS is indeed of the kinematical nature, so that the validity of Eq. 2 does not depend on the Hamiltonian of the system (accordingly, on the critical behavior of the system, at least in dimensions less than the upper marginal dimensions. Concerning(ii), Eq. 2 does not hold when L is sufficiently small. One can easily check that when L < 6, qp(x) is not even a single-valued function for most models such as the 2D and 3D Ising models; for L > 10, however, Eq. 2 starts to be accurate within the numerical error of the very precise measurements. How about the correction to Eq. 2 due to other scaling variables from the irrelevant operators? We believe that this is an artifact of the real space renormalization group equation treatment of the FSS. In particular, we numerically checked this point for the 2D four state Potts model (where the leading PL has a multiplicative logarithmic correction[7]), and observed that the correction due to another scaling variable is indeed negligible[6]. 3. R E S U L T F O R T H E 2D 0 ( 3 ) M O D E L
We applied this method to the 2D standard, nearest neighbor, 0(3) non-linear sigma model. First, we measured ~L(fl) at fl = 1.5, 1.7, 1.8, 1.9, and 2.0 varying L from 20 up to L=200 (for
Table 1 in the 2D Ising model. L Z fL 40 0.430 18.97(7) 60 0.432 25.10(9) 60 0.434 29.62(12) 80 0.436 40.92(21) 150 0.438 73.91(40)
703
23.28(16) 28.80(28) 37.27(15) 53.40(62) 92.85(50)
23.215... 28.60... 37.209... 53.16... 92.87_
/3 = 1.9). Evaluating qf(x) was facilitated by the use of the BVs available at these values of /3[8]. In the range of x thus calculated, Eq. 3 is verified by the observation that q~(x) at different values of fl tend to overlap each other (Figure 1). In spite of some fluctuations depending on the points of interpolation (in the region where qp(x) is steep one needs highly accurate value of x for a reliable interpolation), the BVs obtained by this method agree well with those from the direct measurements in this range of ft. Although our data for fl > 2.2 are more or less preliminary, we present our resultup to fl = 2.4 without providing all the detailed numerics here (Table 2). (A detailed paper will be published elsewhere.) For 2.1 < fl < 2.4, our ~oo(fl) are compared with the analytical prediction, that is, for the 2D O(N) model =
+
+...],
(4)
where C~ = (e/8)l/(N-2)2-5/2e-*/[2(N-2)]r(1 + 1 / ( N - 2))[9] and a: x -0.0915 for N=3 from the 3-loop perturbation calculation[10]. In this range, our values deviate from Eq. 4 by about 11,~ 14 percent (Table 2), which certainly improve the previous report of 20 percent deviation at ~ = 2.05[8]. To check asymptotic scaling we introduce ~ - ~oo/[ e2~(1 + al/fl)/fl ], and the results are: 6~ x 103=1.642(31), 1.715(16), 1.729(41), 1.769(38), and 1.766(28)for ~=2.0, 2.1, 2.2, 2.3, and 2.4 respectively. Asymptotic scaling seems to set in around fl = 2.3, but more data at fl > 2.4 are needed to clarify the issue.
J..-K. Kim /Estimating bulk values based on finite size scaling
704
Table 2 ~oo in the 2D 0(3) model. ~ represents the BVs directly measured for 1.7 < /~ < 2.0, while it represents the analytical prediction for 2.1
80 100 90 120 150 60 100 200 60 60 140 60 120
29.19(5) 31.50(9) 41.37(9) 49.22(14) 54.83(14) 35.03(5) 53.14(12) 86.48(18) 38.29(6) 40.88(6) 93.10(9) 45.61(8) 90.50(7)
34.3(1) 34.4(3) 64.2(4) 65.0(3) 64.6(5) 121.2(4) 120.9(9) 121.7(14) 227.9(17) 423.9(28) 759(10) 1395(18) 2505(27)
1.0
x ~.1.s
%
0 ~b- 1.7
t. p - l . a a p-1.9 o 13-2.o
Q ®
A ~--2.2 v ~-2.4
[D
34.4(1)
(D
64.5(5) 0.5 ®
122.7(11) 224.3(42) 485.5 872.3 1566.9 2819.5
\ 0'%.o
o:s
1.o
Figure 1. q~(x) versus x for the 2D 0(3) model.
4. C O N C L U S I O N A N D D I S C U S S I O N S The method described above can certainly be a way to overcome the difficulties in large scMe simulations. Finally a few comments are in order: * In general, Eq. 2 is violated in the upper marginal dimensions[11]; however, it is only weakly violated. (See, for example, Ref. [12] for the glueball mass in 4D QCD.) • qp(X) characterizes a universality class.
* For the 2D O(N) model (N _> 3) where the critical behavior is of the form P~(/~) ,,~ exp(pfl)/fl d (v and v' for the ~) as fl ~ ~ , Eq. 2 predicts pip' = v/u ~ [6]; this contradicts the analytical prediction based on the perturbation theory[13]. REFERENCES
1. M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28, (1972) 1516 2. J.-K. Kim, Controlling finite size effects in Monte Carlo simulations, Univ. of Arizona Preprint, AZPH-TH/93-36 (1993)
3. W.tt. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes, (Cambridge University Press, 1986) 4. E. Br6zin and J. Zinn-Justin, Nucl. Phys. B 257 (1985) 867 5. M. Liischer, Phys. Lett. B118 (1982) 391; Nucl. Phys. B 219 (1983) 233 6. J.-K. Kim, Finite size scaling for general critical behavior, Univ. of Arizona Preprint, AZPIt-TH/93-38 (1993) 7. J.L. Cardy, M. Nauenberg and D.J. Sealpino, Phys. Rev. B 22 (1980) 2560 8. J. Apostolakis, C.F. Baillie, and J. Fox, Phys. Rev. D 43 (1991) 2687 9. P. Hasenfratz, M. Maggiore and F. Niedermayer, Phys. Lett. B 245 (1990) 522; P. Hasenfratz and F. Niedermayer, Phys. Lett. B 245 (1990) 529 10. M. Falcioni and A. Treves, Phys. Lett. B 159 (1985) 140 11. E. Br6zin, J. de Phys. 43 (1982) 15 12. M. Liiseher, Com. Math. Phys. 104 (1986) 177; Com. Math. Phys. 105 (1986) 153; Nucl. Phys. B 354 (1991) 531 13. E. Br6zin and J. Zinn-Justin, Phys. Rev. B 14 (1976) 3110