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Mass gap of O (N) σ-models in 2D. Support for exact results from 1N -expansion
Physics Letters B 266 ( 1991 ) 92-98 North-Holland
PHYSICS LETTERS B
Mass gap of O (N) a-models in 2D. Support for exact results from 1/N-expansion Henrik Flyvbjerg ' and Finn Larsen 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark Received 25 April 1991
The mass gaps of the non-linear a-models in 2D are given to three leading orders in 1/Nas functions of the bare coupling up to correlation lengths 150. Within the small systematic error introduced by truncating the 1/N-expansion, our results agree with available Monte Carlo results for N>~3. At weak coupling, our results agree with recent exact results for the mass gaps based on the Bethe ansatz and the S-matrices of the models, the two sets of results thus lending support to each other. Our calculation features "Fourier accelerated" numerical evaluation of Feynman diagrams, and the use of finite size scaling to obtain infinite volume results from finite volume results.
I. Introduction
Peter Hasenfatz et al. have recently given exact formulae for the mass gaps o f the O ( N ) - s y m m e t r i c nonlinear a-models in 2D with N>~ 3. Since these results are based on the Bethe ansatz and the m o d e l s ' S-matrices [ 1,2], the question naturally arises, whether, in a d d i t i o n to being exact, they are also correct. This question is e m p h a s i z e d by the observation that the result for O ( 3 ) is identical to an older result, obtained by assuming the path integral is saturated by classical multi-instanton configurations [ 1,3 ]. Such an assumption can only lead to an approximate, semiclassical result, one should think. Even m o r e so, because only instantons and no anti-instantons (or vice versa) were included in the contributing configurations. One answer to this question m a y be o b t a i n e d by c o m p a r i n g the exact results o f questioned correctness with a p p r o x i m a t e results o f unquestionable correctness. This is done below, where we give results for the mass gap to three leading orders in 1IN. The F e y n m a n d i a g r a m s o f the 1/N-expansion are impossible to evaluate analytically, except for leading order diagrams and a few second o r d e r diagrams E-mail address: flyvbj @ nbivax, nbi. dk. 2 E-mail address: flarsen @ nbivax, nbi. dk. 92
[4,5 ]. F o r this reason we evaluate the diagrams o f our third o r d e r calculation numerically on square lattices, and c o m p a r e our results at weak coupling with those ofrefs. [ 1,2]. It is well established that the asymptotic scaling limit o f the lattice O ( 3 )-model with s t a n d a r d action is a p p r o a c h e d in a n o n - m o n o t o n i c way [6,7]. Only for correlation lengths larger than 20 does the asymptotic scaling defects o f the correlation length a n d the magnetic susceptibility a p p r o a c h a constant monotonically for decreasing coupling. Similar, though less pronounced, behaviour is seen for N > 3. We were able to evaluate the necessary F e y n m a n diagrams numerically on lattices with up to 256 X 256 sites. Such lattices are too small to represent an infinite lattice, when a s y m p t o t i c scaling is to be demonstrated. However, since our results on these lattices contain no errors b e y o n d the systematic ones i n t r o d u c e d by truncating the series in 1/N, we can use finite size scaling theory to remove finite size effects in our results. This can be done without introducing any a d d i t i o n a l errors o f significance for correlation lengths up to 150 lattice spacings. This upper limit is due to violations o f finite size scaling. W h e r e Monte Carlo results are available, our resuits agree with them well within the systematic error introduced by truncating the 1/N-expansion after three leading orders.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 266, number 1,2
PHYSICS LETTERS B
Since we have results for correlation lengths as large and larger than Monte Carlo simulations presently provide, we can check the mass gaps given in refs. [ 1,2 ]. We confirm these mass gaps to within the small systematic truncation-errors in our results.
2.1/N-expansion We consider the non-linear a-model on a square lattice with the standard nearest neighbour interaction and inverse coupling ft. As is well known, this model can be expanded in 1/N. The connected, mass renormalized two-point function to three leading orders in 1 / N reads [ 8 ]
c~_~=~ _Z --
x
y
N
x
Y
+~1 x
x
y
Y
x
x
y
x
~
x
+~-
Y
graphs, amounts to one subtraction at zero momentum [ 8 ]. The coupling constant is not renormalized. Monte Carlo simulations conventionally give the dressed mass as a function of the bare coupling, so we do the same, to facilitate comparison of results. We find the renormalized mass and the renormalization factor Z from the Fourier transform of cx:
~o=Z/m~,
(4) Z
C2rr/L =
m~+4sin2(n/L ) +O((2n/L)4),
(5)
L being the linear extent of the lattice. This definition of the renormalized mass is a pragmatic one, which yields a physical quantity of dimension mass on a finite lattice, and is equal to the properly defined mass on an infinite lattice, provided m 2 << 1, which is the case below, after finite size effects have been removed. On a finite lattice there is no such thing as a mass in the sense of an inverse correlation length, or, equivalently, a pole in the propagator, simply because there are no asymptotically large distances, and, equivalently, no continuum of m o m e n t u m values. On the other hand, we need a physical quantity of dimension mass on the finite lattice, to construct the proper mass from by finite size analysis. This is provided by eqs. (4) and ( 5 ). Setting x = y in eq. (1) we find the inverse coupling q as a function of m2R/Z:
~/=Co =
-
-5
-
N
Y
+O(1/N x
y
22 August
3) ,
(1)
y
where r/= fl/N, and, with the notation/3 u = 2 sin (½Pu), the propagators and their m o m e n t u m transforms are
x
y=Gx_y,
~=Dx-y,
(2)
1
Gv= f & + m 2 / Z , ( D - l ) x _ y = ' G x-v--~ 2 .-
x (~[[~
•
+g
(3)
Mass renormalization, denoted by the letter R on
+O(
1/N 3) .
(6) 93
Volume 266, number 1,2
PHYSICS LETTERS B
3. Evaluation of diagrams The Feynman diagrams in eq. (5) were evaluated numerically of finite lattices using "Fourier acceleration", i.e. sub-diagrams forming loops were evaluated as products in configuration space rather than convolutions in momentum space. Once evaluated, such a sub-diagram was Fourier transformed to momentum space to have its external legs multiplied on there, rather than convoluted on in configuration space. The general strategy was to avoid all convolutions, and do only multiplications, at the cost of having to Fourier transform between momentum- and configuration space. In this way we evaluated all Feynman diagrams on lattices with L--256 in less than a CPU-hours per fl-value on an Amdahl VP 1100, which has a theoretical peak performance of 285 MFLOPS. We reached speeds higher than 50 MFLOPS. Details are given in ref. [9]. Since the computational task is O ( L 4 log2 L), a doubling of the lattice size would increase the CPUtime requirements by up to a factor 18. Inclusion of another order in 1/N increases the loop-order of the most demanding diagrams by one, thus increasing the CPU-time requirements by a factor L 2. We therefore believe that we have done what is possible with the 1/N-expansion of the non-linear a-model in 2D, using an ordinary vector computer. And we are rather convinced that there is no way to evaluate the diagrams of the 1/N-expansion of this theory analytically, except in the special cases pointed out in ref. [4].
"b
i
Mass gaps were evaluated on lattices with L = 32, 64, 128 and 256, and finite size scaling was confirmed and used to extrapolate results to their values on an infinite lattice. Figs. la, lb, lc show our results for the correlation length ~ - 1~mR as a function offl for N = 3, 4, and 8, respectively. Dotted and dashed lines show results to leading and two leading orders in 1/N, respectively, evaluated on huge lattices. Partly overlapping full lines show results to three leading orders in 1/N, for a 1282, a 2562, and an effectively infinite lattice, the lower lines corresponding to the smaller lattices. The 94
i
i
i
(a)
l.l
i
:
........ --'"
~
m
............................
i
i
i
i
i
1.3
1.5
1.7
1.9
2.1
i
i
i
J
i
"b
..--
(b)
b
1.2
I
I
I
I
I
1.6
2.0
2.4
2.8
3.2
:3.0
4.0
5.0
6.0
7.0
'b
(c)
°o 2.0
4. Results
22 August
Fig. 1. (a) Correlation length ~= 1/mR versus inverse coupling/~ for N = 3. Dotted and dashed lines are results to leading and two leading orders in 1/N, respectively. Partly overlapping full lines are results to three leading orders in 1IN, on lattices with linear extent L = 128, 256, and ~ , the lower lines corresponding to the smaller lattices. Monte Carlo results are from ref. [6] (circles), ref. [7] (triangles), ref. [ 10] (squares), and ref. [ 11 ] (crosses). (b) Same as (a), except N=4. Monte carlo results are from ref. [ 12] (triangles), ref. [ 13] (squares), ref. [ 14] (inverted triangles), and ref. [15] (circles). (c) Same as (a) and (b), except N = 8. Monte Carlo results are from ref. [ 15 ].
result for the infinite lattice was obtained by a finite size scaling analysis of the results for the finite lattices in the following way:
Volume 266, number 1,2
PHYSICS LETTERS B
For convenience, we use units in which the lattice spacing is one. For given inverse coupling fl and lattice size L, we calculated the correlation length ~L. The same value of fl would result in a correlation length ~ooon an infinite lattice. The value of ~ is related to our result for ~L. For dimensional reasons this relationship can be written as
~o~=~LF(~L/L, I/~L) ,
I
0) .
I
I
(a) +
=
x
=
"
=
32/64 64//128 128/256
X +o
~(x)
X* + +X" X"
(7)
x-b .bxXl +xl
......................
d
where 1/~e is the dimensionless ratio between the lattice spacing and the correlation length. For sufficiently large values of ~c we have to a good approximation
~:~ =~LF(~L/L,
22 August
0.0
I
I
I
I
1
0.1
0.2
0.3
0.4
0.5
,
,
0.6
X i
i
,
o
(b)
(8)
This is the finite size scaling hypothesis, stating that the finite size effect ~ L / ~ depends only on the physical size L/~L of the system, and not on parameters of the system, like fl [ 16 ]. The formulation (8) of this hypothesis is particularly well suited for our purpose. ~ was calculated in a "bootstrap" program using two lattices of side last inequality makes ~L a close approximation to ~ . Then x=~,/:/½L and F(x) =~oo/~L/2. Having found F( ) in a certain range of x-values this way, we used it to find ~ from ~t for ~L/L in that range. Next we used these larger values for ~ to determine F(x) for larger values of x. In short, we iterate F to find its values for larger arguments from its values for smaller arguments. It is necessary to do so, because the requirement i << ~L/2, and the fact that ~, < L, prevents us from finding F using lattices smaller than ½L. It is also perfectly correct to do so, as long as small scaling violations do not compound to significant errors by repeated iteration. We have monitored this source of error by using different values for L in the iteration procedure, and only accepted the results when they did not differ significantly. This was the case for correlation lengths ~o~< 150. Fig. 2a shows our results for F(x) for N = 3, in the range of x-values we could determine F in. The different symbols correspond to different values for L. lengths L and ~L, respectively. We calculated the function F(x) =F(x, O) by calculating ~L and ~L/2for such values offl for which 1 << ~L/2 and ~L<< L. The Consequently, where the different symbols all fall on the same smooth curve, the finite size scaling hypothesis is satisfied. Scaling violations are seen to grow
o
a
F(x)
0.0
I
I
I
I
0.1
0.2
0.3
0.4
0,5
0.6
X Fig. 2.(a) Scaling function F(x) from eq. (8) for N=3. (b) Scaling function F(x) for N= 3, 4 and 8 shown with triangles, squares and octagons, respectively. The full line corresponds to
with x. For this reason, we only use F (x) for x ~<0.30. Fig. 2b shows F for N = 3, 4, 8, ~ . It depends smoothly on N. Finite size effects are larger for larger values of N. This N-dependence depends on the massdefinition chosen, however. As already mentioned, there is no unique definition of the mass gap on a finite lattice. Our definition in eqs. (4), (5) involves, when these equations are solved, the lattice derivative of the inverse propagator at zero momentum. This is how Z - ~ is defined by these equations. As an alternative definition of the mass gap on a finite lattice, we differentiated the inverse propagator with respect to momentum squared, to find an expression for Z - l in terms of Feynman diagrams, before we introduced finite lattices. Since differentiation with respect to momentum, and discretization in a finite volume do not commute, we thus obtained a different mass in finite volumes. Both the function F and F ' s dependence on N were different for this alternative mass parameter. The finite size effects were stronger, but decreased for increasing N. 95
Volume 266, number 1,2
PHYSICS LETTERS B
We would like to emphasize that finite size scaling analysis is a perfect supplement to the 1/N-expansion. The expansion gives numerical results that contain a systematic error due to its truncation, but which are otherwise exact up to machine precision for the lattice size considered. Consequently, the finite size scaling analysis gives the result for the truncated expansion on an infinite lattice to a precision which is limited only by the finiteness of the correlation length in lattice units. On the rather large lattices we use, we obtain the relevant scaling functions with a precision that allows us to iterate them several times, before errors accumulate to a level that cannot be neglected in the figures shown. In fig. 1 we have also plotted Monte Carlo results, most o f which are of per-mille precision. So we can use these Monte Carlo results as benchmarks, against which we measure the precision of the approximation provided by the truncated 1/N-expansion. Only in fig. la is a discrepancy clearly visible, and for all three values of Nconsidered here the discrepancy between our results and Monte Carlo results is well within the systematic error of O( 1 / N 3) we expect from neglecting terms of that and higher orders in
22 August
(a) "'Z ... ".
(3 ~
© o
I
1.0
1,1
1.2
"', I
1.3
1.4
I
I
1.5
1.6
2.0
2.2
1.7
1.8
1.9
2.0
2.1
o
1.2
1.4
1.6
1,8
2.4
2,6
2.0
3.0
1/N. When calculating this discrepancy, one should bear in mind that eq. (6) gives q, or, equivalently, fl as a series in 1 / N for fixed values of mZ/Z. Since Z ~ 1 and ~= m ~, ', the truncation error is mainly on fl for given ~, and not vice versa. Consequently, if one wants the relative error on ( f o r fixed fl, it depends onfl. For fl large, for example, with the asymptotic behaviour given in eqs. (9) and (10) below one obtains a relative error of O ( [ 2 g / ( N - 2 ) I N -3) from neglected fourth orders terms. Notice the extra factor 2 g / ( N - 2 ) on the error on ~ for fixed fl, relative to the error on fl for fixed ~. Fig. 3 shows the asymptotic scaling defects ~(fl)/~,symp~(fl) versusfl for N = 3 , 4, and 8, respectively. Here ~(asymp) :
32-
1/2
(c)
6 2.5
3.0
I
I
I"
4.0
4.5
5.0
I 5,5
6.0
6.5
7,0
7.5
Fig. 3. (a) Asymptoticscalingdefect of mass gap versusfl for N= 3. The symbols are the same as in fig. la. (b) Same as (a), except N=4. The symbols are the same as in fig. lb. (c) Same as (a) and (b), except N=8. Monte Carlo results are from ref. [ 15].
F(I+I/(N-2)) [8 e x p ( ~ z / 2 - I ) ] ' / ~ N - 2 ) A z ~ ,
(9)
( N - 2"] 1/(x-2)
A~'=\~flj
where
2gfl exp(~)
0.486 + 0.089(N-2) +O(1/f12) ) X
96
3.5
1+
2~z(N-2)fl
(10)
Volume 266, number 1,2
PHYSICS LETTERS B
is the perturbative three-loop result for ~ [ 17 ] with the exact prefactor found from the Bethe ansatz by P. Hasenfratz et al. [ 2 ]. In fig. 3 our third order results approach the asymptotic behaviour monotonically, contrary to the behaviour of the Monte Carlo results. The failure of the three first terms of the 1/N-series to reproduce the non-monotonic approach to asymptotic scaling agrees well with our understanding of this approach as being a lattice- and action-specific small-N effect. It is related to the phase transitions in the models having N = 1 and 2, the Ising- and XY-models [ 18 ]. This small-Neffect, non-perturbative in 1/N, also explains to the often observed fact that asymptotic scaling is reached faster for larger values of N - here illustrated in fig. 3. In ref. [ 5 ] the mass gap is determined to two leading orders in 1IN, and its asymptotic scaling defect is seen to exhibit a non-monotonic behaviour. This requires comment in the light of our monotonic results. In the present paper the expansion in 1 / N is limited to the necessary minimum: only the path integral for the two-point function is expanded, yielding the Feynman diagrams in eq. ( 1 ). No further approximations are done, as all quantities of interest are easily determined numerically from the Feynman diagrams. In ref. [5], on the other hand, further expansions in 1/N are done for the purpose of obtaining analytical results. In particular, the square of the mass gap is expanded, and consequently the results obtained differ from our result to O ( l / N ) by terms of O( 1IN 2). In principle one result is as good as another since they differ only by higher order terms, some of which were neglected in both calculations. But since we have pushed our calculation to one additional order in 1 / N here, and still obtain a monotonic approach to asymptotic scaling, we yet have too see a calculation along the lines of ref. [5] yield a non-monotonic result to the next order in 1 IN. The radius of convergence of the 1/N-expansion depends on fl: Inspection of the N-dependence of the terms of the strong coupling expansion shows that the radius of convergence of the 1/N-expansion increases with the coupling at strong coupling, becoming infinite at infinite coupling. At intermediate coupling, we expect the radius of convergence of the 1/N-expansion to be ~ 0.4, since the strong coupling expansion indicates a critical point on the fl-axis for
22 August
N < 2.5 [ 19 ]. According to eqs. (9), (10), asymptotically, at weak coupling, the 1/N-expansion of the mass gap is convergent with radius of convergence ½. So, as we move along the fl-axis, our results are of a fixed order in 1/N, but the radius of convergence of the 1/N-expansion varies with fl, and, as a consequence, so does the precision of our results, in the following way: the precision is perfect at fl= 0, decreases with increasing fl to a m i n i m u m at intermediate fl-values, then increases again as fl~ ~ . This explains the fl-dependence of the discrepancy between our results and Monte Carlo results shown in the figures. It also means that our results are more reliable at large fl-values than they are at intermediate fl-values, where the scaling defects have a minimum. At weak coupling, we see that our results confirm the results by Hasenfratz et al. They in turn confirm the reliability of our results. We conclude that the 1/N-expansion for the non-linear a-model is convergent within a finite radius, that its truncation yields approximations with systematic errors the size of neglected terms, and that the exact, ansatz-based results of ref. [1] and ref. [2] for the mass gap m i~"s-v'np)(fl) are correct to three orders in 1 / N up to errors of relative size O ( [ 2 z ~ / ( N - 2) ] N - 3), or less.
Acknowledgement We thank M. Campostrini and P. Rossi for discussions of this paper. Time on the Danish Amdahl VP1 100 was made available by the Danish Natural Science Research Council through contracts no. 1 18415 and 11-8664.
References [ 1] P. Hasenfratz, M. Maggioreand F. Niederrnayer, Phys. Lett. B245 (1990) 522. [2] P. Hasenfratz and F. Niedermayer, Phys. Lett. B 245 (1990) 529. [ 3 ] Y. Iwasaki, Phys. Lett. B 104 ( 1981 ) 458; Prog. Theor. Phys. 68 (1982) 448. [4] H. Flyvbjerg, Phys. Lelt. B 245 (1990) 533; Nucl. Phys. B 348 (1991) 714. [ 5 ] M. Campostrini and P. Rossi, Phys. Lett. B 242 (1990) 81; P. Biscari, M. Campostrini and P. Rossi, Phys. Lett. B 242 (1990) 225.
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Volume 266, number 1,2
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[6] U. Wolff, Phys. Lett. B 222 (1989) 473; Nucl. Phys. B 344 (1990) 581. [7] P. Hasenfratz, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 427; F. Niedermayer, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 319; P. Hasenfratz and F. Niedermayer, Nucl. Phys. B 337 (1990) 233. [8] H. Flyvbjerg and S. Varsted, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 343; Nucl. Phys. B 344 (1990) 646. [9] F. Larsen and H. Flyvbjerg, preprint NBI-HE-91-19 (May 1991). [ 10] 1. Bender, W. Wetzel and B. Berg, Nucl. Phys. B 269 (1986) 389. [ 11 ] J. Apostolakis, C.F. Baillie and G.C. Fox, Caltech preprint C 3 P-943. 10/6/90. [ 12 ] E. Seiler, I.O. Stamatescu, A. Patrascioiu and V. Linke, Nucl. Phys. B 305 [FS23] (1988) 623.
98
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[ 13 ] U. Heller, Phys. Rev. D 38 (1988) 3834. [ 14] R.E. Edwards and A.D. Sokal, Phys. Rev. D 40 (1989) 1374. [ 15] U. Wolff, Phys. Lett. B 248 (1990) 335. [ 16] M.N. Barber, in: Phase transitions and critical phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983) p. 146. [17] E. Br6zin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Phys. Rev. B 14 (1976) 985; M. Falcioni and A. Treves, Phys. Lett. B 159 (1985) 140; Nucl. Phys. B 265 [FSI5] (1986) 671. [ 18 ] M. Klomfass, U.M. Heller and H. Flyvbjerg, Phys. Lett. B 258 ( 1991 ) 386; Nucl. Phys. B [FS] 360 ( 1991 ) 264. [ 19] P. Butera, M. Comi and G. Marchesini, Phys. Rev. B 41 (1990) 11494.
PHYSICS LETTERS B
Mass gap of O (N) a-models in 2D. Support for exact results from 1/N-expansion Henrik Flyvbjerg ' and Finn Larsen 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark Received 25 April 1991
The mass gaps of the non-linear a-models in 2D are given to three leading orders in 1/Nas functions of the bare coupling up to correlation lengths 150. Within the small systematic error introduced by truncating the 1/N-expansion, our results agree with available Monte Carlo results for N>~3. At weak coupling, our results agree with recent exact results for the mass gaps based on the Bethe ansatz and the S-matrices of the models, the two sets of results thus lending support to each other. Our calculation features "Fourier accelerated" numerical evaluation of Feynman diagrams, and the use of finite size scaling to obtain infinite volume results from finite volume results.
I. Introduction
Peter Hasenfatz et al. have recently given exact formulae for the mass gaps o f the O ( N ) - s y m m e t r i c nonlinear a-models in 2D with N>~ 3. Since these results are based on the Bethe ansatz and the m o d e l s ' S-matrices [ 1,2], the question naturally arises, whether, in a d d i t i o n to being exact, they are also correct. This question is e m p h a s i z e d by the observation that the result for O ( 3 ) is identical to an older result, obtained by assuming the path integral is saturated by classical multi-instanton configurations [ 1,3 ]. Such an assumption can only lead to an approximate, semiclassical result, one should think. Even m o r e so, because only instantons and no anti-instantons (or vice versa) were included in the contributing configurations. One answer to this question m a y be o b t a i n e d by c o m p a r i n g the exact results o f questioned correctness with a p p r o x i m a t e results o f unquestionable correctness. This is done below, where we give results for the mass gap to three leading orders in 1IN. The F e y n m a n d i a g r a m s o f the 1/N-expansion are impossible to evaluate analytically, except for leading order diagrams and a few second o r d e r diagrams E-mail address: flyvbj @ nbivax, nbi. dk. 2 E-mail address: flarsen @ nbivax, nbi. dk. 92
[4,5 ]. F o r this reason we evaluate the diagrams o f our third o r d e r calculation numerically on square lattices, and c o m p a r e our results at weak coupling with those ofrefs. [ 1,2]. It is well established that the asymptotic scaling limit o f the lattice O ( 3 )-model with s t a n d a r d action is a p p r o a c h e d in a n o n - m o n o t o n i c way [6,7]. Only for correlation lengths larger than 20 does the asymptotic scaling defects o f the correlation length a n d the magnetic susceptibility a p p r o a c h a constant monotonically for decreasing coupling. Similar, though less pronounced, behaviour is seen for N > 3. We were able to evaluate the necessary F e y n m a n diagrams numerically on lattices with up to 256 X 256 sites. Such lattices are too small to represent an infinite lattice, when a s y m p t o t i c scaling is to be demonstrated. However, since our results on these lattices contain no errors b e y o n d the systematic ones i n t r o d u c e d by truncating the series in 1/N, we can use finite size scaling theory to remove finite size effects in our results. This can be done without introducing any a d d i t i o n a l errors o f significance for correlation lengths up to 150 lattice spacings. This upper limit is due to violations o f finite size scaling. W h e r e Monte Carlo results are available, our resuits agree with them well within the systematic error introduced by truncating the 1/N-expansion after three leading orders.
0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 266, number 1,2
PHYSICS LETTERS B
Since we have results for correlation lengths as large and larger than Monte Carlo simulations presently provide, we can check the mass gaps given in refs. [ 1,2 ]. We confirm these mass gaps to within the small systematic truncation-errors in our results.
2.1/N-expansion We consider the non-linear a-model on a square lattice with the standard nearest neighbour interaction and inverse coupling ft. As is well known, this model can be expanded in 1/N. The connected, mass renormalized two-point function to three leading orders in 1 / N reads [ 8 ]
c~_~=~
x
y
N
x
Y
+~1 x
x
y
Y
x
x
y
x
~
x
+~-
Y
graphs, amounts to one subtraction at zero momentum [ 8 ]. The coupling constant is not renormalized. Monte Carlo simulations conventionally give the dressed mass as a function of the bare coupling, so we do the same, to facilitate comparison of results. We find the renormalized mass and the renormalization factor Z from the Fourier transform of cx:
~o=Z/m~,
(4) Z
C2rr/L =
m~+4sin2(n/L ) +O((2n/L)4),
(5)
L being the linear extent of the lattice. This definition of the renormalized mass is a pragmatic one, which yields a physical quantity of dimension mass on a finite lattice, and is equal to the properly defined mass on an infinite lattice, provided m 2 << 1, which is the case below, after finite size effects have been removed. On a finite lattice there is no such thing as a mass in the sense of an inverse correlation length, or, equivalently, a pole in the propagator, simply because there are no asymptotically large distances, and, equivalently, no continuum of m o m e n t u m values. On the other hand, we need a physical quantity of dimension mass on the finite lattice, to construct the proper mass from by finite size analysis. This is provided by eqs. (4) and ( 5 ). Setting x = y in eq. (1) we find the inverse coupling q as a function of m2R/Z:
~/=Co =
-
-5
-
N
Y
+O(1/N x
y
22 August
3) ,
(1)
y
where r/= fl/N, and, with the notation/3 u = 2 sin (½Pu), the propagators and their m o m e n t u m transforms are
x
y=Gx_y,
~=Dx-y,
(2)
1
Gv= f & + m 2 / Z , ( D - l ) x _ y = ' G x-v--~ 2 .-
x (~[[~
•
+g
(3)
Mass renormalization, denoted by the letter R on
+O(
1/N 3) .
(6) 93
Volume 266, number 1,2
PHYSICS LETTERS B
3. Evaluation of diagrams The Feynman diagrams in eq. (5) were evaluated numerically of finite lattices using "Fourier acceleration", i.e. sub-diagrams forming loops were evaluated as products in configuration space rather than convolutions in momentum space. Once evaluated, such a sub-diagram was Fourier transformed to momentum space to have its external legs multiplied on there, rather than convoluted on in configuration space. The general strategy was to avoid all convolutions, and do only multiplications, at the cost of having to Fourier transform between momentum- and configuration space. In this way we evaluated all Feynman diagrams on lattices with L--256 in less than a CPU-hours per fl-value on an Amdahl VP 1100, which has a theoretical peak performance of 285 MFLOPS. We reached speeds higher than 50 MFLOPS. Details are given in ref. [9]. Since the computational task is O ( L 4 log2 L), a doubling of the lattice size would increase the CPUtime requirements by up to a factor 18. Inclusion of another order in 1/N increases the loop-order of the most demanding diagrams by one, thus increasing the CPU-time requirements by a factor L 2. We therefore believe that we have done what is possible with the 1/N-expansion of the non-linear a-model in 2D, using an ordinary vector computer. And we are rather convinced that there is no way to evaluate the diagrams of the 1/N-expansion of this theory analytically, except in the special cases pointed out in ref. [4].
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Mass gaps were evaluated on lattices with L = 32, 64, 128 and 256, and finite size scaling was confirmed and used to extrapolate results to their values on an infinite lattice. Figs. la, lb, lc show our results for the correlation length ~ - 1~mR as a function offl for N = 3, 4, and 8, respectively. Dotted and dashed lines show results to leading and two leading orders in 1/N, respectively, evaluated on huge lattices. Partly overlapping full lines show results to three leading orders in 1/N, for a 1282, a 2562, and an effectively infinite lattice, the lower lines corresponding to the smaller lattices. The 94
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Fig. 1. (a) Correlation length ~= 1/mR versus inverse coupling/~ for N = 3. Dotted and dashed lines are results to leading and two leading orders in 1/N, respectively. Partly overlapping full lines are results to three leading orders in 1IN, on lattices with linear extent L = 128, 256, and ~ , the lower lines corresponding to the smaller lattices. Monte Carlo results are from ref. [6] (circles), ref. [7] (triangles), ref. [ 10] (squares), and ref. [ 11 ] (crosses). (b) Same as (a), except N=4. Monte carlo results are from ref. [ 12] (triangles), ref. [ 13] (squares), ref. [ 14] (inverted triangles), and ref. [15] (circles). (c) Same as (a) and (b), except N = 8. Monte Carlo results are from ref. [ 15 ].
result for the infinite lattice was obtained by a finite size scaling analysis of the results for the finite lattices in the following way:
Volume 266, number 1,2
PHYSICS LETTERS B
For convenience, we use units in which the lattice spacing is one. For given inverse coupling fl and lattice size L, we calculated the correlation length ~L. The same value of fl would result in a correlation length ~ooon an infinite lattice. The value of ~ is related to our result for ~L. For dimensional reasons this relationship can be written as
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This is the finite size scaling hypothesis, stating that the finite size effect ~ L / ~ depends only on the physical size L/~L of the system, and not on parameters of the system, like fl [ 16 ]. The formulation (8) of this hypothesis is particularly well suited for our purpose. ~ was calculated in a "bootstrap" program using two lattices of side last inequality makes ~L a close approximation to ~ . Then x=~,/:/½L and F(x) =~oo/~L/2. Having found F( ) in a certain range of x-values this way, we used it to find ~ from ~t for ~L/L in that range. Next we used these larger values for ~ to determine F(x) for larger values of x. In short, we iterate F to find its values for larger arguments from its values for smaller arguments. It is necessary to do so, because the requirement i << ~L/2, and the fact that ~, < L, prevents us from finding F using lattices smaller than ½L. It is also perfectly correct to do so, as long as small scaling violations do not compound to significant errors by repeated iteration. We have monitored this source of error by using different values for L in the iteration procedure, and only accepted the results when they did not differ significantly. This was the case for correlation lengths ~o~< 150. Fig. 2a shows our results for F(x) for N = 3, in the range of x-values we could determine F in. The different symbols correspond to different values for L. lengths L and ~L, respectively. We calculated the function F(x) =F(x, O) by calculating ~L and ~L/2for such values offl for which 1 << ~L/2 and ~L<< L. The Consequently, where the different symbols all fall on the same smooth curve, the finite size scaling hypothesis is satisfied. Scaling violations are seen to grow
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with x. For this reason, we only use F (x) for x ~<0.30. Fig. 2b shows F for N = 3, 4, 8, ~ . It depends smoothly on N. Finite size effects are larger for larger values of N. This N-dependence depends on the massdefinition chosen, however. As already mentioned, there is no unique definition of the mass gap on a finite lattice. Our definition in eqs. (4), (5) involves, when these equations are solved, the lattice derivative of the inverse propagator at zero momentum. This is how Z - ~ is defined by these equations. As an alternative definition of the mass gap on a finite lattice, we differentiated the inverse propagator with respect to momentum squared, to find an expression for Z - l in terms of Feynman diagrams, before we introduced finite lattices. Since differentiation with respect to momentum, and discretization in a finite volume do not commute, we thus obtained a different mass in finite volumes. Both the function F and F ' s dependence on N were different for this alternative mass parameter. The finite size effects were stronger, but decreased for increasing N. 95
Volume 266, number 1,2
PHYSICS LETTERS B
We would like to emphasize that finite size scaling analysis is a perfect supplement to the 1/N-expansion. The expansion gives numerical results that contain a systematic error due to its truncation, but which are otherwise exact up to machine precision for the lattice size considered. Consequently, the finite size scaling analysis gives the result for the truncated expansion on an infinite lattice to a precision which is limited only by the finiteness of the correlation length in lattice units. On the rather large lattices we use, we obtain the relevant scaling functions with a precision that allows us to iterate them several times, before errors accumulate to a level that cannot be neglected in the figures shown. In fig. 1 we have also plotted Monte Carlo results, most o f which are of per-mille precision. So we can use these Monte Carlo results as benchmarks, against which we measure the precision of the approximation provided by the truncated 1/N-expansion. Only in fig. la is a discrepancy clearly visible, and for all three values of Nconsidered here the discrepancy between our results and Monte Carlo results is well within the systematic error of O( 1 / N 3) we expect from neglecting terms of that and higher orders in
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1/N. When calculating this discrepancy, one should bear in mind that eq. (6) gives q, or, equivalently, fl as a series in 1 / N for fixed values of mZ/Z. Since Z ~ 1 and ~= m ~, ', the truncation error is mainly on fl for given ~, and not vice versa. Consequently, if one wants the relative error on ( f o r fixed fl, it depends onfl. For fl large, for example, with the asymptotic behaviour given in eqs. (9) and (10) below one obtains a relative error of O ( [ 2 g / ( N - 2 ) I N -3) from neglected fourth orders terms. Notice the extra factor 2 g / ( N - 2 ) on the error on ~ for fixed fl, relative to the error on fl for fixed ~. Fig. 3 shows the asymptotic scaling defects ~(fl)/~,symp~(fl) versusfl for N = 3 , 4, and 8, respectively. Here ~(asymp) :
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F(I+I/(N-2)) [8 e x p ( ~ z / 2 - I ) ] ' / ~ N - 2 ) A z ~ ,
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where
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0.486 + 0.089(N-2) +O(1/f12) ) X
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(10)
Volume 266, number 1,2
PHYSICS LETTERS B
is the perturbative three-loop result for ~ [ 17 ] with the exact prefactor found from the Bethe ansatz by P. Hasenfratz et al. [ 2 ]. In fig. 3 our third order results approach the asymptotic behaviour monotonically, contrary to the behaviour of the Monte Carlo results. The failure of the three first terms of the 1/N-series to reproduce the non-monotonic approach to asymptotic scaling agrees well with our understanding of this approach as being a lattice- and action-specific small-N effect. It is related to the phase transitions in the models having N = 1 and 2, the Ising- and XY-models [ 18 ]. This small-Neffect, non-perturbative in 1/N, also explains to the often observed fact that asymptotic scaling is reached faster for larger values of N - here illustrated in fig. 3. In ref. [ 5 ] the mass gap is determined to two leading orders in 1IN, and its asymptotic scaling defect is seen to exhibit a non-monotonic behaviour. This requires comment in the light of our monotonic results. In the present paper the expansion in 1 / N is limited to the necessary minimum: only the path integral for the two-point function is expanded, yielding the Feynman diagrams in eq. ( 1 ). No further approximations are done, as all quantities of interest are easily determined numerically from the Feynman diagrams. In ref. [5], on the other hand, further expansions in 1/N are done for the purpose of obtaining analytical results. In particular, the square of the mass gap is expanded, and consequently the results obtained differ from our result to O ( l / N ) by terms of O( 1IN 2). In principle one result is as good as another since they differ only by higher order terms, some of which were neglected in both calculations. But since we have pushed our calculation to one additional order in 1 / N here, and still obtain a monotonic approach to asymptotic scaling, we yet have too see a calculation along the lines of ref. [5] yield a non-monotonic result to the next order in 1 IN. The radius of convergence of the 1/N-expansion depends on fl: Inspection of the N-dependence of the terms of the strong coupling expansion shows that the radius of convergence of the 1/N-expansion increases with the coupling at strong coupling, becoming infinite at infinite coupling. At intermediate coupling, we expect the radius of convergence of the 1/N-expansion to be ~ 0.4, since the strong coupling expansion indicates a critical point on the fl-axis for
22 August
N < 2.5 [ 19 ]. According to eqs. (9), (10), asymptotically, at weak coupling, the 1/N-expansion of the mass gap is convergent with radius of convergence ½. So, as we move along the fl-axis, our results are of a fixed order in 1/N, but the radius of convergence of the 1/N-expansion varies with fl, and, as a consequence, so does the precision of our results, in the following way: the precision is perfect at fl= 0, decreases with increasing fl to a m i n i m u m at intermediate fl-values, then increases again as fl~ ~ . This explains the fl-dependence of the discrepancy between our results and Monte Carlo results shown in the figures. It also means that our results are more reliable at large fl-values than they are at intermediate fl-values, where the scaling defects have a minimum. At weak coupling, we see that our results confirm the results by Hasenfratz et al. They in turn confirm the reliability of our results. We conclude that the 1/N-expansion for the non-linear a-model is convergent within a finite radius, that its truncation yields approximations with systematic errors the size of neglected terms, and that the exact, ansatz-based results of ref. [1] and ref. [2] for the mass gap m i~"s-v'np)(fl) are correct to three orders in 1 / N up to errors of relative size O ( [ 2 z ~ / ( N - 2) ] N - 3), or less.
Acknowledgement We thank M. Campostrini and P. Rossi for discussions of this paper. Time on the Danish Amdahl VP1 100 was made available by the Danish Natural Science Research Council through contracts no. 1 18415 and 11-8664.
References [ 1] P. Hasenfratz, M. Maggioreand F. Niederrnayer, Phys. Lett. B245 (1990) 522. [2] P. Hasenfratz and F. Niedermayer, Phys. Lett. B 245 (1990) 529. [ 3 ] Y. Iwasaki, Phys. Lett. B 104 ( 1981 ) 458; Prog. Theor. Phys. 68 (1982) 448. [4] H. Flyvbjerg, Phys. Lelt. B 245 (1990) 533; Nucl. Phys. B 348 (1991) 714. [ 5 ] M. Campostrini and P. Rossi, Phys. Lett. B 242 (1990) 81; P. Biscari, M. Campostrini and P. Rossi, Phys. Lett. B 242 (1990) 225.
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[6] U. Wolff, Phys. Lett. B 222 (1989) 473; Nucl. Phys. B 344 (1990) 581. [7] P. Hasenfratz, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 427; F. Niedermayer, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 319; P. Hasenfratz and F. Niedermayer, Nucl. Phys. B 337 (1990) 233. [8] H. Flyvbjerg and S. Varsted, Nucl. Phys. B (Proc. Suppl.) 17 (1990) 343; Nucl. Phys. B 344 (1990) 646. [9] F. Larsen and H. Flyvbjerg, preprint NBI-HE-91-19 (May 1991). [ 10] 1. Bender, W. Wetzel and B. Berg, Nucl. Phys. B 269 (1986) 389. [ 11 ] J. Apostolakis, C.F. Baillie and G.C. Fox, Caltech preprint C 3 P-943. 10/6/90. [ 12 ] E. Seiler, I.O. Stamatescu, A. Patrascioiu and V. Linke, Nucl. Phys. B 305 [FS23] (1988) 623.
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[ 13 ] U. Heller, Phys. Rev. D 38 (1988) 3834. [ 14] R.E. Edwards and A.D. Sokal, Phys. Rev. D 40 (1989) 1374. [ 15] U. Wolff, Phys. Lett. B 248 (1990) 335. [ 16] M.N. Barber, in: Phase transitions and critical phenomena, Vol. 8, eds. C. Domb and J.L. Lebowitz (Academic Press, New York, 1983) p. 146. [17] E. Br6zin and J. Zinn-Justin, Phys. Rev. Lett. 36 (1976) 691; Phys. Rev. B 14 (1976) 985; M. Falcioni and A. Treves, Phys. Lett. B 159 (1985) 140; Nucl. Phys. B 265 [FSI5] (1986) 671. [ 18 ] M. Klomfass, U.M. Heller and H. Flyvbjerg, Phys. Lett. B 258 ( 1991 ) 386; Nucl. Phys. B [FS] 360 ( 1991 ) 264. [ 19] P. Butera, M. Comi and G. Marchesini, Phys. Rev. B 41 (1990) 11494.