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Monte Carlo measurement of the mass gap in the two-dimensional SU(3) × SU(3) chiral model with standard and improved action
Volume 141B, number 5,6
PHYSICS LET'FERS
5 July 1984
MONTE CARLO MEASUREMENT OF THE MASS GAP IN THE TWO-DIMENSIONAL SU(3) × SU(3) CHIRAL MODEL WITH STANDARD AND IMPROVED ACTION Mikiharu NAKAGAWARA and Sung-Kil YANGJ
Department of l'hysics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 5 March 1984
The mass gap in the two-dimensional SU(3) X SU(3) chiral model both with the standard and hnproved action is measured by the Monte Carlo method. The scaling behavior is observed in both actions. The improved action shows an earlier onset of scaling than the standard one.
The lattice SU(N) X SU(N) chiral model in two dimensions is asymptotically free [1--3]. Then the physical quantities are expected to follow the scaling law predicted by the renormalization group near the continuum limit. Recently Monte Carlo (MC) simulations of the SU(3) X SU(3) chiral model have been performed [4]. However, the scaling behavior of the mass gap, which is of real physical interest, has not yet been observed. In this note our purpose is to establish the scaling behavior of the mass gap in the two-dimensional SU(3) X SU(3) chiral model by the MC method. The MC computations i,a practice suffer from the effect of the finite lattice spacing a. A recent theoretical proposal by Symanzik is to construct an improved action where this effect is systematically eliminated in perturbation theol2¢ [5]. Using the lattice action improved by Symanzik's procedure one expects a smoother onset of scaling. From this point of view, it is particularly interesting to simulate the chiral model both with the improved (IA) and standard action (SA), and to compare the results with each other. For our purpose it is sufficient ot use the IA which is determined to eliminate the O(a 2) corrections in the chiral invariant two-point function. Such an IA I Postdoctoral Fellow, Japan Society for the Promotion of Science, on leave from Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan.
has been obtained at the tree level [6], and up to the one-loop level by one of us [7]. The chiral model is defined by the action
S = n,~u ~ [Cl(3)trUnU+n+u+Cz(3)trUU+n+2~ + c.c.],
(1) where/3 = l]g 2, Un is the SU(3) matrix on the lattice site n = (x, t) and/l runs from 1 to 2. The action coefficients are given by c1(/3) = 1,
c2(/3) = 0
(2)
for the SA, and c1(/3)
= -43+-]<'
_~ ':~//3,
(3)
with a =~-~
)
d2k
sin4kl
/e
-rr (2rr) 2 )2u 4(sin2ku/2 + ~ sin4k~/2) = 0.01238
(4)
for the IA up to the one-loop order [7]. The internal energy E is defined as the average action
E = ~ (S)/V
(5)
and the specific heat C as the variance of average action
377
Volume 141B, number 5,6
PItYSICS LETTERS
C = V-I( _ (S>2),
(6)
where < > stands for the expectation value with the weight exp (flS) and V for the lattice volume. Note that for the IA the specific heat is not the derivative o f E with respect to/3. The MC simulations have been pertk)rmed on 20 × 20 to 50 × 50 lattice. For both S A a n d IA we employ the Metropolis algorithm to update site variables. At large/3 the updating is repeated 3 - 5 times on the same site before proceeding to the next. We form the SU(3) matrices on the basis of the method explained in ref. [8]. In the weak-coupling region, the SU(3) matrix, by which a site variable is multiplied in the updating process, is weighted toward the identity so that the acceptance rate in the Metropolis algorithm is about 30%. The MC measurement for the IA is more time-consuming than that for the SA by a factor of 1.1-1.4. The internal energy and the specific heat have been measured in order to search for a crossover region. For both SA and IA we took 20 X 20 lattice and made 8 0 0 - 1 0 0 0 sweeps in which the first 4 0 0 - 6 0 0 sweeps were discarded in the weak-coupling region to achieve thermal equilibrium. Let us briefly mention our result for the SA. Our data for the internal energy agree with the strong-coupling expansion for /3 ~ 0.5 and approach the weak-coupling expansion for ~ >> 1.2. We observe the specific heat peak around /3 ~ 0 . 5 - 0 . 8 , which is compatible with the previous result [4]. The results for the IA are summarized in the figures. In fig. 1, one can see that the internal energy data smoothly approach the weak-coupling expansion. In fig. 2, a peak o f the specific heat around /3 ~ 0.5 is observed. At large ~ the data are in qualitative agreement with the weak-coupling expansion. Now we turn to our main task. To obtain the mass gap m, we heave measured the two-point correlation function C(At) = (Re tr ff(t + At)ff+(t)),
5 July 1984
E
i
1.0
0.5
0.0
1
0.0
sweeps on a 30 × 30 lattice and 1900--2300 sweeps o11 a 40 × 40 lattice. The simulation o f SA on a 50 X 50 lattice with 2800 sweeps has also been done. The correlation function was measured after the first 5 0 0 - 9 0 0 sweeps on a 30 X 30 lattice ( 9 0 0 - 1 3 0 0 for 40 X 40, 1800 for 50 X 50) were discarded. In the case o f a larger lattice we have generated a suitable initial configuration in order to achieve a faster convergence. Our MC computations have been checked
C 25.0
I
20.0
15.0
1 10.0
(7) 5.0
C(At)/C(O) = exp ( - m a A t ) ,
0.0
378
I 1.5
Fig. 1. Internal energy for the IA versus (~. The errors are within the data points. The curve shown is the weak-coupling expansion (E= 1.25 + 0.31 t3-x).
where the zero-momentum operator U(t) = Z x U(x,t) has been used [9]. Then by observing the exponential fall-off
one extracts the mass gap. For both SA and IA we have made 1 5 0 0 - 1 9 0 0
I 1.0
0.5
(8)
0.0
I 0.5
1
I
1.0
1.5
I'~
Fig. 2. Specific heat for the IA versus ~. The curve shown is the weak-coupling expansion (C = 4#-2).
Votumc 141 B, number 5,6
PtfYSICS LHTI'IRS
in the following ways: (i) The mass gap o f SA was measured f o r / 3 = 0 . 1 - 0 . 4 on a 20 X 20 lattice. Agreement with the strong-coupling expansion was confirmed. (ii) At ~ = 0.825 for the SA and ~ = 0.52 for the IA we carried out two independent runs using different initial configurations. The results agree within the statistical accuracy. (iii) To check the size effect the measurement for the SA was performed on three different size lattices at/3 = 0.9. We see agreement among three data within the statistical accuracy. The continuum mass gap is expected to exhibit the scaling behavior [2,3]
ma (m/As) (~ 7r/3)1/2 e x p ( -
8 rr/3),
=
s
SA, IA. (9)
=
As shown in fig. 3, this behavior is confirmed for both actions. For the SA the scaling starts at/3 ~ 0.7, where the correlation length [=(ma) - t ] ~ 2.4. The scaling window is large in contrast with the case of the standard 0 ( 3 ) spin system in two dimensions [9, 10]. The most important point for the improvement procedure is the observation that for the IA an early onset of scaling (/3 ~ 0.5) with a shorter correlation length ( ~ 1.6) is actually realized. Fitting the data with eq. (9) we obtain for the SA m / A s A = 72.7 + 1.1,
(10)
and for the IA m/AIA = 24.6 + 0.5.
(ll)
From these the MC value of the ratio o f the A-parameters turns out to be
5 July 1984
AIA/AsA = 2.95 -+ 0.07,
(12)
whereas the one-loop calculation yields [6,7] AIA/AsA = 1.99.
(13)
One may see a discrepancy by about a factor o f 1.5, which is, however, not so striking as seen in the 0 ( 3 ) non-linear sigma model simulation [ 11 ]. This indicates a possibility that non-universal terms in the #function may be necessary to fit the scaling curve. We also note that there may exist unknown systematic errors in our numerical simulations. A definite conclusion will be drawn with more extended simulations. Finally we conclude that the scaling behavior o f mass gap in the chiral model has been established and the one-loop improved action has shown an early onset o f scaling. Monte Carlo measurements of the other physical quantities, such as magnetic susceptibility [ 1 0 - I 2], might give us further insight into the scaling properties of the chiral model. We would like to thank Professor T. Eguchi for enlightening discussions and careful reading of the manuscript, T. Hattori and N. Katayama for their help in computer programming and the other members of the Institute for their help in running the computer. One of us (SKY) is grateful to Professor A. Ukawa for useful suggestions. The numerical calculation was performed on the HITAC M-280H computer at the Computer Centre, University o f Tokyo.
References ma
I
i
i
i
1
\
~.O
1
i
.s,,3o2,
- ~
•
\
, SA(50 ~)
~ ~
c 1A(30') * [A(40 :~)
""
o.,
0.3
J
,
0.4
0.5
:
"
6
7
,
,
,
0.8
0.9
1.0
1.1 j'~
Fig. 3. Mass gap versus #. The solid line represents the scaling curve for the SA [IA] with the value (10) [(11)].
[1] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [2] A. MeKane and M. Stone, Nucl. Phys. B163 (1980) 169. [3] J. Shigemitsu and J.B. Kogut, Nucl. Phys. B190 [FS3] (1981) 365. [4l J. Kogut, M. Snow and M. Stone, Nucl. Phys. B215 IFS3] (1983)45. [5] K. Symanzik, in: Mathematical problems in theoretical physics, eds. R. Schrader et al. (Springer, Berlin, 1982); Nucl. Phys. B226 (1983) 187, 205. [6] M. CaseUe, F. Gliozzi, R. Megna, F. Ravanini and S. Sciuto, Phys. Lett. 130B (1983) 81. [7] M. Nakagawara, University of Tokyo preprint UT-416 (1983), to be published in Phys. Rev. 1). [8] M. Okawa, Phys. Rev. Lett. 49 (1982) 353. [9] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. 13205 [FSS] (1982) 188. 379
Volume 141B, number 5,6
PIIYSICS LF.TTERS
[10] M. Fukugita and Y. Oyanagi, Phys. Lett. 123B (1983) 71. [11 ] B. Berg, S. Meyer, I. Montvay and K. Symanzik, Phys. Lett. 126B (1983) 467; B. Berg, S. Meyer and I. Montvay, DESY preprint DESY 834398 (1983).
380
5 July 1984
[12] G. Martinelli, G. Parisi and R. Petronzio, Phys. Lett. 100B (1981) 485; B. Berg and M. L~scher, Nucl. Phys. I~190 [FS3] (1981) 412; Y. lwasaki and T. Yoshi~, Phys. Lett. 125B (1983) 201.
PHYSICS LET'FERS
5 July 1984
MONTE CARLO MEASUREMENT OF THE MASS GAP IN THE TWO-DIMENSIONAL SU(3) × SU(3) CHIRAL MODEL WITH STANDARD AND IMPROVED ACTION Mikiharu NAKAGAWARA and Sung-Kil YANGJ
Department of l'hysics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 5 March 1984
The mass gap in the two-dimensional SU(3) X SU(3) chiral model both with the standard and hnproved action is measured by the Monte Carlo method. The scaling behavior is observed in both actions. The improved action shows an earlier onset of scaling than the standard one.
The lattice SU(N) X SU(N) chiral model in two dimensions is asymptotically free [1--3]. Then the physical quantities are expected to follow the scaling law predicted by the renormalization group near the continuum limit. Recently Monte Carlo (MC) simulations of the SU(3) X SU(3) chiral model have been performed [4]. However, the scaling behavior of the mass gap, which is of real physical interest, has not yet been observed. In this note our purpose is to establish the scaling behavior of the mass gap in the two-dimensional SU(3) X SU(3) chiral model by the MC method. The MC computations i,a practice suffer from the effect of the finite lattice spacing a. A recent theoretical proposal by Symanzik is to construct an improved action where this effect is systematically eliminated in perturbation theol2¢ [5]. Using the lattice action improved by Symanzik's procedure one expects a smoother onset of scaling. From this point of view, it is particularly interesting to simulate the chiral model both with the improved (IA) and standard action (SA), and to compare the results with each other. For our purpose it is sufficient ot use the IA which is determined to eliminate the O(a 2) corrections in the chiral invariant two-point function. Such an IA I Postdoctoral Fellow, Japan Society for the Promotion of Science, on leave from Department of Physics, Tokyo Metropolitan University, Setagaya, Tokyo 158, Japan.
has been obtained at the tree level [6], and up to the one-loop level by one of us [7]. The chiral model is defined by the action
S = n,~u ~ [Cl(3)trUnU+n+u+Cz(3)trUU+n+2~ + c.c.],
(1) where/3 = l]g 2, Un is the SU(3) matrix on the lattice site n = (x, t) and/l runs from 1 to 2. The action coefficients are given by c1(/3) = 1,
c2(/3) = 0
(2)
for the SA, and c1(/3)
= -43+-]<'
_~ ':~//3,
(3)
with a =~-~
)
d2k
sin4kl
/e
-rr (2rr) 2 )2u 4(sin2ku/2 + ~ sin4k~/2) = 0.01238
(4)
for the IA up to the one-loop order [7]. The internal energy E is defined as the average action
E = ~ (S)/V
(5)
and the specific heat C as the variance of average action
377
Volume 141B, number 5,6
PItYSICS LETTERS
C = V-I(
(6)
where < > stands for the expectation value with the weight exp (flS) and V for the lattice volume. Note that for the IA the specific heat is not the derivative o f E with respect to/3. The MC simulations have been pertk)rmed on 20 × 20 to 50 × 50 lattice. For both S A a n d IA we employ the Metropolis algorithm to update site variables. At large/3 the updating is repeated 3 - 5 times on the same site before proceeding to the next. We form the SU(3) matrices on the basis of the method explained in ref. [8]. In the weak-coupling region, the SU(3) matrix, by which a site variable is multiplied in the updating process, is weighted toward the identity so that the acceptance rate in the Metropolis algorithm is about 30%. The MC measurement for the IA is more time-consuming than that for the SA by a factor of 1.1-1.4. The internal energy and the specific heat have been measured in order to search for a crossover region. For both SA and IA we took 20 X 20 lattice and made 8 0 0 - 1 0 0 0 sweeps in which the first 4 0 0 - 6 0 0 sweeps were discarded in the weak-coupling region to achieve thermal equilibrium. Let us briefly mention our result for the SA. Our data for the internal energy agree with the strong-coupling expansion for /3 ~ 0.5 and approach the weak-coupling expansion for ~ >> 1.2. We observe the specific heat peak around /3 ~ 0 . 5 - 0 . 8 , which is compatible with the previous result [4]. The results for the IA are summarized in the figures. In fig. 1, one can see that the internal energy data smoothly approach the weak-coupling expansion. In fig. 2, a peak o f the specific heat around /3 ~ 0.5 is observed. At large ~ the data are in qualitative agreement with the weak-coupling expansion. Now we turn to our main task. To obtain the mass gap m, we heave measured the two-point correlation function C(At) = (Re tr ff(t + At)ff+(t)),
5 July 1984
E
i
1.0
0.5
0.0
1
0.0
sweeps on a 30 × 30 lattice and 1900--2300 sweeps o11 a 40 × 40 lattice. The simulation o f SA on a 50 X 50 lattice with 2800 sweeps has also been done. The correlation function was measured after the first 5 0 0 - 9 0 0 sweeps on a 30 X 30 lattice ( 9 0 0 - 1 3 0 0 for 40 X 40, 1800 for 50 X 50) were discarded. In the case o f a larger lattice we have generated a suitable initial configuration in order to achieve a faster convergence. Our MC computations have been checked
C 25.0
I
20.0
15.0
1 10.0
(7) 5.0
C(At)/C(O) = exp ( - m a A t ) ,
0.0
378
I 1.5
Fig. 1. Internal energy for the IA versus (~. The errors are within the data points. The curve shown is the weak-coupling expansion (E= 1.25 + 0.31 t3-x).
where the zero-momentum operator U(t) = Z x U(x,t) has been used [9]. Then by observing the exponential fall-off
one extracts the mass gap. For both SA and IA we have made 1 5 0 0 - 1 9 0 0
I 1.0
0.5
(8)
0.0
I 0.5
1
I
1.0
1.5
I'~
Fig. 2. Specific heat for the IA versus ~. The curve shown is the weak-coupling expansion (C = 4#-2).
Votumc 141 B, number 5,6
PtfYSICS LHTI'IRS
in the following ways: (i) The mass gap o f SA was measured f o r / 3 = 0 . 1 - 0 . 4 on a 20 X 20 lattice. Agreement with the strong-coupling expansion was confirmed. (ii) At ~ = 0.825 for the SA and ~ = 0.52 for the IA we carried out two independent runs using different initial configurations. The results agree within the statistical accuracy. (iii) To check the size effect the measurement for the SA was performed on three different size lattices at/3 = 0.9. We see agreement among three data within the statistical accuracy. The continuum mass gap is expected to exhibit the scaling behavior [2,3]
ma (m/As) (~ 7r/3)1/2 e x p ( -
8 rr/3),
=
s
SA, IA. (9)
=
As shown in fig. 3, this behavior is confirmed for both actions. For the SA the scaling starts at/3 ~ 0.7, where the correlation length [=(ma) - t ] ~ 2.4. The scaling window is large in contrast with the case of the standard 0 ( 3 ) spin system in two dimensions [9, 10]. The most important point for the improvement procedure is the observation that for the IA an early onset of scaling (/3 ~ 0.5) with a shorter correlation length ( ~ 1.6) is actually realized. Fitting the data with eq. (9) we obtain for the SA m / A s A = 72.7 + 1.1,
(10)
and for the IA m/AIA = 24.6 + 0.5.
(ll)
From these the MC value of the ratio o f the A-parameters turns out to be
5 July 1984
AIA/AsA = 2.95 -+ 0.07,
(12)
whereas the one-loop calculation yields [6,7] AIA/AsA = 1.99.
(13)
One may see a discrepancy by about a factor o f 1.5, which is, however, not so striking as seen in the 0 ( 3 ) non-linear sigma model simulation [ 11 ]. This indicates a possibility that non-universal terms in the #function may be necessary to fit the scaling curve. We also note that there may exist unknown systematic errors in our numerical simulations. A definite conclusion will be drawn with more extended simulations. Finally we conclude that the scaling behavior o f mass gap in the chiral model has been established and the one-loop improved action has shown an early onset o f scaling. Monte Carlo measurements of the other physical quantities, such as magnetic susceptibility [ 1 0 - I 2], might give us further insight into the scaling properties of the chiral model. We would like to thank Professor T. Eguchi for enlightening discussions and careful reading of the manuscript, T. Hattori and N. Katayama for their help in computer programming and the other members of the Institute for their help in running the computer. One of us (SKY) is grateful to Professor A. Ukawa for useful suggestions. The numerical calculation was performed on the HITAC M-280H computer at the Computer Centre, University o f Tokyo.
References ma
I
i
i
i
1
\
~.O
1
i
.s,,3o2,
- ~
•
\
, SA(50 ~)
~ ~
c 1A(30') * [A(40 :~)
""
o.,
0.3
J
,
0.4
0.5
:
"
6
7
,
,
,
0.8
0.9
1.0
1.1 j'~
Fig. 3. Mass gap versus #. The solid line represents the scaling curve for the SA [IA] with the value (10) [(11)].
[1] A.M. Polyakov, Phys. Lett. 59B (1975) 79. [2] A. MeKane and M. Stone, Nucl. Phys. B163 (1980) 169. [3] J. Shigemitsu and J.B. Kogut, Nucl. Phys. B190 [FS3] (1981) 365. [4l J. Kogut, M. Snow and M. Stone, Nucl. Phys. B215 IFS3] (1983)45. [5] K. Symanzik, in: Mathematical problems in theoretical physics, eds. R. Schrader et al. (Springer, Berlin, 1982); Nucl. Phys. B226 (1983) 187, 205. [6] M. CaseUe, F. Gliozzi, R. Megna, F. Ravanini and S. Sciuto, Phys. Lett. 130B (1983) 81. [7] M. Nakagawara, University of Tokyo preprint UT-416 (1983), to be published in Phys. Rev. 1). [8] M. Okawa, Phys. Rev. Lett. 49 (1982) 353. [9] G. Fox, R. Gupta, O. Martin and S. Otto, Nucl. Phys. 13205 [FSS] (1982) 188. 379
Volume 141B, number 5,6
PIIYSICS LF.TTERS
[10] M. Fukugita and Y. Oyanagi, Phys. Lett. 123B (1983) 71. [11 ] B. Berg, S. Meyer, I. Montvay and K. Symanzik, Phys. Lett. 126B (1983) 467; B. Berg, S. Meyer and I. Montvay, DESY preprint DESY 834398 (1983).
380
5 July 1984
[12] G. Martinelli, G. Parisi and R. Petronzio, Phys. Lett. 100B (1981) 485; B. Berg and M. L~scher, Nucl. Phys. I~190 [FS3] (1981) 412; Y. lwasaki and T. Yoshi~, Phys. Lett. 125B (1983) 201.