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Monte Carlo analysis of the 2D chiral SU(2) × SU(2) lattice theory with a spin-32 action
Volume 178, n u m b e r 2,3
PHYSICS LETTERS B
2 October 1986
MONTE CARLO ANALYSIS OF THE 2D CHIRAL SU (2) X SU (2) LATTICE THEORY WITH A SPIN-3/2 ACTION ~ M. BAIG Departament de Fisica Te6rica, Universitat Autbnoma de Barcelona, Bellaterra (Barcelona), Spain Received 8 July 1986
Using Monte Carlo simulations, the phase structure of the two-dimensional SU(2) x S U ( 2 ) lattice theory with an action defined through the spin-3/2 representation of the SU (2) matrices has been determined. A second-order phase transition has been found, like the SO (3) x SO(3) chiral model, in contrast to the equivalent four-dimensional lattice gauge theory with the 3/2 representation action that exhibits a clear first-order transition.
Introduction. It is known from the work of Polyakov [ 1] and Migdal [2] that there is an interesting similarity between the four-dimensional SU(N) lattice guage theories and the two-dimensional SU (N) × SU (N) chiral models. Subsequent work by Kogut and Shigemitsu [3] and Green and Samuel [ 4 ] has been devoted to the study by numerical simulations of the cross-over of the string tension and other physically relevant properties in the chiral models. Furthermore, they claimed that the twodimensional chiral models must be considered as an interesting laboratory to test the ideas and methods of lattice gauge theories in a less computer-timedemanding environment [ 5 ]. The analysis of the similarities between the SU (2) lattice gauge theory and the two-dimensional chiral models was continued in ref. [6], where the phase diagram of the SU(2) XSU (2) chiral model with an action mixture of the fundamental and adjoint representations of the SU(2) group was analyzed. This showed the existence of a phase diagram very similar to that of the four-dimensional gauge case but with second-order phase-transition lines instead of firstorder. The absence of first-order lines in this case was already predicted in ref. [ 7]. In addition, the work of ref. [8] devoted to the analysis of the SO(3) XSO(3) chiral model, i.e. the adjoint axis of the fundamental-adjoint mixed theory, showed using "~ Work partially supported by research project CAICYT.
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
great statistics the second-order character of the phase transition. In the present paper we go further in the analysis of the similarities between two-dimensional chiral models and four-dimensional gauge theories studying the phase structure of the SU (2) X SU (2) chiral theory with an action defined through the trace over the spin-3/2 representation of the SU(2) group. It has been shown that the corresponding gauge model exhibits a clear first-order phase transition [ 9 ]. Our main purpose is to determine if this chiral model behaves as the adjoint case manifesting also a second-order phase transition.
The model. The lattice action that we have considered for the numerical computations corresponds to the sum over the traces of the spin-3/2 representations of the SU(2) matrices with the following normalization:
S= 2 •3/2 Tr3/2 [ U(x) U+(x+u) ],
(1)
where the U(x) variables are associated to the sites of the two-dimensional lattice, and x and/~ label sites and directions of the lattice. The relation between the trace on the 3/2 representation and that of the fundamental is just T r 3 / z U = (Trl/2U)3-2 Trl/2U.
(2)
To update the lattice we have used the modified 255
Volume 178, n u m b e r 2,3
PHYSICS LETTERS B
2 October 1986
UmmmmmmmmmmmmmmJmRm|mlmlml 0.8
>o dk" Ld Z Ld
0.6
D-2 0.4
SIZE
0.2
• h ot stG rt • cold stGrt
o.
o.
.5
I
0.5
~
1.
ell
I
I
1.5
2.
I
2.5
I
3.
I
3.5
4.
BETA 3/2 Fig. 1. Thermal cycle of the SU(2) × S U ( 2 ) two-dimensional chiral model with a spin-3/2 representation action. The lattice size used is 5 and the number of iterations per point is 3000. Error bars are directly the variances.
Metropolis algorithm with six attempts per point. We have used the complete S U ( 2 ) group instead o f its icosahedral subgroup to a v o i d the presence o f extra phase transitions. The c o m p u t e r program that we have used is a modified version o f that constructed for the numerical work o f ref. [ 6 ]. Since we will deal with a second-order transition a quantity that is necessary to measure is the specific heat. We have measured it through the analysis o f the fluctuations. We define C= ((E3/2) 2 _
(E2/2) )N,
(3)
where N i s the n u m b e r o f links o f the lattice. The E3/2 energy is defined in turn as
E3/2 = 1 - ~ (Tr3/2 [ U(x) U r ( x + / * ) ] ) .
(4)
A second-order phase transition is reflected in a sharp peak o f the specific heat in such a way that its size increases with the lattice size. The specific heat is a very c o m p u t e r - t i m e - d e m a n d i n g quantity and the 256
errors are difficult to estimate. F o r this reason, runs with great statistics have been performed to test explicitly the convergence o f the numerical results for the specific heat with the n u m b e r o f lattice iterations.
Results. We have performed a thermal cycle varying J~3/2 from 0 to 5 in order to obtain a first sight inside the model. The lattice size used has been 52 and we have performed 3000 iterations per point with a step of/xf13/2 = 0.1, starting from both, hot and cold initial configurations. The results o f this thermal cycle are represented in fig. 1. The great increase o f the fluctuations in the region 3 . 0 < f l 3 / 2 < 3 . 6 that m a y correspond to a singularity, is remarkable excluding a first-order transition. To better d e t e r m i n e the nature o f the detected fluctuations the analysis o f the specific heat turns out to be necessary. To this purpose we have p e r f o r m e d m e a s u r e m e n t s with great statistics considering simulations o f 20 000 iterations per point for the lattice
2.
D-2
1.75
1.5
It~
T 0 E
SIZE 5
-
1.25
~
3.2 3.1
1.
(j~) 0.75
o~
t
o.
~ I
0.
2.5
CI
5.
I
--
"
I
7.5
a736 I
10.
12.5
I
I
15.
XI0 3
17.5
ITERATIONS Fig. 2. Specific-heat d e t e r m i n a t i o n in a 52 lattice for different values
off13/2.
The m e a s u r e m e n t s are up to 20 000 iterations per point.
D=2 4
D
SIZE 1 0 t
-1-
3.4
3
C.) LL (,9 LLI Q_ 00
3.2
,.
_ =
°o.
I 2.5
I 5.
I 7.5
I 10.
I 12.5
ITERATIONS Fig. 3. Specific heat m e a s u r e m e n t s in a 102 lattice.
I 15.
3.6 3.1
~ I 17.5
3.0 x 10
Volume 178, number 2,3
PHYSICS LETTERS B
sizes 5 2 and 10 2. Figs. 2 and 3 collect the results o f the m e a s u r e m e n t s o f the specific h e a t eq. (3) as a f u n c t i o n o f the n u m b e r o f lattice iterations. F r o m these figures we can see the great f l u c t u a t i o n s in the v a l u e o f the specific h e a t w h e n it is m e a s u r e d with low statistics. T h e c o n v e r g e n c e o f the results o b t a i n e d with high statistics is v e r y reasonable. Fig. 4 collects the m e a s u r e m e n t s o f the specific heat o b t a i n e d f r o m different lattice sizes with respect to fl3/2. We h a v e i n c l u d e d also the m e a s u r e m e n t s on a 15 2 lattice a l t h o u g h they h a v e b e e n p e r f o r m e d w i t h
2 October 1986
less statistics, for i n s t a n c e 3000 i t e r a t i o n s per point. T h e points in the figure are the results o f the longest runs and the error bars h a v e b e e n e s t i m a t e d d i v i d i n g the run in sets o f 500 i t e r a t i o n s a n d c o n s i d e r i n g the v a r i a n c e o f the averages :). F r o m fig. 4 o n e can conf i r m the s e c o n d - o r d e r n a t u r e o f the phase transition.
:t Although using this simple procedure the error bars of fig. 4 can be overestimated, it is sufficient for determining in an umambiguous way the presence of a second-order phase transition.
c
4
-
2
-
5 2
10 2 15 2
o
J 3.
,3.1
I
3.2
I
3.3
I
3.4
I
3.5
I
3.6 0~23"7
Fig. 4. Summary of the specific-heat measurements as a function off13/2 at three different lattice sizes. Error bars are directly the variances from averages of measures of the specific heat from sets of 500 iterations. 258
Volume 178, number 2,3
PHYSICS LETTERS B
The exact location of this transition is difficult to determine and a complete finite-size analysis of this transition is out of the scope of the present paper. Nevertheless, an estimation of the location of the phase transition can be performed by a simple extrapolation to an infinite-size lattice of the results collected in fig. 4. We estimate, finally, fl~3/2= 3.75.
Conclusions.
In the present paper we have analyzed the phase structure of a two-dimensional SU (2) X SU (2) chiral lattice model with an action defined through the trace over the spin-3/2 representation of the S U ( 2 ) matrices. The scope has been mainly to determine if this model exhibits a secondorder phase transition like the adjoint case, in correspondence with the first-order transitions of the equivalent four-dimensional lattice gauge theories. The numerical results obtained through the Monte Carlo simulations have confirmed the presence of a second-order phase transition. The analysis of the specific heat with runs of high statistics have shown the presence of a sharp peak that increases with the lattice size, confirming the second-order nature of the transition found at fl~3/2= 3.75.
2 October 1986
The author is grateful to Dr. E. Dagotto and to Dr. A. Moreo for usseful c o m m e n t s and suggestions. The numerical work has been performed on the VAX 1 1/780 of the Laboratori de Fisica d'Altes Energies, Universitat A u t 6 n o m a de Barcelona, Bellaterra.
References [ 1] A.M. Polyakov, Phys. Lett. B 59 (1975) 79. [2] A.A. Migdal, Zh. Eksp. Teor. Fiz. (USSR) 69 (1975) 810 [Sov. Phys. JETP 42 (1976) 413]. [3] J. Shigemitsuand J. Kogut, Nucl. Phys. B 190 [FS3] (198l) 365. [4] F. Green and S. Samuel, Nucl. Phys. B 190 [FS3] (1981) 113. [5] J. Kogut, M. Snow and M. Stone, Nucl. Phys, B 215 [FS7] (t983) 45. [6] M. Baig, E. Dagotto and A. Moreo, Phys. Lett. B 165 (1985) 121. [7] S. Duane and M. Green, Phys. Lett. B 103 (1981) 359. [8] M, Caselle, Z. Phys. C28 (1985) 233. [9] C. Ayalaand M. Baig, preprint UAB-FT-140,Phys. Rev. D., to be published.
259
PHYSICS LETTERS B
2 October 1986
MONTE CARLO ANALYSIS OF THE 2D CHIRAL SU (2) X SU (2) LATTICE THEORY WITH A SPIN-3/2 ACTION ~ M. BAIG Departament de Fisica Te6rica, Universitat Autbnoma de Barcelona, Bellaterra (Barcelona), Spain Received 8 July 1986
Using Monte Carlo simulations, the phase structure of the two-dimensional SU(2) x S U ( 2 ) lattice theory with an action defined through the spin-3/2 representation of the SU (2) matrices has been determined. A second-order phase transition has been found, like the SO (3) x SO(3) chiral model, in contrast to the equivalent four-dimensional lattice gauge theory with the 3/2 representation action that exhibits a clear first-order transition.
Introduction. It is known from the work of Polyakov [ 1] and Migdal [2] that there is an interesting similarity between the four-dimensional SU(N) lattice guage theories and the two-dimensional SU (N) × SU (N) chiral models. Subsequent work by Kogut and Shigemitsu [3] and Green and Samuel [ 4 ] has been devoted to the study by numerical simulations of the cross-over of the string tension and other physically relevant properties in the chiral models. Furthermore, they claimed that the twodimensional chiral models must be considered as an interesting laboratory to test the ideas and methods of lattice gauge theories in a less computer-timedemanding environment [ 5 ]. The analysis of the similarities between the SU (2) lattice gauge theory and the two-dimensional chiral models was continued in ref. [6], where the phase diagram of the SU(2) XSU (2) chiral model with an action mixture of the fundamental and adjoint representations of the SU(2) group was analyzed. This showed the existence of a phase diagram very similar to that of the four-dimensional gauge case but with second-order phase-transition lines instead of firstorder. The absence of first-order lines in this case was already predicted in ref. [ 7]. In addition, the work of ref. [8] devoted to the analysis of the SO(3) XSO(3) chiral model, i.e. the adjoint axis of the fundamental-adjoint mixed theory, showed using "~ Work partially supported by research project CAICYT.
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
great statistics the second-order character of the phase transition. In the present paper we go further in the analysis of the similarities between two-dimensional chiral models and four-dimensional gauge theories studying the phase structure of the SU (2) X SU (2) chiral theory with an action defined through the trace over the spin-3/2 representation of the SU(2) group. It has been shown that the corresponding gauge model exhibits a clear first-order phase transition [ 9 ]. Our main purpose is to determine if this chiral model behaves as the adjoint case manifesting also a second-order phase transition.
The model. The lattice action that we have considered for the numerical computations corresponds to the sum over the traces of the spin-3/2 representations of the SU(2) matrices with the following normalization:
S= 2 •3/2 Tr3/2 [ U(x) U+(x+u) ],
(1)
where the U(x) variables are associated to the sites of the two-dimensional lattice, and x and/~ label sites and directions of the lattice. The relation between the trace on the 3/2 representation and that of the fundamental is just T r 3 / z U = (Trl/2U)3-2 Trl/2U.
(2)
To update the lattice we have used the modified 255
Volume 178, n u m b e r 2,3
PHYSICS LETTERS B
2 October 1986
UmmmmmmmmmmmmmmJmRm|mlmlml 0.8
>o dk" Ld Z Ld
0.6
D-2 0.4
SIZE
0.2
• h ot stG rt • cold stGrt
o.
o.
.5
I
0.5
~
1.
ell
I
I
1.5
2.
I
2.5
I
3.
I
3.5
4.
BETA 3/2 Fig. 1. Thermal cycle of the SU(2) × S U ( 2 ) two-dimensional chiral model with a spin-3/2 representation action. The lattice size used is 5 and the number of iterations per point is 3000. Error bars are directly the variances.
Metropolis algorithm with six attempts per point. We have used the complete S U ( 2 ) group instead o f its icosahedral subgroup to a v o i d the presence o f extra phase transitions. The c o m p u t e r program that we have used is a modified version o f that constructed for the numerical work o f ref. [ 6 ]. Since we will deal with a second-order transition a quantity that is necessary to measure is the specific heat. We have measured it through the analysis o f the fluctuations. We define C= ((E3/2) 2 _
(E2/2) )N,
(3)
where N i s the n u m b e r o f links o f the lattice. The E3/2 energy is defined in turn as
E3/2 = 1 - ~ (Tr3/2 [ U(x) U r ( x + / * ) ] ) .
(4)
A second-order phase transition is reflected in a sharp peak o f the specific heat in such a way that its size increases with the lattice size. The specific heat is a very c o m p u t e r - t i m e - d e m a n d i n g quantity and the 256
errors are difficult to estimate. F o r this reason, runs with great statistics have been performed to test explicitly the convergence o f the numerical results for the specific heat with the n u m b e r o f lattice iterations.
Results. We have performed a thermal cycle varying J~3/2 from 0 to 5 in order to obtain a first sight inside the model. The lattice size used has been 52 and we have performed 3000 iterations per point with a step of/xf13/2 = 0.1, starting from both, hot and cold initial configurations. The results o f this thermal cycle are represented in fig. 1. The great increase o f the fluctuations in the region 3 . 0 < f l 3 / 2 < 3 . 6 that m a y correspond to a singularity, is remarkable excluding a first-order transition. To better d e t e r m i n e the nature o f the detected fluctuations the analysis o f the specific heat turns out to be necessary. To this purpose we have p e r f o r m e d m e a s u r e m e n t s with great statistics considering simulations o f 20 000 iterations per point for the lattice
2.
D-2
1.75
1.5
It~
T 0 E
SIZE 5
-
1.25
~
3.2 3.1
1.
(j~) 0.75
o~
t
o.
~ I
0.
2.5
CI
5.
I
--
"
I
7.5
a736 I
10.
12.5
I
I
15.
XI0 3
17.5
ITERATIONS Fig. 2. Specific-heat d e t e r m i n a t i o n in a 52 lattice for different values
off13/2.
The m e a s u r e m e n t s are up to 20 000 iterations per point.
D=2 4
D
SIZE 1 0 t
-1-
3.4
3
C.) LL (,9 LLI Q_ 00
3.2
,.
_ =
°o.
I 2.5
I 5.
I 7.5
I 10.
I 12.5
ITERATIONS Fig. 3. Specific heat m e a s u r e m e n t s in a 102 lattice.
I 15.
3.6 3.1
~ I 17.5
3.0 x 10
Volume 178, number 2,3
PHYSICS LETTERS B
sizes 5 2 and 10 2. Figs. 2 and 3 collect the results o f the m e a s u r e m e n t s o f the specific h e a t eq. (3) as a f u n c t i o n o f the n u m b e r o f lattice iterations. F r o m these figures we can see the great f l u c t u a t i o n s in the v a l u e o f the specific h e a t w h e n it is m e a s u r e d with low statistics. T h e c o n v e r g e n c e o f the results o b t a i n e d with high statistics is v e r y reasonable. Fig. 4 collects the m e a s u r e m e n t s o f the specific heat o b t a i n e d f r o m different lattice sizes with respect to fl3/2. We h a v e i n c l u d e d also the m e a s u r e m e n t s on a 15 2 lattice a l t h o u g h they h a v e b e e n p e r f o r m e d w i t h
2 October 1986
less statistics, for i n s t a n c e 3000 i t e r a t i o n s per point. T h e points in the figure are the results o f the longest runs and the error bars h a v e b e e n e s t i m a t e d d i v i d i n g the run in sets o f 500 i t e r a t i o n s a n d c o n s i d e r i n g the v a r i a n c e o f the averages :). F r o m fig. 4 o n e can conf i r m the s e c o n d - o r d e r n a t u r e o f the phase transition.
:t Although using this simple procedure the error bars of fig. 4 can be overestimated, it is sufficient for determining in an umambiguous way the presence of a second-order phase transition.
c
4
-
2
-
5 2
10 2 15 2
o
J 3.
,3.1
I
3.2
I
3.3
I
3.4
I
3.5
I
3.6 0~23"7
Fig. 4. Summary of the specific-heat measurements as a function off13/2 at three different lattice sizes. Error bars are directly the variances from averages of measures of the specific heat from sets of 500 iterations. 258
Volume 178, number 2,3
PHYSICS LETTERS B
The exact location of this transition is difficult to determine and a complete finite-size analysis of this transition is out of the scope of the present paper. Nevertheless, an estimation of the location of the phase transition can be performed by a simple extrapolation to an infinite-size lattice of the results collected in fig. 4. We estimate, finally, fl~3/2= 3.75.
Conclusions.
In the present paper we have analyzed the phase structure of a two-dimensional SU (2) X SU (2) chiral lattice model with an action defined through the trace over the spin-3/2 representation of the S U ( 2 ) matrices. The scope has been mainly to determine if this model exhibits a secondorder phase transition like the adjoint case, in correspondence with the first-order transitions of the equivalent four-dimensional lattice gauge theories. The numerical results obtained through the Monte Carlo simulations have confirmed the presence of a second-order phase transition. The analysis of the specific heat with runs of high statistics have shown the presence of a sharp peak that increases with the lattice size, confirming the second-order nature of the transition found at fl~3/2= 3.75.
2 October 1986
The author is grateful to Dr. E. Dagotto and to Dr. A. Moreo for usseful c o m m e n t s and suggestions. The numerical work has been performed on the VAX 1 1/780 of the Laboratori de Fisica d'Altes Energies, Universitat A u t 6 n o m a de Barcelona, Bellaterra.
References [ 1] A.M. Polyakov, Phys. Lett. B 59 (1975) 79. [2] A.A. Migdal, Zh. Eksp. Teor. Fiz. (USSR) 69 (1975) 810 [Sov. Phys. JETP 42 (1976) 413]. [3] J. Shigemitsuand J. Kogut, Nucl. Phys. B 190 [FS3] (198l) 365. [4] F. Green and S. Samuel, Nucl. Phys. B 190 [FS3] (1981) 113. [5] J. Kogut, M. Snow and M. Stone, Nucl. Phys, B 215 [FS7] (t983) 45. [6] M. Baig, E. Dagotto and A. Moreo, Phys. Lett. B 165 (1985) 121. [7] S. Duane and M. Green, Phys. Lett. B 103 (1981) 359. [8] M, Caselle, Z. Phys. C28 (1985) 233. [9] C. Ayalaand M. Baig, preprint UAB-FT-140,Phys. Rev. D., to be published.
259