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Complex singularities in the specific heat of the SU(2) lattice gauge model
Volume 102B, number 4
PHYSICS LETTERS
18 June 1981
COMPLEX SINGULARITIES IN THE SPECIFIC HEAT OF THE SU(2) LATTICE GAUGE MODEL M. FALCIONI a, E. MARINARI
a,b,
M.L. PACIELLO a, G. PARISI c and B. TAGLIENTI a
a Istituto Nazionale di Fisica Nucleate, Sezione di Roma, Rome, Italy b lstituto di Fisica dell'Universitd di Roma, Rome, Italy c Istituto Nazionale di Fisica Nucleate, Laboratorio Nazionali di Frascati, Frascati (Rome), Italy Received 19 March 1981
A recent Monte Carlo simulation showing a sharp peak versus/3 in the specific heat for an SU(2) lattice gauge theory is compared with the high temperature expansion. An interpretation based on complex ~-plane singularities is proposed.
In the last years, Monte Carlo computer simulations have provided a powerful tool to study lattice gauge theories in the non-perturbative region [1,2]. These methods were introduced to study the properties of statistical systems [3] and quite recently Wilson [4] suggested an extension to field theory. Computer simulations of lattice gauge theories for different gauge groups have been performed by several authors [5,6] to investigate the occurrence of phase transitions in the intermediate region lying between the high-temperature (strong coupling) regime and the low-temperature (weak coupling) one. In particular, Creutz [5] analyzed the average plaquette energy as a function of/3 - 1/kT = 4/g 2 and found a considerable variation of the slope near/3 2.2. To analyze the nature of the transition between the strong and weak coupling it seems more suitable to study-the specific heat rather than the energY. Recently, Lautrup and Nauenberg have performed a Monte Carlo simulation of the specific heat C(/3) for U(1) [7] and SU(2) [8] gauge groups. In both cases a sharp peak shows up, with a completely different behaviour under the increasing of the lattice size: the peak height increases for U(1), as it is expected in a phase transition, and remains nearly constant for SU(2). While for the U(1) group one expects a transition from a confining phase to a deconfining one, for the SU(2) group such an interpretation seems to be unlikely. 270
The purpose of this letter is to provide an explanation of the peak of ref. [8] in terms of complex/3-singularities rather than a phase transition. In our analysis, we will assume that the available terms of the high-temperature expansion are enough to get a rough picture of the nearby singularities. The high-temperature expansion of the average plaquett e energy is: 10 N=2 the E N are positive numbers given by Wilson [9] * 1, fsu(2) dU l t r U exp(1/3 tr U) j(/3) =
fsu(2) dU exp(½/3 tr U) (2)
= -2//3 +I0(/3)/I1(~) = ~ o Ak/32k+l' 10(/3) and 11(/3) being Bessel functions. From (1) the specific heat is obtained:
C(/3) = _(32 dE/d~3
(3) lO
=/32 dJ (l +N~=2 (2N+ l ) E N ( - 1 ) N j 2 N + o ( j 2 2 ) ) *1 The values ofE N can be found in ref. [6]. Up to the order jlS the EN coefficients can be found in ref. [10].
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
Volume 102B, number 4
PHYSICS LETTERS
18 June 1981
10(
The ratio test applied to the first terms suggests a singularity on the negative real axis at j 2 ~ - 1 / 6 ; as we are trying to reach a singularity at fl ~ 2.2 (corresponding to j 2 ~. 0.22), we push the j 2 ~ - 1 / 6 singularity far away by a conformal transformation. We choose:
a ( z ) = N =~ 2
z = 6 j 2 / ( 6 J 2 + 1).
F(z) contains two complex singularities at z = Zc(1
(4)
The specific heat reads now as: 10 CO3)=j32d-~(1 + ~ r N z N + O ( z l l ) ) N=2
,
(5)
where the coefficients rN are easily derived from E N. As expected the rN are all positive, suggesting a singularity on the real axis. Under this hypothesis, the ratio test gives now a singularity at z -~ 0.59, corresponding to/3 ~ 2.4. This is an important point to remark: the peak of Lautrup and Nauenberg [8] occurs at a value of 13near the point where the high-temperature expansion suggests a singularity, however the precise values of the singularity points do not coincide as it should be. It is possible that this discrepancy is due to the finite length of the high-temperature expansion or to finite size effects in the Monte Carlo simulations. In this letter we explore an alternative point of view i.e.: both the asymptotic behaviour of the hightemperature expansion and the "experimental" peak in the specific heat originate from a pair of complex singularities near the real axis, as suggested in ref. [ 11 ]. The reader may find it interesting that the specific heat of a two-dimensional superconductor in the Random Phase Approximation shows both a marked peak and a pair of complex singularities near the real axis [ 12]. In the RPA approximation this two-dimensional system is similar to gauge theories as far as it has both asymptotic freedom and absence of phase transition. I n t h e hypothesis of the existence of complex singularities a more refined analysis is needed for the SU(2) specific heat. We proceed by looking for a function with the following properties: (i) It has the right high temperature expansion, (ii) It reproduces the simulation data near fl ~ 2.2. We have studied the following simple f o r m :
CO3) = fl2(dJ/d~)[1 + F(z) + a ( z ) ] ,
(6)
where:
Y(z) = [A(1 - zlzc) + B ] / [ ( i
-
zlzJ
+
~>21.r12,
(7)
rN
N! dzNz=o (8)
+ i5); by construction eq. (6) automatically satisfies condition (i). The adjustable parameters A, B, z c, 5, 3' must be chosen in such a way that (i) CO3) =/32(dJ/d13)[1 + F(z) + G(z)] fits the simulation data near fl ~ 2.2. (ii) G(z) must be well approximated by a low-order polynomial in the region of interest near z ~ 0.57; in practice only its first 4 ~ 5 terms must be sensibly 4= 0. (This amount to that the singularities of G(z) are further away.) Two reasonable sets of values we get for the parameters are:
(a): A = 1.52,
B =0.657,
zc
0.570,
:
(9) 6 = 0.0507,
3' = 0.530.
1.5
t
\
\
i
I
I
1
2
3
\
,~
Fig. 1. Specific heat as function of/3; the dots indicate the simulation data, the solid curve is the fit (a), the dashed curve is the fit (b). 271
Volume 102B, number 4
PHYSICS LETTERS
Table 1 CoefficientsgN defined in (8) for the two fits (a) and (b) [see (9)] compared with the rN as defined in (5).
N
rN
gN (fit a)
gN (fit b)
2 3 4 5 6 7 8 9 10
0.5556 0.8519 1.3241 2.1287 3.5196 5.9055 9.9365 16.6354 27.5977
0.330 0.188 0.861 -0.735 -0.805 -0.109 -0.726 0.676 -0.519
0.210 0.109 0.264 -0.530 -0.112 -0.125 -0.731 0.167 -0.631
× 10 -1 X 10 -2 × 10 -1 × 10 -1 10 -2 x 10 -3
×
X 10 -1 × 10 -1 × 10 -3
(b): A =1.04,
B =0.623,
6 --- 0.0532,
zc--0.565,
3' = 0.517.
The reader m a y w o n d e r h o w m a n y terms o f the high-temperature expansion are needed to detect complex singularities. The answer is given in fig. 2, where the ratios
R N _ CN CN+ 1 ' × 10 -1 × 10 -1
(9 c o n ' d )
In fig. 1 we compare the results o f the fit with the simulation data, in table 1 we give the coefficients gN o f G(z) c o m p a r e d w i t h rN.
18 June 1981
CN
_ 1 d _ ~ , F(z)! N ! d(ZlZc)~V Iz=O
(10)
are p l o t t e d versus N . We get R N ~ 1 (as for a real singularity at z = Zc) up to N ~ 30; after this order we have a fast decrease. We conclude that the c o m p l e x singularities hypothesis, although it has m a n y attractive features, is very hard to test by looking at the high-temperature expansion only (unless very long series w o u l d be available). We are i n d e b t e d to K. Wilson for having provided us w i t h the coefficients o f the high-temperature expansion. We thank also C. R e b b i for having provided us w i t h a c o p y o f ref. [6] prior to publication and N. Sourlas for useful discussions.
References
Cli CN+I
Q0 0000gqIQIIQOOOOQIO QOOOBQQQ
I
I
8
I
I
16
I
I
24
I
I
32
N
Fig. 2. Ratios R N = CN/CN+ 1 [see eq. (10)] versus N for the fit (a), a very similax behaviour is obtained for the fit (b). 272
[1] K. Wilson, Corneil preprint CLNS/80/442 (January 1980). [2] For a recent review see: G. Parisi, talk XXth Conf. on High energy physics (Madison, 1980), Frascati preprint LNF-80/52 (P) (September 1980). [3] K. Binder, ed., Monte Carlo methods (Springer, Berlin, 1979). [4] K. Wilson, Carg6se Summer School (1977) (Plenum, New York, 1978). [5] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20 (1979) 1915; M. Creutz, Phys. Rev. Lett. 43 (1979) 553; M. Creutz, Phys. Rev. Lett. 45 (1980) 313; M. Creutz, Phys. Rev. D21 (1980) 2308; C. Rebbi, Phys. Rev. D21 (1980) 3350. [6] G. Bhanot and C. Rebbi, SU(2) string, gluebail mass and interquark potential by Monte Carlo computations •(1980), to be published. [7] B. Lautrup and M. Nauenberg, Phys. Lett. 95B (1980) 63. [8] B. Lautrup and M. Nauenberg, Phys. Rev. Lett. 45 (1980) 1755. [9] K. Wilson, private communication. [10] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D l l (1975) 2104; D19 (1979) 2514 (E). [11] D.J. Gross, Les Houches (1980), to be published in Phys. Rep. [12] W.E. Masker, S. Mar~elja and R.D. Parks, Phys. Rev. 188 (1969) 745.
PHYSICS LETTERS
18 June 1981
COMPLEX SINGULARITIES IN THE SPECIFIC HEAT OF THE SU(2) LATTICE GAUGE MODEL M. FALCIONI a, E. MARINARI
a,b,
M.L. PACIELLO a, G. PARISI c and B. TAGLIENTI a
a Istituto Nazionale di Fisica Nucleate, Sezione di Roma, Rome, Italy b lstituto di Fisica dell'Universitd di Roma, Rome, Italy c Istituto Nazionale di Fisica Nucleate, Laboratorio Nazionali di Frascati, Frascati (Rome), Italy Received 19 March 1981
A recent Monte Carlo simulation showing a sharp peak versus/3 in the specific heat for an SU(2) lattice gauge theory is compared with the high temperature expansion. An interpretation based on complex ~-plane singularities is proposed.
In the last years, Monte Carlo computer simulations have provided a powerful tool to study lattice gauge theories in the non-perturbative region [1,2]. These methods were introduced to study the properties of statistical systems [3] and quite recently Wilson [4] suggested an extension to field theory. Computer simulations of lattice gauge theories for different gauge groups have been performed by several authors [5,6] to investigate the occurrence of phase transitions in the intermediate region lying between the high-temperature (strong coupling) regime and the low-temperature (weak coupling) one. In particular, Creutz [5] analyzed the average plaquette energy as a function of/3 - 1/kT = 4/g 2 and found a considerable variation of the slope near/3 2.2. To analyze the nature of the transition between the strong and weak coupling it seems more suitable to study-the specific heat rather than the energY. Recently, Lautrup and Nauenberg have performed a Monte Carlo simulation of the specific heat C(/3) for U(1) [7] and SU(2) [8] gauge groups. In both cases a sharp peak shows up, with a completely different behaviour under the increasing of the lattice size: the peak height increases for U(1), as it is expected in a phase transition, and remains nearly constant for SU(2). While for the U(1) group one expects a transition from a confining phase to a deconfining one, for the SU(2) group such an interpretation seems to be unlikely. 270
The purpose of this letter is to provide an explanation of the peak of ref. [8] in terms of complex/3-singularities rather than a phase transition. In our analysis, we will assume that the available terms of the high-temperature expansion are enough to get a rough picture of the nearby singularities. The high-temperature expansion of the average plaquett e energy is: 10 N=2 the E N are positive numbers given by Wilson [9] * 1, fsu(2) dU l t r U exp(1/3 tr U) j(/3) =
fsu(2) dU exp(½/3 tr U) (2)
= -2//3 +I0(/3)/I1(~) = ~ o Ak/32k+l' 10(/3) and 11(/3) being Bessel functions. From (1) the specific heat is obtained:
C(/3) = _(32 dE/d~3
(3) lO
=/32 dJ (l +N~=2 (2N+ l ) E N ( - 1 ) N j 2 N + o ( j 2 2 ) ) *1 The values ofE N can be found in ref. [6]. Up to the order jlS the EN coefficients can be found in ref. [10].
0 031-9163/81/0000-0000/$ 02.50 © North-Holland Publishing Company
Volume 102B, number 4
PHYSICS LETTERS
18 June 1981
10(
The ratio test applied to the first terms suggests a singularity on the negative real axis at j 2 ~ - 1 / 6 ; as we are trying to reach a singularity at fl ~ 2.2 (corresponding to j 2 ~. 0.22), we push the j 2 ~ - 1 / 6 singularity far away by a conformal transformation. We choose:
a ( z ) = N =~ 2
z = 6 j 2 / ( 6 J 2 + 1).
F(z) contains two complex singularities at z = Zc(1
(4)
The specific heat reads now as: 10 CO3)=j32d-~(1 + ~ r N z N + O ( z l l ) ) N=2
,
(5)
where the coefficients rN are easily derived from E N. As expected the rN are all positive, suggesting a singularity on the real axis. Under this hypothesis, the ratio test gives now a singularity at z -~ 0.59, corresponding to/3 ~ 2.4. This is an important point to remark: the peak of Lautrup and Nauenberg [8] occurs at a value of 13near the point where the high-temperature expansion suggests a singularity, however the precise values of the singularity points do not coincide as it should be. It is possible that this discrepancy is due to the finite length of the high-temperature expansion or to finite size effects in the Monte Carlo simulations. In this letter we explore an alternative point of view i.e.: both the asymptotic behaviour of the hightemperature expansion and the "experimental" peak in the specific heat originate from a pair of complex singularities near the real axis, as suggested in ref. [ 11 ]. The reader may find it interesting that the specific heat of a two-dimensional superconductor in the Random Phase Approximation shows both a marked peak and a pair of complex singularities near the real axis [ 12]. In the RPA approximation this two-dimensional system is similar to gauge theories as far as it has both asymptotic freedom and absence of phase transition. I n t h e hypothesis of the existence of complex singularities a more refined analysis is needed for the SU(2) specific heat. We proceed by looking for a function with the following properties: (i) It has the right high temperature expansion, (ii) It reproduces the simulation data near fl ~ 2.2. We have studied the following simple f o r m :
CO3) = fl2(dJ/d~)[1 + F(z) + a ( z ) ] ,
(6)
where:
Y(z) = [A(1 - zlzc) + B ] / [ ( i
-
zlzJ
+
~>21.r12,
(7)
rN
N! dzNz=o (8)
+ i5); by construction eq. (6) automatically satisfies condition (i). The adjustable parameters A, B, z c, 5, 3' must be chosen in such a way that (i) CO3) =/32(dJ/d13)[1 + F(z) + G(z)] fits the simulation data near fl ~ 2.2. (ii) G(z) must be well approximated by a low-order polynomial in the region of interest near z ~ 0.57; in practice only its first 4 ~ 5 terms must be sensibly 4= 0. (This amount to that the singularities of G(z) are further away.) Two reasonable sets of values we get for the parameters are:
(a): A = 1.52,
B =0.657,
zc
0.570,
:
(9) 6 = 0.0507,
3' = 0.530.
1.5
t
\
\
i
I
I
1
2
3
\
,~
Fig. 1. Specific heat as function of/3; the dots indicate the simulation data, the solid curve is the fit (a), the dashed curve is the fit (b). 271
Volume 102B, number 4
PHYSICS LETTERS
Table 1 CoefficientsgN defined in (8) for the two fits (a) and (b) [see (9)] compared with the rN as defined in (5).
N
rN
gN (fit a)
gN (fit b)
2 3 4 5 6 7 8 9 10
0.5556 0.8519 1.3241 2.1287 3.5196 5.9055 9.9365 16.6354 27.5977
0.330 0.188 0.861 -0.735 -0.805 -0.109 -0.726 0.676 -0.519
0.210 0.109 0.264 -0.530 -0.112 -0.125 -0.731 0.167 -0.631
× 10 -1 X 10 -2 × 10 -1 × 10 -1 10 -2 x 10 -3
×
X 10 -1 × 10 -1 × 10 -3
(b): A =1.04,
B =0.623,
6 --- 0.0532,
zc--0.565,
3' = 0.517.
The reader m a y w o n d e r h o w m a n y terms o f the high-temperature expansion are needed to detect complex singularities. The answer is given in fig. 2, where the ratios
R N _ CN CN+ 1 ' × 10 -1 × 10 -1
(9 c o n ' d )
In fig. 1 we compare the results o f the fit with the simulation data, in table 1 we give the coefficients gN o f G(z) c o m p a r e d w i t h rN.
18 June 1981
CN
_ 1 d _ ~ , F(z)! N ! d(ZlZc)~V Iz=O
(10)
are p l o t t e d versus N . We get R N ~ 1 (as for a real singularity at z = Zc) up to N ~ 30; after this order we have a fast decrease. We conclude that the c o m p l e x singularities hypothesis, although it has m a n y attractive features, is very hard to test by looking at the high-temperature expansion only (unless very long series w o u l d be available). We are i n d e b t e d to K. Wilson for having provided us w i t h the coefficients o f the high-temperature expansion. We thank also C. R e b b i for having provided us w i t h a c o p y o f ref. [6] prior to publication and N. Sourlas for useful discussions.
References
Cli CN+I
Q0 0000gqIQIIQOOOOQIO QOOOBQQQ
I
I
8
I
I
16
I
I
24
I
I
32
N
Fig. 2. Ratios R N = CN/CN+ 1 [see eq. (10)] versus N for the fit (a), a very similax behaviour is obtained for the fit (b). 272
[1] K. Wilson, Corneil preprint CLNS/80/442 (January 1980). [2] For a recent review see: G. Parisi, talk XXth Conf. on High energy physics (Madison, 1980), Frascati preprint LNF-80/52 (P) (September 1980). [3] K. Binder, ed., Monte Carlo methods (Springer, Berlin, 1979). [4] K. Wilson, Carg6se Summer School (1977) (Plenum, New York, 1978). [5] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. D20 (1979) 1915; M. Creutz, Phys. Rev. Lett. 43 (1979) 553; M. Creutz, Phys. Rev. Lett. 45 (1980) 313; M. Creutz, Phys. Rev. D21 (1980) 2308; C. Rebbi, Phys. Rev. D21 (1980) 3350. [6] G. Bhanot and C. Rebbi, SU(2) string, gluebail mass and interquark potential by Monte Carlo computations •(1980), to be published. [7] B. Lautrup and M. Nauenberg, Phys. Lett. 95B (1980) 63. [8] B. Lautrup and M. Nauenberg, Phys. Rev. Lett. 45 (1980) 1755. [9] K. Wilson, private communication. [10] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D l l (1975) 2104; D19 (1979) 2514 (E). [11] D.J. Gross, Les Houches (1980), to be published in Phys. Rep. [12] W.E. Masker, S. Mar~elja and R.D. Parks, Phys. Rev. 188 (1969) 745.