- Email: [email protected]
Phase structure of three-dimensional quantum chromodynamics with dynamical fermions
Physics Letters B 313 ( 1993 ) 191 - 196 North-Holland
PHYSICS LETTERS B
Phase structure of three-dimensional quantum chromodynamics with dynamical fermions Vicente Azcoiti and Xiang-Qian Luo Departamento de Fistca Terrtca, Facultad de Czenctas, 50009 Zaragoza, Spam Received 11 May 1993 Editor: R. Gatto
We present the numerical results of the first computer simulation of lattice quantum chromodynamlcs in three dimensions (QCD 3 ) with dynamical fermions, obtained with the mzcrocanonlcal fermionic average method. The phase diagram in the (fl, Nf) plane is analysed by means of a mechanism recently developed for understanding the Nf-critlcal phenomenon m the full theory. The approxlmatzons involved are related to an expansion of the fermlonic effective action as a power series of the number of flavours Nf. Our results m the choral limit show the existence of a first order phase transition at very large Nf terminating in a critical point at fimte fl, where the specific heat diverges This suggests a phase structure for QCD 3 similar to that of compact QED 3 The investigation on q u a n t u m gauge theories with dynamical fermions in 2 + 1 dimensions ( Q F T 3) is getting more and more interesting. It is m o t i v a t e d by the following reasons. (1) Recent developments in QFT3 reveal their phenomenological relevance for high temperature superconductivity, q u a n t u m Hall effects, and related phenomena. (2) One o f the most difficult problems in lattice gauge theory IS the inclusion of dynamical fermlons. It would be more economical to investigate various algorithms and methods on a nontrlvial theory which is similar to Q C D in 3 + 1 dzmensions, but simpler. These similarities include asymptotic freedom, quark confinement, spontaneous chlral-symmetry breaking and meson and glueball spectrum. The advantages are lower dimensionality and superrenormahzability. It ts hoped that some o f the discoveries in lower dimensional models do not depend on the s p a c e - t i m e dimensions or can easily be extended to more reahstlc theories like QCD4. In fact, the so called dimensional reduction arguments [ 1 ] allow one to conjecture that at high enough temperature, renormalizable four-dimensional Q F T may be reduced to effective three dimensional models. In this frame, QED3 has been extensively studied both in the continuum [2,3] and on the lattice [ 4 - 1 0 ] . To analyse the effects o f dynamical fermlons in nonabehan models, QCD3 deserves a further research since it has more slmdarltles to QCD4 than QED3 and other lower dimensional models. In the last ten years, numerical results for (2 + 1 )-dimensional SU (2) lattice gauge theory without [ 11,12 ] and with dynamical fermlons [ 13 ] have been obtained. However, in the case o f S U ( 3 ) , only Monte Carlo data for the quenched theory at finite temperature exist in the literature [ 14,15 ] in addition to some preliminary analytic calculations in the continuum [ 16 ] and on the lattice [ 17 ]. The purpose o f this letter is to present the first numerical study of three-dimensional S U ( 3 ) lattice gauge theory with dynamical fermlons at zero temperature. We will apply the same methods and algorithms [8-10] used in the study of QED3 to analyse the phase structure in the (fl, Nf) plane. The aim o f these simulations is to provide information on the qualitative behaviour o f massless QCD3. The dynamics o f QCD3 is d e t e r m i n e d by the action
Elsevier SczencePubhshers B.V.
191
Volume 313, number 1,2
PHYSICS LETTERS B + E-~(x)Ax,yq/(y),
Nc Z R e T r ( U p )
S-
26 August 1993
p
(1)
x,y
where Nc = 3 is the number of colours, Up is the oriented plaquette variable and Ax,y is the fermlonic matrix. Here, four-component staggered formulatmn for fermmns is employed so that the action is chiral-invariant for massless fermlons. The approach to dynamical fermions we employ here has been explained in details m ref. [18]. It is the mlcrocanonical fermmmc average method introduced m [19] and applied to the compact and noncompact formulatmns of QED in three [8-10] and four [20-23 ] dimensmns. Our calculation is based on the defimtion of an effective fermionlc action [ 18 ] e_S~e(E,m.Nf) = f [dUu (x) ] (det A (m, Uu (x) ) ) Nf/2(~(SG ( e,u (X) ) -- 3 v g )
(2)
f [dUu(x) ],~(SG(Uu(x) ) - 3VE) which depends on the pure gauge plaquette energy E, fermion mass m and flavour number Nf and where the integration is over all gauge link variables belonging to the SU (3) gauge group. V is the lattice volume, and the exponent 1/2 m the determinant is due to the species doubling. SG (U) = ~ p Re Tr Up~No and the denominator in (2) is the density of states N ( E ) of fixed pure gauge normahzed energy E. The thermodynamlcal properties of the system can be obtained by differentiating the partition function
Z = /dE
e -Seff(E'm'Nf'fl),
(3)
which is now a one-&menslonal integral over the normalized pure gauge energy. Here Seff(E, m, Nf, fl) is the full effective action related to N(E), SVeff(E, m, Nf) and E by F Serf(E, m, Nf, fl) = - In N ( E ) - 3fiVE + Serf(E, m, Nr).
(4)
Since the effective action diverges hnearly wath the latUce volume V in the infinite volume limit, the thermodynamics of the system m this limxt can be analysed using the saddle point technique. The mean plaquette energy (Ep) = E0 (m, fl, Nf) is obtained by solving the saddle point equation -1 dn(E) n (E') d E
+
OSVerf(E,m, Nf) OE
- 3/~ = 0
(5)
sausfylng the minimum condition (l rd~(~>l ~ n-~j 5 / ~ 1
1
2--F
d2n ( E ) O S e f f ( E , m ,
n ( E ~ )- -dE --W- +
Nf))
OE 2
>0,
(6)
Eo(m,fl,Nf)
where S~rfV(E, m, Nf) in (5) and (6) is the effective fermionic action normalized by the lattice volume, and n (E) is related to the density of states N ( E ) by the expression
n(E) = N ( E ) l/v.
(7)
By differentiating eq. (5), the specific heat C~ = OEo/Ofl can be obtained; the result is
(, C# = 3 ~
ran(E)] 2 [ ~ 1
1 n(E)
2--F d2n (E)OSeff(E,m,
dE 2
+
Nf)) OE 2
- 1
(8)
Eo(m,fl,Nf)"
The phase structure of the system is completely described by the behavlour of the effective action as a function of the energy and of the other parameters. For instance, nonanalytic~ty of the normahzed denmty of states n (E) or the fermmmc effective action S~Vffcould generate phase transitions. This is in fact what happens m the three 192
Volume 313, number 1,2
PHYSICS LETTERS B
26 August 1993
Nf=O 50
~
,
,
,
i
,
- -
,
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-
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.
.
.
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.
. 0 5
.
.
l 1
1 5
2
2 5
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Fig. 1.
(NcCp)-I (Nf
= 0) as a function of the energy. Statistical errors are not visible at thas scale.
and four dimensional noncompact models [ 10,21-23 ]. However, even if n ( E ) and sFff are analytic functions, a phase transition can be dynamically generated by fermlonlc effects. Indeed this will happen if the second energy derivative o f sFrf is negative and large enough to compensate the contribution from the density of states in the saddle point equation (6). In such a case there will be some energy interval where the saddle point equations (5), (6) have no solution. This energy interval will be inaccessible to the system and a first order phase transition will appear. By decreasing the number o f flavours, we will reach a critical value N~ at which the energy interval where eq. (6) is not satisfied becomes a single point. This critical value N~, which must be larger than zero since the quenched model has no phase transinons, will correspond to the end point o f a first order phase transition line in the (fl, Nf) plane, the specific heat C~ (8) being &vergent at this point. Here we will pay special attention to the behaviour o f the plaquette energy and specific heat in the (fl, Nf) plane and will show how this mechanism works in QCD3. Let us first consider the quenched theory, Nf = 0. We have done simulations on the 6 3, 103, 14 3 and 16 3 lattices. The plaquette energy (Ep) and specific heat C~ in the pure gauge theory are calculated using the CablbboMarlnari heat bath algorithm [24] for S U ( 3 ) , c o m b i n e d with a mlcrocanonical process [25 ] in order to destroy correlations. No singular behavlour in these two quantities occurs at any finite value o f the inverse coupling constant ft. This implies that moving fl from zero to ~ , the plaquette energy takes continuously all the values from 0 to 1. Therefore the sum o f the first two terms in the d e n o m i n a t o r of the specific heat (8) will be positive In all the (0,1) energy interval (convexity of the effective action)), as shown in fig. 1 How the phase structure is when dynamical fermions are switched on? Let us consider the number of flavours Nf as a continuous parameter. If the effecnve fermionlc action sFff(E, m, Nf) is an analytic function, a phase transition can only be generated by compensating ~ts second energy derivative with the first two terms in (6). It has been shown in [23] that the effective fermionlc action sVfr (E, m, Nf) diverges linearly with Nf when Nr goes to ~ . Then, if the sign o f the second energy derivative of-SFff(E, m, Nf) is negative m some energy interval and Nf is large enough, the last term in (6) compensates the first two terms and the global sign o f (6) will change. Let us now &scuss our numerical results for the effective fermionlc action defined in (2). The feaslbihty of a direct computation o f the effective fermlonlc action (2) depends on the form o f the probabdlty distribution function o f the logarithm o f the fermlonic determinant at fixed pure gauge energy. It has been shown that it is more reliable to compute sFff by expanding it m a power series of Nf (cumulant expansion) [18]
-SV~rr(E,m, Nr) = 1 N r ( l n d e t d ) E + ] N 2 [ ( ( l n d e t ~ ) z ) E + ~ N f S [ ( ( l n d e t d - (lndetA)E)3)E] + ...
( l n d e t A ) 2] (9) 193
Volume 313, number 1,2
0
6
. . . .
0 4
~u~ ~.~
O~
i
. . . .
0
m=O0
<>
m=005
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m=01
PHYSICS LETTERS B
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. . . .
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26 August 1993
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i
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,
,
2
,
,
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25
,
i
,
,
001
0 005
~ , , , I
0
3
Fig. 2. First contribution to the effective fermiomc action
(9) normalized by the volume versus 3E at several fermlon masses and two flavours, obtained through microcanonical simulations. Statistical errors are invisible at this scale. The open symbols represent the results on the 103 lattice, the sohd symbols stand for those on the 143 lattice, and the squares at 3E = 2 correspond to those on the 163 lattice.
0
. . . .
05
i , , , , i , , , , I , , , , 1 1 , , ,
1
15
2
25
3E
Fig. 3. Second contribution to the effective fermlonic action (9) normalized by the volume versus 3E at zero fermlon mass and two flavours, obtained through mlcrocanomcal simulations. The meanings of the symbols are the same as in fig. 2.
where (O)e stands for the mean value o f the operator O(U~(x)) computed with the probabihty distribution
[dU~(x)]~(So(Uu(x)) - 3 V E ) / N ( E ) . It is ~mportant to do a fimte size scaling analysis o f the effective action before using the saddle point techmque because only terms proportional to V are relevant m the thermodynamlcal hmlt. Simulations on 103, 143 and 163 lattices have been carried out. On each lattice 30000-60000 configurations at each fixed E were generated by the microcanomcal algorithm [25]. The fermiomc matrix at zero mass is exactly diagonahzed for 100-200 decorrelated configurations at each energy by means of a modified Lanczos algorithm [26]. F r o m the eigenvalues obtained m this way we can compute the further terms m (9) for any value of the fermlon mass m and flavour numbers Nf. Fig. 2 shows the (lndetA)e/V as a funcUon o f E for several values o f the fermion mass m. We observe that the first term in (9) is proportional to the lattice volume V, mside statistical errors, for all the energies explored. The second term m (9), normalized by the volume and for Nf = 2 is presented in fig. 3. We find that the relative fluctuations o f the effective fermlomc action in QCD3 (normalized by the number of colours) are smaller than those in QED3. This is natural because there are more degrees of freedom in QCD3. W i t h the increase o f the n u m b e r o f degrees o f freedom, the fluctuations are reduced as expected in [18]. Furthermore, the normalised fluctuaUons decrease as V increases, at least for large energies (3E >/ 2). The results for the first contribution to the effective fermionic action can be well fitted by a eighth degree polynomial (continuous hne in fig. 2). Once we have the effective fermionlc action and the pure gauge specific heat C a as a function o f E, we can do a saddle point analysis. The second energy derivative of the effective pure gauge actmn (sum o f the first two terms in (6)) is, as can be seen m fig. 1, p o s m v e for all E > 0. Even more, when the energy approaches the m a x i m u m value E = 1, this quantity diverges or equivalently the specific heat goes to zero when fl goes to ~ in the pure gauge theory. The second energy derivative o f the effective fermionic action is negauve in the s m a l l - i n t e r m e d i a t e energy region but it becomes positive in the large energy region (see fig. 2). The only possible region where the sign of (6) could change is in this small-intermediate energy interval. We found indeed that this is what happens for Nf large enough and negative inverse square coupling ft. This p h e n o m e n o n generates a first order phase transition line in the (fl, Nf) plane ending at a critical point. The critical number of flavours at the end point of the first order phase transition line is N~ ~ 24 and the critical plaquette energy Ec ~ 0.41, which corresponds to tic = - 0 . 3 7 5 5 . All the results reported here refer to the massless case. When fermmns are massive, the end point of the first 194
Volume 313, number 1,2
PHYSICS LETTERS B
26 August 1993
m=0 50
'
i
~
L
~
0
2
~
--
'°
30 20
-6
-4
-2
4
Fig. 4. Phase diagram in the (fl, Nf) plane. order phase transition line moves to larger values of Nf and this line disappears in the infinite mass limit. This result can be understood by looking at the mass dependence of the fermionic effective action (fig. 2). W h e n the fermlon mass increases, the absolute value o f the second energy derivative of the effective action decreases m the small-intermediate energy region. Therefore we need larger values of Nf m order to compensate the first two contributions m the saddle point equation (6). This feature is c o m m o n to other gauge models. In fact our mvesugations o f the compact U (1) model in four dimensions [18-20 ] have shown that a similar m e c h a m s m to the one described before allows to understand the weakness of the first order phase transition in this model when the fermion mass increases. In the infinite mass limit, the normalized fermlomc effective action behaves like In m, i.e., it is independent o f E. Let us discuss now how our results can be affected by finite size effects and by the approximations m the determination o f the fermionic effective action. Our numerical results for (In det d ) e on 103, 143 and 163 lattices show a perfect hnear scaling w~th the lattice volume V inside the small statistical errors, in all the energy interval explored. On the contrary, the second cumulant contribution to the fermlomc effective action (9), i.e., the fluctuations o f the fermlonlc acUon at fixed pure gauge energy E, does not show any analogous scahng behavlour. The systematic decreasing of the normalized fluctuations with the lattice volume in the large energy region could suggest that this contribution will be irrelevant for the thermodynamic hmlt. However, another possibdity is that we have not reached the asymptotic regime for th~s quantity. In general one can show by a simple analysis [ 23 ] that the fermlonic effective action is a linear function o f Nf in the large Nf limit. This implies that successive terms in the cumulant expansion (9), each one being proportional to Nf~, k = 1,2, .., will conspire in order to give the linear behaviour. Our approximation for the effective action in this work has been to take only the first contribution to (9), which is a linear function of Nr and, as previously shown, free o f finite size effects. O f course our phase diagram (fig. 4) could change when including all the cumulant contributions to the effective action. F o r instance we can not exclude from our results a nonanalitlcity o f the fermlomc effective action in the flavour n u m b e r Nf, which could generate a chiral transition similar to the one claimed m [13] for the SU (2) model. However a change of quahtatlve features, hke the fact that the first order line finishes at some finite 13, is highly unlikely. In fact, in order to get a first order phase transition line ending at fl = ~ , a change of the sign of the second energy derivative of the fermlonic action near E = 1 is needed. However it seems very difficult that such a change happens since the normalized fluctuations of the logarithm of the fermlomc d e t e r m i n a n t go to zero when E ~ 1. Furthermore, even if the sign of the second energy derivative o f the effecuve fermiomc action changed near E = 1, we should need an unbounded second derivative in order to compensate the positive divergence in the E ~ 1 h m l t of the sum of the first two terms in the saddle point equation (6). The most relevant result which emerges from the analysis of our numerical simulations is the (fl, Nf) plane phase diagram for massless fermlons plotted in fig. 4. The striking feature o f this phase diagram is the fact that 195
Volume 313, number 1,2
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the first order phase transition line ends at fimte (negative) ft. The physical picture which emerges from the phase dmgram plotted In fig. 4 is very similar to that of compact QED3 [8,9]. Can we generalize these conclusions to (3 + 1 )-dimensional QCD9 The situation at present is not clear. Although there are many similarities between theories m three and four dimensions, there are some essential differences as well, as mentioned m [ 11 ]. Quenched QCDa at finite temperature experiences a first order phase transition while m QCD3 a second order one occurs [ 14,15 ]. This means that dimensionality plays an important role. Further work on (3 + 1 )-dimensional QCD with dynamical fermions is xn progress. We thank G. D1 Carlo and A.F. Grillo for interesting dtscusstons. This work has been partly supported by MEC (Ministerio de Educacton y Ciencia) and CICYT (Comision I n t e r m l m s t e n a l de Ciencia y Tecnologla).
References [ 1] T. Appelqmst and R. Pisarskl, Phys. Rev. D 23 (1981 ) 2305. [2] T. Appelqmst, D. Nash and L.C.R. Wijewardhana, Phys Rev. Lett. 60 (1988) 1083. [3] D.C. Curtis, M.R Pennmgton and D. Walsh, Phys. Lett. B 295 (1992) 313, and references therein [4] E. Dagotto, J.B. Kogut and A Kocic, Phys. Rev Lett 62 (1989) 1083. [5] E. Dagotto, J.B. Kogut and A. Koclc, Nucl. Phys. B 334 (1990) 279 and references thereto. [6] S. Hands and J.B. Kogut, Nucl. Phys. B 335 (1990) 455 [7] J.B. Kogut and J.F. Lagae, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 737. [8] V. Azcoitl and X.Q. Luo, Nucl Phys. B (Proc. Suppl.) 30 (1993) 741. [9] V. Azcoitl and X.Q. Luo, DFTUZ. 92/25 [10] V. Azcoltl, X Q. Luo, C.E. Piedrafita, G. Dl Carlo, A.F Grlllo and A Galante, Fermlonlc effective action and the phase structure ofnoncompact quantum electrodynamlcs in 2 + 1 dimensions, DFTUZ 93/08 (1993), to appear in Phys. Lett. B. [11] E D'Hoker, Nucl. Phys. B 180 (1981) 341. [12] J. Ambjorn, R. Olesen and C. Peterson, Nucl. Phys. B 244 (1984) 262 [ 13 ] E. Dagotto, A. Koclc and J.B. Kogut, Nucl. Phys. B 362 ( 1991 ) 498. [14] M. Gross and J.F. Wheater, Z. Phys C 28 471. [15] J. Chlastensen, G Thorleifsson, P.H Damgaard and J.F. Wheater, Nucl. Phys. B 374 225. [16] M. Prakash and I. Zahed, Phys. Rev. Lett. 69 (1992) 3282. [17] X Q. Luo and Q.Z. Chen, Phys Rev. D 46 (1992) 814. [18] V. Azcoltl V. Lahena, X Q Luo, C.E Piedrafita G. D1 Carlo, A Galante, A.F Grdlo, L.A. Fernandez and A. Vladikas, Phys. Rev. D 48 (1993) 404. [19] V. Azcottl, G Dx Carlo and A.F. Grdlo, Phys. Rev. Lett 65 (1990) 2239. [20] V Azcoitl, A. Cruz, G. DI Carlo, A.F. Grlllo and A. Vladlkas, Phys. Rev. D 43 (1991 3487. [21 ] V. Azcoltl, G Dl Carlo and A.F. Grdlo, Mod. Phys. Lett. A 7 (1992) 3561. [22] V Azcoitl, G. D1 Carlo and A.F Grlllo, A new approach to noncompact lattice QED with light fermlons, DFTUZ 91/34 (1992), to appear m Int. J. Mod: Phys A. [23] V Azcolti, G. D1 Carlo and A.F. Grallo, Phys Lett. B 305 (1993) 275 [24] N. Cablbbo and E. Marinarl, Phys. Lett. B 119 (1982) 387. [25] R. Petronzlo and E. Vlcan, Phys. Lett. B 248 (1990) 159. [26] I M. Barbour, N.E. Behlhl, P. Gibbs, G Schlerholz and M. Teper, in The recursion method and its apphcatlons, Solid State Sciences vol 58, ed. D. Pettifor and D. Weane (Springer, 1985) p. 149
196
PHYSICS LETTERS B
Phase structure of three-dimensional quantum chromodynamics with dynamical fermions Vicente Azcoiti and Xiang-Qian Luo Departamento de Fistca Terrtca, Facultad de Czenctas, 50009 Zaragoza, Spam Received 11 May 1993 Editor: R. Gatto
We present the numerical results of the first computer simulation of lattice quantum chromodynamlcs in three dimensions (QCD 3 ) with dynamical fermions, obtained with the mzcrocanonlcal fermionic average method. The phase diagram in the (fl, Nf) plane is analysed by means of a mechanism recently developed for understanding the Nf-critlcal phenomenon m the full theory. The approxlmatzons involved are related to an expansion of the fermlonic effective action as a power series of the number of flavours Nf. Our results m the choral limit show the existence of a first order phase transition at very large Nf terminating in a critical point at fimte fl, where the specific heat diverges This suggests a phase structure for QCD 3 similar to that of compact QED 3 The investigation on q u a n t u m gauge theories with dynamical fermions in 2 + 1 dimensions ( Q F T 3) is getting more and more interesting. It is m o t i v a t e d by the following reasons. (1) Recent developments in QFT3 reveal their phenomenological relevance for high temperature superconductivity, q u a n t u m Hall effects, and related phenomena. (2) One o f the most difficult problems in lattice gauge theory IS the inclusion of dynamical fermlons. It would be more economical to investigate various algorithms and methods on a nontrlvial theory which is similar to Q C D in 3 + 1 dzmensions, but simpler. These similarities include asymptotic freedom, quark confinement, spontaneous chlral-symmetry breaking and meson and glueball spectrum. The advantages are lower dimensionality and superrenormahzability. It ts hoped that some o f the discoveries in lower dimensional models do not depend on the s p a c e - t i m e dimensions or can easily be extended to more reahstlc theories like QCD4. In fact, the so called dimensional reduction arguments [ 1 ] allow one to conjecture that at high enough temperature, renormalizable four-dimensional Q F T may be reduced to effective three dimensional models. In this frame, QED3 has been extensively studied both in the continuum [2,3] and on the lattice [ 4 - 1 0 ] . To analyse the effects o f dynamical fermlons in nonabehan models, QCD3 deserves a further research since it has more slmdarltles to QCD4 than QED3 and other lower dimensional models. In the last ten years, numerical results for (2 + 1 )-dimensional SU (2) lattice gauge theory without [ 11,12 ] and with dynamical fermlons [ 13 ] have been obtained. However, in the case o f S U ( 3 ) , only Monte Carlo data for the quenched theory at finite temperature exist in the literature [ 14,15 ] in addition to some preliminary analytic calculations in the continuum [ 16 ] and on the lattice [ 17 ]. The purpose o f this letter is to present the first numerical study of three-dimensional S U ( 3 ) lattice gauge theory with dynamical fermlons at zero temperature. We will apply the same methods and algorithms [8-10] used in the study of QED3 to analyse the phase structure in the (fl, Nf) plane. The aim o f these simulations is to provide information on the qualitative behaviour o f massless QCD3. The dynamics o f QCD3 is d e t e r m i n e d by the action
Elsevier SczencePubhshers B.V.
191
Volume 313, number 1,2
PHYSICS LETTERS B + E-~(x)Ax,yq/(y),
Nc Z R e T r ( U p )
S-
26 August 1993
p
(1)
x,y
where Nc = 3 is the number of colours, Up is the oriented plaquette variable and Ax,y is the fermlonic matrix. Here, four-component staggered formulatmn for fermmns is employed so that the action is chiral-invariant for massless fermlons. The approach to dynamical fermions we employ here has been explained in details m ref. [18]. It is the mlcrocanonical fermmmc average method introduced m [19] and applied to the compact and noncompact formulatmns of QED in three [8-10] and four [20-23 ] dimensmns. Our calculation is based on the defimtion of an effective fermionlc action [ 18 ] e_S~e(E,m.Nf) = f [dUu (x) ] (det A (m, Uu (x) ) ) Nf/2(~(SG ( e,u (X) ) -- 3 v g )
(2)
f [dUu(x) ],~(SG(Uu(x) ) - 3VE) which depends on the pure gauge plaquette energy E, fermion mass m and flavour number Nf and where the integration is over all gauge link variables belonging to the SU (3) gauge group. V is the lattice volume, and the exponent 1/2 m the determinant is due to the species doubling. SG (U) = ~ p Re Tr Up~No and the denominator in (2) is the density of states N ( E ) of fixed pure gauge normahzed energy E. The thermodynamlcal properties of the system can be obtained by differentiating the partition function
Z = /dE
e -Seff(E'm'Nf'fl),
(3)
which is now a one-&menslonal integral over the normalized pure gauge energy. Here Seff(E, m, Nf, fl) is the full effective action related to N(E), SVeff(E, m, Nf) and E by F Serf(E, m, Nf, fl) = - In N ( E ) - 3fiVE + Serf(E, m, Nr).
(4)
Since the effective action diverges hnearly wath the latUce volume V in the infinite volume limit, the thermodynamics of the system m this limxt can be analysed using the saddle point technique. The mean plaquette energy (Ep) = E0 (m, fl, Nf) is obtained by solving the saddle point equation -1 dn(E) n (E') d E
+
OSVerf(E,m, Nf) OE
- 3/~ = 0
(5)
sausfylng the minimum condition (l rd~(~>l ~ n-~j 5 / ~ 1
1
2--F
d2n ( E ) O S e f f ( E , m ,
n ( E ~ )- -dE --W- +
Nf))
OE 2
>0,
(6)
Eo(m,fl,Nf)
where S~rfV(E, m, Nf) in (5) and (6) is the effective fermionic action normalized by the lattice volume, and n (E) is related to the density of states N ( E ) by the expression
n(E) = N ( E ) l/v.
(7)
By differentiating eq. (5), the specific heat C~ = OEo/Ofl can be obtained; the result is
(, C# = 3 ~
ran(E)] 2 [ ~ 1
1 n(E)
2--F d2n (E)OSeff(E,m,
dE 2
+
Nf)) OE 2
- 1
(8)
Eo(m,fl,Nf)"
The phase structure of the system is completely described by the behavlour of the effective action as a function of the energy and of the other parameters. For instance, nonanalytic~ty of the normahzed denmty of states n (E) or the fermmmc effective action S~Vffcould generate phase transitions. This is in fact what happens m the three 192
Volume 313, number 1,2
PHYSICS LETTERS B
26 August 1993
Nf=O 50
~
,
,
,
i
,
- -
,
i
~
-
'
'
i
'
'
'
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,
'
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.
.
.
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.
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Fig. 1.
(NcCp)-I (Nf
= 0) as a function of the energy. Statistical errors are not visible at thas scale.
and four dimensional noncompact models [ 10,21-23 ]. However, even if n ( E ) and sFff are analytic functions, a phase transition can be dynamically generated by fermlonlc effects. Indeed this will happen if the second energy derivative o f sFrf is negative and large enough to compensate the contribution from the density of states in the saddle point equation (6). In such a case there will be some energy interval where the saddle point equations (5), (6) have no solution. This energy interval will be inaccessible to the system and a first order phase transition will appear. By decreasing the number o f flavours, we will reach a critical value N~ at which the energy interval where eq. (6) is not satisfied becomes a single point. This critical value N~, which must be larger than zero since the quenched model has no phase transinons, will correspond to the end point o f a first order phase transition line in the (fl, Nf) plane, the specific heat C~ (8) being &vergent at this point. Here we will pay special attention to the behaviour o f the plaquette energy and specific heat in the (fl, Nf) plane and will show how this mechanism works in QCD3. Let us first consider the quenched theory, Nf = 0. We have done simulations on the 6 3, 103, 14 3 and 16 3 lattices. The plaquette energy (Ep) and specific heat C~ in the pure gauge theory are calculated using the CablbboMarlnari heat bath algorithm [24] for S U ( 3 ) , c o m b i n e d with a mlcrocanonical process [25 ] in order to destroy correlations. No singular behavlour in these two quantities occurs at any finite value o f the inverse coupling constant ft. This implies that moving fl from zero to ~ , the plaquette energy takes continuously all the values from 0 to 1. Therefore the sum o f the first two terms in the d e n o m i n a t o r of the specific heat (8) will be positive In all the (0,1) energy interval (convexity of the effective action)), as shown in fig. 1 How the phase structure is when dynamical fermions are switched on? Let us consider the number of flavours Nf as a continuous parameter. If the effecnve fermionlc action sFff(E, m, Nf) is an analytic function, a phase transition can only be generated by compensating ~ts second energy derivative with the first two terms in (6). It has been shown in [23] that the effective fermionlc action sVfr (E, m, Nf) diverges linearly with Nf when Nr goes to ~ . Then, if the sign o f the second energy derivative of-SFff(E, m, Nf) is negative m some energy interval and Nf is large enough, the last term in (6) compensates the first two terms and the global sign o f (6) will change. Let us now &scuss our numerical results for the effective fermionlc action defined in (2). The feaslbihty of a direct computation o f the effective fermlonlc action (2) depends on the form o f the probabdlty distribution function o f the logarithm o f the fermlonic determinant at fixed pure gauge energy. It has been shown that it is more reliable to compute sFff by expanding it m a power series of Nf (cumulant expansion) [18]
-SV~rr(E,m, Nr) = 1 N r ( l n d e t d ) E + ] N 2 [ ( ( l n d e t ~ ) z ) E + ~ N f S [ ( ( l n d e t d - (lndetA)E)3)E] + ...
( l n d e t A ) 2] (9) 193
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Fig. 2. First contribution to the effective fermiomc action
(9) normalized by the volume versus 3E at several fermlon masses and two flavours, obtained through microcanonical simulations. Statistical errors are invisible at this scale. The open symbols represent the results on the 103 lattice, the sohd symbols stand for those on the 143 lattice, and the squares at 3E = 2 correspond to those on the 163 lattice.
0
. . . .
05
i , , , , i , , , , I , , , , 1 1 , , ,
1
15
2
25
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Fig. 3. Second contribution to the effective fermlonic action (9) normalized by the volume versus 3E at zero fermlon mass and two flavours, obtained through mlcrocanomcal simulations. The meanings of the symbols are the same as in fig. 2.
where (O)e stands for the mean value o f the operator O(U~(x)) computed with the probabihty distribution
[dU~(x)]~(So(Uu(x)) - 3 V E ) / N ( E ) . It is ~mportant to do a fimte size scaling analysis o f the effective action before using the saddle point techmque because only terms proportional to V are relevant m the thermodynamlcal hmlt. Simulations on 103, 143 and 163 lattices have been carried out. On each lattice 30000-60000 configurations at each fixed E were generated by the microcanomcal algorithm [25]. The fermiomc matrix at zero mass is exactly diagonahzed for 100-200 decorrelated configurations at each energy by means of a modified Lanczos algorithm [26]. F r o m the eigenvalues obtained m this way we can compute the further terms m (9) for any value of the fermlon mass m and flavour numbers Nf. Fig. 2 shows the (lndetA)e/V as a funcUon o f E for several values o f the fermion mass m. We observe that the first term in (9) is proportional to the lattice volume V, mside statistical errors, for all the energies explored. The second term m (9), normalized by the volume and for Nf = 2 is presented in fig. 3. We find that the relative fluctuations o f the effective fermlomc action in QCD3 (normalized by the number of colours) are smaller than those in QED3. This is natural because there are more degrees of freedom in QCD3. W i t h the increase o f the n u m b e r o f degrees o f freedom, the fluctuations are reduced as expected in [18]. Furthermore, the normalised fluctuaUons decrease as V increases, at least for large energies (3E >/ 2). The results for the first contribution to the effective fermionic action can be well fitted by a eighth degree polynomial (continuous hne in fig. 2). Once we have the effective fermionlc action and the pure gauge specific heat C a as a function o f E, we can do a saddle point analysis. The second energy derivative of the effective pure gauge actmn (sum o f the first two terms in (6)) is, as can be seen m fig. 1, p o s m v e for all E > 0. Even more, when the energy approaches the m a x i m u m value E = 1, this quantity diverges or equivalently the specific heat goes to zero when fl goes to ~ in the pure gauge theory. The second energy derivative o f the effective fermionic action is negauve in the s m a l l - i n t e r m e d i a t e energy region but it becomes positive in the large energy region (see fig. 2). The only possible region where the sign of (6) could change is in this small-intermediate energy interval. We found indeed that this is what happens for Nf large enough and negative inverse square coupling ft. This p h e n o m e n o n generates a first order phase transition line in the (fl, Nf) plane ending at a critical point. The critical number of flavours at the end point of the first order phase transition line is N~ ~ 24 and the critical plaquette energy Ec ~ 0.41, which corresponds to tic = - 0 . 3 7 5 5 . All the results reported here refer to the massless case. When fermmns are massive, the end point of the first 194
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L
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'°
30 20
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Fig. 4. Phase diagram in the (fl, Nf) plane. order phase transition line moves to larger values of Nf and this line disappears in the infinite mass limit. This result can be understood by looking at the mass dependence of the fermionic effective action (fig. 2). W h e n the fermlon mass increases, the absolute value o f the second energy derivative of the effective action decreases m the small-intermediate energy region. Therefore we need larger values of Nf m order to compensate the first two contributions m the saddle point equation (6). This feature is c o m m o n to other gauge models. In fact our mvesugations o f the compact U (1) model in four dimensions [18-20 ] have shown that a similar m e c h a m s m to the one described before allows to understand the weakness of the first order phase transition in this model when the fermion mass increases. In the infinite mass limit, the normalized fermlomc effective action behaves like In m, i.e., it is independent o f E. Let us discuss now how our results can be affected by finite size effects and by the approximations m the determination o f the fermionic effective action. Our numerical results for (In det d ) e on 103, 143 and 163 lattices show a perfect hnear scaling w~th the lattice volume V inside the small statistical errors, in all the energy interval explored. On the contrary, the second cumulant contribution to the fermlomc effective action (9), i.e., the fluctuations o f the fermlonlc acUon at fixed pure gauge energy E, does not show any analogous scahng behavlour. The systematic decreasing of the normalized fluctuations with the lattice volume in the large energy region could suggest that this contribution will be irrelevant for the thermodynamic hmlt. However, another possibdity is that we have not reached the asymptotic regime for th~s quantity. In general one can show by a simple analysis [ 23 ] that the fermlonic effective action is a linear function o f Nf in the large Nf limit. This implies that successive terms in the cumulant expansion (9), each one being proportional to Nf~, k = 1,2, .., will conspire in order to give the linear behaviour. Our approximation for the effective action in this work has been to take only the first contribution to (9), which is a linear function of Nr and, as previously shown, free o f finite size effects. O f course our phase diagram (fig. 4) could change when including all the cumulant contributions to the effective action. F o r instance we can not exclude from our results a nonanalitlcity o f the fermlomc effective action in the flavour n u m b e r Nf, which could generate a chiral transition similar to the one claimed m [13] for the SU (2) model. However a change of quahtatlve features, hke the fact that the first order line finishes at some finite 13, is highly unlikely. In fact, in order to get a first order phase transition line ending at fl = ~ , a change of the sign of the second energy derivative of the fermlonic action near E = 1 is needed. However it seems very difficult that such a change happens since the normalized fluctuations of the logarithm of the fermlomc d e t e r m i n a n t go to zero when E ~ 1. Furthermore, even if the sign of the second energy derivative o f the effecuve fermiomc action changed near E = 1, we should need an unbounded second derivative in order to compensate the positive divergence in the E ~ 1 h m l t of the sum of the first two terms in the saddle point equation (6). The most relevant result which emerges from the analysis of our numerical simulations is the (fl, Nf) plane phase diagram for massless fermlons plotted in fig. 4. The striking feature o f this phase diagram is the fact that 195
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the first order phase transition line ends at fimte (negative) ft. The physical picture which emerges from the phase dmgram plotted In fig. 4 is very similar to that of compact QED3 [8,9]. Can we generalize these conclusions to (3 + 1 )-dimensional QCD9 The situation at present is not clear. Although there are many similarities between theories m three and four dimensions, there are some essential differences as well, as mentioned m [ 11 ]. Quenched QCDa at finite temperature experiences a first order phase transition while m QCD3 a second order one occurs [ 14,15 ]. This means that dimensionality plays an important role. Further work on (3 + 1 )-dimensional QCD with dynamical fermions is xn progress. We thank G. D1 Carlo and A.F. Grillo for interesting dtscusstons. This work has been partly supported by MEC (Ministerio de Educacton y Ciencia) and CICYT (Comision I n t e r m l m s t e n a l de Ciencia y Tecnologla).
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