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A remark on the phase transitions of modified action spin and gauge models
Volume 126B, number 3,4
A REMARK
PtlYSICS LETTERS
ON THE PHASE TRANSITIONS
OF MODIFIED
30 June 1983
ACTION
SPIN A N D G A U G E M O D E L S Nathan S E I B E R G t The Institute for Advanced Study, Princeton, NJ 08540, USA and Sorin S O L O M O N 2 California Institute of Technology, Physics 452-48, Pasadena, CA 91125, USA Received 14 March 1983
We consider the phase diagrams of modified action gauge and spin models and concentrate on their periphery - infinitely far from their origins (zero temperature - 13-t = 0). In this limit the exact positions of the phase transitions are found by looking for the global minimum of the single plaquette action (for a spin system - the single link energy). As the parameters of the model are varied,the position of such a global minimum is in gener-,d changed. When this change is non-analytic, a phase transition takes place. The phase structure for finite 13is clearly similar, but not identical to the infinite 13one. We discuss several finite 13corrections that should be applied to the exactly known infinite 13picture. We confront our analysis for infinite 132 = r.i13t~ with the Monte Carlo simulations for two four-dimensional gauge systems: an SU(3) gauge model with action S = - R e r p(13t tr Up + 132(tr Up) 1 ) and an SU(2) model with S = - R e Xp [13t { tr Up + 132(-~ tr Up)2 + 133(1 tr Up)31.
Modified action models were first introduced in order to study the role o f the global topology o f the parameter space (the manifold on which the variables are defined) in the dynamics o f spin [ 1] and gauge [2] models. Several authors considered m o d i f e d action models based on two parameters/31 and/32 that enabled the interpolation between the models with th non-trivial t o p o l o g y (/31 = 0) and the u n m o d i f i e d original models (/32 = 0) [2,3]. The various phase transitions that were f o u n d in these models were all related to the topological structure o f the theory. Later phase transitions o f non-topological origin were f o u n d [4,5]. In order to shed some light on all these phase transitions, we focus our attention on the periphery o f the phase diagram and succeed in analytically solving the models there. A typical m o d e l o f this kind is based on the partition function i Supported by the US Department of Energy under Grant No. DE-AC02-76ER02220. 2 Bantrell Fellow; work supported in part by the U.S. Department of Energy under Contract No. DE-AC03-81ER40050.
236
Z =
~ e-S . configurations
(l)
For spin systems, S in eq. (1) is usually referred to as the energy over the temperature. We will a d o p t here the language o f gauge systems and use the term action for S. The dynamical variables o f the spin systems are usually n dimensional unit vectors o s defined on the sites o f the lattice and S is given as a sum over the links s =
(2)
Sdx), links
where x = 1 -~
Iwel 2
.
S~ is a real f u n c t i o n and wQ is the difference between the two spins on the ends o f the link £: wQ=v s-os+~,
x = v s "Os+ ~.
(3)
We limit ourselves to the case that S~ is a polynomial o f its argument
Volume 126B, number 3,4
S~(x) = ~ /3ix i , i
PHYSICS LETTERS (4)
and define the temperature/3 -1 by =
.
(5)
SQ is then given by /3i
S~=/3~i ~ xi=/3~i %.xi.
(6)
A model which is identified by N different coupling constants/3 i has an N dimensional phase diagram. We are interested in the model in the limit/3 ~ oo, where its phase diagram becomes an N -- 1 dimensional sphere ~0N_1 parameterized by the "directions" a i. For/3 ~ ~, the partition function Z [eq. (1)] is clearly dominated by the minimal action configuration because the contributions from all the other configurations are exponentially suppressed. All we have to do now is to find the global minimum of the action S. The equality sign in rainS = m i n ~ SQ>~ ~ minS~, links links
(7)
is achieved when there exists at least one configuration with constant SQ all over the lattice. A stronger requirement is thatwQ itself be constant up to its sign. Such configurations can easily be constructed. Let o and o' be two n dimensional unit vectors obeying o o' = w. Then label the sites of the lattice by integer vectors (nx , ny, ...) and assign ° s = o f o r e v e n n x + ny + ... (8)
= o ' for odd n x + ny + .... Clearly, w~ = -+w for every link in the lattice. In general, there are many configurations with constant Iw~ I. Since the orientation of w may vary from link to link (except for the cases Iw[ = 0 or 2), the entropy of the constant Iw~l configurations (the log of their number) is proportional to the volume of the system. Since constant S~ configurations exist, the equality sign in eq. (7) is indeed achieved and the global minimum of S can be easily determined by the global ininimum ofS~. In order to solve the model for/3 -+
it is, therefore, sufficient to find the global minimum o f the single link action S~.
30 June 1983
The phase diagram for/3 -~ ~ is an N -- 1 dimensional sphere SN_ 1 and it is parameterized by the "directions" a i. As the direction is varied, the position Xmi n of the global minimum of the single link action SQ is in general changed as well. Although every local minimum is changed in a continuous fashion, this is not necessarily the case for the global minimum. In particular, it may happen that for a certain range of directions, S~ has two distinct local minima; in a part of this range, one of these local minima is the global one and in the other part, the other local minimum takes over. Then the position of the global minimum has a "jump" in this range of directions, corresponding to a discontinuity of the order parameter Iwl - a first order transition. Generally such crossing of local minima is not related to the topology of the parameter space and may occur even in topologically trivial models. A continuous but non-differentiable change in the position of the global minimum, which is also possible, represents a second order transition. Although we considered a modified action spin system, a similar simplification in the behavior at the periphery of the phase diagram occurs for modified action chiral models [6] * t or gauge systems. The dynamical variables in the gauge models are matrices U~ defined on the links of the lattice and the action S is given by S=
~ Sp(tr Up), plaquettes
(9)
where Sp is a real function of the complex variable tr U_. U^ is the ordered product of the U~ matrices V around t~e plaquette. A construction of a constant Up (up to hermitian conjugation) configuration was given in refs. [5,7]. Again, the phase transitions at the periphery of the phase diagram (/3 ~ '~) are easily de te rmine d. We reduced our multi-variable problem to the simpler problem of finding the global minimum of a real function S~(x) (or Sp(x) for the gauge system) of one complex number. Sincex is defined on a bounded domain, the global minimum may occur on the boundary of this domain. In that case OSUOx (or 1 Here one should look for the global minimum of the single link action S12.The construction of the constant S~ configuration is similar to eq. (8). 237
Volume 126B, number 3,4
PHYSICS L E T T E R S
3Sp/3X) does not necessarily vanish there. The domain o f x depends, of course, on the parameter space. For the O(n) symmetric spin system described above, o s is defined on S n_l and
x~[-l,l]. For SO(2n) systems,x = tr Up E [ - 2 n , 2n]; for SO(2n + 1) systems,x = trUp ~ [ - 2 n + 1,2n+ 1]; for SU(2) systems,x -- tr Up E [ - 2 , 2]; for SU(n) systems, the domain can be parametrized by exp(i~0i)+ex p \ - i
/=1
j--1
~0j ,
~0jE [0,21r1 ;(10)
for U(n) systems the domain can be parametrized by n
tr Up = ~ exp (i~oi), j=l
~ol E [0, 2rr] .
We define the directions (ai) of the phase transitions at infinite/3 as the "ancestors". Although we know these directions for infinite/3, we are unable to find them for finite/3, nor even are we able to find lim~__,.~/3i = lima-,®/3ai when it is finite. Yet, we expect the actual phase transitions for finite/3 - the "descendants" - to be close to the ancestors, i.e. approximately parallel to them. The phase transitions for/3 ~ ,,~ depend only on the domain on which the argument of S~ (or Sp) is defined and on the particular form of this function. The ancestors do not depend on the dimensionality of the system nor on its nature (gauge, spin, etc.). These important details introduce finite/3 corrections which are responsible to the exact form of the descendants. Before considering these corrections we describe a simple and well known example. This is the four dimensional SU(2) gauge system [2] with the action
s=
~
plaquettes
Sp,
Sp = -(,ill x +/32 x 2 ) = -/3(O~lX + 0~2x 2 )
= -fl(sin tgx + cos Ox 2) , _
x-i
1
trUp.
As O is varied the global minimum of Sp for x G [ - 1 , 11 is 238
(11)
30 June 1983
for 0 < 0 < 7r - O 0 ,
Xmi n = 1 _
1
- -~tgO,
f o r n - O O < O
(12)
for 7r + 0 0 < 0 < 2rr, where tg O0 = 2. The order parameter (x) = (tr Up) becomes identical to Xmi n for/3 -+ oo. It has a jump on the line O =0 corresponding to a first order ancestor. On the lines O = It -+ O0, Xmi n is continuous but dXmin/dt9 is discontinuous, namely there are second order ancestors there. The various corrections make the actual finite/3 phase diagram different from the simple ancestors diagram. There are three possible sources of corrections: (1) Topological corrections. The phase diagram of the previous example has an ancestor at/31 = 0. However, as/32 ~ o~ but/31 remains finite, the system becomes an effective Z 2 gauge system, which is known to have two phase transitions as the coupling constant /31 is varied (one phase transition for positive/31 and one for negative/31). Therefore, the single ancestor/31 = 0 splits into two almost parallel descendants. This phenomenon is not surprising since our analysis at/3 oo is not sensitive to finite changes in/31 for infinite /32" In general, such a splitting takes place when the invariance group admits a discrete center which becomes the symmetry of an effective theory when some parameters become infinite. Another transition line, related to the nontrivial topology, may connect these two descendants [ 1,2]. These new lines can also be related to the periphery of the phase diagram by introducing a term which controls the monopole energy (for the gauge case see ref. [2]) or the vortex energy (for the spin case, see ref. [8]). (2) Gaussian corrections. When/3 is finite, one has to perform the whole partition sum [eq. (1)] taking into account all configurations. For large but finite/3, it is possible to calculate the partition sum by a saddlepoint type expansion around the minimal action configuration. This configuration is different on the two sides & t h e transition, and the corresponding gaussians have in general two different widths. The transition line is, therefore, "pushed" towards the phase with the smaller width. (3) Entropy corrections. We have already mentioned the fact that the lowest action configuration is not unique and it is necessary to consider all such configurations. This amounts to the entropy factor which
Volume 126B, number 3,4
PHYSICS LETTERS
"pushes" the transition line towards me lower e n t r o p y phase. The e n t r o p y corrections may have another important effect. Consider the second order phase transition at infinite 3 in the last example. The configuration on one side of the transition is a c o n s t a n t tr Up = 2 configuration. Since tr Up = 2 implies Up = (1 0), this configuration is u n i q u e up to gauge transformations. On the other side o f the transition - for O > lr - O0 - the configurations are those o f constant tr Up = - t g O. There are m a n y different such configurations and a non-trivial e n t r o p y factor is e n c o u n t e r e d . The multiplicity o f these configurations is given by [f(tr Up)]nP where np is the n u m b e r o f plaquettes in the system and f ( t r Up) is a function (which is in principle calculable) obeying f ( 2 ) = 0. The e n t r o p y is, therefore ,rip In f ( t r Up). Then, roughly speaking, rather than minimizing Sp, we should minimize Sp + l n f ( t r Up). For infinite/3 the e n t r o p y term does not change the previous result. However, for finite/3, it has the effect o f smoothing the variation o f tr Up and it may, therefore, destroy the transition. We conclude that for any finite/3 the second order descendant o f this ancestor may n o t exist as a result o f the e n t r o p y corrections. Indeed, we could not find this transition in Monte Carlo simulations of this model. It is interesting to note that a similar second order phase transition is not destroyed in an analogous four dimensional U(1) gauge theory [3] where such e n t r o p y corrections are not present. To further examine our m e t h o d , we considered the four dimensional SU(3) gauge system [5] based on the action S =-
plaquettes
30 June 1983
c
[
Fig. 1. The phase diagram of the four dimensional gauge model based on eq. [ 13]. The second order ancestor (dotted line) has no descendant. The fkst order ones (thick continuous lines) have first order descendants (thin lines)• l:or 13~ o0, ( tr Up) is exactly calculable. It is equal to 3 in reNo~ A, -1 in phase B, 3e 2ua,'3 in phase C and is given by 2ei~0 + e-~'l¢(~ovaries from grr atB2 = --½P] to zero at ~2 .... g~l) in region D. -
>
/ - j '/2
'..
i I JA%
I I\
'.
i/
\
....
/
. . . . . . . ".
\
/\
\
oi~._._~o-,
......
C-)' .......
~-'x.....
L,I ,.,,D
t
\H_ ;,o \
[ill Re tr Up + 132 Re (tr Up) 2 ]
1
l"J
i
1 I ]
~3
', / / _,.),
~<~-7
(13) =-3
~ [al Re tr Up + a2 Re (tr Up)2] , plaquettes
tr Up can be parametrized according to eq. (10) as trUp = exp (i~01) + exp (i@2) + exp [ - i ( ~ 1 + ~02)],
(14) (Re (tr Up)) and ( l l m (tr Up)l) are exactly determined for 13--~ oo and the ancestors are easily lcoated. Fig. 1 shows these ancestors and their descendants as obtained in Monte Carlo simulations. The phase diagram exhibits the following features that were discussed above. ( I ) Every first order ancestor has a descendant that becomes parallel to it for t3 = oo.
Fig. 2. The ancestor diagram of the model based on eq. (5). The thick lines are first order ancestors (continuous lines are on the visible side of the sphere and dashed lines are on its opposite side). The dotted lines are second order ancestors (full dots are visible and open dots are on the other side of the sphere). O is the center of the sphere - origin of the coordinate system. The value ofx = ~ tr Up is gLven at some typical points on the sphere. The coordinates of" some importam points are: A: t0, l, 01, B: (+3/~'T-O, 0, -- l f i ~ ) ; C: (2A,,~, -1/x/5-. 0); D: ( 3 / ~ , - 3 L ~ N , 1/~/19); E: (0, •
I
.
.
-1 Iv'2, 1 / ~ ) ; F: (- ~, o, ~); G: (~, 0, - ~); H: (0, - 1/V2, - 1/x/2); I: ( - 3 / ~ T ~ , - 3 / ~ , - 1 / v T 9 ) ; J: ( - 2 / , f 5 , -l/x/5", 0); K: (-3/x/T'6, O, l/V']-6). 239
Volume 126B, number 3,4
PtlYSICS LETTERS ~32
-~
L -2
C ~ "
i
o
i
•
b Z33
1•2
3
\
-3
Fig. 3. The ancestors and their descendants as found by Monte Carlo simulations in several two dimensional sections of the three dimensional phase diagram of the model of eq. (15). The second order ancestors (dotted lines) do not have any descendants. The first order ones (thick continuous lines) have first order descendants (thin lines). The ancestors are labeled by the same letters as in fig. 2 [For detailed MC graphs, see ref. 4 ].
30 June 1983
Several particular cases of this model were already studied [2,4,7,9]. The phase diagram of this model is three dimensional and is symmetric under (/31 , ~2,/33) -+ (-/31,/32, -/33)" The ancestor diagram depends on the directions (a t , a 2, a3) only and can, therefore, be drawn on a sphere (fig. 2). The topological corrections split the ancestor (0, 1,0) and introduce a monopole transition; the entropy corrections destroy the descendants of the second order ancestors. Fig. 3 shows the ancestors and their descendants in several two dimensional sections of the three dimensional phase diagram. Our method works for studying topological phase transitions too. For implementing it, one has to consider interaction terms which are defined on different elements of the lattice, e.g., both link and plaquette interactions (see e.g. ref. [8]) (or both plaquettes and cubes). After verifying that a constant action (constant link and constant plaquette) configuration exists, the ancestors are easily found by minimizing the action. We hope that our approach will serve as a useful tool in a systematic understanding of modified action models both in statistical mechanics and in quantum field theory. N.S. would like to thank the LBL theory group for the hospitality extended to him when part of this work was done. He wishes to acknowledge stimulating discussions with D. Amit, R. Dashen and U. Heller. S.S. wants to thank A. Schwimmer and F. Fucito for discussions.
References (2) Neither topological splitting nor monopole phase transitions are observed as the model is topologically trivial. (3) The descendants are shifted towards the low entropy phase. l (4) The second order ancestor/32 = -?,/31 has no descendant. Another example is the four dimensional SU(2) gauge system based on S= -
~ i~lX +/J2 x2 +/33x3 plaquettes
= --/3 ~ t~lX + 0~2X2+ O~3X3 , plaque ttes X--~i trUp .
240
(15)
[ 1] S. Solomon, Phys. Lett. 100B (1981) 492; J. Kogut, N. Snow and M. Stone, Phys. Rev. Lett. 47 (1981) 1767. 121 I. Halliday and A. Schwimmer, Phys. Lett. 101B (1981) 327; I. Greensite and B. Lautrup, Phys. Rev. Lett. 47 (1981) 9; G. Bhanot and M. Creutz, Phys. Rev. D24 (1981) 3212. [31 G. Bhanot, Nucl. Phys. B205 [FS5I (1982) 168. 14] J. Anthony, Phyi. Lett. 110B (1982) 271; L. Caneschi, I. Fox and S. Solomon, W1S-83/7 Feb-Ph. [51 C. Bachas and R. Dashen, IAS preprint. [6] J. Kogut, M. Snow and M. Stone, Phys. Rev. Lett. 47 (1981) 1767. [7] I. Drouffe, Nucl. Phys. B205 [FS5] (1982) 27. [8] S. Solomon, Y. Stavans and E. Domany, Phys. Lett. lI2B (1982) 373; S. Solomon and Y. Stavans, in preparation. [9] K. Bitar, S. Gottlieb and C. Zachos, Phys. Rev. D26 (1982) 2853.
A REMARK
PtlYSICS LETTERS
ON THE PHASE TRANSITIONS
OF MODIFIED
30 June 1983
ACTION
SPIN A N D G A U G E M O D E L S Nathan S E I B E R G t The Institute for Advanced Study, Princeton, NJ 08540, USA and Sorin S O L O M O N 2 California Institute of Technology, Physics 452-48, Pasadena, CA 91125, USA Received 14 March 1983
We consider the phase diagrams of modified action gauge and spin models and concentrate on their periphery - infinitely far from their origins (zero temperature - 13-t = 0). In this limit the exact positions of the phase transitions are found by looking for the global minimum of the single plaquette action (for a spin system - the single link energy). As the parameters of the model are varied,the position of such a global minimum is in gener-,d changed. When this change is non-analytic, a phase transition takes place. The phase structure for finite 13is clearly similar, but not identical to the infinite 13one. We discuss several finite 13corrections that should be applied to the exactly known infinite 13picture. We confront our analysis for infinite 132 = r.i13t~ with the Monte Carlo simulations for two four-dimensional gauge systems: an SU(3) gauge model with action S = - R e r p(13t tr Up + 132(tr Up) 1 ) and an SU(2) model with S = - R e Xp [13t { tr Up + 132(-~ tr Up)2 + 133(1 tr Up)31.
Modified action models were first introduced in order to study the role o f the global topology o f the parameter space (the manifold on which the variables are defined) in the dynamics o f spin [ 1] and gauge [2] models. Several authors considered m o d i f e d action models based on two parameters/31 and/32 that enabled the interpolation between the models with th non-trivial t o p o l o g y (/31 = 0) and the u n m o d i f i e d original models (/32 = 0) [2,3]. The various phase transitions that were f o u n d in these models were all related to the topological structure o f the theory. Later phase transitions o f non-topological origin were f o u n d [4,5]. In order to shed some light on all these phase transitions, we focus our attention on the periphery o f the phase diagram and succeed in analytically solving the models there. A typical m o d e l o f this kind is based on the partition function i Supported by the US Department of Energy under Grant No. DE-AC02-76ER02220. 2 Bantrell Fellow; work supported in part by the U.S. Department of Energy under Contract No. DE-AC03-81ER40050.
236
Z =
~ e-S . configurations
(l)
For spin systems, S in eq. (1) is usually referred to as the energy over the temperature. We will a d o p t here the language o f gauge systems and use the term action for S. The dynamical variables o f the spin systems are usually n dimensional unit vectors o s defined on the sites o f the lattice and S is given as a sum over the links s =
(2)
Sdx), links
where x = 1 -~
Iwel 2
.
S~ is a real f u n c t i o n and wQ is the difference between the two spins on the ends o f the link £: wQ=v s-os+~,
x = v s "Os+ ~.
(3)
We limit ourselves to the case that S~ is a polynomial o f its argument
Volume 126B, number 3,4
S~(x) = ~ /3ix i , i
PHYSICS LETTERS (4)
and define the temperature/3 -1 by =
.
(5)
SQ is then given by /3i
S~=/3~i ~ xi=/3~i %.xi.
(6)
A model which is identified by N different coupling constants/3 i has an N dimensional phase diagram. We are interested in the model in the limit/3 ~ oo, where its phase diagram becomes an N -- 1 dimensional sphere ~0N_1 parameterized by the "directions" a i. For/3 ~ ~, the partition function Z [eq. (1)] is clearly dominated by the minimal action configuration because the contributions from all the other configurations are exponentially suppressed. All we have to do now is to find the global minimum of the action S. The equality sign in rainS = m i n ~ SQ>~ ~ minS~, links links
(7)
is achieved when there exists at least one configuration with constant SQ all over the lattice. A stronger requirement is thatwQ itself be constant up to its sign. Such configurations can easily be constructed. Let o and o' be two n dimensional unit vectors obeying o o' = w. Then label the sites of the lattice by integer vectors (nx , ny, ...) and assign ° s = o f o r e v e n n x + ny + ... (8)
= o ' for odd n x + ny + .... Clearly, w~ = -+w for every link in the lattice. In general, there are many configurations with constant Iw~ I. Since the orientation of w may vary from link to link (except for the cases Iw[ = 0 or 2), the entropy of the constant Iw~l configurations (the log of their number) is proportional to the volume of the system. Since constant S~ configurations exist, the equality sign in eq. (7) is indeed achieved and the global minimum of S can be easily determined by the global ininimum ofS~. In order to solve the model for/3 -+
it is, therefore, sufficient to find the global minimum o f the single link action S~.
30 June 1983
The phase diagram for/3 -~ ~ is an N -- 1 dimensional sphere SN_ 1 and it is parameterized by the "directions" a i. As the direction is varied, the position Xmi n of the global minimum of the single link action SQ is in general changed as well. Although every local minimum is changed in a continuous fashion, this is not necessarily the case for the global minimum. In particular, it may happen that for a certain range of directions, S~ has two distinct local minima; in a part of this range, one of these local minima is the global one and in the other part, the other local minimum takes over. Then the position of the global minimum has a "jump" in this range of directions, corresponding to a discontinuity of the order parameter Iwl - a first order transition. Generally such crossing of local minima is not related to the topology of the parameter space and may occur even in topologically trivial models. A continuous but non-differentiable change in the position of the global minimum, which is also possible, represents a second order transition. Although we considered a modified action spin system, a similar simplification in the behavior at the periphery of the phase diagram occurs for modified action chiral models [6] * t or gauge systems. The dynamical variables in the gauge models are matrices U~ defined on the links of the lattice and the action S is given by S=
~ Sp(tr Up), plaquettes
(9)
where Sp is a real function of the complex variable tr U_. U^ is the ordered product of the U~ matrices V around t~e plaquette. A construction of a constant Up (up to hermitian conjugation) configuration was given in refs. [5,7]. Again, the phase transitions at the periphery of the phase diagram (/3 ~ '~) are easily de te rmine d. We reduced our multi-variable problem to the simpler problem of finding the global minimum of a real function S~(x) (or Sp(x) for the gauge system) of one complex number. Sincex is defined on a bounded domain, the global minimum may occur on the boundary of this domain. In that case OSUOx (or 1 Here one should look for the global minimum of the single link action S12.The construction of the constant S~ configuration is similar to eq. (8). 237
Volume 126B, number 3,4
PHYSICS L E T T E R S
3Sp/3X) does not necessarily vanish there. The domain o f x depends, of course, on the parameter space. For the O(n) symmetric spin system described above, o s is defined on S n_l and
x~[-l,l]. For SO(2n) systems,x = tr Up E [ - 2 n , 2n]; for SO(2n + 1) systems,x = trUp ~ [ - 2 n + 1,2n+ 1]; for SU(2) systems,x -- tr Up E [ - 2 , 2]; for SU(n) systems, the domain can be parametrized by exp(i~0i)+ex p \ - i
/=1
j--1
~0j ,
~0jE [0,21r1 ;(10)
for U(n) systems the domain can be parametrized by n
tr Up = ~ exp (i~oi), j=l
~ol E [0, 2rr] .
We define the directions (ai) of the phase transitions at infinite/3 as the "ancestors". Although we know these directions for infinite/3, we are unable to find them for finite/3, nor even are we able to find lim~__,.~/3i = lima-,®/3ai when it is finite. Yet, we expect the actual phase transitions for finite/3 - the "descendants" - to be close to the ancestors, i.e. approximately parallel to them. The phase transitions for/3 ~ ,,~ depend only on the domain on which the argument of S~ (or Sp) is defined and on the particular form of this function. The ancestors do not depend on the dimensionality of the system nor on its nature (gauge, spin, etc.). These important details introduce finite/3 corrections which are responsible to the exact form of the descendants. Before considering these corrections we describe a simple and well known example. This is the four dimensional SU(2) gauge system [2] with the action
s=
~
plaquettes
Sp,
Sp = -(,ill x +/32 x 2 ) = -/3(O~lX + 0~2x 2 )
= -fl(sin tgx + cos Ox 2) , _
x-i
1
trUp.
As O is varied the global minimum of Sp for x G [ - 1 , 11 is 238
(11)
30 June 1983
for 0 < 0 < 7r - O 0 ,
Xmi n = 1 _
1
- -~tgO,
f o r n - O O < O
(12)
for 7r + 0 0 < 0 < 2rr, where tg O0 = 2. The order parameter (x) = (tr Up) becomes identical to Xmi n for/3 -+ oo. It has a jump on the line O =0 corresponding to a first order ancestor. On the lines O = It -+ O0, Xmi n is continuous but dXmin/dt9 is discontinuous, namely there are second order ancestors there. The various corrections make the actual finite/3 phase diagram different from the simple ancestors diagram. There are three possible sources of corrections: (1) Topological corrections. The phase diagram of the previous example has an ancestor at/31 = 0. However, as/32 ~ o~ but/31 remains finite, the system becomes an effective Z 2 gauge system, which is known to have two phase transitions as the coupling constant /31 is varied (one phase transition for positive/31 and one for negative/31). Therefore, the single ancestor/31 = 0 splits into two almost parallel descendants. This phenomenon is not surprising since our analysis at/3 oo is not sensitive to finite changes in/31 for infinite /32" In general, such a splitting takes place when the invariance group admits a discrete center which becomes the symmetry of an effective theory when some parameters become infinite. Another transition line, related to the nontrivial topology, may connect these two descendants [ 1,2]. These new lines can also be related to the periphery of the phase diagram by introducing a term which controls the monopole energy (for the gauge case see ref. [2]) or the vortex energy (for the spin case, see ref. [8]). (2) Gaussian corrections. When/3 is finite, one has to perform the whole partition sum [eq. (1)] taking into account all configurations. For large but finite/3, it is possible to calculate the partition sum by a saddlepoint type expansion around the minimal action configuration. This configuration is different on the two sides & t h e transition, and the corresponding gaussians have in general two different widths. The transition line is, therefore, "pushed" towards the phase with the smaller width. (3) Entropy corrections. We have already mentioned the fact that the lowest action configuration is not unique and it is necessary to consider all such configurations. This amounts to the entropy factor which
Volume 126B, number 3,4
PHYSICS LETTERS
"pushes" the transition line towards me lower e n t r o p y phase. The e n t r o p y corrections may have another important effect. Consider the second order phase transition at infinite 3 in the last example. The configuration on one side of the transition is a c o n s t a n t tr Up = 2 configuration. Since tr Up = 2 implies Up = (1 0), this configuration is u n i q u e up to gauge transformations. On the other side o f the transition - for O > lr - O0 - the configurations are those o f constant tr Up = - t g O. There are m a n y different such configurations and a non-trivial e n t r o p y factor is e n c o u n t e r e d . The multiplicity o f these configurations is given by [f(tr Up)]nP where np is the n u m b e r o f plaquettes in the system and f ( t r Up) is a function (which is in principle calculable) obeying f ( 2 ) = 0. The e n t r o p y is, therefore ,rip In f ( t r Up). Then, roughly speaking, rather than minimizing Sp, we should minimize Sp + l n f ( t r Up). For infinite/3 the e n t r o p y term does not change the previous result. However, for finite/3, it has the effect o f smoothing the variation o f tr Up and it may, therefore, destroy the transition. We conclude that for any finite/3 the second order descendant o f this ancestor may n o t exist as a result o f the e n t r o p y corrections. Indeed, we could not find this transition in Monte Carlo simulations of this model. It is interesting to note that a similar second order phase transition is not destroyed in an analogous four dimensional U(1) gauge theory [3] where such e n t r o p y corrections are not present. To further examine our m e t h o d , we considered the four dimensional SU(3) gauge system [5] based on the action S =-
plaquettes
30 June 1983
c
[
Fig. 1. The phase diagram of the four dimensional gauge model based on eq. [ 13]. The second order ancestor (dotted line) has no descendant. The fkst order ones (thick continuous lines) have first order descendants (thin lines)• l:or 13~ o0, ( tr Up) is exactly calculable. It is equal to 3 in reNo~ A, -1 in phase B, 3e 2ua,'3 in phase C and is given by 2ei~0 + e-~'l¢(~ovaries from grr atB2 = --½P] to zero at ~2 .... g~l) in region D. -
>
/ - j '/2
'..
i I JA%
I I\
'.
i/
\
....
/
. . . . . . . ".
\
/\
\
oi~._._~o-,
......
C-)' .......
~-'x.....
L,I ,.,,D
t
\H_ ;,o \
[ill Re tr Up + 132 Re (tr Up) 2 ]
1
l"J
i
1 I ]
~3
', / / _,.),
~<~-7
(13) =-3
~ [al Re tr Up + a2 Re (tr Up)2] , plaquettes
tr Up can be parametrized according to eq. (10) as trUp = exp (i~01) + exp (i@2) + exp [ - i ( ~ 1 + ~02)],
(14) (Re (tr Up)) and ( l l m (tr Up)l) are exactly determined for 13--~ oo and the ancestors are easily lcoated. Fig. 1 shows these ancestors and their descendants as obtained in Monte Carlo simulations. The phase diagram exhibits the following features that were discussed above. ( I ) Every first order ancestor has a descendant that becomes parallel to it for t3 = oo.
Fig. 2. The ancestor diagram of the model based on eq. (5). The thick lines are first order ancestors (continuous lines are on the visible side of the sphere and dashed lines are on its opposite side). The dotted lines are second order ancestors (full dots are visible and open dots are on the other side of the sphere). O is the center of the sphere - origin of the coordinate system. The value ofx = ~ tr Up is gLven at some typical points on the sphere. The coordinates of" some importam points are: A: t0, l, 01, B: (+3/~'T-O, 0, -- l f i ~ ) ; C: (2A,,~, -1/x/5-. 0); D: ( 3 / ~ , - 3 L ~ N , 1/~/19); E: (0, •
I
.
.
-1 Iv'2, 1 / ~ ) ; F: (- ~, o, ~); G: (~, 0, - ~); H: (0, - 1/V2, - 1/x/2); I: ( - 3 / ~ T ~ , - 3 / ~ , - 1 / v T 9 ) ; J: ( - 2 / , f 5 , -l/x/5", 0); K: (-3/x/T'6, O, l/V']-6). 239
Volume 126B, number 3,4
PtlYSICS LETTERS ~32
-~
L -2
C ~ "
i
o
i
•
b Z33
1•2
3
\
-3
Fig. 3. The ancestors and their descendants as found by Monte Carlo simulations in several two dimensional sections of the three dimensional phase diagram of the model of eq. (15). The second order ancestors (dotted lines) do not have any descendants. The first order ones (thick continuous lines) have first order descendants (thin lines). The ancestors are labeled by the same letters as in fig. 2 [For detailed MC graphs, see ref. 4 ].
30 June 1983
Several particular cases of this model were already studied [2,4,7,9]. The phase diagram of this model is three dimensional and is symmetric under (/31 , ~2,/33) -+ (-/31,/32, -/33)" The ancestor diagram depends on the directions (a t , a 2, a3) only and can, therefore, be drawn on a sphere (fig. 2). The topological corrections split the ancestor (0, 1,0) and introduce a monopole transition; the entropy corrections destroy the descendants of the second order ancestors. Fig. 3 shows the ancestors and their descendants in several two dimensional sections of the three dimensional phase diagram. Our method works for studying topological phase transitions too. For implementing it, one has to consider interaction terms which are defined on different elements of the lattice, e.g., both link and plaquette interactions (see e.g. ref. [8]) (or both plaquettes and cubes). After verifying that a constant action (constant link and constant plaquette) configuration exists, the ancestors are easily found by minimizing the action. We hope that our approach will serve as a useful tool in a systematic understanding of modified action models both in statistical mechanics and in quantum field theory. N.S. would like to thank the LBL theory group for the hospitality extended to him when part of this work was done. He wishes to acknowledge stimulating discussions with D. Amit, R. Dashen and U. Heller. S.S. wants to thank A. Schwimmer and F. Fucito for discussions.
References (2) Neither topological splitting nor monopole phase transitions are observed as the model is topologically trivial. (3) The descendants are shifted towards the low entropy phase. l (4) The second order ancestor/32 = -?,/31 has no descendant. Another example is the four dimensional SU(2) gauge system based on S= -
~ i~lX +/J2 x2 +/33x3 plaquettes
= --/3 ~ t~lX + 0~2X2+ O~3X3 , plaque ttes X--~i trUp .
240
(15)
[ 1] S. Solomon, Phys. Lett. 100B (1981) 492; J. Kogut, N. Snow and M. Stone, Phys. Rev. Lett. 47 (1981) 1767. 121 I. Halliday and A. Schwimmer, Phys. Lett. 101B (1981) 327; I. Greensite and B. Lautrup, Phys. Rev. Lett. 47 (1981) 9; G. Bhanot and M. Creutz, Phys. Rev. D24 (1981) 3212. [31 G. Bhanot, Nucl. Phys. B205 [FS5I (1982) 168. 14] J. Anthony, Phyi. Lett. 110B (1982) 271; L. Caneschi, I. Fox and S. Solomon, W1S-83/7 Feb-Ph. [51 C. Bachas and R. Dashen, IAS preprint. [6] J. Kogut, M. Snow and M. Stone, Phys. Rev. Lett. 47 (1981) 1767. [7] I. Drouffe, Nucl. Phys. B205 [FS5] (1982) 27. [8] S. Solomon, Y. Stavans and E. Domany, Phys. Lett. lI2B (1982) 373; S. Solomon and Y. Stavans, in preparation. [9] K. Bitar, S. Gottlieb and C. Zachos, Phys. Rev. D26 (1982) 2853.