Series analysis in four-dimensional Zn lattice gauge systems

Nuclear Physics B170 [FS1] (1980) 91-97 O North-Holland Publishing Company

SERIES A N A L Y S I S IN F O U R - D I M E N S I O N A L Z,, LATI"ICE G A U G E SYSTEMS J.M. DROUFFE CEN-Sacla y, Bofte Postale no. 2-91190 Gif-sur-Yvette, France

Received 27 December 1979 (Final version received 4 February 1980)

The strong coupling series for the free energy of four-dimensional Zn lattice gauge systems are analyzed. The energy per plaquette is displayed as a function of the coupling constant and the nature of the transition(s) is discussed.

Lattice gauge theories [1] provide a promising approach to the study of strong coupling regimes. The continuous field theory is expected to be obtained by shrinking the lattice spacing, which has been introduced as a regulator. However, little is known about this continuous limit. According to the conventional lore, this process is related to the existence of second-order transitions in the lattice system. Recently Monte Carlo experiments have been performed [2] for some gauge groups in four dimensions. They show evidence of a first-order transition in Z2 and Z3 systems, while an intermediate phase appears between two second-order transition points in Zn systems with n > 4. Therefore these systems do not present the simplest (and thus desired) pattern of a single second-order transition point; nevertheless this behaviour is predicted by theoretical speculations such as by the somewhat controversial mean-field analysis for the first-order transition, and by analogies with the xy-model for the third phase [3]. All these results may be derived directly from the strong coupling expansions for lattice systems. The formalism of these expansions has been known for a long time, as well as the corresponding series for usual gauge groups up to 16th order [4]. However they have never been used, to our knowledge, for obtaining these results. The aim of this paper is to supply the missing analysis for Zn gauge groups. First, we display the series to be used and expose the method of analysis. Then we give our conclusions successively for Z2, Z3, Z5 and Z6 groups. We do not consider the case of continuous groups (U(1), SU(2), SU(3) . . . . ); their richer structure as well as the lack of duality relations require other techniques which are postponed to a forthcoming work. The formalism of strong coupling expansions has been published [4, 1] as well as the series for the free energy for some usual groups. However, we display here an unpublished version of these series, valid for any gauge group and for any dimension. 91

92

J.M. Drouffe / Lattice gauge systems

The action S, which generally appears as the sum over all plaquettes of the lattice of a class function on the gauge group G for the product Up of gauge fields along the plaquette p, is exponentiated and expanded in Fourier series (i.e, in terms of characters X, of all irreducible representations): expS=~a[l+

r#O ~

Vr~rXr(Up)]

(1)

Here, the contribution A of the trivial representation has been factored out and/'Pr is the dimension of the representation r. The coefficients A and/3r are computable functions of the coupling constant which depend on the precise form chosen for the action. Using methods of ref. [4], the free energy F (logarithm of the partition function per site in the thermodynamic limit) is given up to and including 16 plaquettes diagrams by

F = ~d(d - 1){ln A + (d -

2)[~S6 + (2d - 5)S~o + ~ ( d - 3)S~2

+ (20d 2 - 106d + 143)$14 + (84d 2 - 504d + 757)$16 + ) ( d - 3)T16 - (2d - ~)S~ - (44d 2 - 226d + 294)$6Sxo + (2d - 5)0551 + 2(d - 3)(2d - 5)(0555 + ~:5551)+ ~(d - 3)(3 0842+ 3 01042 -}-20933 + ~444 "~111~)+ (20d 2 - 108d + 149)(20951 + ~'51415)+" " " ]} .

(2)

In this expression, each symbol introduced corresponds to a particular topology of graph. For the sphere and the torus,

Sn = 2

1 /2r ~nr ,

(3)

r#0

T,, = E BT, r#O

(4)

and for graphs with singular branch lines, Oijk : ~ .

NrstVrVsV g i'rfl ]sfl tk ,

(5)

rst (i]klm =

¢ijkl = ) lmn

~ NrstStuvl:rl/sl.'ul~v~'r~s[3t~u~ i j k l vm , rstuo E rstuo i

j

i jsB k,[3,, l NrstuVrVsVtPufl rB k

l

rn

(6)

(7)

n

ul)w rst

X

f Xr(RS-1)X,(ST-1)x,(TR-~)x~(R)x~(S)Xw(T) DR DS DT, J

(8)

where Nrs.-. is the n u m b e r of times the trivial representation is contained in

J.M. Drouffe/ Lattice gaugesystems

93

r ® s ®" • •. The use of formulae (1) to (8) allows the computation of the strong coupling expansion in the inverse coupling constant/3 for any gauge group. From now on, we restrict to the four-dimensional case d = 4. The numerical analysis of these series must take into account some remarks. In formula (2), the first term, ½d(d - 1) In A, is perfectly regular and therefore we only consider the term between brackets. The magnitude of the consecutive terms of these series fluctuates wildly; in particular, a four-term period seems to appear (this corresponds to the threshold of a new dimension for the graphs [5]; a pictorial description is obtained by sticking a cube on the surface of a given graph, a process which needs four plaquettes). As a consequence, the method of locating the nearest singularity by studying the ratio of two consecutive terms is not well adapted and shows large and wild fluctuations. Furthermore, one expects singularities out of the real positive/3 axis, for real negative and even complex/3 (for large dimension, the expansion reduces to an expansion in/34d [5], that is, gives four singularities on the /3-plane at equal distance from the origin). All these problems are already known in spin systems, where one prefers to use the susceptibility rather than the free energy when studying the location of the transition point. Unfortunately, we do not have anything similar to the susceptibility in gauge systems and the only available quantity is the free energy. The technique of Pad6 approximants [6] has been used here. We expect that the poles of an approximant will simulate the singularities and thus a good representation of the function will be obtained. It can be noted in particular that localizing the singularities by this method generalizes the ratio technique, which does nothing but localize the poles of a Pad4 approximant with degree 1 denominator. Furthermore, diagonal Pad6 approximants contain built-in homographic conformal transformations which may be used to remove the effects of unwanted complex or negative singularities. Therefore we expand the series as continued fractions (a version of diagonal Pad6 approximants for which very efficient numerical algorithms exist). The location of singularities is determined by the nearest real positive pole of the continued fraction, either for F or for its logarithmic derivative. Zz and Z3 groups. The actions are

Sz~ =/3 ~., o'i,'o'~kO'kzO'li,

or//= + 1,

(9)

P

Sz~ =/3 X cos (g,tr(o'ij+O'ik+O'kt+O'~i)), 2 p

O'q = 1, 2, 3,

O'ji + O'q = 3,

(10)

and the corresponding free energies obey the duality relations [7] Fz2(/3 ) = Fz2 ( - 1 In tanh 13) + 3 In sinh 2/3, Fz~(B)

[

2

e 3~/2"1~

Fz~ \ - - ~ l n ~ j + 2 1 n ( e 3 °

+2e-3B/2-3)--31n3.

(11) (12)

94

J.M. DroUffe / Lattice gauge systems 1

U

0.8

0.6

0./, m

0.2 I 0.1

0

I 0.2

I 0.3

I 0./,

" ~ l l t

0.5

............ I............

0.6

0.7

0.8

13c Z2

Fig. 1. Average energy per plaquette for the Z2 gauge group. The points are the Monte Carlo results of ref. [2].

These relations exchange the strong- and low-coupling regions and have a fixed point at fl~2 = 1 In (x/2+ 1) ~- 0.4407,

(13)

fl~3 = 2 In ( x ~ + 1) = 0.6700.

(14)

The Pad6 analysis shows that the nearest real positive singularity is far beyond these points (above 0.55 for Z2 and 0.9 for Z3). Therefore second-order transitions are completely excluded*. We compute the energy per plaquette U = 1 - ~ dF/d/~ by using the corresponding continued fraction up to the fixed point/3c and the duality relations beyond this point. The corresponding curves are displayed in figs. 1 and 2 1

I

I

I

ic Z3

0.8

U O.B

0.6

0.4

02 ••~..

0

I

I 0.2

I

I O.L.

I

I 0.6

.......... | .............

13

Fig. 2. Average energy per plaquette for the Z3 gauge group. The points are the Monte Carlo results of ref. [2]. * Corresponding two-dimensional spin systems (Ising and 3-components Potts models) as well as three-dimensional gauge and spin systems (dual to each other) show a completely different behaviour in which the singularity coincides with the expected second-order transition•

95

ZM. Drouffe / Lattice gauge systems

(together with the points of Monte Carlo experiments [2] which perfectly fit). They show the characteristic jump of the first order transitions with a discontinuity A U z : = 0.425,

(15)

d Uz3 = 0.541.

(16)

Z , groups, n >-4. The standard action is written as

(17) but it is better to use a more general action S = ~ In

3', exp •

(orii-~Orjk -~-Orkl -~-Orli)

,

with y,_, = 3',.

(18)

1

This action obeys the duality relation [7] F(3',) = F

3'jcos

+3 In n,

(19)

]=1

which exchanges, in the .parameter space 3'1/3". . . . . 3"tn/21/3'., the two regions separated by a hyperplane of fixed points 3'...~1 3t_ " • • ..~ 3"n--1 = ~ n - -

Yn

1.

(20)

Yn

The strong coupling expansion (2) is an expansion around the origin of this parameter space and allows computation in one of the two regions; then eq. (19) gives the result in the other one. Finally the standard action (17) lies along the line in the parameter space given by

Y"

=1

n j=l

(2;,)

exp /3 cos

cos 21rlr. n

(21)

The parameter space is only two-dimensional for the Z5 group. For the sake of simplicity, we first discuss this case, illustrated by fig. 3. The expansion (2) appears as a double series around the origin O, and is valid in the region I limited by the line of singularities ABCDE. We compute this line point by point by the following process. On a straight line starting from O (i.e., for a fixed ratio 3 " 2 / Y l ) , the expansion reduces to a series in only one variable and we compute, as previously, the limiting point as the nearest real positive pole of the corresponding continued fraction. By duality, we obtain the low-coupling region II, around the zero-coupling point O', limited by the curve A'BC'DE'. Therefore, there are two candidates for the free energy in the common region BCDC'B, which coincide on the dual symmetric straight line BD. Maximizing the free energy leads to a first-order transition line BD separating the

96

ZM. Drouffe / Lattice gauge systems 1

0'.x _.

0.8

:..: A'

0.6 -~ \

.' 1T

111"

'~

0.2 -

:"

/ i~. ~C "X/..------"

\

.."

%//....

0 O ............. I " ...... E" ' ; ~ ",I

2

\ ' \ ~ .I'~

.4

I

~

.e

"~/~'o

I i.

Fig. 3. The phase diagram for the Zs gauge group.

two regions I and II. There remains a third (non-connected) region III in which we cannot compute by using the expansion (2) and which is limited by the second-order transition lines A B A ' and EDE'. The conventional Z5 model (17) implies a relation between the ~/i's and lies in the diagram along the dotted line OO'. Fig. 4 displays the average energy per plaquette, together with Monte Carlo results [2], which again fit. There are two second-order transitions; the dotted line in fig. 3 cuts the two arcs D E and DE'. However the position of these transitions cannot be made very precise because the intersections _

l

0.8 -

~

o.6-

,i,

-,.

I

,i

I!

i-'"~!

O4-_ --

I " I I

I

I1"". ""'.

I

M. "-...

°o

I o.2

i o.4

i a~

J o.8

i I 1 ~ 2 ;os0.2 1.4

1.6

i 1.B~2

Fig. 4. Average energy per plaquette for the Zs gauge group. The points are the Monte Carlo results oI ref. [2].

J.M. Drouffe / Lattice gauge systems

97

1 0.8

0.6

I i

iX', ~ I

"t.

,

,

i

1

0.2 I

OI

I

0.2 0.4

I

I

0.8 0.8

iI ~

1.

1.2~S1.4 1.6 13z6

i

1.8

I

2

Z2 13 2_4

Fig. 5. Average energy per plaquette for the Z6 gauge group. The points are the Monte Carlo results of ref. [2].

occur with small angles. This uncertainty is reflected in fig. 4 by zones of transition rather than precise points. We cannot compute with our method in the intermediate phase III; however, as this region is particularly restricted in the Z5 case, the continued fraction apparently provides a reasonable interpolation for the curve in this phase. Z4 group leads to similar curves. However the point D, where the first-order transition line splits into two second-order transition lines, lies (up to the precision of our method) on the line OO' corresponding to the conventional model (17). Therefore we were unable to choose between three different possibilities: (i) two second-order transitions; (ii) one first-order transition; and the limiting case (iii) one-second order transition. For n greater than 5, we conclude that there are two second-order transitions, as in Z5 case. The parameter space has more than two dimensions and this forbids an easy display of the phase diagram. As an example, fig. 5 gives the average energy per plaquette in the Z6 case. I am grateful to C. Itzykson and J.B. Zuber for their critical advice and support. References [1] J.M. Drouffe and C. Itzykson, Phys. Reports 38 (1975) 133. [2] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rev. Lett. 42 (1979) 1390; Phys. Rev. D20 (1979) 1915. [3] J.L. Cardy, Santa Barbara preprint 723-79TH30 (1978); A. Ukawa, P. Windey and A.H. Guth, Princeton preprint; S. Elitzur, R.B. Pearson and J. Shigemitsu, Phys. Rev. D19 (1979) 3698. [4] R. Balian, J.M. Drouffe and C. Itzykson, PhyL Rev. D l l (1975) 2104; D19 (1979) 2514. [5] J.M. Drouffe, G. Parisi and N. Sourlas, Nucl. Phys. B161 (1979) 397. [6] G.A. Baker, Jr., Essentials of Pad6 approximants, (Academic Press, 1975). [7] R. Balian, J.M. Drouffe and C. Itzykson, Phys. Rev. D l l (1975) 2098; J.M. DroufFe, Phys. Rev. D18 (1978) 1174; R. Savit, preprint UMHE 79-8.