Mean-field analysis of SU(2) lattice gauge theory at finite temperature

Nuclear Physics B285 [FS19](1987) 173-182 North-Holland, Amsterdam

MEAN-FIELD ANALYSIS OF SU(2) L A T T I C E GAUGE T H E O R Y AT F I N I T E TEMPERATURE+ R.M. MARINHOJr.* Laboratoire de Physique Th~orique et Hautes Energies* *, Universit~ de Paris XI, BStiment 211, 91405 Orsay, France

Received 20 October 1986 (Revised 8 December 1986)

Mean-field approximation is used to study the lattice SU(2) gauge field theory at finite temperature, with an improvedtechnique that includes finite temperature effectsin the saddle-point approximation. We obtain good agreement with the Monte Carlo data.

1. Introduction It is a well known fact that standard mean-field methods, when applied to lattice gauge theories at a finite temperature, are sensitive to the periodic boundary conditions in the euclidean time direction related to finite temperature effects only at the level of one-loop corrections to the mean-field (saddle point) approximation. A n alternative method has been proposed, in which finite temperature effects are already present at the level of the saddle point approximation to the euclidean path integral. In this approach, standard mean field techniques are used only to integrate the gauge degrees of freedom sitting on the links in spacial direction, that is to say, to obtain an effective action for the thermal Wilson loops. Up to now, this method has been used only in the simple case of a Z(2) lattice gauge theory at a finite temperature, because the gauge variables remain coupled in the euclidean time direction, and an effective one-dimensional lattice gauge theory with periodic boundary condition has to be solved to start with. The purpose of this paper is to examine the case of a SU(2) lattice gauge theory at finite temperature and to show that even in this case the finite temperature mean-field methods of ref. [1] are quite useful to understand the phase diagram of *Work partially supported by CAPES- Coordena~ao de Aperfei~oamento de Pessoal de Nivel Superior- Brasil. * Permanent address: Instituto de Estudos Avanqados, Divis~o de Fisica Te6rica - CTA, 12200 S~o Jos6 dos Campos - SP, Brasil. ** Laboratoire associ6 au Centre National de la Recherche Scientifique. 0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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this theory. In sect. 2 we present the finite temperature mean-field theory of ref. [1]. In sect. 3 the phase diagram is studied at the highest temperature supported by the lattice, and the critical values of the gauge coupling constant are obtained. The deconfining phase transition is found to be of the second order. In sect. 4, the stability of the confining phase against a second-order transition is examined for any value of the temperature. In sect. 5 we develop a low temperature approximation which naturally matches the standard mean-field methods at a zero temperature. Finally, putting all these results together, we discuss the phase diagram in sect. 6 and present the conclusions.

2. Mean-field approximation Let us divide the ( d + 1)-dimensional euclidean space-time in N t layers of dimension d. We have labelled the sites by x and the links by (x, #), /~ running from 1 to d. We are going to call Ux,~,r the space-like gauge variable of the r th layer and ~2f the time-like gauge variable that join the r t h and (r + 1)th layers. In this way we have for the SU(2) Wilson action

S [ U , ~ ] = ¼fl E

Tr(U~,~Ux+., ~Ux+","r+ Ux,,r+ + h.c.)

• x,lz,v

r=l

p,
+¼fie E T - - r \(~•x , ~ r=l

r-o- x + ~ t r.~ x , p . (r+l)+.Qr+.h.c.) --x --

,

(2.1)

x,/~

where fl = 4/g 2 is the gauge coupling constant, and the physical temperature is given by aT = 1/Nt, a being the lattice spacing (which we set henceforth equal to 1). The finite temperature effects arise from the periodic boundary condition,

U~, U,+l= Ux , ~ 1 "

(2.2)

In the case of a continuous group it is always possible to choose a gauge in which the thermal Wilson loop W ( x ) = Tr(~21x... ~2~,) is uniformly spread in the time direction, i.e. W(x) = Tr(Y/x) N,. However, we prefer to choose a gauge in which all the thermal Wilson loops are concentrated in a given layer, for example, the one corresponding to r = 1, in which case we set ~2f = I for r = 2, 3... N t. In order to treat the mean-field equations in a translationally invariant manner in the time direction, we could take the partition function that is the arithmetic mean of all partition functions, differing from one another in the location of the layer where the thermal Wilson loop concentrates. This is not necessary, for all these integrals are the same. So we can treat only one of these integrals in which the link differing from 1 is between the first and the second layers.

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175

The partition function then becomes

z = fDS~ DUe stU' el,

(2.3)

where DO = H D~2x, x

DU=

I-I DU~,~,r X~p., r

with DI2~ and DU~,~r the Haar measure of the gauge group. Applying now the usual method of mean-field theory, introducing the non-compact mean fields Vx,~r and the external fluctuating fields Ax,~r for the fields not coupled to ~2x, (these are the only terms of the action sensitive to periodic boundary conditions) [1], we obtain the expression for the partition function ,/

Z = f D U ~ V ~ A e s' ,

(2.4)

where the non-compact fields Ax, r and Vx,gr a r e SL(2, C) matrices, ~ A = I-[ "~Ax'~ x,~,r ( 2 ~ ) 8 ' ~v=

FI ev~,~ ~

(2.5)

and N, S , = Y'~ E 113 Tr( Vx,~Vx +~,~V~ +u,gr+ Vx,vr+ + h.c.) r=l

x,g

N~ N, - ¼ E ETr(A~,.~V~,.r+h.c.) + ¼ E ~_.Tr(A~,.~+Ux,~ +h.c.) r=l

x,/L

r=l

x,g

N, + ¼ B ] ~ T r ( "V x , p , xO 2 +H . xQ + + h . c . ) + l B 2 " ' x + l ~ V xtl ,l~ x,g

r=2

]~Tr(Ux, tU~,/~+~)++h.c.) x,/~

(2.6) As we can see from the last equation, the integration over Ux,/zr factorizes in (x,/~),

R.M. MarinhoJr. / Mean fieM analysis

176

so following ref. [1] we introduce the N t link integral

Wu= Wu( ~2x,, ~x, Ax,,r; ~ ) = ln ZN, ,

(2.7)

where

=f

I-I D u r r=l

exp

¼ Tr( A r+U r q- b . c . ) ~ r=l

+ ¼ f l T r ( V r ~ x , , U (r+l)+~x + q- h . c . ) + lfl E Tr( UrU(r+l)++

h.c.)

.

(2.8)

r~l

In this way we can write our partition function as N, 1 r=l

r

r

r+

Vx,.r + +h.c.)

x,~

N, -¼ E ETr(Ax,,r+Vx,, •+h'c') r=l

x,~

+ E Wu(I2x+,,g2x, A~,,r, fl)) •

(2.9)

x,g

As was pointed out in ref. [1] this way of proceeding introduces temperature dependent 1/d effects already at the level of the saddle point approximation to (2.9). Periodic boundary conditions are included in the definition of the N t link integral (2.7). The thermal Wilson loop variables can now be thought of as a d-dimensional SU(2) spin system coupled to the d-dimensional gauge system (A, V). In order to treat this spin system it is convenient to introduce a mean field Wx and a fluctuating external field H~ for the I2x's. After doing this we can search for a uniform saddle point of the functional (2.9): H~, W~ independent of x; Ax, p,,•Vx, l~r independent of x, #, r; real and proportional to the unit matrix, as usual,

Ax'~'r = A t

Vx,;=v] Hx=H I Wx = WJ

Vx,/L, r ,

Vx.

(2.10)

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177

The free energy of the system reads

Ix(H)

HW

F = ½ f l ( d - 1)V 4 + - -

U,d

1

Uta

A V + Nt ITVu( W, A , fl ) ,

(2.11)

where the functions ff'v(W, A; t ) and IX(H) are given by

A; B) = l n f DUexp{ ¼A Tr(U r + U "+) + ¼flW2Tr(U1U2++ U2U 1+)

+ ¼fl Y'. Tr(Uru(r+l)++ U(r+I)U r+) , (2.12) r=2 IX(H) = L f D~2exp( ¼H Tr( 9 + a2+) ),

(2.13)

respectively. The saddle point equations are

OF OA =0'

OF OH = O,

OF OV = O,

OF OW = 0.

(2.14)

It is possible to eliminate the V and W variables from these equations, with the result:

F=

f l ( d - 1) AWuA HIX'(H) 2Nt 4 WUA4+ N t + N t ~

IX(H) Ntd

Wv Nt'

(2.15)

where lower index A (and in the following also H ) means the partial derivative with respect to A (and H), IX' is the derivative of IX with respect to its argument A (or H). The results of the preceding sections are analogous for any gauge group. We only need to change the integrals over the group manifold into a sum over all its elements if it is discrete.

3. High temperature region

The saddle point equations (2.14) are easily handled at high temperatures ( N t = 1, 2) because in this case the N t link integral is elementary. If N t = 1 we find

Wu= IX(A) + flix'( H) 2.

(3.1)

R.M. Marinho Jr. / Mean field analysis

178

The free energy becomes F = - 1 / 3 ( d - 1)/.d(A) 4 + Abt'(A ) + - d

d

.(A)

2. (3.2)

We observe that the gauge and spin systems decouple, and that we end up with a gauge theory in d dimensions and a d-dimensional spin system. In this case the deconfining phase transition is the second-order phase transition of the spin system, and it is obtained by setting Fitit= 0 at H = 0, with the result /3~2= 2. The first-order phase transition of the gauge system is found numerically and occurs at /3~1= 4.23, but of course this transition has nothing to do with deconfinement at finite temperature. When N t = 2, the link integral reduces to, 21nfo~dOsinZOexp{AcosO+t~[A+/3(#'(H)2+

1)cos0]}.

(3.3)

As soon as N t > 1 the gauge and spin system are coupled. However, the saddlepoint equations can be investigated numerically. We obtain in this way that, as/3 increases, the second order phase transition of the spin system at /3c2= 1.726, and that the first-order phase transition of the gauge system /3cl = 2.33. Again, we emphasize that it is the first one which is related to deconfinement for N t = 2, the coupling between the gauge and the spin system is rather weak and therefore does not qualitatively change the results obtained for N t = 1. From the high temperature results, which show that the second order transition of the spin system comes first as /3 increases, we can investigate the stability of the strong coupling confining phase against a second-order transition for any temperature.

4. Points of second-order phase transitions (13c2) After eliminating V and W from the system of equations (2.14), we can search for a minimum of the free energy (2.15). In order to do this we need to know the solution of the system of equations,

FA =

{ Fit =

Nt4

2fl(d-1) N4

WvA 3 +

N~} WUA3 +

WUA H +

WvA A = O,

H~'(H) N td

Wv Nt

O.

(4.1)

We can determine where a second-order phase transition occurs by analysing the

R.M. Marinho Jr. / Mean field analysis

179

stability of the free energy (2.15) at A = 0, H = 0. These are obtained by imposing the determinant of the hessian matrix to be zero, i.e.

FAAFHH- FfN = 0.

(4.2)

Next we need to know the N t link integral (2.12) at the origin. This is done by the character expansion of the exponential in (2.12). We find,

Wu= (N t

-

1)/z(fl),

WuA = O, w v . = O,

WvAA = Nt, WUA H ~- O,

WVHH = ~Pl~~' ,, ,~N,)-l~,p

,

(4.3)

where

/~(fl) = l n - - - - g ~ , P

I~'(fl)

Iz(fl) Ii(fl) '

(4.4)

(4.5)

where 11 and 12 are the modified Bessel functions. Hence we obtain from eq. (4.2) 2 d

fl[£t(fl)Nt-l=o"

(4.6)

The solution fl¢2 of this equation as a function of N t is shown in table 1 for d = 3.

5. Approximation for the N t link integral at low temperature F r o m our experience in calculating the preceding cases we have tried to find an approximation to the N t link integral valid at low temperatures ( N t large). This is possible because the solutions of the self-consistent equations in the preceding cases yield large values for the external field A. In this case A drives the gauge degrees of freedom U in the N t link integral to the weak coupling region. We can then set all U r equal to the non-compact variable V r, except the one which is coupled to the

R.M. MarinhoJr. / Meanfield analysis

180

TABLE 1 Solution of eq. (4.6) for d = 3 and various Nt

N~

~2

1

2 3

2 3 4 8 16

1.726 2.539 3.224 5.451 9.091

Wilson loop variables ~2x and which satisfies the periodic boundary condition (2.2). With this approximation, the N t link integral becomes

W U= ( N t - 1)/~(A) + ~ ( A + flV+ f l w Z v ) .

(5.1)

The free energy is

F =- - ½ f l ( d - 1)V 4 + A V +

HW -Ntd

I~(H) Ntd

( U t -- 2)

--flY Nt

(Ut -

1)

2- --I~(A) Nt

~( A -.}-~V-.~- ~ W 2 V )

(5.2)

iv,

It is now clear that when N t ~ oo we recover the gauge system in an axial gauge at d + 1 dimensions. The self-consistent equations of F are,

#'( A + flV + f l w Z v )

Nt-1

FA= V

--I.t'(A)-

N,

=0,

Ut

Fv= A - 2fl( d - 1)V 3

2 ( N t - 2)

N,

By

~ (1 + W2)~,'(A + ~v + 3W2V) =0, FH

W

I~'(H)

Ntd

Utd

O,

H Fw

Nd

WV 2fltL'(A + flY+ flW2V)

N,

= O. (5.3)

R.M, MarinhoJr. / Mean fieM analysis

181

TABLE 2 First-order points obtained from eq. (5.3)

2 3 4 8 16 oo

2.22 2.03 1.94 1.81 1.75 1.68

The points of the first order phase transition are shown in table 2 for various N t. Notice that this matches the mean-field results at T = 0, ref. [2], and that even for N t = 2 we have an excellent agreement with the exact result of the N t link integral (2.12). We expect that these results are accurate for any N t > 2. It can be easily checked that this first-order transition drives the behaviour of the spin system, in the sense that H jumps to a non-vanishing value when this transition arises by increasing ft. 6. Results and conclusions

We have studied the phase diagram of lattice SU(2) gauge field at a finite temperature. The results are shown in fig. 1. This diagram shows the lines of the first and second-order phase transitions. At a high temperature the spin system is in a confining phase and undergoes a deconfining transition on the line A 2C. This line and the dashed line are calculated exactly from the character expansion of the N t link integral. The gauge system is responsible for the first order line AxI. The line CI is also a line of deconfinement. In principle this transition has nothing to do with deconfinement, but as the systems are coupled, the jump of A to a value different from zero induces the spin system to go to the ordered phase. The points A 1 and B 1 are computed exactly, and for N t >/3 we have approximated the N t link integral by its saddle point. We have computed the point B~ in this approximation and the deviation from the exact result is less than 5%. We expect, and this is the case, that the agreement will increase with N t. For N t ~ ~ we recover the gauge system in four dimensions in the axial gauge. Our results are in excellent agreement with Monte Carlo data of ref. [3] available for Nt = 1,2. In spite of that we have obtained good agreement for the second-order phase transition line, that of the first order have a slope of wrong sign. We would expect the line IC to have negative tangent, for as we raise the temperature the tendency of the system is to go to a disordered state. We can justify our results by observing that not all the 1/d corrections are contained in our

182

R.M. Marinho Jr. / Mean fieM analysis

II~

l

A2

AI

B'

I/:,

1/3

I/4 I18 1/16 I

I

!

T

2

3

4

I

5

Fig. 1. Mean-field prediction for the phase diagram of SU(2) lattice gauge theory at finite temperature aT = N(-1. The solid line A2C is the boundary of the confining phase.

approximation: there are still 1/d corrections to be made in the gauge degrees of freedom lying in the spatial directions. The author thanks Professor V. Alessandrini for many enlightening discussions. The hospitality of the LPTHE is also gratefully acknowledged. References [1] V. Alessandrini and Ph. Boucaud, Nucl. Phys. B235 [FS11] (1984) 599 [2] H. Flyvbjerg, B. Lautrup and J.M. Zuber, Phys. Lett. 110B (1982) 279 [3] J. Kuti, J. Polonyi and K. Szlachanyi, Phys. Lett. 98B (1981) 199