THE NATURE OF THE TRANSITION IN d = 4 U(I) LATTICE GAUGE THEORY
Rajan GUPTA 1; Mark A. NOVOTNY and Robert CORDERY 2
Physics Department, Northeastern University, Boston, MA 0211S, USA
Received 31 October 1985; revised manuscript received 24 January 1986
The phase transition for the d = 4 compact U(1) lattice gauge theory has been studied using the Monte Carlo renormaliza- tion-group method. A single relevant eigenvalue is observed on the Wilson axis. The M C R G estimate for the exponent v changes with the coupling fl; an extrapolation towards f l~ provides an estimate v ~ 0.42 rather than - 0.33 as obtained by finite-size scaling. It is confirmed that loops of monopole current are the mechanism driving the transition. It is shown that at the transition the largest loop of monopole current undergoes a discontinuity in size and begins to span the lattice. On the basis of these findings, it is conjectured that on the Wilson axis the discontinuity is a finite-size effect.
1. Introduction. Wilson loops in lattice gauge theo. ties exhibit an area law in the strong coupling limit [1]. Thus in this limit both abelian and non-abelian theo- ries are confining. Since QED does not confine, the corresponding continuum limit of compact U(1) gauge theory should be in a different phase with a massless photon. The existence of such a phase was rigorously proven by Guth and by Fr61ich and Spencer [2] and has been amply corroborated by Monte Carlo data [3,4]. The resulting phase diagram along the Wilson axis consists of a confining phase for/~ = 1/e 2 </3 c and a massless Coulomb phase for/~ > ¢1 c. It is the nature of the transition and the mechanism responsi- ble for it that are not fully understood and these fea- tures will be addressed in this paper. The numerical tool used is Monte Carlo combined with a x/3 scale factor renormalization-group (MCRG) method. We find that the convergence in the MCRG is very slow and the critical exponent v changes as a function of/3. A simple extrapolation of the results for v to infinite
a o
lattice critical coupling,/~c , gives v ~ 0.42 rather than 0.33 as obtained from finite size scaling [5]. We con- firm that the mechanism driving the transition is the nucleation and growth of closed loops of monopole
i Present address: MS-B276, Los Alamos National Laboratory, Los AMmos, N M 87545, USA.
2 Present address: Pitney Bowes, Walter H. Wheeler Jr. Drive, Stamford, CT 06810, USA.
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current [1,4,6-8]. We associate the two-state struc- ture (hysteresis) observed on the Wilson axis [7,9] with a growth in the largest monopole current loop. However, we find that at the transition the largest loop in the disordered state spans the lattice and con- jecture that the observed hysteresis is a finite-size ef- fect. Further details of this calculation are contained in ref. [7 ].
2. Background and known results. The Wilson action for the U(1) gauge theory is [1]
s = ! t , ) e 2
= a COS O.v, (1)
where Un,n+ u is the link variable from site n to n +/z represented by a complex number (mod 1) and Our are the plaquette angles. The phase diagram in a two- coupling space defined by the action
S = t~ ~ c o s O.~ + 3' ~[~ cos 2 0 . ~ , (2)
is given in fig. 1 [ 10,11 ]. For 7 > 0.15 there is a dis- tinct first-order line separating the confining phase from the spin-wave phase. There is a discontinuity AE in the average plaquette (hysteresis) which de- creases along the line CD and there may be a point X where it goes to zero. Evertz et al. [11] find that X
i ' ' ' I ' ' ' a l l . . . . I . . . . I . . . . I . . . .
~ c B - ~ Spin Wave
Confined X
Z2
-1.5 . . . . I . . . . I . . . . I . . . . I . . . . . . 0.25 0.5 0.75 1 1.25 1.5
Fig. 1. The phase boundary for d = 4 U(1) is shown. See the text for a complete description.
lies below the Wilson axis and is a tricritical point (TCP). By a scaling analysis o f the latent heat, AE, as a function o f 3'
AE = constant('), -- 3'TCl')t3u, (3)
they locate X at/3 = 1.09 + 0.04 and 3' = - 0 . 1 1 + 0.05. In the fit, eq. (3), they used 5 points in the range 3' = (0.2, 0.5) (see fig. 1). The phase boundary in this
range is sufficiently flat to make a three-parameter fit unreliable for such a large extrapolat ion. The data points with 3' < 0.2 were not used in their analysis since A E becomes very sensitive to the lattice size and the infinite lattice values could not be estimated. For example, in fig. 2 we show a sample plot o f the flips between the two states on a 94 lattice and the large MC lifetime. Including data from ref. [11], the results for A E at ? = 0 are
0.057 -+ 0.003 L = (3x/~) 4 , (4a)
0.055 + 0.003 L = 64 , (4b)
0.042 -+ 0.002 L = 94 , (4c)
0.030 + 0.004 L = 164 . (4d)
This discontinui ty is associated with a jump in the density o f monopoles and it is not established if it sur- fives in the infinite-volume limit. Consequently, the location of the TCP is not well known. The exponents flu and ~o which determine the shape o f the phase boundary were found [11] to be/3 u = 1.7 + 0.2 and ~0 = 1.5 -+ 0.3. The exponents/3 u and ~o together with ~'t should satisfy the hyperscaling relation [12]
~u = ~(vt d _ 1), (5)
in the absence o f logarithmic corrections. However,
f.fl
0
I:Z- °
OJr.fl
C_
>
<3r. o
0
1 0 . 0 0
I I I I I f
' 1 2 . 0 0 2 4 . 0 0 3 B . O 0 4 B . O 0 - BO.O0 2 . 0 0
Number " o f MC C 0 n f - 1 0 ~
Fig. 2. The behavior of the average plaquette as a function of MC configurations at # = 1.009 and 3~ = 0 is shown. Each configura- tion is separated by 10 update sweeps and 2000 MC configurations were discarded for equilibrium to be reached. The two-state structure on this 94 lattice is obvious.
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this hyperscaling relation provides a value of v close to the classical value ½ rather than ~½. It is possible that logarithmic corrections invalidate eq. (5) since d = 4 is the upper critical dimension for this model. Lastly, the phase boundary near the claimed TCP has very little curvature. This justifies the large extrapolation to find the TCP but it also suggests classical behavior.
Previous finite-size scaling analysis on the Wilson axis by Nauenberg and Lautrup, Bhanot, and Caldi [5] found evidence for a second-order transition with v
] for the simple Wilson action. These results were derived from small lattices and the authors had missed the hysteresis due to inadequate statistics. Similarly, Jersghk, Neuhaus and Zerwas [13] did not find a dis- continuity in the renormalized charge or the string tension, and claim that the transition is first order on the basis of the hysteresis. Their scaling analysis of the string tension yielded a value v ~ 0.31. The explana- tion they proposed for this apparent contradiction (first order versus v ~ 0.31) is the presence of a tri- critical point close to the transition point on the Wilson axis.
We feel that the existing Monte Carlo studies are not conclusive and the questions that eventually need to be answered are:
(1) Is the phase boundary DZ still first order for the action given in eq. (2)? A very small latent heat would be consistent with a fluctuation-induced first-order transition. In this case, the critical fluctuations asso- ciated with the appearence of a massless photon would be prematurely aborted. The photon mass would change discontinuously at the transition and the first- order transition would be brought about by the bind- hag of monopole current loops as is the case along the line CD.
(2) Does the line of first-order transitions have an endpoint X? Is the point X a tricritical or a multicrit- ical point? What are the associated exponents? What is the nature of the continuum theory at this point - gaussian or non-trivial? What is the phase diagram in a many-coupling constant space?
(3) What is the nature of the transition along the line XZ? If the point X is a multicritical point, then can one calculate the crossover exponent and show a change in exponents along XZ?
(4) What are the .renormalization-group flow lines in the spin-wave phase?
(5) What is the infinite-volume limit of the mono-
pole density at the transition and the thermodynamics of the closed loops of monopole current.
Since the location of the TCP, critical exponent v and the behavior of monopoles are very sensitive to finite.volume effects, we resort to a MCRG calculation. In previous studies of TCP Landau and Swendsen [14] showed that whereas finite.size scaling of Monte Carlo data could miss extremely subtle finite-size dependence of the hysteresis, a MCRG calculation was able to ob- tain estimates for the TCP and the TCP exponents.
3. The critical coupling ~c. Earlier studies [9] have shown that the critical coupling for the U(1) gauge theory depends strongly on the lattice size. Using the results obtained by Jersfik et al. [9] and by us [7] for the finite-volume ~c L
0.9980 + 0.0010 L = (3Wf3) 4 , (6a)
1.0020 + 0.0005 L = 64 , (6b)
1.0085 + 0.0005 L = 94 , (6c)
1.0106 +- 0.0018 L = 164 , (6d) o o
in the scaling form 1//3 L = c/L 1/v + 1/[3 e we estimate . o o
/3 c . The result of this finite-size scaling is/3 e = 1.0111 + 0.0005. However, it must be pointed out that the extrapolation of the data in eq. (6) is insensitive to the value of v between 0.32 and 0.42.
4. MCRG using the x/ff block transformation. The details of the X/~ block transformation and the MCRG method are given in ref. [15]. The starting lattice used was 94 and the 3 blocking steps were 94 ~ (3x/3) 4 -+ 34 ~ (Vt3) 4. To improve the statistics all possible block lattices at level 34 (9 such) and (vc3) 4 (81 such) were constructed and the block site on the (3x/~) 4 lattice was selected randomly. The operators evaluated were the simple plaquette in the charge-l, -2, and -3 repre- sentations and the three 6-link operators 6p (planar), 6£ (L shaped) and 6t (twisted). The number of con- figurations at each coupling are shown in table 1. Each configuration is separated by 10 sweeps through the lattice. The MC update for the Wilson action was done using a 6.hit (8 for/3 = 1.0083 and 1.009) Metropolis algorithm. The acceptance was adjusted to be ~-30%.
In the MCRG analysis, the complete data sample at each value of ~ was used. This is correct irrespective of the order of the transition. In the case of a first-
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Table 1 Eigenvalues at 3 levels of blocking for 4 different couplings. The 5 operators included in the linearized transformation matrix are the simple plaquette in the charge-1 and -2 representations and the three 6-1ink operators. The statistical errors are much smaller than the systematic errors associated with the different levels of blocking and the dependence on #.
order transition the leading eigenvalue should be (scale factor) d = (Vr3) 4 = 9 as dictated by a discontinuity fixed point [16]. To find the discontinuity fixed point, the data sample has to be large enough to include a few flips between the two states. This condition is barely satisfied by our data sample as shown in fig. 2.
The exponent v is found from the leading eigenvalue of the linearized transformation matrix. The results
are summarized in table 1. We do not present any sta- tistical errors since the systematic errors due to the lack of convergence in the eigenvalues (dependence on /3 and the blocking level n) are much larger and are the quantities to keep in focus.
For n = 2 and n = 3, we note a decrease in the lead- ing eigenvalue with increasing/3. A simple linear ex- trapolat ion using the four values at n --- 2 and 3 con- verges to a value X ~ 3.65 at/3 = 1.011 + 0.001 as shown in fig. 3. This immediately suggests that the lack of convergence is due to not doing the simulation at t ic . Also, the value ~ = 3.65 is very close to the clas- sical value 3 (i.e. ~ = 0.5) and significantly far from finite-size scaling values ~ = 5.2 (v = 0.33) and the discontinuity value ~ = 9.
Some other features of the eigenvalue analysis are: (1) The second largest eigenvalue is irrelevant and
is principally associated with the 6-link planar (6p) operator.
(2) The charge-2 coupling 3' had very little effect on the leading eigenvalue.
(3) In addit ion to the decrease in the magnitude of the eigenvalues with increasing/3, we find that the de- pendence on the number of operators used in the diagonalization also decreased. For example, at/3 =
1.0075 the leading eigenvalue changed from 4.42 (1 operator - simple plaquette) to 5.66 (5 operators of table 1). The corresponding numbers at/3 = 1.009 were 4.72 ~ 4.81. This is an additional pointer to the
o o
need to perform a simulation at ~3 c . One surprise in our results is the indication of a
o o
~ 8
ro > c o~
i j j co
4~ t - ro > o
0~ r r
oo
o o
i
~o.o7 ~'o.oa l o . oe l o .~2 B e t a
Fig. 3. The critical exponents from the MCRG are shown for the values of fl studied. The first, second and third iterations are indicated respectively by the symbols o, zx and +. The lines are linear extrapolations toward/~e"
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value v ~ 0.42 even though finite-size scaling results favor ~, ~ 0.33. One possible explanation consistent with the data is that the point X lies above the Wilson axis and v ~ 0.33 is the tricritical exponent. Then, after going through the crossover region, p changes to the value ~0.42 (or even 0.5 which would imply clas- sical behavior) associated with the second order line.
5. Topological excitations. The topological excita- tions in d = 4 are world lines of d = 3 monopoles. For brevity we shall call them monopole strings. At any given point on the lattice one can define the monopole current M u as the flux flowing through a d = 3 surface orthogonalto the current [4] i.e. 27rM~ = eU~PoavOpo where avOp~ is the finite difference of the plaquette angle. For each face of a 3-cube Opo is constrained to lie between - r r and n. Since topological charge is con- served, i.e. ~ M u = 0, the current forms a dosed loop. If all plaquette angles are constrained to lie between -Tr and lr, then the number of monopoles inside any 3-cube is the sum of the 6 plaquette angles modulo 2rr. This definition is identical to the one used by DeGrand and Toussaint [4], and the maximum num- ber of monopoles (anti-monopoles) allowed in a 3- cube is 2. This truncation does not affect the results since at the couplings explored, the probability of producing monopoles with charge >13 is negligible. In fact even cubes with 2 charges were present <0.2% of the time.
In addition to the total number of monopoles and antimonopoles, another quantity of interest is the size of the largest string. We define a connected monopole string to be made up of all monopoles or antimono- poles that lie in at least one of the 20 3-cubes adjoin- ing any given 3-cube in the string. The minimum size that a string can have is 4 charges. With this definition we do not distinguish between a single closed loop and many that are connected either by sharing 3-cubes with 2 monopoles in them and/or those that lie side by side. We also analysed strings with just monopoles (anti- monopoles) separately. The one large string broke up into several smaller ones. However, since both mono- poles and antimonopoles are present in a single string and disorder the system in the same way, the relevant variable is the charge-blind string. Therefore we have only used the combined number in the analysis.
The data in fig. 4 consists of 131 configurations separated by 500 Monte Carlo sweeps at/~ = 1.0075
8 a
8
C)
c o
8 8
0.5B
• @
:'~. z'-.
o l p
i i i i a
0 . B 0 0 . B 2 O . B 4 O . B B
Average P2acluette
i
O.BB
g b
8 8
o
r -
~g
I--
g
o
× x ' q f l o ~
eo
x • e~
o e @
• ~e~ e
e x K S
i i i O. 10 0 .20 0.30 0 .40 0 .50
Average Plaquette
e
i
O.BO
Fig. 4. The total number of topological charges (monopoles + antimonopoles) are shown as a function of the average plaquette for # = 1.0075 (x) and # = 1.009 (o). The lattice size in (a) is 94, and in (b) is the 34 lattice obtained from the 94 lattice after two RG iterations.
and 1.009. To justify combining the two values of/~ we invoke the microcanonical formulation [17] in which the coupling is the derived quantity and energy is the independent variable. Fig. 4a shows a linear cor-
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relation between the value of the average plaquette and the total monopole charge on the 94 lattice. This relationship is preserved by two RG transformations, as shown in fig, 4b. The other main features of the data are:
(a) Configurations with a small average plaquette (~0.6) have a large number of monopoles; ~2500 for the 94 lattice. Almost all of these monopoles are part of a single large string. The next largest string had ~<5% of the charges. The total number of strings was ~20. The largest string survives the RG transforma- tions and on the 34 lattice all monopoles are in a single large string and the density changes from ~,0.05 (94) to ~0.18 (34).
(b) Configurations with a large average plaquette (~>0.65) have a small number of monopoles (~,1000). These monopoles do not form a single large string. The strings on the renormalized 34 lattice are small and often more than one is present.
(c) The total number of strings decreased with the average plaquette. Most of the monopoles become part of one large string when the average plaquette ~<0.63. If we interpret this as critical fluctuations then the analogue of the unbinding of vortex-antivortex pairs in the d = 2 X - Y model is the growth in the size of the monopole strings. However, this may be due to an intriguing possibility - the system may be near the percolation transition. An examination of the largest string for the average plaquette <0.6 showed that it spanned the lattice, Another way to state this is that the monopole string acts as a topo- logical defect (domain) whose contribution increases with repeated blocking. This is suggested by the in- crease in the monopole density on the renormalized 34 lattice (an expected increase in the domain's sur- face to volume ratio with blocking). As a consequence, the discontinuity AE increases with the number of RG transformations. This effect on AE due to the monopole strings is reminiscent of the effect that domains have in spin systems. For example in the d = 2 Baxter-Wu model, which is rigorously known to have a second-order transition [18], a MC analysis demonstrated a two-state internal energy associated with the nucleation and the growth of domains [19]. The difference in energy was the contribution of the domain wall. However, in that model both finite-size scaling [19] and a MCRG analysis [20] found the transition to be second order.
6. Conclusions. Our results on the nature of the phase transition on the Wilson axis can be summarized as follows:
(1) From a finite-size scaling we estimate the in- finite-lattice critical/3 to be/3 c = 1.011 + 0.0005.
(2) The MCRG analysis yields only one relevant eigenvalue. The value depends on the number of RG iterations and on/~. The convergence should improve for a simulation done closer to/3 c and on a larger lattice. The values of ~ investigated here are either within the crossover regions or else they are too far from the tricritical point to observe the second rele- vant exponent.
(3) The mechanism responsible for this transition is closed loops of monopoles (strings) [4,6]. We show that the monopole density and the average plaquette have a linear relationship which is preserved by the RG transformation. We also show that the number of monopoles in the largest string and the number of strings depend strongly on the average plaquette. The size of the largest monopole loop undergoes a transi- tion and gives rise to the small latent heat. To check if this is a finite.size effect requires simulations of larger lattices.
To end, we propose that the transition on the Wilson axis is akeady second order. The discontinuity AE is due to the formation of a topological domain that spans the lattice due to the periodic boundary conditions and will vanish in the finite-volume limit.
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