We introduce the mutual information to the Zn lattice gauge theory for testing the degree of correlations. It is shown that our simulation result for the plaquettes gives a clear peak on the critical point

HASHIMOTO, Takaaki

KOBAYASHI, Yoshihito

KUDO, Kiyoshi

TAMURA, Yoshiteru

Community Repository of Fukui

Application of the Mutual Information 
to the Zn Gauge Model 
Yoshiteru TAMURA·, Kiyoshi KUDO·, Takaaki HASHIMOTO·, 
and Yoshihito KOBAYASHI* 
(Recieved Feb. 29, 1996) 
We introduce the mutual information to the Zn lattice gauge 
theory for testing the degree of correlations. It is shown that our 
simulation result for the plaquettes gives a clear peak on the critical 
point. 
1. Introduction 
When we know the Hamiltonian of a system, we can calculate a physical 
quantity of the system using a Monte CaIro method. We will apply the 
method to a system of elementary particles. 
The method proposed by Wilson [1] is as follows. With a path integral 
theory an expectation value of operator F is written as 
J VAF[A]exp{iS[A]} 
< F >= J ' VAexp{iS[A]} (1.1 ) 
where S is an action. Regarding the time variable as an imaginary one, and 
to look for a Lagrangian on a time-space lattice, Eq. (1.1) translates 
L F[A] exp{ -S[A]} 
< F >= ....::A~ ________ _ 
Lexp{-S[A]} (1.2) 
A 
It means that the local field follows a canonical distribution. It is known that 
there is a phase transition in the system of glUOll or photon. A new signal of 
the transition has been investigated by many physicists. 
On the other hand, Matuda verified that a mutual information [3] is a 
good signal [2] to find a critical point of the Ising system. So, we investigate 
the validity of the mutual information to the lattice gauge theory. 
This report is organized as follows. We introduce Wilson's formulation of 
the gauge fields on a space-time lattice in section 2. The mutual information 
-Department of Applied Physics 
15 
16 
is explained in section 3. And our result on the mutual information for lattice 
gauge theory by the Monte CaIro simulation is shown in section 4. Section 
5 is devoted to discussion. 
2. Zn Lattice Gauge Model 
We work on the space-time lattice with lattice spacing a. For each pair 
of nearest-neighbor sites i and j we have a gauge or link variable Vij in the 
group Zn, 
(2.1 ) 
where 90 stands for the coupling constant. The dynamics of this gauge system 
follows from the action, 
(2.2) 
The sum in Eq. (2.2) is over all elementary squares or so called plaquettes 
which contribute 
(2.3) 
where the site i, j, k, and l circulate around the square. When a lattice 
spacing a tends to 0, the action in Eq. (2.2) becomes 
for a U( 1) gauge system, where FIJ.II is field strength, 
under the condition, 
3. Mutual Information 
Information is defined as 
S=-LPilnPi , 
i 
(2.4) 
(2.5) 
(2.6) 
(3.1 ) 
where Pi stands for a probability of state i. When the system is not deter-
ministic, that is to say all the probability fill with 
1 
PI =, .. ',Pn =-, 
n 
(3.2) 
the information becomes maximum. When the system is deterministic, i.e., 
the probability of the system fill with 
(3.3) 
the information becomes O. 
To consider the amount of the information, the mutual information [3] is 
defined as 
1M -" P··lnp·· - '"' P·lnp· - '"'P·lnp· 
- ~ lJ lJ ~ l l ~ J J' (3.4 ) 
ij i 
where Pij stands for a probability of state i on a system 1, and of state j on 
a system 2. 
The mutual information behave as follows. If there is no correlation 
between the system 1 and 2, 1M becomes O. If two system are correlated, 1M 
becomes finite value. 
4. Result 
For simplicity, we choose the Z2 gauge model. But it is a non-trivial 
model for the reasons it has a respectable phase transition like U(l) gauge 
theory. We show the average plaquette P versus f3 in fig. 1 for the Z2 group 
on a four dimensional lattice with lattice size L = 4. This figure tells us that 
a phase transition is occurred at f3 ~ 0.47 in the system by the discontinuity 
in P. 
We calculate the temporal mutual information lit for the plaquette, which 
is gauge invariant. We choose two system which is far separated from each 
other in Me time T on each. lit versus f3 is drawn in fig. 2 for T = 1. It 
shows that lit has a maximum value at a critical point f3c. We checked that 
lit has a maximum at f3c for any values of n. There is the reason why lit has 
a maximum value on the critical point next. If f3 is smaller than f3c, the local 
state has almost random. So, even if the information of each system is large, 
the mutual information is small. If f3 is lager than f3c, the local state has 
almost the same value, and freezes. In such a case the mutual information is 
small. On the critical point the local state shows complex behavior. 
5. Conclusion 
Using a Zn lattice gauge model we checked that lit is an effective method 
for finding a critical point in lattice gauge theory. lit applied to the plaquette, 
which is gauge invariant, is a good signal to find the critical point. But lit 
applied to a link variable, which is not a gauge invariant quantity, there is 
no guarantee lit has a maximum at f3c. 
17 
18 
p 
0.9 fo •• 
..•.. 0.4 
"1 .. 
0.5 
·· .. l., 0.3 
.Y··} 1 0.2 
0.1 
.' I 
I·' '. . . 
o .~ .:: .. :::~ ................. >3t.u .... . 
0.1 
0.1 0.3 0.5 0.7 0.9 f3 0.1 0.4 0.7 f3 
fig. 1. The average of plaquette versus {3. fig. 2. lit versus (3. 
References 
[1] K. Wilson. : Phys. Rev. DIO(1974) 2445. 
[2] Hiroyuki. Matuda : Mutual Information of the Ising System. 
Thesis of Fukui University (1994) 29. 
[3] Erica Jen : 1989 Lectures in Complex Systems. 
Addison-Wesley Publishing Company (1990). 
[4] M. Creutz. : Quarks, gluons and lattices. 
Cambride University Press (1983). 


Application of the Mutual Information to the Zn Gauge Model

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