Generalized Potts-Models and their Relevance for Gauge Theories
We study the Polyakov loop dynamics originating from finite-temperature Yang-Mills theory. The effective actions contain center-symmetric terms involving powers of the Polyakov loop, each with its own coupling. For a subclass with two couplings we perform a detailed analysis of the statistical mecha...
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| Cite this: | Generalized Potts-Models and their Relevance for Gauge Theories / A. Wipf, T. Heinzl, T. Kaestner, C. Wozar // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 25 назв. — англ. |
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| citation_txt | Generalized Potts-Models and their Relevance for Gauge Theories / A. Wipf, T. Heinzl, T. Kaestner, C. Wozar // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 25 назв. — англ. |
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| description | We study the Polyakov loop dynamics originating from finite-temperature Yang-Mills theory. The effective actions contain center-symmetric terms involving powers of the Polyakov loop, each with its own coupling. For a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and antiferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents ν and γ at the continuous transition between symmetric and antiferromagnetic phases are the same as for the 3-state spin Potts model.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 006, 14 pages
Generalized Potts-Models and their Relevance
for Gauge Theories?
Andreas WIPF †, Thomas HEINZL ‡, Tobias KAESTNER † and Christian WOZAR †
† Theoretisch-Physikalisches Institut, Friedrich-Schiller-University Jena, Germany
E-mail: wipf@tpi.uni-jena.de
URL: http://www.personal.uni-jena.de/∼p5anwi/
‡ School of Mathematics and Statistics, University of Plymouth, United Kingdom
E-mail: thomas.heinzl@plymouth.ac.uk
Received October 05, 2006, in final form December 12, 2006; Published online January 05, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/006/
Abstract. We study the Polyakov loop dynamics originating from finite-temperature Yang–
Mills theory. The effective actions contain center-symmetric terms involving powers of the
Polyakov loop, each with its own coupling. For a subclass with two couplings we perform
a detailed analysis of the statistical mechanics involved. To this end we employ a modified
mean field approximation and Monte Carlo simulations based on a novel cluster algorithm.
We find excellent agreement of both approaches. The phase diagram exhibits both first and
second order transitions between symmetric, ferromagnetic and antiferromagnetic phases
with phase boundaries merging at three tricritical points. The critical exponents ν and γ at
the continuous transition between symmetric and antiferromagnetic phases are the same as
for the 3-state spin Potts model.
Key words: gauge theories; Potts models; Polyakov loop dynamics; mean field approxima-
tion; Monte Carlo simulations
2000 Mathematics Subject Classification: 81T10; 81T25; 81T80
1 Introduction
Symmetry constraints and strong coupling expansion for the effective action describing the
Polyakov loop dynamics of gauge theories lead to effective field theories with rich phase struc-
tures. The fields are the fundamental characters of the gauge group with the fundamental
domain as target space. The center symmetry of pure gauge theory remains a symmetry of the
effective models. If one further freezes the Polyakov loop to the center Z of the gauge group
one obtains the well known vector Potts spin-models, sometimes called clock models. Hence
we call the effective theories for the Polyakov loop dynamics generalized Z-Potts models. We
review our recent results on generalized Z3-Potts models [1]. These results were obtained with
the help of an improved mean field approximation and Monte Carlo simulations. The mean field
approximation turns out to be much better than expected. Probably this is due to the existence
of tricritical points in the effective theories. There exist four distinct phases and transitions of
first and second order. The critical exponents ν and γ at the second order transition from the
symmetric to antiferromagnetic phase for the generalized Potts model are the same as for the
corresponding Potts spin model.
Earlier on it had been conjectured that the effective Polyakov loop dynamics for finite
temperature SU(N) gauge theories near the phase transition point is very well modelled by
?This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and
Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). The full collection is available at
http://www.emis.de/journals/SIGMA/LOR2006.html
mailto:wipf@tpi.uni-jena.de
http://www.personal.uni-jena.de/~p5anwi/
mailto:thomas.heinzl@plymouth.ac.uk
http://www.emis.de/journals/SIGMA/2007/006/
http://www.emis.de/journals/SIGMA/LOR2006.html
2 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
3-dimensional ZN spin systems [2, 3]. For SU(2) this conjecture is supported by universality
arguments and numerical simulations. The status of the conjecture for SU(3) gauge theories is
unclear, since the phase transition is first order such that universality arguments apparently are
not applicable.
2 Recall of planar Potts models
The q-state Potts model [4, 5, 6] is a natural extension of the Ising model. On every lattice site x
there is a planar vector with unit length which may point in q different directions (Fig. 1) with
angles
Figure 1. Angles in a q-state Potts model.
θx ∈
{
2π
q
,
4π
q
, . . . , 2π
}
= Zq.
Only nearest neighbors interact and their contribu-
tion to the energy is proportional to the scalar prod-
uct of the vectors, such that
H = −J
∑
〈xy〉
cos (θx − θy) . (2.1)
The Hamiltonian H is invariant under simultaneous
rotations of all vectors by a multiple of 2π/q. These
Zq symmetries map a configuration w = {θx|x ∈ Λ}
into w′ = {θx + 2πn/q}, n ∈ {1, . . . , q}. In two and
higher dimensions the spin model shows a phase transition at a critical coupling Kc = βJc > 0
from the symmetric to the ferromagnetic phase. In two dimensions this transition is second
order for q ≤ 4 and first order for q > 4. In three dimensions it is second order for q ≤ 2 and
first order for q > 2.
Figure 2. Coloring of neighboring lattice
sites.
There is another phase transition at negative cri-
tical coupling K ′
c from the symmetric to the anti-
ferromagnetic phase. For q ≥ 3 and negative K the
number of ground states of (2.1) increases rapidly
with the number of lattice sites. It equals the num-
ber of ways one can color the vertices with q colors
such that two neighboring sites have different colors
(Fig. 2). In the antiferromagnetic case the dege-
nerate ground states contribute considerably to the
entropy [7]
SB(P ) = −
∑
P (w) log P (w),
where one sums over all spin configurations and P is
the probability of w. We use the well known varia-
tional characterization of the free energy,
βF = inf
P
(β〈H〉P − SB) , 〈H〉P =
∑
w
P (w)H(w),
where the minimum is to be taken on the space of all probability measures. The unique mini-
mizing probability measure is the Gibbs state
PGibbs ∼ e−βH (2.2)
Generalized Potts-Models and their Relevance for Gauge Theories 3
belonging to the canonical ensemble. In the variational definition of the convex effective action
one minimizes on the convex subspace of probability measures with fixed mean field,
Γ[m] = inf
P
(
β〈H〉P − S(P )
∣∣〈eiθx〉P = m(x)
)
. (2.3)
The field m(x) which minimizes the effective action is by construction the expectation value of
the field eiθx in the thermodynamic equilibrium state (2.2).
In the mean field approximation to the effective action one further assumes that the measure
is a product measure [8, 9],
P (w) = P ({θx}) =
∏
x
px(θx), (2.4)
where the single site probabilities are maps px : Zq → [0, 1]. The approximate effective action is
denoted by Γmf [m].
The symmetric and ferromagnetic phases are both translationally invariant. In the mean field
approximation all single site probabilities are the same, px = p, and the mean field is constant,
m(x) = m. The effective potential is the effective action for constant mean field, divided by the
number of sites. Its mean field approximation is
umf(m) = inf
p
(
−Kmm∗ +
∑
θ
p(θ) log p(θ)
∣∣∣∑
θ
p(θ) eiθ = m
)
, K = dJ. (2.5)
It agrees with the mean field approximation to the constraint effective potential, introduced by
O’Raifeartaigh et al. [10, 11]. In the antiferromagnetic phase there is no translational invariance
on the whole lattice Λ but on each of two sublattices in the decomposition Λ = Λ1 ∪ Λ2. The
sublattices are such that two nearest neighbors always belong to different sublattices. Thus the
single site distributions px in (2.4) are not equal on the whole lattice, but only on the sublattices,
px = p1 on Λ1 and px = p2 on Λ2. (2.6)
The minimization of the effective action on such product states is subject to the constraints∑
θ∈Zq
p1(θ)eiθ = m1 and
∑
θ∈Zq
p2(θ)eiθ = m2
and yields the following mean field effective potential
umf (m1,m2) =
1
2
(
K|m1 −m2|2 +
∑
i
umf(mi)
)
,
where umf is the effective potential (2.5). For K > 0 the minimum is attained for m1 = m2
and translational invariance is restored. In the symmetric and ferromagnetic phases the single
site probabilities p1 = p2 = p. In the symmetric phase p = 1/q for every orientation and in the
ferromagnetic phase p is peaked at one orientation. Hence there exist q different ferromagnetic
equilibrium states related by Zq symmetry transformations. In the antiferromagnetic phase the
probabilities p1 and p2 are different.
For the 3-state Potts model one can calculate the single site probabilities explicitly. On one
sublattice it is peaked at one orientation and on the other sublattice it is equally distributed
over the remaining 2 orientations. Hence there are 6 different antiferromagnetic equilibrium
states related by Z3 symmetries and an exchange of the sublattices. The results for single site
distributions of the 3-state Potts model are depicted in Fig. 3.
4 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
Figure 3. Single site distributions of the 3-state Potts model.
3 Polyakov-loop dynamics
We consider pure Euclidean gauge theories with group valued link variables Ux;µ on a lattice with
Nt sites in the temporal direction. The fields are periodic in this direction, Ut+Nt,x ;µ = Ut,x ;µ.
We are interested in the distribution and expectation values of the traced Polyakov loop variable
Lx = trPx , Px =
Nt∏
t=1
Ut,x ;0, (3.1)
since 〈Lx 〉 is an order parameter for finite temperature gluodynamics (see [12] for a review).
In the low-temperature confined phase 〈Lx 〉 = 0 and in the high-temperature deconfined phase
〈Lx 〉 6= 0. The effective action for the Polyakov loop dynamics is
e−Seff [P] =
∫
DUδ
(
Px ,
Nt∏
t=1
Ut,x ;0
)
e−Sw[U ], DU =
∏
links
dµHaar(Ux;µ), (3.2)
with gauge field action Sw. In this formula the group valued field Px is prescribed and the delta-
distribution enforces the constraints (3.1). In the simulations we used the Wilson action for the
gauge fields. Gauge invariance of the action Sw[U ] and measure DU implies Seff [P] = Seff [L].
In addition there is the global center symmetry, under which all Px are multiplied by the same
Figure 4. Fundamental domain of L.
center element of the group. For SU(N) the center
consists of the N roots of unity, multiplied by the
identity matrix. Hence for SU(N) theories we have
Seff [L] = Seff [z · L], zN = 1.
In Fig. 4 on the left we plotted the domain of the
traced Polyakov loop variable for SU(3). The values
of L at the three center elements are 3, 3z and 3z2
with z = e2πi/3. They form the edges of the trian-
gle. What is needed is a good ansatz for the effective
action Seff in (3.2). To this aim we calculated the
leading terms in the strong coupling expansion for
Seff in gluodynamics [1, 13]. As expected one finds
a character expansion with nearest neighbor inter-
actions,
Seff = λ10S10 + λ20S20 + λ11S11 + λ21S21 + ρ1V1 + O(β3Nt),
where Spq depends on the character χpq belonging to the representation (p, q) of SU(3). Ex-
pressing the characters as function of the fundamental characters χ10 = L and χ01 = L∗ the
Generalized Potts-Models and their Relevance for Gauge Theories 5
different center-symmetric contributions have the form
S10 =
∑
〈xy〉
(
LxL∗
y + h.c.
)
,
S20 =
∑
〈xy〉
(
L2
xL∗2
y − L2
xLy − L∗
xL∗2
y + L∗
xLy + h.c.
)
,
S11 =
∑
〈xy〉
(
|Lx |2|Ly |2 − |Lx |2 − |Ly |2 + 1
)
,
S21 =
∑
〈xy〉
(
L2
xLy + L2
yLx − 2L∗
xLy + h.c.
)
, V1 =
∑
x
(
|Lx |2 − 1
)
.
The target space for L is the fundamental domain inside the triangle depicted above. The
functional measure is not the product of Lebesgue measures but the product of reduced Haar
measures on the lattice sites.
In the following we consider the two-coupling model
Seff = (λ10 − 2λ21)
∑(
LxL∗
y + h.c.
)
+ λ21
∑(
L2
xLy + L2
yLx + h.c.
)
(3.3)
which contains the leading order contribution in the strong coupling expansion. For vanishing λ21
it reduces to the Polonyi–Szlachanyi model [17].
4 Gluodynamics and Potts-model
There is a direct relation between the effective Polyakov loop dynamics and the 3-state Potts
model: If we freeze the Polyakov loops to the center of the group,
Px −→ zx1 ∈ center
(
SU(3)
)
⇐⇒ θx ∈
{
0,
2π
3
,
4π
3
}
then the effective action (3.3) reduces to the Potts-Hamiltonian (2.1) with J = 18(λ10 + 4λ21).
The same reduction happens for all center-symmetric effective actions with nearest neighbor
interactions. Only the relation between the couplings λpq and J is modified.
For the gauge group SU(2) the finite temperature phase transition is second order and the
critical exponents agree with those of the 2-state Potts spin model which is just the ubiquitous
Ising model. The following numbers are due to Engels et al. [14]
β/ν γ/ν ν
4d SU(2) 0.525 1.944 0.630
3d Ising 0.518 1.970 0.629
and support the celebrated Svetitsky–Yaffe conjecture [2]1. For the gauge group SU(3) the finite
temperature phase transition is first order and we cannot compare critical exponents. But the
effective theory (3.3) shows a second order transition from the symmetric to a antiferromagnetic
phase and we can compare critical exponents at this transition with those of the same transition
in the 3-state Potts spin model.
In a first step we study the ‘classical phases’ of the Polyakov loop model with action (3.3).
The classical analysis, where one minimizes the ‘classical action’ Seff , shows a ferromagnetic,
antiferromagnetic and anticenter phase. The classical phase diagram is depicted in Fig. 5. For
1For the relations between 3-dimensional gauge theories at the deconfining point and 2-dimensional Potts-
models, the so-called gauge-CFT correspondence, we refer to the recent paper [15].
6 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
Figure 5. Classical phase diagram.
small couplings the quantum fluctuations will disorder the system and entropy will dominate
energy. Thus we expect a symmetric phase near the origin in the (λ10, λ21)-plane and such
a phase was inserted by hand in the diagram. There exists one unexpected phase which cannot
exist for Potts spin models. It is an ordered phase for which the order parameter is near to the
points opposite to the center elements, hence we call it anticenter phase. They are marked in the
fundamental domain in Fig. 4. We shall see that the classical analysis yields the qualitatively
correct phase diagram.
4.1 Modified mean field approximation
In a second step we calculate the effective potential for the Polyakov loop model with action (3.3)
in the mean field approximation. Here we are not concerned with the relevance of this model
for finite temperature gluodynamics and just consider (3.3) as the classical action of a field
theoretical extension of the Potts spin models. We use the variational characterization (2.3)
for the effective action where we must minimize with respect to probability measures on the
space of field configuration {Px |x ∈ Λ} with fixed expectation values 〈χpq〉 of all characters
χpq(Px ) showing up in the Polyakov loop action. As outlined in Section 2, in the mean field
approximation we assume the measures to have product form,
DP −→
∏
x
dµred(Px ) px (Px ) .
For further details the reader is referred to our earlier paper [16]. Here µred is the reduced Haar
measure of SU(3). Since we expect an antiferromagnetic phase we only assume translational
invariance on the sublattices in the decomposition (2.6). This way one arrives at a non-trivial
variational problem on two sites.
We illustrate the procedure with the simple Polyakov loop model studied in [17],
Seff = λS10 = λ
∑(
LxL∗
y + h.c
)
. (4.1)
To enforce the two constraints 〈Lx 〉 = Li for x ∈ Λi one introduces two Lagrangian multipliers.
For the minimal model (4.1) one arrives at the following mean field effective potential
2umf(L1, L
∗
1, L2, L
∗
2) = −dλ|L1 − L2|2 +
∑
vmf(Li, L
∗
i )
with vmf(L,L∗) = dλ|L|2 + γ0(L,L∗).
Generalized Potts-Models and their Relevance for Gauge Theories 7
Figure 6. The order parameter ` for (a) mean field approximation and (b) Monte Carlo simulation.
Here γ0 is the Legendre transform of
w0(j, j∗) = log z0(j, j∗), z0(j, j∗) =
∫
dµred exp (jL + j∗L∗) .
The last integral has an expansion in terms of modified Bessel functions [18, 19],
z0(j, j∗) =
∑
n∈Z
einNα det
In In+1 In+2
In−1 In In+1
In−2 In−1 In
(2|j|) , j = |j|eiα.
As order parameters discriminating the symmetric, ferromagnetic and antiferromagnetic phases
we take [20]
L =
1
2
(L1 + L2), M =
1
2
(L1 − L2), ` = |L|, m = |M |. (4.2)
Fig. 6 shows the value of the order parameter ` as function of the coupling λ10 near the phase
transition point from the symmetric to the ferromagnetic phase. The value of the critical cou-
pling and the jump ∆` of the order parameter in the mean field approximation and simulations
8 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
Figure 7. The order parameter m for (a) mean field approximation and (b) Monte Carlo simulation.
agree astonishingly well. In Fig. 6b the order parameter ` for the ferromagnetic phase is plotted
against its probability distribution given by the shaded area. In the vicinity of the first order
transition we used a multicanonical algorithm on a 163 lattice to calculate the critical coupling
to very high precision. For further details on algorithmic aspects I refer to our recent paper [1].
Since the model has the same symmetries and dynamical degrees as the 3-state Potts model
in 3 dimensions we expect the model to have a second order transition from the symmetric
to the antiferromagnetic phase. To study this transition we calculated the order parameter m
in (4.2) in the modified mean field approximation and with Monte Carlo simulations. The order
parameter m is sensitive to the transition in question.
Again the expectation value and probability distribution of m near the transition is plotted
in Fig. 7. To get a clear signal we have chosen a large lattice with 283 sites and evaluated
5 × 105 sweeps. The mean field approximation and Monte Carlo simulations together with
the calculation of Binder cumulants in [1] indicate that the transition to the antiferromagnetic
phase is second order. We plan to use a finite size scaling method to confirm this result in our
upcoming work.
With the cumulant method we have calculated the critical exponents γ and ν and compared
our results with the same exponents for the 3-state Potts model at the second order transition
from the symmetric to the antiferromagnetic phase [20]. Within error bars the critical exponents
are the same.
Generalized Potts-Models and their Relevance for Gauge Theories 9
exponent 3-state Potts minimal Seff
ν 0.664(4) 0.68(2)
γ/ν 1.973(9) 1.96(2)
This is how the conjecture relating finite temperature gluodynamics with spin models is at
work: The Polyakov loop dynamics is effectively described by generalized Potts models with the
fundamental domain as target space, for SU(3) it is the triangularly shaped region in Fig. 4. The
first order transition of gluodynamics is modelled by the transition from the symmetric to the
ferromagnetic phase in the generalized Potts models. These generalized Potts models are in the
same universality class as the ordinary Potts spin models; they have the same critical exponents
at the ‘unphysical’ second order transition from the symmetric to the antiferromagnetic phase.
It is astonishing how good the mean field approximation is. The reason is probably, that the
upper critical dimension of the (generalized) 3-state Potts model is 3 and not 4 as one might
expect. This is explained by the fact, that the models are embedded in systems with tricritical
points, see below, and for such systems the upper critical dimension is reduced [21, 22].
5 Simulating the effective theories
We have undertaken an extensive and expensive scan to calculate histograms in the coupling
constant plane (λ10, λ21) of the model (3.3). Away from the transition lines we used a standard
Metropolis algorithm giving results within 5 percent accuracy. Near first order transitions
we simulated with a multicanonical algorithm on lattices with up to 203 lattice sites. Most
demanding have been the simulations near second order transitions. For that we developed a new
cluster algorithm [1] which improved the auto-correlation times by two orders of magnitude on
larger lattices. We found a rich phase structure with 4 different phases with second and first
order transitions and tricritical points. As for the minimal model the Monte Carlo simulations
are in good and sometimes very good agreement with the mean field analysis.
Fig. 8 shows the phase structure in the generalized MF approximations and the corresponding
results of our extended MC simulations. The results of the simulations are summarized in the
following phase portrait (Fig. 9), in which we have indicated the order of the various transitions.
The calculations were done on our Linux cluster with the powerful jenLaTT package. In 3000
CPU hours we calculated 8000 histograms in coupling constant space.
Figure 8. Phase structure for MF approximation and MC simulation.
10 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
Figure 9. Phase portrait of the model (3.3). Histograms for the transitions are marked with arrows
and figure numbers.
Figure 10. Histograms of L which show a clear signal for the first order transition from the symmetric
to the ferromagnetic phase.
For each marked transition in Fig. 9 a set of 6 histograms is displayed. These histograms
(and many others, see [1]) have been used to localize the phase transition lines and to investigate
the nature of the transitions. The results are summarized in the phase portrait on page 10. The
critical exponents ν and γ given above have been determined at the second order transition
indicated with arrow 13 in the portrait.
Generalized Potts-Models and their Relevance for Gauge Theories 11
Figure 11. Histograms of L for the continuous transition from the symmetric to the ferromagnetic
phase.
Figure 12. Histograms of L for the continuous transition from the symmetric to the anticenter phase.
12 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
Figure 13. Histograms of M for the second order transition from the symmetric to the antiferromagnetic
phase.
6 Conclusion
The strong coupling expansion results in a character expansion for the Polyakov-loop dynamics.
The leading terms are center symmetric nearest neighbor interactions containing the characters
of the smallest representations of the gauge group. We have performed an extensive modified
mean field analysis which includes antiferromagnetic states without translational invariance on
the whole lattice. A new and efficient cluster algorithm has been developed and applied to
study the second order transitions from the symmetric to the antiferromagnetic phase. The
autocorrelation times were improved by 2 orders of magnitude. We discovered an unexpectedly
rich phase structure of the simple 2-coupling Polyakov loop model (3.3). This model is in the
same universality class as the 3-state Potts spin model. The mean field results are surprisingly
accurate that seems to indicate that the upper critical dimension of the generalized Potts models
is 3. This is attributed to the existence of tricritical points [21, 22].
To relate our results to gluodynamics we have to calculate the effective couplings λpq gover-
ning Polyakov-loop dynamics as functions of the Wilson coupling β in gluodynamics. We have
done this successfully for SU(2) gauge theory with inverse Monte Carlo techniques [16, 23] and
plan to publish our results for SU(3) very soon [24]. For the inverse Monte Carlo simulations to
work one needs simple geometric Schwinger Dyson equations for the Polyakov loop dynamics.
Such equations have been derived very recently in [19]. It would be interesting to see whether
the antiferromagnetic phase of the Polyakov loop models plays any role at all for gluodynamics.
In the present paper it was needed to show that certain critical exponents of the Polyakov loop
models are the same as of the q = 3 Potts spin model. Finally one would like to include heavy
fermions in the effective Polyakov-loop dynamics [25]. To that end one needs to add center
Generalized Potts-Models and their Relevance for Gauge Theories 13
symmetry breaking terms to the effective actions studied in the present paper. This will lead
to a proliferation of additional terms in the effective action which renders a systematic study
more difficult as compared to pure gluodynamics.
Acknowledgements
Andreas Wipf would like to thank the local organizing committee of the O’Raifeartaigh Sympo-
sium on Non-Perturbative and Symmetry Methods in Field Theory for organizing such a stimu-
lating and pleasant meeting on the hills of Budapest to commemorate Lochlainn O’Raifeartaigh
and his important contributions to physics. This contribution is very much in the tradition of
his research on the role of symmetries and effective potentials in field theory.
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14 A. Wipf, T. Heinzl, T. Kaestner and C. Wozar
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1 Introduction
2 Recall of planar Potts models
3 Polyakov-loop dynamics
4 Gluodynamics and Potts-model
4.1 Modified mean field approximation
5 Simulating the effective theories
6 Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-147809 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-28T16:03:24Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wipf, A. Heinzl, T. Kaestner, T. Wozar, C. 2019-02-16T08:36:02Z 2019-02-16T08:36:02Z 2007 Generalized Potts-Models and their Relevance for Gauge Theories / A. Wipf, T. Heinzl, T. Kaestner, C. Wozar // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 25 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81T10; 81T25; 81T80 https://nasplib.isofts.kiev.ua/handle/123456789/147809 We study the Polyakov loop dynamics originating from finite-temperature Yang-Mills theory. The effective actions contain center-symmetric terms involving powers of the Polyakov loop, each with its own coupling. For a subclass with two couplings we perform a detailed analysis of the statistical mechanics involved. To this end we employ a modified mean field approximation and Monte Carlo simulations based on a novel cluster algorithm. We find excellent agreement of both approaches. The phase diagram exhibits both first and second order transitions between symmetric, ferromagnetic and antiferromagnetic phases with phase boundaries merging at three tricritical points. The critical exponents ν and γ at the continuous transition between symmetric and antiferromagnetic phases are the same as for the 3-state spin Potts model. This paper is a contribution to the Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory (June 22–24, 2006, Budapest, Hungary). Andreas Wipf would like to thank the local organizing committee of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory for organizing such a stimulating and pleasant meeting on the hills of Budapest to commemorate Lochlainn O’Raifeartaigh and his important contributions to physics. This contribution is very much in the tradition of his research on the role of symmetries and ef fective potentials in field theory. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Potts-Models and their Relevance for Gauge Theories Article published earlier |
| spellingShingle | Generalized Potts-Models and their Relevance for Gauge Theories Wipf, A. Heinzl, T. Kaestner, T. Wozar, C. |
| title | Generalized Potts-Models and their Relevance for Gauge Theories |
| title_full | Generalized Potts-Models and their Relevance for Gauge Theories |
| title_fullStr | Generalized Potts-Models and their Relevance for Gauge Theories |
| title_full_unstemmed | Generalized Potts-Models and their Relevance for Gauge Theories |
| title_short | Generalized Potts-Models and their Relevance for Gauge Theories |
| title_sort | generalized potts-models and their relevance for gauge theories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147809 |
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