On the type of the temperature phase transition in O(N) models
The temperature induced phase transition is investigated in the O(N) models by using graphics processing units (GPU) for Monte Carlo simulations on a lattice. General purpose computing on GPU (GPGPU) technology allows to collect a huge amount of data that gives a possibility to investigate the type...
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irk-123456789-1069792016-10-11T03:02:15Z On the type of the temperature phase transition in O(N) models Bordag, M. Demchik, V.I. Gulov, A.V. Skalozub, V.V. Section A. Quantum Field Theory The temperature induced phase transition is investigated in the O(N) models by using graphics processing units (GPU) for Monte Carlo simulations on a lattice. General purpose computing on GPU (GPGPU) technology allows to collect a huge amount of data that gives a possibility to investigate the type of the phase transition for a wide interval of coupling values. It is found that for the small values of λ a weak-first-order phase transition happens. It converts into a second order phase transition with the increase of λ. A comparison with analytic calculations in continuum field theory and lattice simulations obtained by other authors is given. Исследован температурный фазовый переход в O(N)-моделях с помощью использования графических видеокарт (GPU) для Монте-Карло симуляций на решетке. Технология расчетов общего назначения на видеокартах (GPGPU) сделала возможным собрать огромное количество данных, позволивших исследовать тип фазового перехода в широком интервале значений константы связи. Найдено, что для малых значений величин λ наблюдается фазовый переход первого рода. С ростом константы связи λ фазовый переход становится фазовым переходом второго рода. Представлено сравнение результатов с результатами, полученными другими авторами с помощью аналитических вычислений в континуальной теории и решеточных симуляций. Досліджено температурний фазовий перехід в O(N)-моделях за допомогою використання графічних відеокарт (GPU) у Монте-Карло симуляціях на решітці. Технологія розрахунків загального призначення на відеокартах (GPGPU) зробила можливим зібрати величезну кількість даних, які дозволили дослідити тип фазового переходу в широкому інтервалі значень константи зв'язку. Знайдено, що для малих значень величин λ спостерігається фазовий перехід першого роду. З ростом константи зв'язку λ фазовий перехід стає фазовим переходом другого роду. Представлено порівняння результатів з результатами, отриманими іншими авторами за допомогою аналітичних обчислень у континуальній теорії та симуляцій на решітках. 2012 Article On the type of the temperature phase transition in O(N) models / M. Bordag, V.I. Demchik, A.V. Gulov, V.V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 43-47. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 11.15.Ha, 05.30.Rt http://dspace.nbuv.gov.ua/handle/123456789/106979 en Вопросы атомной науки и техники Leipzig University, Institute for Theoretical Physics |
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Section A. Quantum Field Theory Section A. Quantum Field Theory |
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Section A. Quantum Field Theory Section A. Quantum Field Theory Bordag, M. Demchik, V.I. Gulov, A.V. Skalozub, V.V. On the type of the temperature phase transition in O(N) models Вопросы атомной науки и техники |
| description |
The temperature induced phase transition is investigated in the O(N) models by using graphics processing units (GPU) for Monte Carlo simulations on a lattice. General purpose computing on GPU (GPGPU) technology allows to collect a huge amount of data that gives a possibility to investigate the type of the phase transition for a wide interval of coupling values. It is found that for the small values of λ a weak-first-order phase transition happens. It converts into a second order phase transition with the increase of λ. A comparison with analytic calculations in continuum field theory and lattice simulations obtained by other authors is given. |
| format |
Article |
| author |
Bordag, M. Demchik, V.I. Gulov, A.V. Skalozub, V.V. |
| author_facet |
Bordag, M. Demchik, V.I. Gulov, A.V. Skalozub, V.V. |
| author_sort |
Bordag, M. |
| title |
On the type of the temperature phase transition in O(N) models |
| title_short |
On the type of the temperature phase transition in O(N) models |
| title_full |
On the type of the temperature phase transition in O(N) models |
| title_fullStr |
On the type of the temperature phase transition in O(N) models |
| title_full_unstemmed |
On the type of the temperature phase transition in O(N) models |
| title_sort |
on the type of the temperature phase transition in o(n) models |
| publisher |
Leipzig University, Institute for Theoretical Physics |
| publishDate |
2012 |
| topic_facet |
Section A. Quantum Field Theory |
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http://dspace.nbuv.gov.ua/handle/123456789/106979 |
| citation_txt |
On the type of the temperature phase transition in O(N) models / M. Bordag, V.I. Demchik, A.V. Gulov, V.V. Skalozub // Вопросы атомной науки и техники. — 2012. — № 1. — С. 43-47. — Бібліогр.: 13 назв. — англ. |
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Вопросы атомной науки и техники |
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AT bordagm onthetypeofthetemperaturephasetransitioninonmodels AT demchikvi onthetypeofthetemperaturephasetransitioninonmodels AT gulovav onthetypeofthetemperaturephasetransitioninonmodels AT skalozubvv onthetypeofthetemperaturephasetransitioninonmodels |
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2025-07-07T19:16:48Z |
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2025-07-07T19:16:48Z |
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| fulltext |
ON THE TYPE OF THE TEMPERATURE PHASE
TRANSITION IN O(N) MODELS
M. Bordag 1, V.I. Demchik 2, A.V. Gulov 2, and V.V. Skalozub 2∗
1Leipzig University, Institute for Theoretical Physics, 04109, Leipzig, Germany
2Dnipropetrovs’k National University, 49010, Dnipropetrovs’k, Ukraine
(Received October 31, 2011)
The temperature induced phase transition is investigated in the O(N) models by using graphics processing units
(GPU) for Monte Carlo simulations on a lattice. General purpose computing on GPU (GPGPU) technology allows
to collect a huge amount of data that gives a possibility to investigate the type of the phase transition for a wide
interval of coupling values. It is found that for the small values of λ a weak-first-order phase transition happens.
It converts into a second order phase transition with the increase of λ. A comparison with analytic calculations in
continuum field theory and lattice simulations obtained by other authors is given.
PACS: 11.15.Ha, 05.30.Rt
1. INTRODUCTION
Scalar field models with orthogonal symmetry O(N)
are applied in various fields of physics, like quantum
field theory, collective phenomena, quantum dots,
high-temperature superconductivity, etc. In three
spatial dimensions no analytic solutions exist, so dif-
ferent type approximations are used to estimate their
physical relevance [1,2]. In the literature various cal-
culation schemes — daisy and super-daisy resumma-
tions, the optimized perturbation theory [3], the two-
particle-irreducible (2PI) formalism [4], 1/N expan-
sion [5] and renormalization group flow [2] — have
been applied to investigate the thermodynamic be-
havior of models.
The temperature induced phase transition in the
O(N) models with a spontaneous symmetry breaking
(SSB) was studied either by analytic methods or in
lattice simulations (see Refs. [1, 2, 6] and references
therein). It was observed in the daisy, super-daisy
and some type beyond resummations [7] that the first
order phase transition could occur in the O(1)-model.
However, the lack of the expansion parameter hap-
pens near the phase transition temperature T ∼ Tc
for various kind resummations. So, it is impossible to
draw a reliable conclusion about the transition type
even for small values of the coupling constant λ. In
Ref. [8] some extended kind of resummations was
used for the O(N) model, and the phase transition
of the second order was determined independently of
the coupling value. The same result was also derived
by applying the renormalization group approach [9].
Analogous observations have been obtained in Monte
Carlo (MC) simulations on a lattice. On the con-
trary, in recent paper [10] within the 2PI formalism
in the double-bubble approximation the first order
phase transition was determined. As a result, nowa-
days the general believe is that the phase transition
is of the second order and the perturbation theory
fails in this problem. However, in the O(N)-models,
the results of perturbation theory calculations coin-
cide with the lattice MC ones in the limit of N → ∞,
only [7].
Recently, a new powerful computational platform
— General Purpose computing on Graphics Process-
ing Units (GPGPU) technology — has been put in
force [11, 12] that gives a possibility to generate ex-
tremely large amount of MC data. Therefore the ac-
curacy of calculations can be essentially increased and
it becomes possible to shed light upon hidden pecu-
liarities and details of different processes of interest
studied by MC simulations. One of such unsolved
problems is the kind of the temperature phase tran-
sition in the O(N)-models for small values of coupling
constant λ and the reliability of the perturbation the-
ory results.
In the present paper we investigate the tempera-
ture induced phase transition in the O(N)-models in
a wide interval of the coupling constant λ using the
GPGPU technology. We test our approach on O(1)-
model as a simplest member of O(N)-models class. In
order to determine the type of the phase transition,
we apply a procedure known in the lattice quantum
chromodynamics and compare the MC simulations
obtained with the hot and cold starts. For small λ,
λ ≤ λ1 � 10−3, an order parameter shows a hystere-
sis behavior near the phase transition temperature.
Such kind behavior means the phase transition of the
first order. With further increasing of λ the hysteresis
behavior becomes less pronounced and disappears at
all, reflecting the phase transition of the second order.
Thus, the type of the phase transition is dependent
∗Corresponding author E-mail address: vadimdi@yahoo.com
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 43-47.
43
on the value of coupling λ.
The paper is organized as follows. In Sec. 2 we
describe the model and its realization on a lattice. In
Sec. 3 a necessary information on the MC simulations
is given. Sec. 4 summarizes the results obtained.
2. THE MODEL
In order to construct a self-consistent lattice version
of the O(N) φ4-model we start with quantum field
theory in the continuous space. The thermodynami-
cal properties of the model are described by the gen-
erating functional
Z =
∫
Dϕ e−S[ϕ], (1)
where ϕ is a real N -component scalar field, and the
action is
S[ϕ] =
∫
dx
(1
2
∂μϕi∂μϕi − 1
2
m2ϕ2 +
λ
4
(ϕ2)2
)
, (2)
where
ϕ2(x) =
∑
i
ϕi(x)ϕi(x). (3)
The standard realization of generating functional in
MC simulations on a lattice assumes a space-time
discretization and the probing random values of fields
in order to construct the Boltzmann ensemble of field
configurations. Then any macroscopic observable can
be measured by averaging the corresponding micro-
scopic quantity over this ensemble.
A direct lattice implementation of (1) encounters
an evident problem: the fields ϕi(x) are distributed
uniformly in the infinite interval (−∞;∞). However,
a random number generator suitable in this case does
not exist. Usually, one cuts the interval off, since the
tails ϕ → ± appear to be exponentially suppressed in
actual simulations in any Metropolis algorithm. The
cut scale is chosen in a way separating the unessential
tails from the interval of physically important values
of ϕ. But such kind scale cannot be predetermined
being the result of an interplay between all the pa-
rameters entering the action. As a result, one has
to adjust the cut scale manually for every set of the
parameter values.
Since we are going to investigate the phase tran-
sition in a wide interval of couplings and tempera-
tures, we prefer to rewrite the initial φ4 model in
the continuum space-time in the form allowing a fur-
ther self-consistent lattice realization without manu-
ally adjusted cuts.
First we separate the absolute value from the di-
rection of the vector ϕi:
ϕi(x) = R(x)ni(x), (4)
where R(x) ∈ [0;∞), and the vector ni(x) runs over
the surface of the unit sphere:
n2(x) =
∑
i
ni(x)ni(x) = 1. (5)
In the spherical coordinates:
n1 = sin θ1... sin θN−1,
ni = sin θ1... sin θN−i, cos θN−i+1,
(i = 2, ..., N − 1)
nN = cos θ1 (6)
with angles θ1,...,N−2(x) ∈ [0; π] and θN−1(x) ∈
[0; 2π]. These measure in the integral in (1) can be
written in terms of new variables as
Dϕ =
∏
x
N∏
i=1
dϕi(x) (7)
=
∏
x
RN−1dR(x)
N−1∏
i=1
sin θN−1−i
i dθi(x).
The second step is to introduce one-to-one trans-
formation R(U) to a new field variable U(x) defined
in the finite interval [0; 1). In what follows we as-
sume that U = 0 corresponds to R = 0 (ϕ = 0). The
measure in the integral in (1) becomes
Dϕ =
∏
x
R(U)N−1R′(U)dU(x)
×
N−1∏
i=1
sin θN−1−i
i dθi(x), (8)
where prime denotes the derivative, R′(U) = dR/dU .
Now the generating functional can be expressed
in terms of field variables defined in finite intervals:
Z =
∫ ∏
x
dU(x)
N−1∏
i=1
dθi(x) exp(−S̃[U, θ]), (9)
with the action
S̃[U, θ] = S[ϕ(U, θ)] (10)
−
∑
x
log
[
R(U)N−1R′(U)
N−1∏
i=1
sin θN−1−i
i
]
.
The first term in the action is just the initial action
with ϕ substituted by new field variables, whereas the
second term arises from the measure transformation.
For MC simulations we introduce a hypercubic lattice
with hypertorous geometry. We use an anisotropic
lattice with a spatial and a temporal lattice spacing
as and at = as/ζ with ζ > 1, respectively. The scalar
field is defined in the lattice sites.
In the case of pure condensate field in some fixed
direction in ϕ-space the action is determined by the
potential
Ṽ [U ] =
a4
s
ζ
(
−m2
2
R(U)2 +
λ
4
R(U)4
)
− log
[
R(U)N−1R′(U)
]
. (11)
In order to get finite value of the potential at zero
field (U = 0), we assume
R ∼ U1/N , U → 0. (12)
44
Then, the potential has one local maximum at U = 0
and one global minimum at U0. The spread between
the values of the potential at the local maximum and
the global minimum is
ΔV = Ṽ [0] − Ṽ [U0]. (13)
The quantities U0 and ΔV play a crucial role in MC
simulations. Being equivalent in theory, different
choices of these parameters can produce drastically
different results in actual simulations. The reason is
the finite number of simulations in an actual com-
puter experiment. If rare but important events could
be missed, then the MC algorithm will not converge
to the Boltzmann ensemble of configurations. In case
of ∼ 103 iterations all the important probabilities
have to be greater than 103.
Considering the phase transition, one must guar-
antee that the MC algorithm meets the field values
compatible with both the phases to choose. If U0 →
0, then the broken phase can be missed since the cor-
responding field values are extremely rare events. On
the other hand, U0 → 1 washes the unbroken phase
out. It is also important to ensure finite probability
of transition between those field values. The accep-
tance of non-zero condensate values of the field is
ruled approximately by exp(ΔV ) at each lattice site.
If ΔV � 1, then the unbroken phase never occurs
in actual simulations. If ΔV → 0, then there is no
broken phase. To study the phase transition in the
model, we choose the following conditions:
U0 = 0.25, ΔV = 1. (14)
Thus, the valuable part of generated field values re-
alizes the global minimum of the ‘effective’ potential,
and no phase will be accidentally missed. The prob-
ability to prefer condensate or non-condensate values
will be of order ∼ 10−1 ensuring the fast convergence
of MC algorithm. Of course, the choice (14) is not
optimal for temperatures far away from the critical
temperature.
To satisfy two conditions (14) we use a convenient
two-parameter function
R[U ] = mξK[U ], (15)
K[U ] =
[
N
(
− log(1 − U) +
η
2
log2(1 − U)
)]1/N
with ξ > 0 and η > 0. R[U ] ensures the limit (12).
Finally the lattice action S̃[U, θ] is
S̃[U, θ] =
∑
x
∑
μ
[
Y
√
z
ζλ
(K′[U(x)])2 ×
(
U(x + aμμ̂) − U(x)
aμ/as
)2]
+
∑
x
∑
μ
2Y
(aμ/as)2
√
z
ζλ
×
(
1 −
N∑
i=1
ni(x + aμμ̂)ni(x)
)
K[U(x)]
(
K[U(x)] −K′[U(x)]
(
U(x + aμμ̂) − U(x)
))
+
∑
x
[
−1
4
G[U(x)] − Y K2[U(x)] + Y 2zK4[U(x)] + V0
]
, (16)
where
Y =
F(U0) − G(U0)
2K2[U0]
, (17)
V0 = −ΔV − log(mξ)N − log π,
F [U ] = N − 1 +
K′′[U ]K[U ]
(K′[U ])2
,
G[U ] = 4[log
(K′[U ]KN−1[U ]
)− ΔV ];
here the values of ξ and η have to be found as the
solution of equations Ṽ ′
0 [U0] = 0 and (13).
3. MONTE-CARLO SIMULATIONS
In this section we consider the results of MC simu-
lations of O(1) model. To determine the type of the
phase transition we use the field condensate as an or-
der parameter. It is non-zero in the broken phase and
vanishing in the high-temperature phase. In case of
the first-order transition, the overheated and super-
cooled states are possible. So, the MC simulations
with the hot and cold starts have to lead to different
phases near the critical temperature. Combining the
MC simulations for the hot and cold starts we will
see an exfoliation of data like a hysteresis plot. Such
type procedure was successfully applied to determine
the type of the phase transition in the lattice QCD.
The exfoliation of the simulated data in the vicin-
ity of the critical temperature is a tiny effect. A large
amount of simulation data must be prepared to ob-
serve it. In this regard, achieving the highest per-
formance of computational hardware is a problem of
great importance. To speed up essentially the simu-
lation process we apply a GPU cluster of AMD/ATI
Radeon GPUs: HD6970, HD5870, HD5850, HD4870
and HD4850 [13]. The peak performance of the clus-
ter is up to 11 Tflops. The low-level AMD Intermedi-
ate Language (AMD IL) is used in order to obtain the
maximal performance of the hardware. Some techni-
cal details of MC simulations on the ATI GPUs and
the review of the AMD Stream SDK are given in
Ref. [11] and references therein.
The MC simulations are realized at hypercubic
lattices up to 644. Most of the obtained statistics
come from the lattice 164. The lattice data are
stored with the single precision. Updating the MC
45
configurations are also performed with the single pre-
cision, whereas all the averaging measurements are
carried out with the double precision to avoid the
accumulation of errors. The system is thermalized
by passing 5000 MC iterations for every run. For
measuring we use 1024 MC configurations separated
by 10 bulk updates.
Temperature dependence of the absolute value
of the averaged field |ϕ̄| in O(1) model for the
lattice 164 at z = 0.35 for ζ = [1.5; 2.4] and differ-
ent λ = 5·10−5 (top), 10−4 (center), 10−2 (bottom)
We collect the data for the absolute value of the
averaged field |ϕ̄| representing the field condensate.
The temperature dependence of |ϕ̄| for the lattice
164 at z = 0.35 for ζ = [1.5; 2.5] is shown in the
figure. The whole data set for every plot is divided
into 15 bins. Different initial conditions are marked
with different colors: the hot start is depicted in red
(lower bins) and the cold start is represented in blue
(upper bins). The mean values and the 95% confi-
dence intervals are shown for each bin. Every bin
contains 150 simulated points. As it is seen from the
figure, for λ = 0.01 the temperature dependence of
the field condensate is insensitive to the start config-
uration chosen. Both the cold and hot starts lead to
the same behavior of the field condensate for various
ζ. This means the phase transition to be of the sec-
ond order, and this result is in agreement with the
common opinion on the type of the phase transition
stated in Refs. [1,2,8,9]. However, therein this value
of the coupling is considered as a small one.
Then, for smaller values of λ the overheated con-
figurations occur in the broken phase for the hot
start, and the supercooled states can be found for
the cold start. That is, the exfoliation of the simu-
lated data in the vicinity of the critical temperature
is observed for different start configurations. Such
a hysteresis behavior corresponds to the phase tran-
sition of the first order. With further decreasing of
λ to the values of order λ0 ∼ 10−5 the behavior of
hot- and cold-started simulations becomes completely
separated and independent of the temperature. Such
type property means that the SSB does not happen
even at zero temperature.
4. CONCLUSIONS
As it was discovered in the MC simulations, the tem-
perature phase transition in the O(1) φ4 model is
strongly dependent on a coupling value λ. There
is the low bound λ0 ∼ 10−5 determining the range
where the SSB is not realized. Close to this value in
the interval 10−5 ≤ λ ≤ 10−3 the phase transition
is of the first order. For larger values of λ a sec-
ond order phase transition happens. These types of
the behavior have been determined on the lattices of
different sizes. Our calculation procedure was devel-
oped to accelerate the MC procedure in the domain
of parameters close to the phase transition for a wide
range of coupling. For usually considered values of
λ ∼ 0.01... 0.1 it gives the results coinciding with
the existing literature and signaling the second order
phase transition.
Our observations, in particular, may serve as a
guide for applicability of different kind resummations
in perturbation theory. In fact, we see that the daisy
and super daisy resummations give qualitatively cor-
rect results for small values of λ. For larger values
they become non-adequate to the second-order na-
ture of the phase transition. In this case other more
complicated resummation schemes should be used.
The change of the phase transition type following
due to the change of the coupling value is not a new
phenomenon. For example, in the standard model of
elementary particles it is well known that the elec-
troweak phase transition is of the first order for small
λ and it converts into the cross-over or the second
order one for sufficiently large values of λ. Our inves-
tigation has shown that this takes also place even in
the simple model with one coupling.
The research of order of the temperature phase
transition depending on λ values for O(N)-models
with N > 1 is in progress.
46
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