A lattice Monte Carlo study of inverse symmetry breaking in a two-scalar model in three dimensions

Abstract

We carry a Monte Carlo study of the coupled two-scalar λf12f22 model in three dimensions. We find no trace of inverse symmetry breaking in the region of negative λ's for which the one-loop effective potential predicts this phenomenon. Moreover, for λ's negative enough, but still in the stability region for the potential, one of the transitions turns out to be of first order, both for zero and finite temperature.

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arXiv:hep-lat/9707029v2 7 Jan 1998
A lattice Monte Carlo study of
Inverse Symmetry Breaking
in a two-scalar model in three dimensions.
G. Bimontea,b, D. I˜ nigueza, A. Taranc´ onaand C.L.Ulloda
February 1, 2008
a) Departamento de F´ ısica Te´ orica, Facultad de Ciencias,
Universidad de Zaragoza, 50009 Zaragoza, Spain
e-mail: david, tarancon, clu@sol.unizar.es
b) Dipartimento di Scienze Fisiche, Universit´ a di Napoli,
Mostra d’Oltremare, Pad.19, I-80125, Napoli, Italy
e-mail: bimonte@napoli.infn.it
Abstract
We carry a Monte Carlo study of the coupled two-scalar λφ2
model in three dimensions. We find no trace of Inverse Symmetry
Breaking in the region of negative λ’s for which the one-loop effec-
tive potential predicts this phenomenon. Moreover, for λ’s negative
enough, but still in the stability region for the potential, one of the
transitions turns out to be of first order, both for zero and finite tem-
perature.
1φ2
2
DFTUZ preprint 97/17 Napoli preprint 38/97hep-lat/9707029
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1Introduction
It is well known today that the strong and electroweak interactions at the low
temperatures and energies characteristic of the present-day Universe are well
described by a spontaneously broken relativistic gauge theory. According
to this picture, a region of empty space resembles a ferromagnet below the
Curie temperature, since it is characterized by a number of non-vanishing
order parameters (one or more, depending on the elementary particle model)
which break some of its internal symmetries, an analogue of spontaneous
magnetization for ferromagnets. Another piece of wisdom, from statistical
mechanics this time, is that an ordered system, when its temperature is
raised, gradually looses its order undergoing one or more phase transitions
until it reaches a temperature above which no order at all is present anymore.
Long ago Kirzhnits and Linde [1] argued that something similar might have
occurred in the early Universe, when high temperatures were present, so that
the internal symmetries that appear broken today might in fact have been
manifest at that time. Since then this intuitive picture has been confirmed
by detailed computations [2, 3] and computer simulations. The idea that
the ground state of the Universe was symmetric in the early times and that
later, when the Universe cooled down during its expansion, there took place a
series of phase transitions that eventually led to the present day asymmetric
vacuum, is of the greatest importance for the structure of the Universe as it
influences such important issues as baryogenesis, the formation and dynamics
of topological defects [4] like monopoles and cosmic strings, just to mention
a few.
Is this scenario the only possible one? Maybe not. In fact, already in the
classic paper by S. Weinberg [2] it was pointed out that the degree of order
of the vacuum may increase when the temperature is raised, contrary to any
intuition, in models with a sufficiently reach scalar sector. This phenomenon
was called Symmetry Non-Restoration (SNR) or Inverse Symmetry Breaking
(ISB), depending on whether the vacuum is ordered or disordered at zero
temperature. It is due to the possibility that some of the scalar fields acquire
a negative Debye mass, via negative quartic interactions among themselves
and with the other scalars, something which is allowed in multiscalar models
without causing any instability in the potential. To make this seemingly
paradoxical statement more plausible, the author observed that in Nature
there exists at least one substance, the Rochelle salt, which does exhibit
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this type of “inverse” behavior in a certain range of temperatures and thus
the expectation that more heat necessarily means more disorder is clearly
not always true.In any case, these remarks of Weinberg passed totally
unnoticed, until some authors resumed this idea, applying it to important
cosmological questions like the domain wall and monopole problems [5, 6,
7, 8], the breaking of the CP symmetry [9], baryogenesis [10], inflation [11]
and the breaking of the P, Strong CP and the Peccei-Quinn symmetries [12].
Recently, the possibility of ISB and SNR also in supersymmetric models has
been explored [13].
Despite the potentially important consequences for Cosmology, in the
literature on ISB and SNR there is a certain amount of scepticism about the
very existence of these phenomena. The example of the Rochelle salt is not
really persuading, because it is obvious that if a piece of it is heated enough
it eventually melts, while in a system with ISB or SNR there is order at
arbitrarily large temperatures. Moreover the result of Weinberg was based
on a one-loop perturbative computation and so there is the possibility that
it is an artifact of this approximation. Since then, many authors have tried
to improve the one-loop result, by resorting to a number of techniques all
including some amount of non-perturbative physics. It seems fair to say that
the current situation is perplexing: while some studies have confirmed that
ISB and SNR exist [15, 16, 17, 18], even though the region of parameter
space for which they occur appears to be reduced in size compared to the
lowest-order result, others have come to the conclusion that these phenomena
disappear once the non-perturbative information is put in [19, 20]. In any
case, no one of these results appears as conclusive, because all of them rely on
approximations, with various degree of applicability. The only exact result
known so far regards the lattice, where it can be shown that no order is
possible at sufficiently high temperatures for arbitrary models, gauge or not
[21]. Unfortunately these theorems have been proven only for lattices with
finite spacing in the space-directions (but with continuum euclidean-time
axis) and their impact for the continuum is thus unclear.
In this paper we present the results of the first Monte Carlo study of ISB.
We have considered a two-scalar model in 2+1 euclidean dimensions, for
which a one-loop computation predicts the possibility of ISB and SNR. We
have studied its phase diagram for various choices of the coupling constants
as a function of the temperature. Concretely, we have simulated the model
on various asymmetric lattices Nt×N2
s, where Ntand Nsare respectively the
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number of sites in the ”euclidean time” and space directions, the temperature
being related to Ntby T = 1/(aNt), with a the lattice spacing. In order to
avoid finite-size effects we have taken the thermodynamic limit Ns→ ∞ (for
any fixed Nt) using Finite Size Scaling (FSS) techniques. The high level of
precision required to clearly separate the transitions corresponding to the
various values of Ntrequired very large statistics and correspondingly long
simulation times. We have found that for all the values of Ntthat we have
considered, ISB seems ruled out, irrespective of the values of the couplings.
In fact, it appears that the size of the disordered region of the phase diagram
increases when the temperature is raised (i.e. when Nt is reduced), as it
happens in normal cases (for example in the φ4
decrease as required for ISB to take place. As for SNR the results of our
simulations show that, if one starts at T = 0 from an ordered phase, the
system disorders above a certain critical temperature Tc(coherently with the
results of [21]), again as it happens in normal cases. In principle, this does
not exclude the possibility of SNR, because it might happen that Tcdiverges
in the continuum limit. In order to see if this possibility really occurs, one
would have to carry a detailed study of the continuum limit, something that
we have not done.
Another interesting result of our simulations is that one of the transitions
turned out to be of first order, both at T = 0 and at finite temperature,
when the coupling among the two fields is negative and large in absolute
value, being of second order otherwise. This result is in qualitative agreement
with refs.[23, 24], where our model is considered for the particular case when
an extra φ1 ↔ φ2 symmetry is present (this case of course excludes the
possibility of ISB or SNR). Using the method of the Average Effective Action
[26], the authors found in the phase diagram surfaces of first order phase
transitions, but there this occurred for all negative values of the coupling
among the two scalars, while we seem to find this behavior only for couplings
strongly negative.
The organization of the paper is the following. In Sec. 2 we present
the continuum model, and discuss the main features of its lattice version,
while in Sec. 3 we review the predictions of perturbation theory on its high-
temperature behavior. The simulation, the techniques used and the results
obtained are presented in Sec. 4. Finally Sec. 5 is devoted to the conclusions.
3model [22]) , and does not
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2The model and its lattice formulation
We consider the theory for two real scalar fields in 3 euclidean dimensions,
described by the bare (euclidean) action:
S =
?
d3x
?1
2
?
i=1,2
?
(∂µΦi)2+1
2m(i)2
BΦ2
i+gi
4!Φ4
i
?
+1
4gΦ2
1Φ2
2
?
.(1)
What will be essential for ISB and SNR, in the above action the quartic
coupling g can be negative, as the condition of boundedness from below of
the potential is satisfied if:
g < 0 ,
g1g2 > 9 g2,g1> 0,g2> 0.(2)
As it is well known scalar models with quartic couplings, like (1), in three
dimensions are superrenormalizable. In the perturbative expansion of the
zero-temperature Green’s functions there is only a finite number of prim-
itively divergent diagrams. Depending on the regularization scheme, UV
divergencies are found only in two classes of diagrams: the tadpole dia-
grams, at one loop, and the sunset diagrams (for zero external momenta),
at two loops, both contributing to the self-energies. As a consequence, in
(1) the only quantities that require an infinite renormalization are the bare
masses m(i)2
the renormalized ones. It is for this reason that in (1) we have written the
latter without the suffix B.
When regularized on an infinite three-dimensional cubic lattice of points
Ω with lattice spacing a, the above action is replaced by its discretized version
B, while the fields and coupling constants can be identified with
SL =
?
x∈Ω
a3??
i=1,2
+g
??
4Φ2
µ
1
2(∆(a)
µΦi,L)2(x) +1
2m(i)2
BΦ2
i,L(x) +gi
4!Φ4
i,L(x)
?
+
1,L(x)Φ2
2,L(x)
?
, (3)
where ∆(a)
µ
is the lattice derivative operator in the direction µ:
∆(a)
µΦi,L(x) ≡Φi,L(x + ˆ µa) − Φi,L(x)
a
.(4)
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