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Domain Walls and Vortices with Non­Symmetric Cores
Minos Axenides 1
Institute of Nuclear Physics, N.C.S.R. Demokritos 153 10, Athens, Greece
Leandros Perivolaropoulos 2
Department of Physics, University of Crete 71 003 Heraklion, Greece
We review recent work on a new class of topological defects which possess a non­symmetric
core. They arise in scalar field theories with global symmetries, U(1) for domain walls and
SU(2) for vortices, which are explicitly broken to Z 2 and U(1) respectively. Both of the
latter symmetries are spontaneously broken. For a particular range of parameters both
types of defect solutions are shown to become unstable and decay to the well known stable
walls and vortices with symmetric cores.
1 Introduction
Topological defects (Vilenkin & Shellard (1985), Vilenkin (1985), Preskill (1984)) are stable field
configurations (solitons: Rajaraman (1987)) that arise in field theories with spontaneously broken
discrete or continuous symmetries. Depending on the topology of the vacuum manifold M they
are usually identified as domain walls (Vilenkin (1985), Preskill (1984)) (kink solutions: Rajaraman
(1987)) when M = Z 2 , as strings (Nielsen & Olesen (1973)) and one­dimensional textures (ribbons:
Bachas & Tomaras (1994), Bachas & Tomaras (1995)) when M = S 1 , as monopoles (gauged: tHooft
(1974), Polyakov (1974), Dokos & Tomaras (1980), or global: Barriola & Vilenkin (1989), Perivolaro­
poulos (1992)) and two dimensional textures (O(3) solitons Belavin & Polyakov (1975), Rajaraman
(1987)) when M = S 2 and three dimensional textures (Turok (1989)), skyrmions: Skyrme (1961))
when M = S 3 . They are expected to be remnants of phase transitions (Kibble (1976), Kolb & Turner
(1990)) that may have occurred in the early universe. They also form in various condensed matter
systems which undergo low temperature transitions (Zurek (1985), Zurek (1996)). Topological defects
appear to fall in two broad categories. In the first one the topological charge becomes non­trivial due
to the behaviour of the field configuration at spatial infinity. The symmetry of the vacuum gets re­
stored at the core of the defect. Domain walls, strings and monopoles belong to this class of symmetric
defects.
In the second category the vacuum manifold gets covered completely as the field varies over the
whole of coordinate space. Moreover its value at infinity is identified with a single point of the vacuum
manifold. Textures (Turok (1989)) (skyrmions: Skyrme (1961)), O(3) solitons (Belavin & Polyakov
(1975)) (two dimensional textures: Turok (1989)) and ribbons (Bachas & Tomaras (1994)) belong to
this class which we will call for definiteness texture­like defects. The objective of the present discussion
is to present examples of defects which belong to neither of the two categories, namely the field variable
covers the whole vacuum manifold at infinity with the core remaining in the non­symmetric phase.
For definiteness we will call these non­symmetric defects.
Examples of non­symmetric defects have been discussed previously in the literature. Vilenkin &
Everett (1982) in particular, pointed out the existence of domain walls and strings with non­symmetric
cores which are unstable though to shrinking and collapse due to their string tension. A particular case
of non­symmetric gauge defect was recently considered by Benson & Bucher (1993), who pointed out
that the decay of an electroweak semi­local string leads to a gauged ``skyrmion'' with non­symmetric
core and topological charge at infinity. This skyrmion however, rapidly expands and decays to the
vacuum.
In the present talk we review recent work where we presented more examples of topological defects
that belong to what we defined as the ``non­symmetric'' class. We will study in detail the properties of
1 mailto:axenides@gr3801.nrcps.ariadne­t.gr
2 http://www.edu.physics.uch.gr/~leandros
1

global domain walls in section 2 and of global vortices in section 3. In both cases we will identify the
parameter ranges for stability of the configurations with either a symmetric or a non­symmetric core.
For the case of a domain wall wall we will discuss results of a simulation for an expanding bubble of
a domain wall.
Finally, in section 4 we conclude, summarise and discuss the outlook of this work.
2 Domain Walls with Non­Symmetric Cores
We consider a model with a U(1) symmetry explicitly broken to a Z 2 . This breaking can be realized
by the Lagrangian density (Vilenkin & Shellard (1985), Axenides & Perivolaropoulos (1997))
L= 1
2 @ ¯ \Phi \Lambda @ ¯ \Phi + M 2
2 j\Phij 2 + m 2
2 Re(\Phi 2 ) \Gamma h
4 j\Phij 4 ; (1)
where \Phi = \Phi 1 + i\Phi 2 is a complex scalar field. After a rescaling
\Phi ! m
p
h
\Phi; (2)
x ! 1
m x; (3)
M ! ffm: (4)
The corresponding equation of motion for the field \Phi is
¨
\Phi \Gamma r 2 \Phi \Gamma (ff 2 \Phi + \Phi \Lambda ) + j\Phij 2 \Phi = 0: (5)
The potential takes the form
V (\Phi) = \Gamma m 4
2h
`
ff 2 j\Phij 2 +Re(\Phi 2 ) \Gamma 1
2 j\Phij 4
'
: (6)
For ff ! 1 it has the shape of a ``saddle hat'' potential i.e. at \Phi = 0 there is a local minimum in
the \Phi 2 direction but a local maximum in the \Phi 1 (Fig.1). For this range of values of ff the equation of
motion admits the well known static kink solution
\Phi 1 = \Phi R j \Sigma(ff 2 + 1) 1=2 tanh
'' `
ff 2 +1
2
' 1=2
x
#
; (7)
\Phi 2 = 0: (8)
It corresponds to a symmetric domain wall since in the core of the soliton the full symmetry of the
Lagrangian is manifest (\Phi(0) = 0) and the topological charge arises as a consequence of the behaviour
of the field at infinity (Q = 1
2
(\Phi(\Gamma1) \Gamma \Phi(+1))=(ff 2 + 1) 1=2 ).
For ff ? 1 the local minimum in the \Phi 2 direction becomes a local maximum but the vacuum
manifold remains disconnected, and the Z 2 symmetry remains. This type of potential may be called a
``Napoleon hat'' potential in analogy to the Mexican hat potential that is obtained in the limit ff !1
and corresponds to the restoration of the S 1 vacuum manifold.
The form of the potential however implies that the symmetric wall solution may not be stable for
ff ? 1 since in that case the potential energy favours a solution with \Phi 2 6= 0. However, the answer
is not obvious because for ff ? 1, \Phi 2 6= 0 would save the wall some potential energy but would cost
additional gradient energy as \Phi 2 varies from a constant value at x = 0 to 0 at infinity. Indeed a
stability analysis was performed by introducing a small perturbation about the kink solution reveals
the presence of negative modes for ff ? ff crit =
p
3 ' 1:73 For the range of values 1 ! ff ! 1:73 the
potential takes the shape of a ``High Napoleon hat''. We study the full non­linear static field equations
obtained from (6) for a typical value of ff = 1:65 with boundary conditions
\Phi 1 (0) = 0 lim
x!1
\Phi 1 (x) = (ff 2 + 1) 1=2 (9)
\Phi 0
2
(0) = 0 lim
x!1
\Phi 2 (x) = 0: (10)
2

(a)
(b)
a<1
F)
a>1
V( F)
F F
1 2
V(
Figure 1: (a) The domain wall potential has a local maximum at \Phi = 0 in the \Phi 1 direction. (b) For
ff ? 1 (ff ! 1) this point is a local maximum (minimum) in the \Phi 2 direction.
Figure 2: Field configuration for a symmetric wall with ff = 1:65.
Figure 3: Field configuration for a non­symmetric wall with ff = 1:8.
3

Figure 4: Initial field configuration for a non­symmetric spherical bubble wall with ff = 3:5.
Using a relaxation method based on collocation at Gaussian points (Press et al. (1993)) to solve the
system (6) of second order non­linear equations we find that for 1 ! ff !
p
3 the solution relaxes to the
expected form of (7) for \Phi 1 while \Phi 2 = 0 (Fig.2). For ff ?
p
3 we find \Phi 1 6= 0 and \Phi 2 6= 0 (Fig.3) obeying
the boundary conditions (13), (14) and giving the explicit solution for the non­symmetric domain wall.
In both cases we also plot the analytic solution (7) stable only for ff !
p
3 for comparison (bold dashed
line). As expected the numerical and analytic solutions are identical for ff !
p
3 (Fig.2).
We now proceed to present results of our study on the evolution of bubbles of a domain wall.
We constructed a two dimensional simulation of the field evolution of domain wall bubbles with both
symmetric and non­symmetric core. In particular we solved the non­static field equation (6) using a
leapfrog algorithm (Press et al. (1993)) with reflective boundary conditions. We used an 80 \Theta 80 lattice
and in all runs we retained dt
dx
' 1
3
thus satisfying the Cauchy stability criterion for the timestep dt
and the lattice spacing dx. The initial conditions were those corresponding to a spherically symmetric
bubble with initial field ansatz
\Phi(t i ) = (ff 2 + 1) 1=2 tanh
'' `
ff 2 +1
2
' 1=2
(ae \Gamma ae 0 )
#
+ i0:1e \Gammajjxj\Gammaae 0 j x
jxj (11)
where ae = x 2 + y 2 and ae 0 is the initial radius of the bubble. Energy was conserved to within 2% in
all runs. For ff in the region of symmetric core stability the imaginary initial fluctuation of the field
\Phi(t i ) decreased and the bubble collapsed due to tension in a spherically symmetric way as expected.
For ff in the region of values corresponding to having a non­symmetric stable core the evolution
of the bubble was quite different. The initial imaginary perturbation increased but even though
dynamics favoured the increase of the perturbation, topology forced the Im\Phi(t) to stay at zero along
a line on the bubble: the intersections of the bubble wall with the y­axis (Figs 4 and 5). Thus in the
region of these points, surface energy (tension) of the bubble wall remained larger than the energy on
other points of the bubble. The result was a non­spherical collapse with the x­direction of the bubble
collapsing first (Fig.5).
4

Figure 5: Evolved field configuration (t = 14:25, 90 timesteps) for a non­symmetric initially spherical
bubble wall with ff = 3:5.
3 Vortices with Non­symmetric Core
We have generalised our analysis for domain walls to the case of a scalar field theory that admits
global vortices. We consider a model with an SU(2) symmetry explicitly broken to U (1). Such a
theory is described by the Lagrangian density:
L= 1
2 @¯ \Phi y @ ¯ \Phi + M 2
2 \Phi y \Phi + m 2
2 \Phi y Ü 3 \Phi \Gamma h
4 (\Phi y \Phi) 2 (12)
where \Phi = (\Phi 1 ; \Phi 2 ) is a complex scalar doublet and Ü 3 is the 2 \Theta 2 Pauli matrix. After rescaling as in
equations (2)--(4) we obtain the equations of motion for \Phi 1;2
@¯ @ ¯ \Phi 1;2 \Gamma (ff 2 \Sigma 1)\Phi 1;2 + (\Phi y \Phi)\Phi 1;2 = 0 (13)
where the +(\Gamma) corresponds to the field \Phi 1 (\Phi 2 ).
Consider now the ansatz
\Phi =
`
\Phi 1
\Phi 2
'
=
`
f(ae)e i`
g(ae)
'
(14)
with boundary conditions
lim ae!0 f(ae) = 0; lim
ae!0
g 0 (ae) = 0; (15)
lim ae!1 f(ae) = (ff 2 + 1) 1=2 ; lim
ae!1
g(ae) = 0: (16)
This ansatz corresponds to a global vortex configuration with a core that can be either in the symmetric
or in the non­symmetric phase of the theory. Whether the core will be symmetric or non­symmetric
is determined by the dynamics of the field equations. As in the wall case the numerical solution of
the system (21) of non­linear complex field equations with the ansatz (22) for various values of the
5

Figure 6: Field configuration for a symmetric­core global string with ff = 2:6.
Figure 7: Field configuration for a non­symmetric­core global string with ff = 2:8.
6

parameter ff reveals the existence of an ff cr ' 2:7 For ff ! ff cr ' 2:7 the solution relaxed to a lowest
energy configuration with g(ae) = 0 everywhere corresponding to a vortex with symmetric core (Fig.6).
For ff ? ff cr ' 2:7 the solution relaxed to a configuration with g(0) 6= 0 indicating a vortex with
non­symmetric core (Fig. 7). Both configurations are dynamically and topologically stable and consist
additional paradigms of the defect classification discussed in the introduction.
4 Conclusions
We have studied the existence and stability properties of defects (domain walls and vortices) with
non­symmetric core and non­trivial winding at infinity. These defects arise in scalar field theories
that exhibit an explicit breaking of a global symmetry, U(1) for domain wall and SU(2) for vortices.
In their spectrum and for a particular range of parameters topologically stable and unstable defects
appear with either symmetric or non­symmetric cores. Possible implications for the cosmology of
the early universe are the following: With regard to the case of domain walls with a symmetric core
(saddle and high Napoleon hat potentials) a possible embedding of such configurations in a realistic
2Higgs electroweak model may realize a new mechanism for baryogenesis at the electroweak phase
transition. Defect mediated baryogenesis has been so far only successfully implemented at scales
introduced near or above the electroweak one (Brandenberger et al. (1996)). These mechanisms are
based on unsuppressed B+L violating sphaleron transitions taking place in the symmetric core of the
defects during scattering processes (Cohen et al. (1993)). As a result of our work the question of
existence of electroweak domain walls with a symmetric core now translates to whether in the most
general electroweak Lagrangian with two Higgs doublets potential energies of the ``saddle hat'' or ``high
Napoleon hat'' type exist for an appropriate range of parameters. As the parity symmetry in these
models is broken both spontaneously an explicitly the expected domain walls in their spectrum are
certainly of the non­topological type. Moreover it would be of interest to see if such defects arise at a
second order electroweak phase transition. Our observation of non­spherical collapse of wall bubbles
with non­symmetric core may imply that the domain wall network simulations need to be re­examined
for parameter ranges where a non­symmetric core in energetically favoured.
With regard to our demonstration of existence of vortices with non­symmetric core it becomes
immediately suggestive the existence of a new kind of a bosonic superconducting string, possessing
massive charge carriers (Witten (1985)). This would be the case if our model is properly coupled to
a U(1) gauge field.The physics of fermions introduced to such a system is also open for investigation.
The astrophysical and cosmological role of superconducting strings has been extensively investigated
in the literature (Ostriker et al. (1986), Davis & Shellard (1988), Davis & Shellard (1988)).
Acknowledgements
This work was supported by the E.U. grants CHRX \Gamma CT93 \Gamma 0340, CHRX \Gamma CT94 \Gamma 0621 and
CHRX \Gamma CT94 \Gamma 0423 as well as by the Greek General Secretariat of Research and Technology grants
95E\Delta1759 and \PiENE\Delta1170=95. We are particular thankful to D.A.M.T.P. of the University of
Cambridge and Anne Davis for their hospitality and during our stay at Cambridge.
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