Kinks in the Hartree approximation

7.2 MB·English

by  Mischa Salle

Kinks in the Hartree approximation
Mischa Sall´e∗
Helsinki Institute of Physics, P.O.Box 64,
FIN-00014 University of Helsinki, Finland
(Dated: February 1, 2008)
The topological defects of λφ4 theory, the kink and antikink, are studied in the Hartree approx-
imation. This allows us to discuss quantum effects on defects in both stationary and dynamical
systems. The kink mass is calculated for a number of parameters, and compared to classical, one
loop and Monte Carlo results known from the literature. We discuss the thermalization of the sys-
tem after a kink-antikink collision. A classical result, the existence of a critical speed, is rederived
and shown for the first time in the quantum theory. We also use kink-antikink collisions as a very
simpletoymodelforheavyion collisions anddiscussthedifferencesandsimilarities, forexamplein
the pressure. Finally, using the Hartree Ensemble Approximation allows us to study kink-antikink
4 nucleation starting from a thermal (Bose-Einstein) distribution. On a qualitative level, our results
0 show only few dissimilarities with the classical results, but on a quantitative level there are some
0
important differences.
2

n PACSnumbers: 03.70.+k,05.45.Yv,11.10.Kk,11.27.+d
a
J
0 I. INTRODUCTION average of Gaussian expectation values. The equations
3 of motion are derived using the Hartree approximation
in each of the “realizations.” The partial “mean fields”

2 Inrecentyearstheincreasingnumericalpowerofcom- ofthese different“realizations,”i.e., the expectationval-
v puters has made it possible to follow quantum fields in uesofthefieldinthedifferentGaussianinitialstates,are
0
real time, studying phenomena such as thermalization, typicallyinhomogeneousandtheinteractionofthequan-
8
phase transitions and the formation of topological de- tummodeswiththeFouriermodesofthese“meanfields”
0
fects. Untilrecentlythis wasmainly doneusingthe clas- leads to approximately thermal distributions. However,
7
0 sicalapproximation,whichcanbejustifiedwhenoccupa- since these “mean fields” are typically inhomogeneous,
3 tion numbers are large. However, this is never the case topologicaldefects canandwillbe presentifthe symme-
0 for allmomentum modes, and for systems far from equi- triesofthetheoryallowforthem. Thesewillbethetopic
/ librium, this approximation may also break down. The of this paper, we will look at the φ4 theory in 1+1 di-
h
p Hartreeandlarge-napproximationsdocapturequantum mensions and its defects, kinks, and make a comparison
- effects but, until recently, the available numerical power betweenthe classicaltheory andthe Hartreeapproxima-
p
did not allow for the simulation of inhomogeneous sys- tion.
e
tems, thereby excluding the study of many physically
h
: interesting phenomena such as topological defects. It
v
alsohasproblems withdescribing thermalization,due to
i
X the lack of direct scattering. Recently, approximations The rest of the paper in organisedas follows. We will,
based on an expansion of the 2PI effective action (see in the remainder of this introduction, briefly review the
r
a Ref. [1] and references therein) and also on a truncation equations of motion and our definition of instantaneous
ofSchwinger-Dysonequations(seeRef.[2]andreferences particle number, used in studying the distribution func-
therein), which do include direct scattering have been tion. InSec.IIwewillderivetheclassicalkinksolutions,
showntoleadtoaquantumthermalequilibrium. Unfor- as this will facilitate comparison with the Hartree ap-
tunately, these methods are still numerically expensive proximation. Then, in Sec. III, we will discuss Hartree
andarenotyetableto describe inhomogeneoussystems. kink-antikink pairs, initially at rest and compare with
In two papers [3, 4] we introduced the Hartree ensem- the classicalapproximation. Astudy ofthe Hartreekink
ble approximation, which tries to combine the good fea- mass will be made. In Sec. IV kink-antikink collisions
tures of both the classical approximation, which is able will be studied. We will look at the critical speed: be-
to thermalize (classically), and the Hartree approxima- lowacertainspeedthepairannihilates,whileabovethey
tion, being aquantumapproximationandthus free from bounceback. Thishasbeenextensivelydiscussedforthe
the Rayleigh-Jeans divergencies. In the Hartree ensem- classicaltheory, but not yet for the quantum theory. We
bleapproximation,theinitialdensitymatrixisexpanded will study whether the system thermalizes after the col-
on a (overcomplete) basis of Gaussian density matrices. lision and use the collisions as a toy model for heavy ion
Expectation values can then be written as a weighted collisions, looking at the central region and its depen-
dency on coupling and initial speed. As a final topic we
will, in Sec. V, look at thermal kink nucleation, starting
from a Bose-Einstein thermal ensemble, as in Ref. [4].
∗Electronicaddress: mischa.salle@helsinki.fi The conclusions are in Sec. VI.
2
A. Equations of motion B. Observables: Particle number
InthissectionweverybrieflydiscusstheHartreeequa- In order to study thermalization we use the same def-
tions ofmotionfortheφ4 theory,asusedinthe separate inition for particle number as before [3, 4, 5], using the
“realizations”of the initial ensemble. For more informa- 2-point functions.
tion see Refs. [3, 4, 5].
The Lagrangian density for our 1+1 dimensional sys- S(t,x y)= φˆ(t,x)φˆ(t,y) φˆ(t,x) φˆ(t,y) , (5a)
tem is given by1 − h i−h ih i
U(t,x y)= πˆ(t,x)πˆ(t,y) πˆ(t,x) πˆ(t,y) , (5b)
1 1 1
= ∂ φ∂µφ µ2φ2 λφ4, (1) − h i−h ih i
µ
L −2 − 2 − 4
where the overline denotes an average over the center
givingrisetothe Heisenberg(operator)equationsofmo-
ofmasscoordinateandoptionallya time intervaland/or
tion
ensemble of initial conditions:
∂ ∂µφˆ+µ2φˆ+λφˆ3 =0. (2)
µ
−
Byintroducingexpectationvaluesofproductsoffieldop- φˆ(t,y+z)φˆ(t,y) =
h i
eratorsoneobtainsahierarchyofequations: theequation 1 1 t+δ/2 L
for the n-point function will contain also the (n + 2)- dt′ dy φˆ(t′,y+z)φˆ(t′,y) , (6)
N Lδ h i
function. Unless the initialdensitymatrix,whichisused ens. Zt−δ/2 Z0
X
in the calculation of the expectation values, is Gaussian,
this hierarchy will be infinite. The Hartree approxima- where N is the number of ensemble members, L the lat-
tion truncates it self-consistently by imposing Gaussian- tice size and δ a small time interval. In principle the
ity, thus factorizingalln-pointfunctions with n>2 into (asymmetric part of the) φπ correlation function could
1- and 2-point functions. It is convenient to expand the also be used, but as it vanishes in equilibrium, we will
2-point function in a set of mode functions, which in the refrain from doing that.
free theory are simply plane waves. This expansion is We cansplitthe 2-pointfunctions inthe separatecon-
most easily implemented in the field operator itself2 tributions from the mean field and the mode functions:
φˆ(x,t)=φ(x,t)+ f (x,t)ˆb +h.c. (3)
α α
Xα S(t,x y)=Sc(t,x y)+Sq(t,x y), (7)
It is important to note that, due to the Hartree approx- − − −
imation, the creation and annihilation operatorsˆbα and where
ˆb† can be chosen time-independent. They also satisfy
α
the usual commutation relations. The complex mode Sc(t,x y)=φ(t,x)φ(t,y) φ(t,x)φ(t,y), (8a)
functions f (x,t) form a complete and orthogonal ba- − −
α Sq(t,x y)=C(t,x;t,y), (8b)
sis with a Klein-Gordon-type inner product. However,
−
the α-label in generaldoes not coincide with the Fourier
label in time-dependent systems. and similarly for U(t,x y). These expressions only de-
−
TheHeisenbergequationsofmotionintheHartreeap- pendontherelativecoordinatex y. WetaketheFourier
−
proximation can thus be written as transform
φ¨=∆φ [µ2+ λφ2+3λC]φ, (4a) 1 L
− S (t)= dz e−ikzS(t,z), (9)
f¨ =∆f [µ2+3λφ2+3λC]f , (4b) k L
α α− α Z0
with
with k = (0, 1, 2, )(2π/L). Analogously to their
C = (2n0α+1)|fα|2, n0α =hˆb†αˆbαi. (4c) expressions fo±r a f±ree·t·h·eory we can use them to define
α instantaneous particle numbers:
X
In1+1-dimensions,onlythemassneedsrenormalization,
due to the divergence of the mode sum C(x,t). We will 1 1
S (t)= n (t)+ , (10a)
do this by absorbingits divergentvacuum partin µ2. In k k 2 ω (t)
terms of the renormalizedµ2 we have obtained a finite (cid:20) (cid:21) k
ren 1
theory. In the rest of the paper we will only consider U (t)= n (t)+ ω (t). (10b)
k k k
2
states for which n0 =0. (cid:20) (cid:21)
α
From these relations we can extract n (t) and ω (t) as
k k
afunctionoftime. Comparedtootherdefinitionsofpar-
1 Weusethemetric(−1,1). ticle number, this one has the advantage of being man-
2 WeassumealargefinitelatticesizeLandusefinitesizenotation, ifestly real, thus making them particularly suitable for
i.e.,adiscretespectrum. studying out-of-equilibrium systems.
3
II. CLASSICAL KINK SOLUTIONS
v λv4 / 2
Whenlookingforstablesolutionstotheclassicalequa-
tionsofmotionoftheφ4 theoryinaninfinitevolume,the 0
field has to be in the vacuum at spatial infinity in order
-v
tokeepthe energyfinite. Inthe “brokenphase”this still
0
leaves some important freedom. For the trivial solution -10 -5 0 5 10 -10 -5 0 5 10
both ends will be in the same vacuum. However, there
is also a nontrivial solution, where the field is in the dif- FIG. 1: Classical kink solution (13) and its energy density
ferent vacua in the different spatial infinities and hops (14) as a function of x in units m−1.
over the central barrier somewhere in between. This is
the so-called (anti)kink solution. The exact location of
the kinkdoesnotinfluencethe energyandaninfinite set Both the solution (13) and the energy density (14) are
of degenerate solutions exists, but transformingthe kink plotted in Fig. 1.
solution into the trivial solution would cost an infinite Thekinkmass,definedastheenergyofthestatickink
amount of energy, and hence the topology has rendered or antikink, can be found either by integrating (14) or
it stable. See Ref. [6] for a standard introduction into directly from the boundary term in (11)
the topic.
It is possible to calculate this lowest non-trivial solu- M = 2√2λv3 = 2mv2 = m3. (15)
kink
tion using a very elegant and general method, the Bogo- 3 3 3λ
mol’nyi equation [7]. In the broken phase the classical
Note that it diverges in the limit λ 0, showing the
static Hamiltonian (energy) is given by →
nonperturbative nature of the configuration.
E = dx1(φ′)2+ λ φ2 v2 2 From the static kink solution (13) we can easily find a
2 4 − moving kink solution by a Lorentz transformation:
Z
(cid:0) (cid:1) 2
1 λ λ m
= dx φ′ φ2 v2 φ′ φ2 v2 φ(x,t)=v tanh γ (x x ut) , (16)
Z (cid:26)2(cid:18) ±r2 − (cid:19) ∓ r2 − (cid:27) 2 − 0−
(cid:0) (cid:1)2 (cid:0) (cid:1) h i
1 λ
= dx φ′ φ2 v2 where γ =1/√1 u2. Note that this is not asolutionof
Z 2(cid:18) ±r2 − (cid:19) theBogomol’nyi−equation(12),sinceitisnotstatic,how-
λ 1 (cid:0) φ(∞) (cid:1) ever it is a solution to the manifestly Lorentz covariant
φ3 v2φ , (11) field equations. The total energy of this moving solution
∓ 2 3 −
(cid:20)r (cid:12)φ(−∞) is equal to E =γM , as expected for a relativistic
(cid:0) (cid:1)(cid:12) kink kink
where φ′ = ∂ φ. The bo(cid:12)undary term is equal to a finite particle. In summary, a Lorentz boost has the effect of
x (cid:12)
constantanddeterminesthe topologicalsector,trivialor increasing the kink mass while decreasing its width.
nontrivial. For given boundaries the energy is therefore
bounded from below by the first term which is larger
than, or equal to zero, the so-calledBogomol’nyibound. A. Lattice (arte)facts
Thelowestenergystatecanbefoundbysolvingthe sim-
ple first order differential equation We end this section by mentioning some consequences
of the use of a finite lattice.
λ
φ′ = φ2 v2 , (12) The topological sector is determined by the type of
∓r2 − boundary conditions: when imposing periodic boundary
which has solutions (cid:0) (cid:1) conditions the net kink number is 0, but a kink-antikink
pair canbe considered. Although sucha configurationis
v tanh m(x x ) ,
φ= ± 2 − 0 (13) inherently unstable, its survivaltime depends heavilyon
( v. h i the relative distance, as a result of the exponential at-
±
tractionbetweenthem. When using antiperiodic bound-
Here x is the point where φ goes through 0 and m =
0 ary conditions, the kink number is 1. By shifting the
√2λv2isthequasiparticleor“meson”massinthebroken ±
kinkover1latticesizeLthroughthe boundary,the kink
vacuum. In the top line, the + solution corresponds to
numberchangessign. Thismeansonecannotdistinguish
a kink, the solution to an antikink. The width of the
− between kink and antikink. However, the defect itself is
kink is inversely proportional to the mass.
stable, as expressed by the nonzero kink number.
Thekinkenergydensitycanbefoundfromthetopline
An unwanted side effect of the use of a lattice is radi-
of (11) using (12)
ation,resultingfromdiscretizationerrorsandinterfering
λ 1 m 2 with the kinks themselves. Some solutions have been
e = (φ2 v2)2 = λv4 tanh2 (x x ) 1 .
kink 2 − 2 2 − 0 − proposed,seeforexampleGleiserandSornborger[8]and
(cid:16) h i ((cid:17)14) Speight and Ward [9, 10, 11]. In Ref. [8] a damping
4
A. Initial condition
Unlikeintheclassicaltheory,wecannotusethe Bogo-
mol’nyiargumenttoderiveastaticsolitonsolutiontothe
Hartreeequations(4). However,ifthecouplingisnottoo
large, or more precisely, if the two vacua are well sepa-
ratedandthebarrierbetweenthemishigh,thequantum
corrections will be relatively small. Furthermore, away
FIG. 2: Kink-antikinkinitialisation procedure. fromthephysicalpositionofthe kink,thefieldresidesin
one of the vacua, where an exact solution of the Hartree
equationsisknown. Thereforeusingtheclassicalkinkso-
method is used to remove the radiation before it can in- lution as initial condition for the mean field, while using
terfere. Speight and Ward [9, 10, 11] propose a lattice the free field plane wave solutions for the mode func-
discretizationpreservingtheBogomol’nyibound,thereby tions will be close to a Hartree kink solution, provided
alsoremovingtheradiationresultingfromthediscretiza- the coupling is not too close to the phase transition, i.e.,
tion. However, according to Ref. [12] this latter solution the vacuum expectation value v is (substantially) larger
isonlysuitableforfreekinks,whileindynamicalsystems than 1. After the configuration has evolved, using the
a standard discretization would be simpler with effec- Hartree equations of motion, from such an initial con-
tively the same accuracy. We will not bother too much dition, it will, at least for a while, oscillate around the
about this issue. If it will be necessary to accurately actual stationary solution.
follow an approaching pair over a long period of time, A way to improve on this initial condition is to add a
large lattices will be used and the kink and antikink will damping term
be put close together initially: the radiation is mainly
emitted backwards and therefore reenters via the peri- Γ∂ φ (18)
t
−
odic boundary condition. Thus, by setting the soliton
pair close together, we canlet it take a long time for the totherighthandsideofthemeanfieldequation3(4a)and
radiation to reach them again. evolvefromtheaforementionedinitialconditiontoobtain
Finally, the use of (anti)periodic boundary conditions anapproximatelystationarysolution,whichcanthen be
in a finite volume introduces a small mismatch in the usedasanewinitialcondition. Wewilldiscussthissetup
derivative of the field at the boundaries,especially if the further in Sec. IV, when considering colliding kinks. In
kink and antikink are close to them. We therefore use Sec. IIIC we will use a damping term to determine the
the initialization setup sketched in Fig. 2: we take sev- quantum kink mass.
eral configurations in several adjacent lattices and use Another possibility would be to start with the initial
the resulting configuration in one of them, conditions as proposed in Ref. [13]. However, these still
arenotstationaryundertheequationsofmotionandthe
advantageisthereforeonlyminor,sowechoosetousethe
φ (x,t) = v+
KK t=0,a0 − simpler vacuum mode functions.
(cid:12)
N (cid:12) m The numerical results presented in the next sections
(cid:12)
v tanh γ (x+[x ut]+n L) are all obtained using the simple initial condition, with
0
2 − ·
n=X−N(cid:26) h i (17) mode functions of free field form and Eq. (17) for the
m mean field.
tanh γ (x [x ut]+n L) ,
0
− 2 − − ·
h i(cid:27)
where N is typically around 8. B. Numerical results: Static kink decay
In this subsection we are discussing the evolution of
III. HARTREE KINKS AT REST
a kink-antikink configuration, initially at rest at a max-
imum distance in the periodic volume. A comparison
Althoughinterestinginitselftheclassicaltheoryisnot will be made between Hartree and classical dynamics,
thefullstoryanditisthereforeimportanttostudyquan- using identical initial conditions, in the sense that the
tized kink solutions. In order to study dynamical pro- mean field in the Hartree simulation is equal to the field
cesses, such as collisions, one has to use approximation in the classical simulation. Hartree simulations at small
schemes,whicharenonperturbative,real-timeandwhich and large coupling will also be compared. In all cases
can handle inhomogeneous configurations. For this pur-
pose the inhomogeneousHartree approximationas stud-
ied in Ref. [3] and [4] should be particularly appropri-
ate,beinganonperturbativesemiclassicalapproximation 3 Note that it is not possible to do this at the level of the action
scheme. orHamiltonian.
5
N=512, Lm=32, 0 modes, E/Lm2=[0.01:0.19], λ/m2=1/1.25, u=0.0 N=512, Lm=32, 512 modes, E/Lm2=[0.01:0.1], λ/m2=1/1.25, u=0.0
15 15
10 10
5 5
m 0 m 0
x x
-5 -5
-10 -10
-15 -15
0 2000 4000 6000 8000 10000 12000 14000 16000 0 5 10 15 20 25 30
tm tm
(a)Classical,λ/m2=1/1.25. Forclaritythex-coordinateis (b)Hartree,λ/m2=1/1.25.
shiftedoverL/2throughtheperiodicboundaryconditions.
N=256, Lm=32, 256 modes, E/Lm2=[0.02:0.18], λ/m2=1/1.25, u=0.0 N=256, Lm=32, 256 modes, E/Lm2=[0.08:1.6], λ/m2=1/12, u=0.0
15 15
10 10
5 5
m 0 m 0
x x
-5 -5
-10 -10
-15 -15
0 100 200 300 400 500 600 0 2000 4000 6000 8000 10000 12000
tm tm
(c)Hartree,λ/m2=1/1.25, withdampingΓ=0.4mtill (d)Hartree,λ/m2=1/12.
tm=15. Plottedenergyismeanfieldonly.
FIG.3: Energydensity contourplot for kink-antikinkannihilation from rest, fordifferentcouplings, classical andHartree. All
plots, except 3(c), show total energy density. The vertical energy density range is chosen such as to emphasize the deviation
from vacuum which in some of the plots will cut-off thepeak at thecenterof thekink.
the same physicalvolumewillbe used. Finally the effect A comparison of classical and Hartree dynamics,
of damping on the lifetime of the kink-antikink pair is Figs.3(a)and3(b),respectively,showsanenormousdif-
discussed. The combined results are shown in Fig. 3, in ference in annihilation time between them. The high
the form of contour plots of the energy density for the instability of the Hartree kink antikink pair is causedby
4 different simulations. We have made plots of the en- the fact that at this strong coupling, the average energy
ergydensityinsteadofthe(mean)field,astheyshowthe densitye/m2 =0.025ishigherthanthepotentialbarrier
position of the kink more clearly. ∆e/m2 =0.019betweenthe threelocalminima(remem-
berthattheHartreeapproximationincorrectlypredictsa
Fig.3(a)showsaclassicalsimulationatthestrongcou- first order phase transition instead of a crossover),so by
pling λ/m2 = 1/1.25. Fig. 3(b) shows the same simula- exciting particles and spreading the energy the barrier
tion, using Hartree dynamics. Fig. 3(c) shows the same effectively disappears. The fact that we do not use an
Hartree simulation, with a damping Γ = 0.4m switched actualstationarysolution to the equations ofmotion (4)
onduringtm 15. FinallyFig.3(d)showstheresultfor therefore causes the KK¯ pair to evaporate very quickly.
a Hartree simu≤lationat a weakercoupling λ/m2 =1/12, It is important to mention that, although it seems from
without a damping term in the equations of motion.
6
Fig. 3(b) that the kink and antikink each split into two, most of the kinetic energy of the mean field has disap-
this is not actually the case. By making an animationof peared, followed by a very slow decrease, draining the
themeanfield,wefoundthatitjust“contracts”overthe energy of the modes. When switching off the damping
barrier, thereby annihilating both kink and antikink. In after the initial stage, as we have done in the simulation
thisprocesssomeenergyissendoffinoppositedirections, of Fig. 3(c), the kinetic energy of the mean field starts
but this cannot be seen as kink-antikink pairs. growing again, be it slowly, and a new energy equilib-
It is also interesting to note that forming a Bose Ein- rium is established. When the damping is not switched
stein distribution with the same average energy density off, the system evolves exponentially slowly towards its
wouldhaveaninversetemperatureβm=3.5,just below finalstate. Wewillcomebacktothisinthenextsection,
the phase transition at which the three minima become in the study of the Hartree kink mass.
degenerate. Combined with the low barrier this makes By comparing Fig. 3(b) at λ/m2 = 1/1.25 with
it profitable for the configuration to disintegrate, in or- Fig.3(d)atλ/m2 =1/12,weseethatattheweakercou-
der to lower the gradient energy, and settle down in the pling,theuseoftheclassicalkinkformforthemeanfield
symmetric minimum. andplanewavesforthemodesismuchclosertoastation-
Comparing the Hartree results in Figs. 3(b) and 3(c) ary solution, even without damping. The configuration
confirmstheaboveinterpretation. Bydampingtheequa- isoscillatingaroundanapproximatelystablesolutionfor
tions of motion we establish two things. We make the anextensiveperiodoftime,beforeannihilation. Bycom-
configuration closer to a truly stationary one, thus mak- paring Fig. 3(d) with the damped strong coupling result
ing it more difficult to excite particles, and by lowering Fig. 3(c), we see that the kink is broader at the stronger
the average energy density the corresponding Bose Ein- coupling,duetoachangeintheeffectivepotentialcaused
stein temperature moves awayfrom the phase transition by the quantum modes (we also checkedthis broadening
point. directlyinaplotofthemeanfield). Classicallythewidth
Afterthedampingisswitchedoffattm=15,thekink- onlydependsonthemassm,whichbyconstructionisthe
antikink pair survives until tm 450. Switching off the same for both couplings. This change of width gives an-
damping at a later instant incr≈eases the survival time: other interpretation for the fact that at larger coupling,
using tm = 40 instead of tm = 15 results in a survival the kink-antikink configuration is less stable: relative to
time of tm 500, i.e., the survival time increases by 50 their own size, the separation is smaller.
≈
due to an increase in “switch-off time” of 25. When we As a final remark, it is important to note that the
do not switch off the damping, the pair survives until annihilation in Figs. 3(b)–3(d) is not due to tunnelling
at least tm = 2000. We have not simulated this system but is an “over the barrier” annihilation. In Fig. 3(a)
longer, but we do not think it will ever decay. this is clear as it is a purely classical simulation, but
Wealsocheckedtheinfluenceoftheprecisevalueofthe one could suspect that the much shorter decay time in
dampingconstantΓ,bymakingadditionalsimulationsat the quantum Hartree simulations is due to tunnelling.
Γ=0.2mandΓ=0.8m. Atthesmallervalue,Γ=0.2m, However, in all cases, including the classical, the decay
the pair survival time decreases slightly, ∆tm= 20, at is due to the attractive force between the kink and the
−
a larger damping, Γ = 0.8m, the survival time increases antikink. The attractive potential falls off exponentially
by only ∆tm = +2. Apparently, using a damping con- with the separation, and small deviations from the sta-
stant Γ = 0.2m until tm = 15 already removes most tionary state only lead to decay after very long times.
of the fluctuations around the stationary configuration. At strong coupling the deviations are larger and, as we
However, the remaining configuration is still much less have seen, in the Hartree simulations the effective rela-
stable than its classical counterpart. This is confirmed tive distance becomes smaller due to the broader kinks.
by a comparisonof the total energy density with the de- In all runs in Figs. 3, except Fig. 3(b), the kink and an-
picted mean field energy density: it turns out that the tikink are moving towards each other before the decay,
total energy density is much smoother than that of the due to this attractive force, until they bounce and anni-
mean field. Apparently the modes in partcancelout the hilate. After that the mean field ends up in one of the
kink-antikink inhomogeneities, thus facilitating the final two “broken minima”. In Fig. 3(b), the barrier is so low
annihilation of the pair. that the field just moves over it and decays in that way.
There is an important subtlety with the implementa- In this run, the field oscillates around the “symmetric
tion of the damping term. Since it is far from easy to minimum” after the decay, as already discussed before.
add such a term to the equations of motion of the mode
functions – even in the vacuum their unrenormalized ki-
netic term is not vanishing – we only implement one in
C. Numerical results: Kink mass
the mean field equation of motion. However, this makes
theactualdecreaseoftheenergydensitymuchslower,as
it involves the transfer of energy from the modes to the In this subsection we will discuss the kink mass, the
mean field. We have seen in the simulations that there energyofthe static solitonic configuration. Classicallyit
are basically two timescales, a rapid initial decrease, ap- is given by Eq. (15). In the full theory this expression
proximatelywiththedampingrateexpected,afterwhich needs corrections. The first corrections were found by
7
Dashen et al.[14], see also Rajaraman [6],4 and more re-
cently by Alonso Izquierdo et al.[15]. A numerical study N=64, a /a=0.1, Γ/m=0.4, tm=90×103…100×103
0
was done in Ref. [16]. The result is obtained using a 1
semiclassicalperturbativeexpansionaroundtheclassical Hartree, m.f. a2µ2ren=-1.0
kink solution. The kink mass is then givenby the lowest Classical
0.8 Hartree, a2µ2 =-1.0
energy level. This mass has to be renormalized and the ren
Semi-classical
net result is
0.6 Hartree, a2µ2=-1.0
nk 0
m3 √3 3 Mki
Mkink = +m + (λ). (19) a 0.4
3λ 12 − 2π O
(cid:16) (cid:17)
A partial calculation of the order λ correction can be 0.2
found in Ref. [17]. In this reference and independently
also in Ref. [18] a full calculation is done for the order λ
0
correction of the kink mass in the sine-Gordon model. 0 1.2 2.4 3.6 4.8 6
More recently lattice Monte Carlo studies of the kink a2λ
mass have been carried out [19, 20]. In these papers
two methods of calculating the kink mass were used: (a)a2µ2=−1.0.
first using the fluctuation-fluctuation 2-point function
hµ(t)µ(0)i introduced in Refs. [21, 22], which in imag- N=64, a0/a=0.1, Γ/m=0.4, tm=90×103…100×103
inary time should decay as exp( M t). The second
kink 4
−
uses the difference inthe groundstate energywhenperi- Classical
odic or antiperiodic boundary conditions are used. Both Hartree, m.f. a2µ2ren=-2.2
studies focus on a range of parameters, where λa2 and 3 Hartree, a2µ2ren=-2.2
µ2a2 are of order unity, i.e., at large lattice distances. Semi-classical
− 0 Hartree, a2µ2=-2.2
Very recently, Bergner and Bettencourt [23] studied nk 0
thekinkmassintheHartreeapproximation. Theirequa- Mki 2
a
tionsarederivedfroma2PIeffectiveaction,butapplying
theHartreeapproximationtoitresultsinthesameequa- 1
tions of motion we are considering here. Using a numer-
ical relaxation method they have obtained the Hartree
kink mass at λ/m2 =0.5 to be 0.4691 (translated in our 0
0.5 1 1.5 2
units),whichisabout70%oftheclassicalvalue. Wehave a2λ
reproduced their value using antiperiodic boundary con-
ditions and a damping term (18) in the mean field equa- (b)a2µ2=−2.2.
tion of motion. On a lattice with N =64 and Lm=16,
a/a =10usingadampingcoefficientΓ/m=0.4wefind
0
atatimetm=9 103 10 103avalueMkink/m=0.47. N=64, a0/a=0.1, Γ/m=0.4, tm=90×103…100×103
Doubling the latt×ice si−ze o×r decreasing the temporal lat- 4 Hartree, m.f. a2µ2 =-4.0
ren
tice distance increases this value by about 1%, while in-
Classical
creasingthe simulation time with an orderof magnitude 3 Hartree, a2µ2ren=-4.0
decreases it by about 2%. Unlike Bergner and Betten- Semi-classical
court we have not fixed the center of the kink, but the Hartree, a2µ2=-4.0
nk 0
fact that the mass turns out equal shows that the zero Mki 2
mode can haveat most a smalleffect, in agreementwith a
Ref. [20].
1
In order to make a detailed comparison with the full
theory, i.e., the Monte Carlo data of Refs. [19, 20]5, we
have done simulations for a range of parameters. The
0
3 3.5 4 4.5 5 5.5
a2λ
(c)a2µ2=−4.0.
4 Notethattherearetwoerrorsin[6]: InEq.(5.69) p2+1(p2+
4), in the numerator of the first term in the integral, should
p
be(p2+1) p2+4and p2+2inthenumeratorofthesecond
termintheintegralshouldbe p2+4. Thisseconderrorcomes FIG.4: Comparison of classical, semiclassical (oneloop) and
fromthetrpansitionof2kp2→p2. Hartree results. The Hartree results are plotted for constant
5 Notethat thehorizontalaxis,pλˆ,inFig.4ofRef.[20]shouldbe renormalized µ2ren and unrenormalized µ20. Also see thetext.
multiplied by 6, as can be derived from the curve of the semi-
classicalkinkmass.
8
resultsareshowninFig.4. We showresultsforconstant Ref. [14] gives a value 0.08, about 1/5 of the classical
renormalizedµ2 aswellasconstantunrenormalizedµ2. value. Note that at λ/m2 1 the semiclassical approxi-
ren 0 ≈
For the constant µ2 we also show separately the mean mation breaks down, showing up by a change of sign in
ren
field contribution, defined as in Eq. (16), Ref. [5]. The the kink mass, the order λ0 correction becomes equal to
data in Refs. [19, 20] is shown as a function of the un- the leading λ−1 term. In the Hartree approximation we
renormalized µ2, in order to study the renormalization can in principle continue till λ/m2 1.2, where the zero
0 ≈
effects. However,to make a meaningfulcomparisonwith temperature phase transition occurs.
the classical and semiclassical 1-loop results, it seems
more proper to compare at constant µ2 . The arrows
ren
in the plots show the zero temperature phase transition IV. MOVING HARTREE KINKS
point,atwhichthethreeminimabecomedegenerate,cal-
culated using the static effective potential, cf. Ref. [3].
A. Initial condition
The dots indicate the point at which the “broken” min-
ima vanish altogether. In order to make the compari-
In order to describe kink-antikink collisions, it is nec-
son with the Monte Carlo data easier, lattice units have
essary to have a description of a moving Hartree kink as
been used. Most simulations where done with N = 64,
well. In the full quantum theory this poses a difficult
a /a=0.1withadampingtermΓ/m=0.4whiletheen-
0 problem, since we do not know how to Lorentz trans-
ergy was measured at tm 100 103. We have checked
≈ × form the “quantum cloud” surrounding the kink. In the
for lattice artefacts (the influence of N and a /a), their
0 Hartree approximationwe are savedby the fact that the
combined error is not larger than about 1%. Also the
field is completely expressed in terms of ordinary func-
effectsofthe precisevalueofthe dampingtermandtime
tions of x and t, and we can find a moving kink by per-
atwhichthe massis measuredaresmallandofthe same
forming a simple Lorentz transformation:
order of magnitude as that of lattice artefacts.
We see that for all renormalized masses, the Hartree f(x,t) f′(x,t)=f(γ[x ut],γ[t ux]), (20)
resultsarebetweentheclassicalandsemiclassicalvalues. → − −
Usingtheunrenormalizedmassastheparametertocom- both on the mode functions and the mean field.
pare with, the Hartree results are much lower than the However, there are still a number of difficulties with
semiclassical values. Comparing these with Figs. 4–6 of findingaproperinitialconditionforakink-antikinkpair,
Ref. [20] we find that the Monte Carlo results are above moving toward each other. First of all, from Eq. (20) it
the Hartree results. Note that the semiclassical line in follows that we need the mode functions on a backwards
Fig. 6 of Ref. [20] is approximately0.3 too high. For the space-time line, while we do not know them in analytic
twohighervaluesof|µ2ren|,wherethedatatakenisfurther form, only from numerical simulations. For the classical
awayfromthefirstorderphasetransition,themeanfield kink solution this problem does not exist, since it is not
contribution is almost equal to the classical value. The only stationary, but also static and known analytically.
difference between the Hartree and classical kink mass Second, we have to boost the kink and antikink, with
only comes fromthe modes. For the lowera2|µ2ren|=1.0 their respective mode-functions, in opposite directions.
the“meanfieldmass”behavesveryinterestingly. Wesee This means that they have to be combined in a nontriv-
that,comingfrombelow,itgoesupagainnearthephase ialwayinthemiddle,they“shiftintoeachother”. Italso
transition. This is caused by the emergence of the sym- means that we have to double the density of mode func-
metric minimum. More data would be needed to study tions in k-space, since the physical space doubles (the
the precise behaviour at the phase transition, but from best one can do is determine the mode functions for a
its definition, Eq. (16a) in Ref. [5], it seems it only goes single kink). We have to decide how to do this in a con-
up a final amount, due to this symmetric minimum and sistentway. Finallythereistheproblemthattheboosted
the change in the effective potential. It is also not clear coordinates γ(x ut) and γ(t ux) will in general not
to us at this stage whether it is only an artefact of the fallontospace-tim−elatticepoin−tsintheunboostedframe
way we split the total energy in a mean field and mode or vice versa.
contribution or is an actual physical phenomenon.
In order to circumvent these problems, we will just
Finally, we have measured the mass at a coupling use the vacuum form as initial conditions for the mode
λ/m2 =1/1.25, used for example in Figs. 3(b) and 3(c), functions. Since the vacuum is invariant under Lorentz
usingLm=16,N =64atatimetm=20 103. Inprin- transformations,wecanusethesamemodefunctionsina
×
ciple we could find the value also in Fig. 4(a), but those boostedframe. Asmentionedbefore,themodefunctions
data where obtained a large lattice distance and a sepa- willbe closeto the vacuumform,if the vacuumexpecta-
rate test is useful. The resultis M /m=0.22,or54% tion value v is large. Furthermore by boosting the kink
kink
of the classical value 0.42. This value for the kink mass solutions we reduce the width of the kinks. However, in
is consistent with Fig. 4(a), showing that we are indeed the center of mass frame, we keep the lattice size the
in the scaling region, as claimed by Ref. [20] (the same same thus effectively increasing the relative distance be-
is also found for the weaker coupling λ/m2 = 0.5, con- tween them. This leads to much more stable solutions,
sidered in Ref. [23]). The semiclassical value (19) from which is even further enhanced by the time dilatation.
9
Before we proceed we will outline a set of solutions fieldonabackwardspace-timeline. Thenoneconstructs
for the aforementioned problems, which will yield a very the mean field from the kink and its mirror image. The
accurate, boosted, Hartree kink-antikink solution and modes are constructed by combining f (x,t) with itself
k
will therefore also be usable at strong coupling and low and with f (x,t)=f (L x,t), thereby doubling their
−k k
−
speeds. Inthesimulationswewillingeneral,forsimplic- number,
ity, only use the vacuum form mode functions.
The way to solve the stability problem of a (single) f (x,t), 0 x L,
f (x,t)= k ≤ ≤ (22a)
kinkistousedampedequationsofmotion,aswealready k,1
(fk(x L,t), L x 2L,
usedinthe precedingsubsections,especially incombina- − ≤ ≤
tion with antiperiodic boundary conditions on the mean f (x,t), 0 x L,
f (x,t)= k ≤ ≤ (22b)
field to make the kink absolutely stable. The damping k,2
(f−k(x L,t), L x 2L.
will give us stationary, but time-dependent solutions for − ≤ ≤
the mode functions in the background of a kink, which
Thelatter,symmetric,combinationcanleadtofunctions
subsequently can be Lorentz boosted.
inwhichthesecondderivativedivergesas1/a: theenergy
The problem of the boosted coordinates not resulting
density, depending only on the squareof the firstderiva-
from discrete lattice points in the unboosted coordinate
tive,isfiniteandcontinuousinthecontinuumlimit,how-
can be adequately solved by linear interpolation, pro-
ever it might cause problems in the equations of motion.
vided the lattice distance is not too large. The related
matching problem is not a real problem: we have to
match the two functions at two consecutive time steps
B. Numerical results: Thermalization from
and when they approach each other with speed u, they
colliding kink-antikinks
shift into each other over a distance ua a, i.e., the
0
≪
mismatch is less than a lattice distance and the linear
We start the discussion of our numerical results with
interpolation automatically solves the problem.
thethermalizationpropertiesofacollidingkink-antikink
The remaining problem is to combine the two sets of
pair,usingthedefiningrelation(10)fortheparticlenum-
modefunctions,oftheseparatekinkandantikinkconfig-
ber n . We used a small coupling, λ/m2 = 1/12, in
urations, into one set which is twice as large. One possi- k
a volume Lm = 32 at a speed for which γ = 2. The
ble solution is to determine both kink and antikink in a
classical kink mass at this coupling is M /m = 4.
volume that alreadyhas the size of the combinedconfig- kink
Since we initialize the mean field as a classical kink and
urationand then use anaveragingprocedure to combine
the modes in the vacuum, the initial energy density is
the modes. This has the disadvantage that oscillatory
E/Lm2 = 2 (4+4)/32 = 0.5. At this energy density
functions with different phases are added. Another way, ·
and coupling we found, in Ref. [3], both in dynamical
which we suggest,is to determine the two configurations
simulationsandaneffective potentialcalculation,atem-
in their original volume and combine them afterwards
perature T/m 1.0.
into one set in the double volume. When using periodic ≈
In order to enable us to correctly interpret the result,
mode functions,itfollowsfromthe antiperiodicityofthe
we also measured the “particle number” at tm = 0. Of
mean field that at all times
course at this time the system is so far from equilibrium
f (x,t)=f (L x,t), (21) thatwe cannotinterpretthe resultas quasiparticles,but
k −k
− it does give an idea of the energy distribution as a func-
providedthis relationholds att=0 andt=a , a condi- tion of k. The result is plotted in Fig. 5. The similar-
0
tionwhichistruefortheplanewaveinitialconditionswe ity with a Bose-Einstein distribution is remarkable and
use. Notethatatlatertimestheklabelisnolongerequal we have to be careful in interpreting the results at later
totheFourierlabel,forwhichtherelationholdstrivially. times. One of the indications, that it is not a truly ther-
We can therefore combine each mode function f with mal distribution, is its very low “temperature”: at an
k
either itself or with f , since both combinations yield energydensityE/Lm2 =0.5weexpecttofindatemper-
−k
continuous functions. The derivative will in general be aturearoundT/m=1.0,whileFig.5seemstoindicatea
discontinuous by changing sign, possibly causing trouble valuewhichisalmostafactorof3smaller. Furthermore,
in the equation of motion. However, the energy density the particle number mainly comes from the even modes,
will be finite and even continuous, as it only depends on as a result of the symmetric initial condition.
the square of the first derivative, which is continuous. Fig.6showstheactualparticlenumber,atlatertimes,
We end this subsection summarizing the above steps when the kink-antikink have already annihilated. Then,
one could follow to obtain a better initial state for a wedofindatemperaturearound1,conformtheresultsof
colliding kink-antikink pair. First, using antiperiodic Ref.[3]. Asacomparisonwealsoplotted,inthesamefig-
boundary conditions for the mean field and a damping ure,the resultobtainedinthatpaperusingaflatensem-
term Γ∂ φ, one evolves the Hartree equations, start- ble simulation. It clearly shows that the temperatures
t
−
ing from a classical kink with free field mode functions. are equal: T is uniquely determined by the coupling and
One thus obtains a stationary Hartree solutionfor a sin- energy,notbytheinitialcondition. Theinitialcondition
gle kink, but has to keep track of the modes and mean still has an influence in determining which modes have
10
→ N=256, Lm=32, λ/m2=1/12, E/Lm2=0.5 C. Numerical results: Critical speed
n) k
+1/ We start this section with a study of classical kink-
1
g( antikink collisions. An extensive study of this can be
o
l
found in Ref. [24].
20 Figs. 7 show energy contour plots of Lorentz boosted
classical kink-antikink configurations. The results in
15 these plots demonstratehow,atverylow incidentspeed,
they just annihilate, while at a certain range of higher
10 speeds, they bounce a few times and then either annihi-
late or escape againto infinity, at a slightly lower speed.
5 β=2.82, µ=0 At high enough speeds, when u is larger than a certain
Kink-antikink uc,whichhasavaluebetween0.25and0.30,thepairim-
0 mediately escapes after the collision, again with a lower
0 1 2 3 4 5 6 7 8 9 ωk → speed. Part of the kinetic energy is transferred to an in-
ternal vibrational mode of the kink, which can be seen
FIG. 5: Initial form (tm=0) of the particle distribution for as a wiggling in the contour plots.
a Hartree kink-antikink collision. The initial form already
These results are consistent with Ref. [24] in which
seems close to a Bose-Einstein distribution.
a critical speed u = 0.2598 is found, above which col-
c
liding kinks always bounce back, while below a speed
0.193, they always annihilate immediately. Between
these speeds Ref. [24] found reflection and annihilation
→ N=256, Lm=32, λ/m2=1/12, tm=9000..10000, E/Lm2=0.5 bands,causedbyaresonancebetweenthecenterofmass
n) k motion and the vibrational mode. This was confirmed
+1/ β=0.94, µ=0 in Ref. [25], also showing that, when zooming in on the
1
g( Kink-antikink bands, their structure behaves fractally. This behaviour
o
l Flat ensemble
could even be reproduced using a collective coordinates
5
approach, reducing the infinite-dimensional system to a
4 two dimensional Hamiltonian one.
ExceptforFig.7(d)allshownresultsareinaccordance
3 with Table I in Ref. [24]. Since we used different dis-
cretizations,etc.,andgiventheverynarrowwidthofthe
2
stability bands already found by Ref. [24] and the frac-
tal behaviour found in Ref. [25], we can certainly expect
1
small differences in the precise position of these bands,
0 explaining the discrepancy. The authors of Ref. [24]
0 0.5 1 1.5 2 2.5 3 3.5 ωk → found that the final speed, for an initial speed above u ,
c
satisfies the following relation
FIG. 6: Late time form of the particle distribution for a
Hartree kink-antikink collision. tm ≈ 10000, after annihi- u2 u2 u2, (23)
lation. f ∝ i − c
which is consistent with Fig. 7(e), from which we derive
a final speed u 0.13 0.14. For u = 0.30 relation
f i
≈ −
(23) predicts u =0.15.
f
It is important to note that although breatherlike
thermalized. We see that starting from an annihilating states seem to emerge after a collision, truly stable
KK actually allows the system to thermalize faster, as breathers do not exist in the λφ4 theory [26]. However,
it has a more favourable energy distribution, cf. Fig 5. very long-lived and almost stationary configurations do
However, the KK initial condition has a delta function exist [27] as we have shown here again. The radiation
initial density matrix, which is not very suitable for our causesthe energy to decay as 1/log(t), as found analyti-
Hartree ensemble approximation: we do not reduce the callybyRef.[26]andconfirmednumericallybyRef.[27].
statisticalerrorsbyaveragingovermultiple initialcondi- We would like to compare these classical results with
tions. the Hartreeapproximationto seeif, inthe quantumthe-
ory, a critical speed still exists. In the classical theory,
Having showed that this initial condition also leads one can always use units such that µ = λ = 1, and one
to an approximately thermal Bose-Einstein distribution expects a unique critical speed, while in the quantum
after annihilation, we will now leave the thermalization theory, including the Hartree approximation, this is not
topic and continue with a discussion of the actual colli- the case. However, we will not go much further into the
sions. questionofthecouplingdependenceofthecriticalspeed.