Form factor and width of a quantum string

0.38 MB·English

by  Arttu Rajantie

Imperial/TP/2012/AR/4
HIP-2012-22/TH
Form factor and width of a quantum string
Arttu Rajantie
∗
Department of Physics, Imperial College London SW7 2AZ, U.K.
Kari Rummukainen and David J. Weir
† ‡
Department of Physics and Helsinki Institute of Physics,
PL 64 (Gustaf Ha¨llstr¨omin katu 2), FI-00014 University of Helsinki, Finland
(Dated: October 4, 2012)
In the Yang-Mills theory, the apparent thickness of the confining string is known to grow loga-
rithmically when its length increases. The same logarithmic broadening also happens to strings in
otherquantumfieldtheories and domain walls in statistical physicsmodels. Even inquantumfield
theories, the observables used to measure and characterise this phenomenon are largely borrowed
2 from statistical physics. Inthispaper, wedescribeit usingthestringform factor, which isamean-
1 ingful quantum observable, and show how the form factor can be obtained from field correlation
0 functionscalculated in lattice MonteCarlo simulations. Weapplythis method to2+1-dimensional
2 scalartheoryinthestrongcouplinglimit,whereitisequivalenttothe3DIsingmodel,andthrough
duality also to 2+1-dimensional Z2 gauge theory. Wemeasure thestring form factor bysimulating
t
c the Ising model, and demonstrate that it displays the same logarithmic broadening as statistical
O physicsobservables.

3 PACSnumbers: 05.50.+q,11.27.+d

]
t
a I. INTRODUCTION arenaturalinstatisticalphysicsmodels,suchasmeasur-
l ing expectation values of the plaquette near the string.
-
p String-like excitations play an important role in many The observables available in a quantum field theoretical
e quantum field theories, for example the confining string system are generally quite different. If we really had ac-
h
[ inYang-Millstheory[1]andcosmicstringsinsomeGrand cess to a real, physical string, we would probably probe
Unified Theories [2]. String excitations are also closely it experimentally by scattering particles off it. This mo-
1 relatedtothephysicsofinterfacesinstatisticalphysics[3, tivates us to focus on the string form factor, which is
v 4]. related to the corresponding scattering amplitude and is
6
0 In the semiclassical approximation, the string is de- a well-defined quantum observable. Building on previ-
1 scribedbyasolutionofthefieldequationswhichistime- ous work on kinks and monopoles [18, 19], we show how
1 independent and translation invariant along the string, the form factor of a string-liketopologicalsolitoncan be
. and describes the properties of the string on all length calculated in Monte Carlo simulations.
0
1 scales. Quantum mechanically, the picture is very differ- Because quantum mechanical strings in Euclidean
2 ent because the string carries massless Goldstone modes spacetime are equivalent to domain walls in statistical
1 whose quantum fluctuations dominate the dynamics on mechanics, we apply the method in practice to calcu-
: small scales. For example, the string width appears to latethedomainwallformfactorinthethree-dimensional
v
i dependlogarithmicallyonitslengthbecauseofthesefluc- Ising model near the critical point. This theory is in the
X
tuations. sameuniversalityclassasthe2+1-dimensionalrealscalar
r BecausethebehaviouroftheGoldstonemodesisinde- fieldtheoryandhasthereforethesamecriticalbehaviour.
a
pendentofmicroscopicdetails,thislogarithmicbroaden- It is also exactly dual to the confining three-dimensional
ingisageneralpropertyofanystringsirrespectiveofthe Z gauge theory, so our conclusions should also be valid
2
theory,andithasbeenstudiedextensivelyinYang-Mills for confining strings, at least qualitatively.
theory and spin models, both analytically [5–9], and nu-
merically[10–16]. Inthecontextoftheconfiningstringin
QCDorYang-Millstheory,itisgenerallyseenasaprob-
lem,becauseitmakesitimpossibletomeasure“intrinsic” II. STRING SOLUTION IN SCALAR THEORY
propertiesof the string whichone wouldliketo knowfor
phenomenological descriptions of confinement [8, 17].
Let us start by considering the 2+1-dimensional real
The previous numerical studies of strings in quantum
scalar field theory with the Lagrangian
fieldtheoryhavebeengenerallybasedonapproachesthat
1 1 1
= ∂ φ∂µφ m2φ2 λφ4, (1)
µ
L 2 − 2 − 4
∗ [email protected]
† kari.rummukainen@helsinki.fi where µ 0,1,2 . In the broken phase, which semi-
‡ david.weir@helsinki.fi classically∈c{orrespo}nds to m2 < 0, the theory has two
2
vacua t
m2
φ= v= | |. (2)
± ± λ
r
Classically, there is also a topologically stable solution
y
m(y y )
0
φ(t,x,y)=vtanh − , (3)
√2
(cid:18) (cid:19)
whichwehavechosentobe perpendiculartothey direc- −
tion. In 2+1 dimensions, this solution can be thought of
either as a string (because it is a one-dimensional object +
on any time slice) or a domain wall (because it divides T
spacetime into two pieces). Since we are principally in-
x
terested in using these objects to model strings in 3+1-
dimensional theories, we will refer to it as a string when Lx Ly
we discuss it in the context of the 2+1-dimensional the-
ory. FIG. 1. System setup. The coordinate x runs along the do-
There is a general result [3], applicable to any string- mainwall(shadedgray)onatimeslice,thetwistedboundary
likeobjects,thatfluctuationsbroadenthe stringsothat, conditions are in the y direction and t parameterises the di-
rection along which correlators are measured.
whatever its initial shape, its width w is given by
1 L
w2 = ln (4)
2πσ cξ hasalinearlydivergentenergy,wecansafelyrestrictour-
selves to only non-relativistic motion.
where σ is tension of the string.
For any local operator ˆ, suchas the field operator φˆ,
However, the arguments used to demonstrate this are O
we can now define the corresponding form factor as
basedonthe dependence onthe expectationvalue ofthe
energy density on the transverse position relative to the
f(p ,p )= p ˆ(0)p , (6)
string, or other similar quantities which are sensitive to 2 1 h 2|O | 1i
the fluctuations of the string position. There has, there-
whereby ˆ(0)wemeantheoperatorincoordinatespace
fore, been discussion on whether there is some other ob- O
at position (x,y)=(0,0).
servable which would measure the “intrinsic width” of
the string [8, 9].
B. Semiclassical Limit
III. STRING FORM FACTOR
In the semiclassical limit [20], the form factor is given
A. Definition by the Fourier transformofthe classicalprofile (y) of
cl
O
the quantity . Taking matrix elements of the Heisen-
O
A stable quantum state corresponding to Eq. (3) also berg equation of motion for the classical field we find
exists in the quantum theory. We can define it as the
f(p ,p )= p ˆ(0)p
lowesttopologicallynon-trivialenergyeigenstate,andwe 2 1 h 2|O | 1i
denote it by |0i. To be precise, there is a degenerate set = dy ei(p2−p1)y cl(y)
of such states corresponding to different orientations of O
Z
thestring,andweagainchoosethestringthatisoriented =O˜ (p p ).
cl 2 1
along the x direction as shown in Figure 1. −
(7)
The string ground state 0 has zero momentum and,
| i
intheinfinite-volumelimit,itsenergyisE =σL,where
0 This expressionis valid only inthe non-relativisticlimit,
the constantσ isthe stringtension. We canobtainmov-
when p , p σL. In this limit the form factor is a
1 2
ing string states by boosting the ground state 0 in the | | | | ≪
| i function of the momentum difference k p2 p1 only,
y direction. We denote these states by p , where p is ≡ −
| i as a direct consequence of the Galileaninvariance, so we
the momentum of the string in y direction. The energy
willdenoteitbyf(k). Althoughourstudiesofkinksand
of such a state is E = p2+E2. We normalise these
p 0 monopolesusedfullyrelativisticsemiclassicalexpressions
states as
p fortheformfactor,this isareasonableapproximationin
thecurrentpapersinceanyresultsobtainedwithastring
p′ p =2πδ(p′ p). (5)
h | i − moving relativistically are most probably due to finite-
This normalisation is not Lorentz invariant under rela- size effects (the amount of energy required to accelerate
tivistic boosts in the y direction, but because the string a string to relativistic velocities on macroscopic scales is
3
too great; the finite box size does not correctly capture where Uˆ(t)= exp( Hˆt) is the Euclidean time evolution
−
this). operator, and Z = Tr Uˆ(T) the partition function for
In particular, choosing ˆ = φˆ, we find from Eq. (3) the worldsheet, not the full Z of the field theory with
tw
O
the semiclassical result for the string form factor twisted boundary conditions. In other words, only when
T is the situation of Eq. (11) realised; otherwise
i√2πv the→str∞ing’s worldline is restricted.
f(k)= . (8)
msinh(kπ/√2m) With twisted boundary conditions, the states α , α
′
| i | i
musthaveanoddnumber ofstrings(inpractice,exactly
At low momenta k m, this behaves asymptotically as one due to heavy volume suppression factors e 2σLT
≪ −
in the partition function; this also suppresses opposite-
2i
f(k)=v , (9) spin bubble formation). Because of momentum conser-
k
vation they must also have opposite overall momentum
which corresponds to an infinitesimally thin string (kx,ky) = (qx,qy). If the momentum is in the y di-
−
rection, i.e., k = 0, then the lowest such state is the
x
φ (y)=vSign(y). (10) moving string state k with momentum k , as defined
cl y y
| i
in Section IIIA. It has energy
Becauseanystringwouldlooklikethiswhenviewedfrom
longdistances,weexpectthelow-kasymptoticbehaviour
k2
(9) to be valid for any string, irrespective of microscopic E = k2+E2 E + y (13)
details and even in quantum theory. Therefore it serves ky y 0 ≈ 0 2E0
q
as a useful benchmark for our calculations.
where E is the ground state energy of the string.
0
Incontrast,ifthemomentumhasanon-zeroxcompo-
C. Correlator and form factor
nent, k = 0, then the interpolating states must be ex-
x
6
citedstateswithsomeexcitationcarryingthemomentum
It was shown previously in Ref. [18] how the form fac- in the x direction. The lightest such states are massless
tors of point-like solitons such as kinks and monopoles Goldstone modes whichexist becausethe positionof the
canbecalculatedfromthefieldcorrelationfunction. The string breaks translation invariance in the y direction.
same technique works also for strings. Because we are only interested in the unexcited states
Weworkinmomentumspace,takingtheFouriertrans- k , we do not consider this possibility, and instead we
form in space but not in the time direction. |simiply restrictourselvesto k =0fromnow on. Tosim-
x
After Wick rotation to Euclidean time, the two-point plify our notation, we therefore suppress the unneeded
correlator of the operator ˆ has the spectral expansion. argument k , and write
O x
(0,k ,k ) (t,q ,q )
hO x y O x y i ˆ(ky) ˆ(0,ky), (14)
0 ˆ(k ,k )α α ˆ(q ,q )0 O ≡O
= h |O x y | ih |O x y | ie−t(Eα−E0), (11)
00 and
α h | i
X
where E is the energy of the single string ground state.
0
(0,k ) (t,q ) (0,0,k ) (t,0,q ) . (15)
A small added complication is that the spacetime is hO y O y i≡hO y O y i
necessarily finite in actual Monte Carlo simulations. We
have periodic boundary conditions in the t-direction (as
Thenextstatesinthe spectrumcontainpairsofmass-
well as the x-direction along the string), and twisted
less Goldstone modes, with opposite quantisedmomenta
boundary conditions in the y-direction. This periodic-
k = 2nπ/L. These propagate along the string per-
x
ity leads us to write the correlator as
pendicular to our chosen direction for time evolution of
the system. The energy of the lowest such state, with
hO(0,kx,ky)O(t,qx,qy)i kx =2π/L,isEky+4π/L. Therefore,ifwechooset&L,
1 these states are strongly suppressed relative to the mov-
= TrUˆ(T t)ˆ(q ,q )Uˆ(t)ˆ(k ,k )
Z − O x y O x y ing string state in the spectral expansion (12). Conse-
1 quently,theproblembecomespracticallyidenticaltothe
= α′ ˆ(qx,qy)α α ˆ(kx,ky)α′ e−Eα′(T−t)−Eαt, 1+1-dimensionalkink case[18], andwecanapproximate
Z h |O | ih |O | i
αX,α′ Eq. (12) by an integral over moving string states ky
(12) | i
only,
4
hO(0,ky)O(t,qy)i= Z1 d2kπy′ d2kπy′′hky′|Oˆ(qy)|ky′′ihky′′|Oˆ(ky)|ky′ie−Eky′(T−t)−Eky′′t
Z
= LZ22πδ(qy+ky) d2kπy′ |f(ky′,ky′ −ky)|2e−Eky′(T−t)−Eky′−kyt, (16)
Z
where we have used the result point k by minimising Eq. (19) , and we obtain
0
hky′|Oˆ(qy)|ky′′i= dxdyeiqyyhky′|Oˆ(x,y)|ky′′i f(k0,k0 ky)= i (0,ky) (t, ky)
Z − ± hO O − i
= dxdyeiqyyei(ky′−ky′′)yhky′|Oˆ(0)|ky′′i 1 Eq0S′′(k0) 1/4e(S(k0)−E0T)/2 (22)
Z × L T
=2πLδ(qy+ky′ −ky′′)f(ky′,ky′′). (17) (cid:18) (cid:19)
for odd. Note that the saddle point k still depends
0
The choice t & L obviously requires a relatively long O
on t.
lattice in the time direction. The dynamics of the
The result (22) should be compared with the corre-
longest-wavelength modes then becomes essentially one-
sponding result for kinks given in Eq. (19) of Ref. [18]
dimensional.
(althoughwe have correcteda typographicalerrorhere).
Similarly, we can write the partition function as
In that case, the interesting length scales were compa-
dk dk rable to the inverse kink mass. Therefore the motion of
Z = 2πy′ hky′|Uˆ(T)|ky′i=L 2πy′ e−Eky′T the kink was relativisticfor the correspondingmomenta,
Z Z and it was natural to express the result in terms of ra-
L dkye−(cid:18)E0+2kEy20(cid:19)T =L E0 e E0T. (18) pidities. In the current case, the interesting momenta
≈ Z 2π r2πT − ky ∼ √σ are much lower than the string mass σL, and
its motion is therefore highly non-relativistic. We can
This partition function is the individual contribution to
therefore simplify Eq. (22) by taking the non-relativistic
the partition function from string’s classical worldsheet.
limit. In that case, k = (t/T)k and S (k ) = T/E ,
0 y ′′ 0 0
To calculate the integral (16), we use the saddle point
and the form factor also becomes a function of the the
approximation. Thesaddlepointk isfoundbyminimis-
0 momentum difference k only,
y
ing the action
f(k ) f(k ,k k )
y 0 0 y
≡ −
S(ky′)=Eky′(T −t)+Eky′−kyt (19) = i (0,k ) (t, k ) 1 exp ky2 t(T −t) .
y y
forgivent. ByapproximatingtheintegralbyaGaussian ± hO O − iL 4E0 T !
q
around the saddle point, we obtain
(23)
L2 Correspondingly Eq. (21) simplifies to
(0,k ) (t,q ) = 2πδ(q +k )
y y y y
hO O i Z ×
d2kπy′ |f(ky′ −ky,ky′)|2e−S(k0)−12S′′(k0)(ky′−k0)2, (20) hO(0,ky)O(t,qy)i k2 t(T t)
Z 2πLδ(k +q )f(k)2exp y − . (24)
y y
This is not a Gaussian approximation of the correlation ≈ | | −2E0 T !
function, nor is it a semiclassicalstationary phase calcu-
Wecanalsousethisresulttodeterminethegroundstate
lation; it does, however, allow us to rearrange Eq. (16)
and solve for f(k ,k k )2 in exchange for imposing energy E0 from the correlator measurements.
| y′ y′ − y |
minor restrictions on what we can measure. In the limit
of large t and T t that we already require, the Gaus-
− D. Linear fluctuations
sianapproachesadeltafunctionandwecancalculatethe
integral
While the primary purpose of this paper is to mea-
sure the form factor fully non-perturbatively using lat-
(0,k ) (t,q )
y y
hO O i tice Monte Carlo simulations, it is useful to look at the
1/2
T leading-order quantum effects analytically. The lowest
2πδ(k +q )L f(k ,k k )2e S(k0)+E0T.
≈ y y E S (k ) | 0 0− y | − excitations in the system are massless Goldstone modes
(cid:18) 0 ′′ 0 (cid:19)
(2o1n)the string worldsheet, which are present because the
string breaks translation invariance in the y direction
WecanuseEq.(21)todeterminetheformfactorfrom spontaneously. To calculate their effect, we assume a
thefieldcorrelator. Forgivenk andt,wefindthesaddle string that moves in the normal direction y without
y
5
changing its profile. If the profile of the string is φ (y), with k = 2πn/L. This is obviously IR divergent, but
c n
and the position of the string is y(t,x), then the field we can use it to write
configuration is
(y(0,0) y(t,x))2 =2 y(0,0)2 2 y(0,0)y(t,x) (34)
h − i h i− h i
φ(t,x,y)=φ (y y(t,x)). (25)
c which yields
−
Taking the Fourier transform over x and y, we find (y(0,0) y(t,x))2 = 1 t+2 ∞ 1−e−kntcos(knx) .
h − i σL" kn #
φ (t,k ,k )=φ˜ (k ) dxei(kxx+kyy(t,x)), (26) nX=1
c x y c y (35)
Z Fort&L,theexponentialsareallverysmallandwecan
whereφ˜ (k )istheFouriertransformoftheclassicalpro- approximate
c y
file,
1 ∞ 1
(y(0,0) y(t,x))2 t+2 (36)
φ˜ (k )= dyeikyyφ (y). (27) h − i≈σL" n=1kn#
c y c X
Z 1 L ∞ 1
= t+ . (37)
We can now write the correlationfunction as σL π n
" #
n=1
X
φ(t,k ,k )φ(t,k ,k ) =φ˜ (k )φ˜ (k ) This diverges in the UV, but a minimum length scale l
h x y ′ x′ y′ i c y c y′ provides a cutoff n<L/ℓ. Then for L ℓ one has
× dxdx′ei(kxx+kx′x′) ei(kyy(t,x)+ky′y(t′,x′)) . (28) L/ℓ 1 L ≫
Z D E =ln +γ, (38)
n ℓ
Writing ∆x=x′−x and ∆t=t′−t, we have nX=1
and we find (for k =0),
hφ(t,kx,ky)φ(t′,kx′,ky′)i=2πδ(kx+kx′)φ˜c(ky)φ˜c(ky′) x
×Z d∆xe−ikx∆xDei(kyy(0,0)+ky′y(∆t,∆x))E. (29) hφ(t,0,=kyL)φ2(φt˜′c,(0k,y−)k2ye)−i2kπy2σ(ln(L/ℓ)+γ)e−(ky2/2σL)t. (39)
| |
Translation invariance in the y direction gives a delta
Inserting this into Eq. (23), we find the form factor
function also for the k component, so we can assume
y
ky′ =−ky and we have f(ky)=φ˜c(ky)e−4kπy2σ(ln(L/ℓ)+γ). (40)
φ(t,k ,k )φ(t,k , k ) =2πδ(k +k )φ˜ (k )2 ComparingwithEq.(7),wecanseethattheeffectofthe
h x y ′ x′ − y i x x′ | c y |
fluctuations is to suppress the form factor at high mo-
× d∆xe−ikx∆x ei(ky(y(0,0)+y(∆t,∆x))) . (30) menta ky & 2πσln(L/ℓ). For large L, this washes out
Z D E any structure the classical string solution has at short
p
distances, and using the asymptotic low-momentum be-
Assuming that y(t,x) is Gaussian, this is equal to
haviourofthe classicalsolutionφ˜ (k ) 2iv/k ,we find
c y y
∼
hφ(t,kx,ky)φ(t′,kx′,−ky)i=2πδ(kx+kx′)|φ˜c(ky)|2 f(k)= 2ive−4kπ2σ(ln(L/ℓ)+γ). (41)
k
d∆xe−ikx∆xe−12ky2h(y(0,0)−y(∆t,∆x))2i. (31)
× If interpret, in line with Eq. (7), this as the Fourier
Z
transform of the quantum-corrected domain wall profile
However, if we consider an elongated lattice with spa- φ (y), we findthatin coordinatespace the domainwall
eff
tial size L and time separation ∆t L, then the cor- has broadened and has width
≫
relator is very different. Let us start with the two-point
1
correlator of the field y, which is given by the Fourier w2 = (ln(L/ℓ)+γ), (42)
2πσ
transform of the propagator
in perfect agreement with Eq. (4).
1 dk dωei(kx+ωt)
y(0,0)y(t,x) = , (32)
h i σ 2π 2π k2+ω2
Z Z IV. ISING MODEL
and in finite spatial volume, the integralover k becomes
a sum over n, so we have A. The model
1 ∞ dωei(knx+ωt)
y(0,0)y(t,x) = (33) To demonstrate the use of Eq. (21) we use it to calcu-
h i σLn= Z 2π kn2 +ω2 latethedomainwallformfactornon-perturbativelynear
X−∞
6
the critical point. This is an interesting calculation be- where G 0.29, and the domain wall tension σ cor-
≈
causethe theorybecomes stronglycoupledandtherefore responds directly to the string tension in the 2+1-
perturbation theory is not valid. In practice, we do not dimensional scalar theory.
actuallysimulatethescalarfieldtheorybutratherthe3D
Isingmodel, whichis knownto be inthe sameuniversal-
ity class and which will therefore give identical results B. Real-space width measurements
near the critical point. From a computational point of
view, the Ising model is much more convenient because Previous measurements of the width of walls and
highlyefficientnumericalalgorithmsareavailabletosim- strings have worked in real space. Here we summarise
ulate it. oneofthemoresuccessfulapproachesandnotethatlater
The worldsheet of the string corresponds to a domain developments aregeneralisationsof the same idea to dif-
wallin the 3D Ising model. In this sectionwe will there- ferent systems or improvements in numerical technique.
fore use the term domain wall, but it should be under- In Ref. [22], the domain wall width was studied. The
stood to refer to the same physical object as the term results were based on linear functions of local spin op-
stringelsewhereinthepaper. The3DIsingmodelisalso erators; in addition the t-direction was not privileged in
exactly dual to the confining 2+1-dimensional Z2 gauge this study. For these reasons, this calculation cannot be
field theory. This duality maps the domain wall to the consideredadirectanalogyofaparticlescatteringoffthe
worldsheet of the confining string of the gauge theory, domain wall.
and therefore our calculation also describes the proper- Note that the geometry used was rather different to
tiesoftheconfiningstringinthissomewhatsimplegauge ours. In addition, the discussion here follows our own
theory. conventionsestablishedinSectionI.Firstthe totalmag-
The logarithmic broadening (4) of the domain wall is netisationfor slicesparallelto the domainwallwasmea-
well known also in the Ising model. In the rough phase sured,
(between β and β ), the domain wall width w has a
R C
logarithmic divergence with domain wall length L [3] of 1
m(y)= s(t,x,y). (47)
the form given in Eq. (4). LT
t,x
The Ising model has Hamiltonian X
For each configuration, m(y) was calculated, and the
H =− kx,x′s(x)s(x′) (43) minimum value y0 of y obtained. Then the ‘normalised
hXx,x′i magnetisation gradient’ is
where x and x refer to position vectors in the 3D Eu-
′ 1
acllildneeaanresspta-cnee,igahnbdohuxr,lxin′iksdeinnottheestlahtatticteh.eTshuemsrpuinnssso(vxe)r ρ(y)= 2M |m(y−y0+1)−m(y−y0)| (48)
takethevalues 1. Wecandenotethesizeofeachdirec- where
±
tion L , L and T respectively, but for our own calcula-
x y
1
tions we shall alwaystake L =L =L. It has partition
x y M = m(y) (49)
function L
y
X
Z = exp( βH), (44)
0 − is the average magnetisation. For a very sharp and
{s(xX)=±1} straight domain wall, ρ(y) will only be nonzero for one
and we impose periodic boundary conditions in all three value of y. In reality, fluctuations will smear the result,
directions, with kx,x′ = 1 for translational invariance. even for individual configurations. There may also be
However, to create a domain wall, a twist is introduced bubbles of spin in the bulk, or (for small lattice sizes)
so that kx,x′ = 1 in the y-direction for one value of y. other odd numbers of domain wall. The height h can be
−
Various techniques exist to measure the free energy of defined as
suchanobject;almostalldependonmeasuringtheratio
h= yρ(y), (50)
y
Ztw X
F = ln , (45)
− Z0 andthenthevarianceinthisexpressiongivesusthemean
squared width w2,
where Z is the partition function on a fully periodic
0
latticeandZ isthepartitionfunctionwithantiperiodic
tw 2
boundary conditions in the y-direction.
w2 = ρ(y)y2 ρ(y)y ; (51)
The free energy of the domain wall can be obtained * − ! +
y y
as a function of L and T. For a cubic system with X X
x
T =Lx =L, a perturbative calculation yields [21] theanglebracketsdenoteanensembleaverage. Notethat
1 1 if the shifting procedure does not introduce a systematic
Fcubic =σL2− 2lnσ+G−4σL2 +O (σL2)−2 , (46) bias inhhi (say,the value ofy0 is chosenrandomlywhen
(cid:0) (cid:1)
7
there are multiple degenerate minima of m(y)), then we
have h =0 which we shall assume shortly when study-
ing hohwithis estimate of w2 behaves. 0.0115
Anyway, using this data for various lattice sizes (for β=0.226
Nambu-Goto
example,inRef.[13]thesystemunderconsiderationhad
0.011
L = L and L up to 27, so the width either side of
x t y
the domainwallwasfixed), afitto the ansatztakingthe
same form as Eq. (4) can be made. 0.0105
L
We assume h = 0, as discussed. Now L)/
h i E(0
0.01
w2 = ρ(y)y2 (52)
* y + 0.0095
X
y2
= (s(t,x,y) s(t,x,y 1)) (53)
* 2LM (cid:12) − − (cid:12)+ 0.0090 50 100 150 200 250 300
Xy (cid:12)Xt,x (cid:12) L
(cid:12) (cid:12)
(cid:12) (cid:12)
In other words(cid:12), this measurement techn(cid:12)ique is
weighted towards long-distance fluctuations of the do- FIG. 2. Energy per unit length of the string measured using
2
main wall. It does not and cannot directly probe the the correlator with β = 0.226. Units are a ; the error bars
are jackknife estimates.
physics of any intrinsic width, nor does it allow for the
possibility of different behaviour being visible at short
distances as such effects will be washed out. One way of
the surface tension undetermined. This approach is in
evading this problem is the rescaling approach adopted
line with our previous studies along the same lines.
in Ref. [15].
Notethatathighermomenta,thebulkscalarcontribu-
The approachdiscussed above is equally applicable to
tion e √m2+k2t decaysmuchmorerapidly with t than
cthaerriceodnfionuitnignsRtreinf.g[1in4].SUO(fNc)ougrasueg,ethteheeonrdyp,oaisntwsaosffitrhset the s∼trin−g contribution e−2kE20t, so discarding the first
∼
string must be fixed as well so there is not even transla- few points at either end of the lattice allows us to ignore
tional invariance in such a calculation. this issue when measuring the form factor; Figure 3 also
shows this effect.
Asexpected,weseeisevidencefortheLu¨schertermin
V. RESULTS our string tension measurements. For a closed Nambu-
Goto string, the ground state energy is
WeusethestandardIsingsingle-clusteralgorithm[23],
π
as well as Metropolis updates to obtain the results pre- E (L)=σL 1 . (54)
0 − 3σL2
sented here. To show the volume-dependence of the r
ground state energy we use β = 0.226; however the
Unfortunately, we cannot rule out the massless world-
domain wall is so light here that finite-size effects for
sheet modes affecting the quality of the data for the
the form factor for higher k are severe. Therefore the
smallest of our lattice sizes. The results are shown for
form factor results are presented for values of β that are
β =0.226 in Figure 2, along with a fit to Eq. (54).
slightly further into the rough phase.
Nonetheless, the quality of this fit gives us confidence
that it is principally the ground state of the string that
gets excited when we carry out measurements at large
A. Stationary string energy time separation; this is in line with the assumptions of
Section IIIC.
Inordertodirectlyrecoverthestringformfactorfrom
correlator measurements, we need an independent esti-
mate of the string tensionσ. This is obtainedfrommea- B. String form factor
surementsofthe the lowestmomentum(k =π/L)corre-
lator in the presence of the string. Theprevioussectionpresentedevidencetosupportour
TheexpressioninEq.(24)isused,alongwithacontri- approximation that the correlator method we employ in
butionfrombulkscalarparticles;thesehavethestandard this paper picks up only the ground state of the string.
cosh-typelongdistancebehaviourandtheircontribution Weproceednowtomeasuretheformfactorandinterpret
tothecorrelatorcanbeeasilyaccountedfor. Toimprove this as a way of measuring the interactions of the string
fitting performance,the scalarmassis obtainedfromfits on various length scales.
tocorrelatorswithperiodicboundaryconditions,leaving Thestringmovesnon-relativisticallyk σLinalmost
≪
onlytherelativeamplitudesofthetwocontributionsand all the cases studied, so while the expressions derived in
8
40 11.8 1000
π/64 3π/64
39.96 11.6 L=32
L=64
L=128
if/v39.92 if/v11.4 SLt=e2p5 f6unction
39.88 11.2 100
39.8460 16 3t2 48 64 1410 16 3t2 48 64 σL f(k)/v
i
5π/64 7π/64
5.6 3.5 10
if/v 5.2 if/v 3
4.8 2.5
4.4 2 1
0 16 32 48 64 0 16 32 48 64 0.001 0.01 0.1 1
t t k/σL
FIG. 3. Typical plots of the string form factor for β =0.23, FIG. 4. String form factor for β = 0.23 and various L,
L = 64, as a function of distance for several k. The error withfitsgivenbytheansatzin Eq.(55),assumingtheshort-
bars here are bootstrap estimators for the random error in distance behaviour takes theform φ˜(k)=2/k, implying that
the correlator measurement; they are therefore correlated at thestringhas nodiscernible intrinsic width. In thisplot and
different values of t. Only the results for 16 . t . 48 are those that follow, the systematic measurement error in σ is
independentofseparationanditisresultsinthatregionthat not shown on thex-axis.
are used in theother plots presented later in this paper.
Section IIIC allow for this, there is no significant de-
10
pendence on the form factor with time separation; there
areonlyshort-distanceeffectsthatcould,inprinciple,be L=32
L=64
removed. L=128
L=256
Let us first demonstrate the applicability of Eq. (21). Step function
Figure 3 shows the form of the processed form factor
qreusaunlttsita(wtiveealylsoidteensttiecdaltrheesutletcshwnieqrueeowbtitahinTed,=al4bLeitanadt σ f(k)/v 1
√
considerably greater computational cost). It is assumed i
that the form factor is pure imaginary, we have no way
of determining its phase. We cannot subtract the bulk
scalar contribution; this is the principal source of devia-
tions from the expected behaviour at short and interme-
diate distances. 0.1
0.1 1 10
Takingresultsatlongdistancefromcorrelatorssimilar k/√σ
to those shown in Figure 3, we arrive at Figure 4. Here,
appropriately scaled form factor measurements are plot-
FIG. 5. As for for Figure 4 but scaled instead by √σ. One
tedasafunctionofthedimensionlesscombinationk/σL.
would expect the combination k/√σ to parameterise the in-
For comparison, the step-function result from Eq. (9) is trinsic width of a string. The data do not collapse onto a
shown. A thin string of this form would be independent single curve.
of volume L, and it is clear that at long distances (small
k), the behaviour of the form factors is independent of
volume.
for all k. This is clearly not the case, and we instead
While Figure 4 is scaled in such a way that the re-
have dependence on volume. Irrespective of the scaling
sults can be compared with the form factors of pointlike
employed we see that for larger L, the form factor falls
solitons (as the x-axis plots total momentum over rest
off faster with increasing k, suggesting that the string
energy) and to demonstrate that the string moves non-
indeed becomes broader at larger volumes.
relativistically in most of the results presented here, to
see whether the string form factor were independent of Finally, Figure 6 shows that because w2 (as defined
volumeadifferentscalingcombinationisrequired. With- in Section II) actually depends logarithmically on the
out logarithmic broadening, we would expect that the correlationlength ξ, there is no way to collapse the data
data points in Figure 5 would collapse onto a single line onto a single curve.
9
10 40
Step function
β=0.228
β=0.23
β=0.232 30
1
σ f(k)/v 2w20
√
i
0.1
10
0.01 0
0.1 1 10 10 100 1000
k/√σ L
FIG. 6. As for Figure 5 but for fixed L = 48 and various β. FIG. 7. The values of w2 obtained when fitting Eq. (55)
As can be seen, the data do not collapse onto a single line to the data shown in Figure 4; this shows the width of a
2
evenforconstantLduetotheweakdependenceofw onthe stringasafunctionofLforβ =0.23. Alinearrelationshipis
bulk correlation length. This supports ouransatz, Eq. (55). apparentandafitofthesepointstoEq.(42)isshown,yielding
σ = 0.0191 0.0003 – a result that is fairly close to but not
±
in excellent agreement with the literature, perhaps due to
C. Broadening unaccounted systematic effects. Nonetheless, the behaviour
is exactly that of Eq. (4).
The analytic approximation in Eq. (41) suggests that
the string form factor should decay as
conclude that the behaviour observed is entirely due to
f(k) φ˜(k)e−21k2w2, (55) the fluctuations in the string.
| |≈| |
where w2 is the ‘width’ of the string due to fluctuations
asderivedinSectionIIID,andφ˜(k) is the intrinsic form VI. CONCLUSIONS
factor of the unbroadened string. Unfortunately, when
the string is moving non-relativistically, it is difficult to We have demonstrated how to study the properties of
distinguishbetweendifferentchoicesofφ˜(k)–allofwhich topological strings and domain walls in quantum field
must fall off as 2iv/k at small k. For the time being we theoryusing two-pointfieldcorrelationfunctions. In the
shall assume that the simplest model – that of a step actualnumericalcalculationwe used the 3D Ising model
function at short distances, given in Eq. (10) – is valid which is in the same universality class as the 2+1D real
and take φ˜(k) 1/k. scalarfield theory,and thereforehas the samebehaviour
∝
If the string had intrinsic width and took the form nearthe criticalpoint. The model is alsoexactlydualto
of a smooth kink then φ˜(k) would be given by Eq. (8). the confining Z2 gauge theory.
However, for k/(σL) 1 at fixed L the difference in The measured ground state energy E shows finite-
0
≪
the two curves is insignificant and therefore we favour sizebehaviourwhichiscompatiblewiththeLu¨scherterm
the simpler fit arising from step-function model, even if generated by quantum fluctuations of Goldstone modes.
it does not capture all the physics on the shorter length In the string form factor, these same Goldstone modes
scales. We shall discuss the prospects for measuring an suppress short-wavelength modes leading to behaviour
intrinsic width in the conclusions. that can be interpreted as logarithmic broadening of the
It is not possible to establish conclusively based on effective string width. This is consistent with previous
Figure 4 that logarithmic broadening is taking place; in- studiesofothermeasuresofstringwidthsordomainwall
deed the high-k behaviour of Eq. (8) is rather similar. widths in statistical physics models. However, we stress
However, in Figures 5 and 6 we can see that even when that in contrast with the quantities studied before, the
correctly scaled the data do not collapse onto a single string form factor is a well-defined quantum observable.
line for different β or L. Although, in principle, it would be possible to use the
Finally,theresultsoffittingEq.(55)tothedatashown formfactor to probe the intrinsic structure of the string,
in Figures 4 or 5 are shown in Figure 7 (although addi- thiswouldrequireexcellentdataandmaynotbefeasible.
tional values of L are shown in the latter). There is a Any deviations from zero intrinsic width in the results
clear linear relationship between w2 and logL, consis- presented here were within the errors of our fits..
tent with Equations (4) and (39). We must therefore For zero-dimensional topological solitons there are no
10
massless fluctuations and the spectrum of Goldstone ACKNOWLEDGMENTS
modes is relatively straightforward. For example, in the
caseofaλφ4kink,theonlyGoldstonemodeisthatwhich
we exploit in measuring the kink’s mass. In the present
model, this situation is complicated considerably. How- Our simulations made use of facilities at the Finnish
ever we have successfully demonstrated that our tech- CentreforScientificComputingCSC.ARwassupported
nique is again of use. by STFC grant ST/J000353/1; KR acknowledges sup-
We would expect our approach to be applicable, with portfromtheAcademyofFinlandproject1134018. This
some modification, to confining strings in SU(N) gauge workwasfacilitatedbyRoyalSocietyInternationalJoint
theory. Project JP100273.
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