Effects of boundary conditions and gradient flow in 1+1 dimensional lattice $\phi^4$ theory

0.9 MB·

by  A. Harindranath

Effects of boundary conditions and gradient flow in 1+1 dimensional lattice φ4 theory
A. Harindranath1,∗ and Jyotirmoy Maiti2,†
1Theory Division, Saha Institute of Nuclear Physics
1/AF Bidhan Nagar, Kolkata 700064, India
2Department of Physics, Tehatta Government College,
Tehatta, Nadia, West Bengal 741160, India
(Dated: January 17, 2017)
In this work we study the effects of gradient flow and open boundary condition in the temporal
directionin1+1dimensionallatticeφ4 theory. Simulationsareperformedwithperiodic(PBC)and
open(OPEN)boundaryconditionsinthetemporaldirection. TheEffectsofgradientflowandopen
boundaryonthefieldφandthesusceptibilityarestudiedindetailalongwiththefinitesizescaling
analysis. Inbothcases,atagivenvolume,thephasetransitionpointisshiftedtowardsalowervalue
of lattice coupling λ for fixed m2 in the case of OPEN as compared to PBC with this shift found
0 0
to be diminishing as volume increases. We compare and contrast the extraction of the boson mass
7
from the two point function (PBC) and the one point function (OPEN) as the coupling, starting
1
frommoderatevalues,approachesthecriticalvaluecorrespondingtothevanishingofthemassgap.
0
2 Inthecriticalregion,boundaryartifactsbecomedominantinthelatter. Ourstudiespointtowards
theneedforadetailedfinitevolume(scaling)analysisoftheeffectsofOPENinthecriticalregion.
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∗ [email protected]
† [email protected]
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I. INTRODUCTION AND MOTIVATION
The quantum field theory of 1+1 dimensional φ4 interaction, in spite of its apparent simplicity, has a very rich
structureandhencehasbeenthetestinggroundforvariousnewnon-perturbativeapproachestowardsthefieldtheory.
It also provides ample opportunity to study newly proposed algorithms and calculational techniques. Extensive
numerical results have been presented in this theory in an earlier work [1]. In this work we present numerical studies
in this theory in the context of comparison between periodic (PBC) and open boundary conditions (OPEN) in the
temporal direction and the effects of gradient flow (also known as Wilson flow).
In addition to the periodic boundary conditions in both temporal and spatial directions, one can have other types
of boundary conditions. For the scalar field, for example, one can have anti-periodic boundary condition in the
spatial direction (APBC). In the latter case one can study quantum kinks [2]. Lattice Quantum ChromoDynamics
(LQCD) conventionally uses PBC in both the temporal and spatial directions for the gauge field. However, in this
case, the spanning of gauge configurations over different topological sectors becomes more and more difficult as the
continuum limit is approached. As a remedy, open boundary condition in the temporal direction has been proposed
[3–5]. Numerical studies in pure Yang-Mills theory [6–10] and QCD [11] have yielded encouraging results.
InorderforOPENtobeeffective,boundaryartifactsshouldbenegligiblesothatonehasabulkregionofconsider-
ableextent. Thisispossibleaslongthesystemisnotgapless(critical)[12]. ThesuccessofOPENinpureYang-Mills
theory and QCD hinges on the existence of a mass gap in these systems, namely glueball and pion respectively. On
the other hand, 1+1 dimensional lattice φ4 theory offers an opportunity to investigate in detail the artifacts induced
by the open boundary condition as the system approaches criticality.
Forextractingvariousobservablesinlatticefieldtheories,smoothingoflatticefieldsisessential,inordertoovercome
lattice artifacts. The gradient (Wilson) flow [13–15] provides a very convenient tool for smoothing, with a rigorous
mathematicalunderpinning. Forsomerecentstudiesofgradientflowinthecontextofscalartheory,seeRefs. [16–18].
It will be very interesting to study this in 1+1 dimensional lattice φ4 theory.
II. BOUNDARY CONDITIONS AND GRADIENT FLOW
In the continuum, the Euclidean Lagrangian (density) for φ4 theory is given by
1 1 λ
L= ∂ φ∂ φ + m2φ2 + φ4. (1)
2 µ µ 2 4!
Onaperiodic(onallspace-timedirections)lattice,theEuclideanactionin‘d’space-timedimensionisconventionally
written as [1]
S =−(cid:88)(cid:88)φ˜ φ˜ +(cid:18)d+ m20(cid:19)(cid:88)φ˜2 + λ0 (cid:88)φ˜4 (2)
x x+µ 2 x 4! x
x µ x x
where all the parameters and fields have been made dimensionless by multiplying them with appropriate powers of
lattice spacing ‘a’.
However, in order to impose open boundary condition in the temporal direction we follow the construction of the
(cid:80)
transfermatrix. ForthispurposethelatticeactioniswrittenintermsoftimesliceactiondensityE(t)asS = E(t)
t
andthekinetictermintemporaldirectionisdistributedsymmetricallyaroundthetimeslice‘t’. Thedetailsaregiven
below.
First, with the aid of forward and backward lattice derivatives
1 1
∂fφ= (φ −φ ) and ∂bφ= (φ −φ ) , (3)
µ a x+µ x µ a x x−µ
we write the symmetrized expression for kinetic term as
1∂ φ∂ φ= 1(cid:0)∂fφ∂fφ+∂bφ∂bφ(cid:1)
2 µ µ 4 µ µ µ µ
(cid:32) (cid:33)
1 (cid:88)
= 2 φ2 + φ2 + φ2 − 2φ φ − 2φ φ . (4)
4a2 x x+µ x−µ x x+µ x x−µ
µ
This enables one to write down the time slice action density for periodic lattice as
1 1
E (t)=T + V + T + T (5)
PBC t t 2 t,t+1 2 t−1,t
3
where
(cid:88) (cid:18) 1 a2λ (cid:19)
V = a2m2φ2 + φ4 ,
t 2 (cid:126)x,t 4! (cid:126)x,t
(cid:126)x
1(cid:88)(cid:16) (cid:17)
T = 2 φ2 + φ2 + φ2 − 2 φ φ − 2 φ φ ,
t 4 (cid:126)x,t (cid:126)x+kˆ,t (cid:126)x−kˆ,t (cid:126)x,t (cid:126)x+kˆ,t (cid:126)x,t (cid:126)x−kˆ,t
(cid:126)x,k
1(cid:88)(cid:16) (cid:17)
T = φ2 + φ2 − 2 φ φ ,
t,t+1 2 (cid:126)x,t (cid:126)x,t+1 (cid:126)x,t (cid:126)x,t+1
(cid:126)x
1(cid:88)(cid:16) (cid:17)
T = φ2 + φ2 − 2 φ φ
t−1,t 2 (cid:126)x,t (cid:126)x,t−1 (cid:126)x,t (cid:126)x,t−1
(cid:126)x
with kˆ being the unit vector in an arbitrary spatial direction.
Following this definition, we denote e−Hm(t,t+1) as the general transfer matrix element between the time slices t
and t+1 where
1 1
H (t,t+1)= (T +V ) + (T +V ) + T . (6)
m 2 t t 2 t+1 t+1 t,t+1
Particularly for a lattice of temporal extent ‘T’, the transfer matrix element between the boundary time slices
t=T −1 and t=0 is determined by
1 1
H (T −1,0) = (T +V ) + (T +V ) + T
m 2 T−1 T−1 2 0 0 T−1,0
Now, if the temporal boundary becomes open, corresponding term drops out from the partition function. This
leads us to relate the actions for lattices with two different boundary conditions in temporal direction (periodic and
open) as S =S +∆S where ∆S =H (T −1,0).
PBC OPEN m
The absence of the term ∆S from the action for lattice with open boundary (temporal direction), in turn, also
modifies the expressions for action densities at the temporal boundaries. They are given by
1 1
E (t=0)= (T + V ) + T (7)
OPEN 2 0 0 2 0,1
1 1
and E (t=T −1)= (T + V ) + T . (8)
OPEN 2 T−1 T−1 2 T−2,T−1
Within the bulk (0<t<T −1), E (t) = E (t).
OPEN PBC
Inordertosmooththelatticeφfield,gradientflowisused. Forφ4 theoryintheEuclideanspace,inthecontinuum,
the flow equation is given by
∂ψ(x,τ) δS[ψ]
=−
∂τ δψ(x,τ)
λ
=∂ ∂ ψ(x,τ)−m2ψ(x,τ) − ψ3(x,τ) (9)
µ µ 6
where ψ(x,τ =0)=φ(x) with ‘τ’ being the flow time.
We numerically solve this equation on the lattice using the 2nd order Runge-Kutta method.
III. EXTRACTION OF BOSON MASS FROM LATTICES WITH DIFFERENT BOUNDARIES
For the extraction of boson mass we have used the simplest and the most familiar scalar operator - the time sliced
field φ(t)= 1 (cid:80)φ((cid:126)x,t) where V is the spatial volume of the lattice. The mass can be easily extracted from the two
V
(cid:126)x
point correlation function for this scalar operator which, in case of periodic boundary in temporal direction, behaves
as
(cid:104) (cid:105)
G(t)=(cid:104)φ(t)φ(t=0)(cid:105) ≈C + C e−mt + e−m(T−t)
PBC 0 1
(cid:16)T (cid:17)
=C + 2C e−mT/2 coshm −t (10)
0 1 2
4
where
|(cid:104)0|φ(0)|B(cid:105)|2
C = (11)
1 2m
with|B(cid:105)beingtheonebosonstate. Toimprovestatistics,onecanaverageoverthesourcetimeaswell. Theeffective
mass can be evaluated by solving the equation F(m)=0
(cid:104) (cid:105) (cid:104) (cid:105)
where F(m) = (r −1) coshm(∆t−1) − coshm∆t + (1−r ) coshm(∆t+1) − coshm∆t (12)
1 2
G(t−1) G(t+1)
with r = , r = and ∆t=T/2−t . (13)
1 G(t) 2 G(t)
Incaseofopenboundaryintimedirection,toavoidtheboundaryeffectsoneneedstobewellwithinthebulkwhile
computing the two point correlation function. However, the second exponential will be absent from the expression
of the two point function due to the loss of periodicity. On the other hand, as the time translational invariance is
also lost in this case, one cannot average over the source time as well. However, within the bulk, well away from the
boundaryregiontranslationalinvarianceisrecovered. Soonecantakeaverageoverfewtimeslicestoregainstatistics.
However, this effort breaks down as one approaches the critical region. It will be shown later on in this study
that, the effect of open boundary starts to engulf the whole bulk region as we move towards the critical point. Mass
extraction from two point function becomes almost impossible.
Surprisingly, the open boundary itself opens up new pathways to extract the mass. Following Ref. [9], in this
section we review how the boson mass can be extracted from a one-point function in open boundary using a generic
time sliced scalar operator O(t).
We start from
(cid:82)
Dφ O(t) e−SOPEN
(cid:104)O(t)(cid:105) = (cid:82)
OPEN
Dφ e−SOPEN
(cid:82) (cid:82)
(cid:46)
Dφ O(t) e−SPBC+∆S Dφ e−SPBC
= (cid:82) (cid:82)
(cid:46)
Dφ e−SPBC+∆S Dφ e−SPBC
(cid:10)O(t) e∆S(cid:11)connected
=(cid:104)O(t)(cid:105) + PBC (14)
PBC (cid:104)e∆S(cid:105)
PBC
1(cid:68) (cid:69)connected
=(cid:104)O(t)(cid:105) + O(t) eHm(T−1,0) (15)
PBC r PBC
where r =(cid:104)e∆S(cid:105)PBC =(cid:104)eHm(T−1,0)(cid:105)PBC. As eHm(t,t+1) is also a scalar operator, from eq. 15 we have
(cid:16)T (cid:17)
(cid:104)O(t)(cid:105) ≈(cid:104)O(t)(cid:105) + 2C(cid:48) e−mT/2 coshm −t . (16)
OPEN PBC 1 2
where m is the scalar boson mass.
Thus we find that one can extract certain two-point correlators computed with periodic boundary condition in the
temporaldirectionbyanalyzingthedataforthefunctionalaverageofascalaroperator(one-pointfunction)computed
withopenboundary(inthetemporaldirection)intheregionoftwhereitdiffers, duetothebreakingoftranslational
invariance, from the same computed with periodic boundary. Now due to time translational invariance in the case of
PBC, we can assume (cid:104)O(t)(cid:105) to be constant and the evaluation of effective mass can then be done again following
PBC
the eqns. 12 and 13.
IV. NUMERICAL RESULTS
As we have restricted ourselves into 1+1 dimension within this study, the fields are dimensionless here. The
notations for the bare parameters of the theory on the lattice are chosen to be m2 = a2m2 and λ = a2λ. As we
0 0
know, for the stability of the theory one must have λ ≥0 and the phase transition associated with the spontaneous
0
breaking (or restoration) of Z(2) symmetry takes place only for m2 <0.
0
For the study with periodic boundary in both directions, following the method of Brower and Tamaya [19], we
have used Wolff’s single cluster algorithm [20, 21] blended with the standard metropolis algorithm in 1:1 ratio for
5
1.132
1.131
flow level = 80
1.13
1.008
>
> ---
(t)|φ1.007
<|
<--- 1.006 flow level = 40
0.893
0.892 flow level = 0
0 20 40 60 80 100 120
t --->
FIG. 1. Plot of expectation value of |φ(t)| for m2 =−0.5, λ =1.65 and L=128 with PBC for three different gradient flow
0 0
levels.
the generation of field configurations. For the details of the whole procedure, see [1]. However, in the study with
open boundary in the temporal direction (periodic in spatial direction), we resort only to the standard metropolis
algorithm for configuration generation.
Followingthediscussionsin[1], wehaveused(cid:104)|φ|(cid:105)astheorderparametertoinvestigatethephasestructure. Here,
(cid:80)
φ= φ(x)/lattice volume. Theabsolutevalueistakentoavoidtheeffectoftunnellingenforcedbythealgorithm
sites x
of configuration generation. For the phase diagram we mainly resorted to the former study [1]. Here, we will study
the effects of gradient flow and boundary conditions on the phase diagram in due course.
In this work, we explored the phase structure and the spectrum of the theory for two different sets of bare pa-
rameters given by m2 = −0.5 and m2 = −1.0. For each set of parameters (i.e., m2 and λ ), we have first discarded
0 0 0 0
106 configurations for thermalization and then generated another 108 configurations for the measurement purpose.
Measurementsaredoneononeineverythousandconfigurations. Inordertostudythefinitesizeeffectsontheresults,
whole investigations have been done with four different lattice volumes such as 482, 642, 962 and 1282.
Configurations chosen for measurements are placed under gradient flow which is run for one hundred steps, to be
called as flow level, with a stepsize of δτ =0.02. Measurements are done after every ten flow levels in addition to the
measurement done before the flow is started.
Now we discuss the effects of gradient flow and boundary conditions on various observables of interest. We start
with the basic observable- time sliced scalar field φ(t). For the reasons stated above, here too, we take absolute value
before evaluating the configuration average.
In Figs. 1 and 2, respectively for periodic and open boundary conditions in temporal directions, we present the
expectation value of |φ(t)| in three different subdiagrams- one without any gradient flow and two others with two
different levels of gradient flow all for a particular set of lattice parameters m2 =0.5, λ =1.65 and 1282 lattice. The
0 0
figures clearly show the smoothening effect with the increase of flow level. Note that, values of (cid:104)|φ(t)|(cid:105) are gradually
rising up with increasing flow level. The widths of windows for the values of (cid:104)|φ(t)|(cid:105) are taken to be same in all
the three subdiagrams instead of taking them to be proportional to the respective average values. This actually has
reduced the manifestation of smoothening effect to some extent.
To make exhibition of boundary effect clearer, in Fig. 3, we compare the behaviour of (cid:104)|φ(t)|(cid:105) for m2 =−0.5 and
0
L = 128 for periodic and open boundary conditions without any gradient flow for three different values of λ all in
0
the broken symmetric phase. For the smallest value of the coupling which is far away from the critical point, effects
ofopenboundaryarefoundtobeonlyattheedgesleavingalongbulkregionmatchingwiththecounterpartinPBC.
As the coupling increases one gets closer to the region of phase transition and effect of open boundary extends on
both side squeezing the bulk. For λ =1.86, we are already in the critical region and we observe that the bulk region
0
is almost vanished. The same is presented in Fig. 4 for gradient flow level 50. We notice that it only smoothens the
data (which is, although, hardly visible here because of the wide scale of values for (cid:104)|φ(t)|(cid:105) covered in the figure)
leaving the boundary effects unchanged.
In Fig. 5 we plot (cid:104)|φ|(cid:105) versus λ for different values of the gradient flow level for 1282 lattice at m2 = −0.5. The
0 0
6
1.132
1.13
flow level = 80
1.128
> 1.008
> ---
(t)|φ 1.006
<|
<--- 1.004 flow level = 40
0.894
0.892
flow level = 0
0.89
0 20 40 60 80 100 120
t --->
FIG. 2. Plot of expectation value of |φ(t)| for m2 = −0.5, λ = 1.65 and L = 128 with OPEN for three different levels of
0 0
gradient flow.
FIG. 3. Comparison of (cid:104)|φ(t)|(cid:105) between PBC and OPEN for m2 =−0.5, L=128 and three different values of λ without any
0 0
gradient flow.
7
FIG. 4. Comparison of (cid:104)|φ(t)|(cid:105) between PBC and OPEN for m2 =−0.5, L=128 and three different values of λ at gradient
0 0
flow level 50.
monotonous rise in the value of (cid:104)|φ|(cid:105) with increasing gradient flow level is consistent with the behaviour of (cid:104)|φ(t)|(cid:105).
Details of the behavior in the critical region are shown in the inset of the figure. The trend is seemed to be retained
across the phase transition.
In Fig. 6, we present the comparison between (cid:104)|φ|(cid:105) computed using PBC and the same obtained with OPEN for
differentlatticesizesatm2 =−0.5beforeapplyingthegradientflowonthefields. Deepinboththebroken-symmetric
0
andthesymmetricphases,forallthelatticevolumes,theresultsforPBCandOPENarefoundtobematchingwithin
our statistical error. However, inside the critical region a clear disagreement is observed. Phase transition appears
to take place at smaller values of λ in case of OPEN compared to PBC. However, the gap between the transition
0
point in two different boundaries seems to be vanishing with the increase of lattice size. This behavior is consistent
with the expectation that the effect of boundary surface diminishes in infinite volume limit. The picture remains to
be almost unaltered by the application of gradient flow other than raising the values universally little bit. This has
been emphasized in Fig. 7.
Now we study the effects of gradient flow and the boundary conditions on the behaviour of another observable
(cid:80)
of interest, susceptibility χ = (cid:104)φ(x)φ(0)(cid:105)/lattice volume. In Fig. 8, we show the behaviour of susceptibility as a
x
functionofλ form2 =−0.5withPBCon1282 latticefordifferentlevelsofgradientflow. Consistentwiththetrends
0 0
asobservedincaseof(cid:104)|φ|(cid:105),thepeakofsusceptibilitybecomeshigherandhigherasthelevelofgradientflowincreases
leaving the peak position unchanged.
Analogoustothestudydonefor(cid:104)|φ|,wecomparethesusceptibilitybetweenperiodicandopenboundaryconditions
fordifferentlatticesizes(L)atm2 =−0.5andgradientflowlevel50inFig. 9. Heretoo,weobservethatcomparedto
0
thecaseofPBC,thepeakisshiftedtowardsthesmallervaluesofλ incaseofOPENatafixedLandthemagnitude
0
of the shift decreases as L increases.
8
4
0.8
3.5
0.6
3
0.4
2.5
> 0.2
|φ 2
<|
1.88 1.92 1.96 2 2.04 2.08
1.5
1
flow level = 0
flow level = 30
0.5 flow level = 60
flow level = 90
0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
λ
0
FIG. 5. Plot of (cid:104)|φ|(cid:105) versus λ for different values of the gradient flow level for m2 =−0.5 and L=128 in case of PBC.
0 0
1.5 periodic
open
> 1
|φ
<|
0.5
L = 48 L = 64
0
1.5
> 1
|φ
<|
0.5
L = 96 L = 128
0
1 1.2 1.4 1.6 1.8 2 2.2 1 1.2 1.4 1.6 1.8 2 2.2
λ λ
0 0
FIG. 6. Plot of (cid:104)|φ|(cid:105) versus λ for PBC and OPEN without gradient flow for different L at m2 =−0.5.
0 0
ThisshiftinthepeakpositionofsusceptibilitybetweenPBCandOPENseemstobeunaffectedbygradientflowas
per our expectation. This has been demonstrated in Figs. 10 and 11 for the smallest and largest lattices sizes L=48
and L=128 respectively both with m2 =−0.5.
0
The fact that the value of (cid:104)|φ|(cid:105) changes with gradient flow even in the critical region raises an interesting question.
Does the determination of critical coupling in a finite size scaling (FSS) analysis gets affected by gradient flow? Here
we study the possible effect of gradient flow in the FSS of the data for (cid:104)|φ|(cid:105) to determine the critical coupling. We
follow the discussion of the main aspects of Finite Size Scaling [22–24] given in Ref. [1]. FSS assumes that, in a finite
system, out of the three length scales involved, namely, the correlation length ξ, the size of the system L and the
microscopic length a (lattice spacing), the last one drops out near the critical region due to universality.
For any observable P computed on a lattice of finite extent L, the nonanaliticity near the critical point in the
L
infinite volume limit can be expressed in the form of a scaling law P (τ)=A τ−ρ where τ = (λc −λ )/λc and ρ is
∞ P 0 0 0
the critical exponent associated with the observable. Following the arguments of FSS analysis, it can be shown that
(cid:104) (cid:105)
Lρ/ν/P (τ)= A−1 Aρ/ν C + D A−1/ν τ L1/ν + O(τ2) (17)
L P ξ P P ξ
where ν is the critical exponent associated with the correlation length ξ. Eq. (17) implies that if we plot Lρ/ν/P (τ)
L
9
1.5 periodic
open
> 1
|φ
<|
0.5
L = 48 L = 64
0
1.5
> 1
|φ
<|
0.5
L = 96 L = 128
0
1 1.2 1.4 1.6 1.8 2 2.2 1 1.2 1.4 1.6 1.8 2 2.2
λ λ
0 0
FIG. 7. Plot of (cid:104)|φ|(cid:105) versus λ for PBC and OPEN at gradient flow level 50 for different L at m2 =−0.5.
0 0
800
flow level = 0
flow level = 30
700 flow level = 60
flow level = 90
600
500
χ 400
300
200
100
0
1.7 1.8 1.9 2 2.1 2.2
λ
0
FIG. 8. Comparison of susceptibility for different levels of gradient flow with PBC at m2 =−0.5 and L=128.
0
versus the coupling λ for different values of L, all the curves will pass through the same point where τ = 0 or
0
equivalently λ =λc [24].
0 0
The critical behavior of (cid:104)φ(cid:105), the susceptibility χ and the mass gap m=1/ξ may be written as
(cid:104)φ(cid:105) = A−1 τβ, χ = A τ−γ and m = A−1 τν. (18)
φ χ ξ
From the general expectation that in 1+1 dimensions, φ4 theory and Ising model belong to the same universality
class, we have used the Ising values for the corresponding exponents as inputs in our FSS analysis. Thus, β =0.125,
γ =1.75 and ν =1.
Theplotof(cid:104)|φ|(cid:105)L0.125 versusλ fordifferentvaluesofthegradientflowlevelatm2 =−0.5withperiodicboundary
0 0
in the temporal direction is presented in Fig. 12. In spite of the fact that (cid:104)|φ|(cid:105) changes with gradient flow level, the
critical coupling λc is found to be unaffected by gradient flow. The critical coupling λc is found to be little less than
0 0
1.94 for m2 =−0.5. Similar FSS analysis has been done at m2 =−1.0 leading to the same conclusion along with an
0 0
estimated value of λc ≈4.46.
0
A similar FSS analysis of the susceptibility calculated at m2 = −0.5 for various values of gradient flow level (for
0
PBC) is presented in Fig. 13. Here, the smoothening effect of gradient flow has helped to pinpoint the location of
critical coupling λc which matches with the estimation done from the FSS analysis for (cid:104)|φ|(cid:105). Similar FSS analysis
0
10
150 200
periodic L = 48 L = 64
open
150
100
χ 100
50
50
0 0
400
600
L = 96 L = 128
300
400
χ 200
200
100
0 0
1.7 1.8 1.9 2 2.1 2.2 1.7 1.8 1.9 2 2.1 2.2
λ λ
0 0
FIG. 9. Comparison of susceptibility between PBC and OPEN for different L at m2 =−0.5 and gradient flow level 50.
0
150
flow level = 0 flow level = 30
periodic
open
100
χ
50
0
150
flow level = 60 flow level = 90
100
χ
50
0
1.6 1.8 2 2.21.6 1.8 2 2.2
λ λ
0 0
FIG.10. ComparisonofsusceptibilitybetweenPBCandOPENfordifferentlevelsofgradientflowatm2 =−0.5withL=48.
0
done at m2 =−1.0 also gives the corresponding value of λc matching with the same obtained from FSS analysis for
0 0
(cid:104)|φ|(cid:105).
Using the same ansatz, we have tried an FSS analysis for both (cid:104)|φ|(cid:105) and χ in case of open boundary in temporal
direction. The results are respectively presented in Figs. 14 and 15. It is obvious from the figures that the critical
pointisnotpinpointedthatclearlylikeinthecaseofPBC.Astrangepatternisfoundforsusceptibilityforwhichthe
curves with different L, instead of passing through a point, appear to overlap within statistical errors at and between
the values 1.92 and 1.94 of λ . Anyway, the effect of surface in case of open boundary needs to be taken into the
0
ansatz for finite size scaling analysis. For that purpose a more precise numerical study is necessary which is beyond
the scope of this study.
Finally, we present the results for the mass spectrum of the theory. The boson mass has been extracted from the
plateau along time direction for effective mass which is computed from two point and one point correlation function
of the time sliced scalar field φ(t) respectively for PBC and OPEN. For this purpose, we first find out the gradient
flowlevelforwhichtheplateauforeffectivemassisfoundtobethemoststableoneseparatelyforeachcoupling(also
for PBC and OPEN separately). The computation of mass has been done only for the two largest lattices namely
for L = 96 and L = 128 as for the smaller lattices, the temporal extent is not sufficiently long to get the fall of
correlationfunctionparticularlywithinthecriticalregion. Becauseofthis, itisnotmeaningfultoperformtheformal