the spectrum of 3d 3-states potts model and universality mario gravina univ. della calabria &...
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The spectrum of 3d The spectrum of 3d 3-states3-states
Potts model and Potts model and universalityuniversality
Mario GravinaMario Gravina
Univ. della Calabria & Univ. della Calabria & INFNINFN
SM & FT 2006, BariSM & FT 2006, Bari
collaborators: R. Falcone, R.Fiore, A. Papacollaborators: R. Falcone, R.Fiore, A. Papa
OUTLINEOUTLINE
SM & FT 2006, BariSM & FT 2006, Bari
introductionintroduction
3d 3q Potts 3d 3q Potts modelmodelnumerical resultsnumerical results
conclusionsconclusions
1) Svetitsky-Yaffe conjecture1) Svetitsky-Yaffe conjecture
2) Universal spectrum 2) Universal spectrum conjectureconjecture
UniversalityUniversality
SM & FT 2006, BariSM & FT 2006, Bari
Theories with different microscopic Theories with different microscopic interactions but possessing the interactions but possessing the
same underlying global symmetry same underlying global symmetry have common long-distance have common long-distance
behaviourbehaviour
1) SVETITSKY-YAFFE 1) SVETITSKY-YAFFE CONJECTURECONJECTURE
SU(N) SU(N) d+1d+1confinement-confinement-
deconfinementdeconfinement
Z(N) dZ(N) dorder-disorderorder-disorder
if transition is 2nd orderif transition is 2nd order
finite temperaturefinite temperature
what about 1st order phase what about 1st order phase transition?transition?
2) universal mass 2) universal mass spectrumspectrum
rm1ji
1eapp)r(C
1m1
SM & FT 2006, BariSM & FT 2006, Bari
r ,rji ...eaeaeapp)r(C rm3
rm2
rm1ji
321
mm11, m, m22,, mm3 3
……
local order local order parameterparameter
ipcorrelation function correlation function ofof
universality conjectureuniversality conjecture
56.2m
m
0
2
SM & FT 2006, BariSM & FT 2006, Bari
mm
11
mm
22
mm
33
mm
44
mm
11
mm
22
mm
33
mm
44
mm
11
mm
22
mm
33
mm
44
theory 1theory 1 theory 2theory 2 theory 3theory 3
mm22
mm
11
mm22
mm
11
mm22
mm
11
mm44
mm
11mm33
mm
11
mm33
mm
11
mm44
mm
11 mm33
mm
11
mm44
mm
11
==
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==
==
== ==
Ising 3dIsing 3d dd SU(2) 4dSU(2) 4dCaselle at al. 1999Caselle at al. 1999 Fiore, Papa, Provero Fiore, Papa, Provero
20032003
83.1m
m
0
0
Agostini at al. Agostini at al. 19971997
SM & FT 2006, BariSM & FT 2006, Bari
CLUSTER ALGORITHMCLUSTER ALGORITHMto reduce autocorrelation to reduce autocorrelation timetime
We want to test these two aspects of We want to test these two aspects of universalityuniversality
3d 3q POTTS MODEL3d 3q POTTS MODEL
1) 1st order 1) 1st order transitiontransition2) 3d Ising point2) 3d Ising point
?mm
0
2
MONTE CARLO simulationsMONTE CARLO simulations
L=4L=488L=7L=700 h
c
hc
Potts modelPotts model
SM & FT 2006, BariSM & FT 2006, Bari
iji
hijihH 2,1,0i
Z(3) breakingZ(3) breaking
order-disorderorder-disorder
PHASE TRANSITIONPHASE TRANSITION
Phase diagramPhase diagram
SM & FT 2006, BariSM & FT 2006, Bari
h
c
hc
1st order critical lines
2nd order critical endpoint
h=0h=0
weak 1st order transition weak 1st order transition pointpoint
2nd order critical ISING 2nd order critical ISING endpointendpoint
h=0h=0Z(3) symmetric phaseZ(3) symmetric phase
Z(3) broken phaseZ(3) broken phaseDoes universality hold Does universality hold also for weak 1st order also for weak 1st order
transition?transition?
Is the mass spectrum Is the mass spectrum universal?universal?
comparison with comparison with SU(3)SU(3)
(work in progress)(work in progress)Falcone, Fiore, Gravina, Falcone, Fiore, Gravina, PapaPapa
h=0 – 1st order transition h=0 – 1st order transition
.const.c.css32
Hji
ijji
ji
SM & FT 2006, BariSM & FT 2006, Bari
2,1,0i i3
2i
i es
order parameter is the order parameter is the magnetizationmagnetization
i
i3 sL1
M
global global spinspin
h=0 – 1st order transition h=0 – 1st order transition at finite volume at finite volume
t
SM & FT 2006, BariSM & FT 2006, Bari
tunneling effectstunneling effects
between symmetric between symmetric and broken phaseand broken phase
between degenerated between degenerated broken minimabroken minima
0.5505650.550565
0.55080.5508
complex M complex M planeplane
SM & FT 2006, BariSM & FT 2006, Bari
h=0 – 1st order transition h=0 – 1st order transition at finite volume at finite volume
To remove the tunneling To remove the tunneling between broken minima we between broken minima we apply a rotation apply a rotation
3
2i
i es
3
2i
i es
only the real only the real phase is presentphase is present
Masses’ computationMasses’ computation
r ,rji
)1r(C)r(C
ln)r(meff
SM & FT 2006, BariSM & FT 2006, Bari
...eaeaeass)r(C rm3
rm2
rm1jiij
321
VARIATIONAL METHODVARIATIONAL METHODto well separate masses contributions to well separate masses contributions
in the same channelin the same channel
by summing over the y and z slicesby summing over the y and z slicesZERO MOMENTUM PROJECTIONZERO MOMENTUM PROJECTION
MASS CHANNELSMASS CHANNELS
by building suitable combinations of by building suitable combinations of the local variablethe local variable
1eff m)r(m
(Kronfeld (Kronfeld 1990)1990)
)ss(ssziyii
)0(i
)ss(ssziyii
)2(i
(Luscher, Wolff (Luscher, Wolff 1990)1990)
SM & FT 2006, BariSM & FT 2006, Bari
0+ CHANNEL0+ CHANNEL
=0.5508 =0.5508 h=0h=0
2+ CHANNEL2+ CHANNEL
mm0+0+=0.1556(3=0.1556(36)6)mm2+2+=0.381(17)=0.381(17)
rr
mmeffeff
masses’ computationmasses’ computation
t
SM & FT 2006, BariSM & FT 2006, Bari
0.55080.5508
0.600.60
0+ channel0+ channel2+ channel2+ channel
1st order transition1st order transition
=1/3=1/3
)(m)(m
20
10
t2
t1
in the scaling in the scaling regionregion
t2
t1
mm000.1550.1556611=0.5508=0.5508
tt=0.550565=0.550565
mm00mm0+0+
((
0.550565 – 0.56 at0.550565 – 0.56 at leastleast
0.5505650.550565
mm0+0+mm2+2+
mass ratiomass ratio
SM & FT 2006, BariSM & FT 2006, Bari
)10(43.2mm
0
2
prediction of 4d SU(3) prediction of 4d SU(3) pure gauge theory at pure gauge theory at
finite temperature finite temperature screening mass ratio screening mass ratio at finite temperature?at finite temperature?
mm2+2+
mm0+0+
2nd order Ising endpoint2nd order Ising endpoint
hMEH
M~
E~
H iji
hijihH
SM & FT 2006, BariSM & FT 2006, Bari h
c
hc
rMEE~
sEMM
~
temperature-temperature-likelikeordering field-likeordering field-like
ISING ptISING pt
((cc,h,hcc)=)=
(0.54938(2),0.000775(10))(0.54938(2),0.000775(10))((cc,,cc)=)=
(0.37182(2),0.25733(2))(0.37182(2),0.25733(2))ssrr-0.69-0.69
Karsch, Stickan (2000)Karsch, Stickan (2000)
h
c
hc
Pc
2nd order endpoint2nd order endpoint
37182.0c
SM & FT 2006, BariSM & FT 2006, Bari
0.372330.37233 0.372480.37248
M~
M~
M~
local variablelocal variable
3
3i iihi 2
sm~
i ij
jihissEMM
~
i
i3 m~L1
M~
SM & FT 2006, BariSM & FT 2006, Bari
jiij m~m~)r(C
order order parameterparameter
Correlation Correlation functionfunction
mass spectrummass spectrum
)37(51.2mm
0
2
SM & FT 2006, BariSM & FT 2006, Bari
We separated contributions from We separated contributions from two picks and calculated massestwo picks and calculated masses
0+ CHANNEL0+ CHANNEL2+ CHANNEL2+ CHANNEL
right-pickright-pick
=0.37248 =0.37248
0.0749(63)0.0749(63)0.188(12)0.188(12)
56,2m
m
0
2
3d ISING VALUE3d ISING VALUE
mm2+2+mm0+0+
rr
CONCLUSIONSCONCLUSIONS
SM & FT 2006, BariSM & FT 2006, Bari
We used 3d 3q Potts model as a We used 3d 3q Potts model as a theoretical laboratory to test some theoretical laboratory to test some
aspects of universalityaspects of universality
1) Ising point1) Ising point evidence found of evidence found of universal spectrumuniversal spectrum
2) weak 1st 2) weak 1st order tr. pt.order tr. pt.
prediction of SU(3) prediction of SU(3) screening screening spectrum?spectrum?
THANK YOUTHANK YOU
)10(43.2mm
0
2
left-pick?left-pick?
SM & FT 2006, BariSM & FT 2006, Bari
1st order transition1st order transition
SM & FT 2006, BariSM & FT 2006, Bari
Tt
discontinous order discontinous order parameterparameter
weakweak
the jump is the jump is smallsmall
Phase diagramPhase diagram
SM & FT 2006, BariSM & FT 2006, Bari
h
c
hc
1st order critical lines
2nd order critical endpoint
h=0h=0
weak 1st order transition weak 1st order transition pointpoint
2nd order critical ISING 2nd order critical ISING endpointendpoint
h=0h=0Z(3) symmetric phaseZ(3) symmetric phase
Z(3) broken phaseZ(3) broken phaseUniversality also holds Universality also holds
for weak 1st order for weak 1st order transition?transition?
Mass spectrum is Mass spectrum is universal?universal?
UniversalityUniversality
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Critical Critical exponentsexponents
order parameterorder parameterTc
susceptibilitysusceptibilityTc
correlation lenghtcorrelation lenght
Tc