1. The spectrum of 3d 3-states Potts model and universality
Mario Gravina
Univ. della Calabria & INFN
collaborators: R. Falcone, R.Fiore, A. Papa
SM & FT 2006, Bari

2. OUTLINE introduction 3d 3q Potts model numerical results conclusions
1) Svetitsky-Yaffe conjecture
2) Universal spectrum conjecture
3d 3q Potts model
numerical results
conclusions
SM & FT 2006, Bari

3. 1) SVETITSKY-YAFFE CONJECTURE
Universality
1) SVETITSKY-YAFFE CONJECTURE
SU(N) d+1
confinement-deconfinement
Theories with different microscopic interactions but possessing the same underlying global symmetry have common long-distance behaviour
Z(N) d
order-disorder
finite temperature
what about 1st order phase transition?
if transition is 2nd order
SM & FT 2006, Bari

4. 2) universal mass spectrum
correlation function of
local order parameter
m1, m2, m3 …
SM & FT 2006, Bari

5. universality conjecture
Ising 3d
theory 1
theory 2
lF4 3d
SU(2) 4d
theory 3
Agostini at al. 1997
Caselle at al. 1999
Fiore, Papa, Provero 2003 m1 m2 m3 m4 m1 m2 m3 m4 m4 m1 m4 m1 m4 m1 = m1 m2 m3 m4 = m3 m1 m3 m1 m3 m1 = = m2 m1 m2 m1 m2 m1 = =
SM & FT 2006, Bari

6. We want to test these two aspects of universality
3d 3q POTTS MODEL h b bc hc
L=48
1) 1st order transition
L=70
2) 3d Ising point
MONTE CARLO simulations
CLUSTER ALGORITHM
to reduce autocorrelation time
SM & FT 2006, Bari

7. Potts model Z(3) breaking order-disorder PHASE TRANSITION
SM & FT 2006, Bari

8. Phase diagram h=0 weak 1st order transition point
Is the mass spectrum universal?
1st order critical lines
2nd order critical endpoint b
Z(3) broken phase
2nd order critical ISING endpoint
h=0
Does universality hold also for weak 1st order transition?
comparison with SU(3)
(work in progress) bc
Falcone, Fiore, Gravina, Papa
Z(3) symmetric phase h hc
SM & FT 2006, Bari

9. h=0 – 1st order transition
order parameter is the magnetization
global spin
SM & FT 2006, Bari

10. h=0 – 1st order transition at finite volume
tunneling effects
0.5508
between symmetric and broken phase
between degenerated broken minima
complex M plane
SM & FT 2006, Bari

11. h=0 – 1st order transition at finite volume
To remove the tunneling between broken minima we apply a rotation
only the real phase is present
SM & FT 2006, Bari

12. Masses’ computation MASS CHANNELS
by building suitable combinations of the local variable
ZERO MOMENTUM PROJECTION
by summing over the y and z slices
VARIATIONAL METHOD
to well separate masses contributions in the same channel
(Kronfeld 1990)
SM & FT 2006, Bari
(Luscher, Wolff 1990)

13. 0+ CHANNEL 2+ CHANNEL b=0.5508 h=0 m0+=0.1556(36) m2+=0.381(17) meff r
SM & FT 2006, Bari

14. masses’ computation 1st order transition n=1/3 0+ channel 2+ channel
0.60
m0+(b1)
m0+(b2)
n=1/3
m0+ (b1)=0.1556
0.5508
b1=0.5508
bt= b
0+ channel
2+ channel
in the scaling region b2
SM & FT 2006, Bari
– at least

15. mass ratio
m2+
m0+
prediction of 4d SU(3) pure gauge theory at finite temperature screening mass ratio at finite temperature? b
SM & FT 2006, Bari

16. 2nd order Ising endpoint
Karsch, Stickan (2000) h b bc hc z t Pc
ISING pt h b bc hc
temperature-like
(bc,hc)= ( (2), (10))
(tc,xc)= ( (2), (2))
ordering field-like
SM & FT 2006, Bari

17. 2nd order endpoint
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18. local variable
Correlation function
order parameter
SM & FT 2006, Bari

19. We separated contributions from two picks and calculated masses
mass spectrum
right-pick
0+ CHANNEL
2+ CHANNEL
t= x=0
We separated contributions from two picks and calculated masses
m0+
m2+
0.0749(63)
0.188(12)
3d ISING VALUE r
SM & FT 2006, Bari

20. CONCLUSIONS
We used 3d 3q Potts model as a theoretical laboratory to test some aspects of universality
THANK YOU
evidence found of universal spectrum
1) Ising point
2) weak 1st order tr. pt.
prediction of SU(3) screening spectrum?
left-pick?
SM & FT 2006, Bari

21. SM & FT 2006, Bari

22. weak 1st order transition discontinous order parameter
the jump is small Tt
SM & FT 2006, Bari

23. Phase diagram h=0 weak 1st order transition point
Mass spectrum is universal?
1st order critical lines
2nd order critical endpoint b
Z(3) broken phase
2nd order critical ISING endpoint
h=0
Universality also holds for weak 1st order transition? bc
Z(3) symmetric phase h hc
SM & FT 2006, Bari

24. Universality order parameter susceptibility correlation lenght
Critical exponents Tc
order parameter Tc
susceptibility
correlation lenght Tc
SM & FT 2006, Bari