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
OUTLINE
introduction
1) Svetitsky-Yaffe conjecture
2) Universal spectrum conjecture
3d 3q Potts model
numerical results
conclusions
SM & FT 2006, Bari
1) SVETITSKY-YAFFE
Universality
CONJECTURE
Theories
with different microscopic
SU(N)
d+1
Z(N) d
interactions but possessing the
same
order-disorder
symmetry have
common long-distance behaviour
confinement-deconfinement
finiteunderlying
temperature global
what about
if transition
1st order
is 2nd
phase
order
transition?
SM & FT 2006, Bari
2) universal mass spectrum
correlation function of
local order parameter
pi
m
 
m
m2r
m3r
1r
1r
C
(
r
)

p
p

a
e
C(r)  pi pij j a1e1
 a2e i  j ra, 3er    ...
1
m1, m
m12, m3 …

SM & FT 2006, Bari
universality conjecture
theory
Ising
3d1
theory
lF4 3d 2
Agostini at al. 1997
Caselle at al. 1999
m4
m1
m3
m1
m2
m1
SM & FT 2006, Bari
m4
m4
m1
=
m3
m2
=
m1
=
m=4
m3
 1.83
m
m
2
3
m0
=
m1
m1
m0 

m2m2
 2.56=
m m1
0
SU(2)
theory4d
3
Fiore, Papa, Provero 2003
m4
m1
m4
m3
m1
m3
m2
m1
m2
m1
We want to test these two aspects of
universality
3d 3q POTTS MODEL
1) 1st order transition
2) 3d Ising point
b
bc
MONTE m
CARLO
simulations
2
?
L=48
L=70
hc
CLUSTER
m0 ALGORITHM

to reduce autocorrelation time
SM & FT 2006, Bari
h
Potts model
H  b  i j  h  ih
i
ij
i  0,1,2
Z(3) breaking
order-disorder
PHASE TRANSITION
SM & FT 2006, Bari
Phase diagram
b
h=0
Is the mass spectrum
universal?
1st order
critical
lines point
weak
1st order
transition
Z(3)
broken
phase endpoint
Does
universality
hold
2nd
order
critical
2nd
order
critical
comparison with ISING
SU(3)
also forendpoint
weak 1st order
(work
in progress)
transition?
Falcone, Fiore,
h=0Gravina, Papa
Z(3) symmetric phase
bc
hc
SM & FT 2006, Bari
h
h=0 – 1st order transition
2

H


b



b
s

i j
j s
i  c.c.  const .
order 
parameter
is
the
magnetization
3 ij
ij
si  e
2
i i
3 M
1
 3
s
i 0,1,2


i
L i
global spin
SM & FT 2006, Bari
h=0 – 1st order transition
at finite volume
tunneling effects
0.5508
bt
0.550565
between symmetric and
broken phase
between degenerated
broken minima
complex M plane
SM & FT 2006, Bari
h=0 – 1st order transition
at finite volume
To remove the tunneling between
broken minima we apply a rotation
si  e
2
i 
3
only the real phase
is present
2
i 
si  e 3
SM & FT 2006, Bari
Masses’ computation
MASS CHANNELS
Cij (rby
) building
si sj suitable
a1e

a
e

a
e

...
2
3
combinations of the

m1r
m2r
m3r
local variable
C(r)
( 0meff
) (r)  ln
MOMENTUM
s
 s (sC(r PROJECTION
1) s )
ZERO
i
i i ŷ
iẑ
by summing over the y and z slices
s

s
(
s

s
)
meff
(
r
)

m
i

j

r
i
i
1
iẑ , r  
VARIATIONALi ŷMETHOD
(2  )
to well separate masses contributions in
the same channel (Kronfeld 1990)
SM & FT 2006, Bari
(Luscher, Wolff 1990)
2+
0+ CHANNEL
h=0
b=0.5508
meff
=0.1556(36)
m2+
0+=0.381(17)
r
SM & FT 2006, Bari
masses’ computation
m2+
0+
0.60
1st order transition
 b1  b t 


b b 
t 
 2
n
m0(b1)
m0+(b2)
n=1/3
bt
m0 (b1)0.1556
0.5508
0.550565
SM & FT 2006, Bari
n
b1=0.5508
 b1  bt 
b
m0  (b1 )bt=0.550565


 b  b   m (b )
t 
0
2
 2
0+scaling
2+
channelregion
in the
0.550565 – 0.56 at least
b2
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
m2
 2.43(10)
m0
2nd order Ising endpoint
Karsch, Stickan (2000)
b   
~H  ~
H  E  M ij
i j
~
E  E  rM
~
M
M
H
bEsE
 hM


temperature-like
ordering field-like
SM & FT 2006, Bari
b
 h  ih
i
bc
b
z
Pc
ISING pt

hc
h
b
c
(b ,h )= (0.54938(2),0.000775(10))
c
c
(c,c)= (0.37182(2),0.25733(2))
s@[email protected]
hc
h
2nd order endpoint
~
M
c  0.37182
SM & FT 2006, Bari
~
M
0.37233
~
M
0.37248
local variable
Correlation function
s 3
~
mi  ih   ii
2 3
order
parameter
SM & FT 2006, Bari
~
~
Cij (r)  mimj
~ 1
~
M  3 m
i
L i
~
M  M  sE   ih  s ij
i
ij
right-pick
mass
spectrum
=0.37248 0
0+ CHANNEL
2+
We separated contributions from
m0+
2+
two picks and calculated masses
m2
m0 
m2
 2.51(37)
m0.0749(63)
0
 2,56
3d ISING VALUE
SM & FT 2006, Bari
0.188(12)
r
CONCLUSIONS
We used 3d 3q Potts model as a
theoretical laboratory to test some
aspects of universality
THANK YOU
1) Ising point
2) weak 1st
order tr. pt.
SM & FT 2006, Bari
evidence found of
universal spectrum
m2
prediction
of(10SU(3)
left-pick?
 2.43
)
m0
screening
spectrum?
SM & FT 2006, Bari
weak
1st order transition
discontinous
order
parameter
the jump
is small
Tt
SM & FT 2006, Bari
Phase diagram
b
h=0
Mass spectrum is
universal?
weak
1st order
transition
1st order
critical
lines point
Z(3)
broken
phase ISING
2ndUniversality
order
critical
alsoendpoint
holds
for endpoint
weak 1st order
transition?
h=0
Z(3) symmetric phase
bc
hc
SM & FT 2006, Bari
h
Universality
Critical exponents
b
order parameter
Tc

susceptibility
Tc
n
correlation lenght
SM & FT 2006, Bari
Tc