Disorder-induced rounding of the phase transition
in the large-q -state Potts model
M.T. Mercaldo
Università di Salerno
J-C. Anglès d’Auriac
CNRS - Grenoble
F. Iglói
SZFKI - Budapest
2
Motivations
.
• C RITICAL P ROPERITES OF D ISORDERED S YSTEMS
• E FFECTS OF D ISORDER ON F IRST O RDER P HASE T RANSITIONS
• ⇒ R ANDOM B OND P OTTS M ODEL (RBPM), ESPECIALLY IN THE LARGE -q - LIMIT, IS A PERFECT
GROUND TO ANALIZE THIS ASPECT
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
2
Motivations
.
• C RITICAL P ROPERITES OF D ISORDERED S YSTEMS
• E FFECTS OF D ISORDER ON F IRST O RDER P HASE T RANSITIONS
• ⇒ R ANDOM B OND P OTTS M ODEL (RBPM), ESPECIALLY IN THE LARGE -q - LIMIT, IS A PERFECT
GROUND TO ANALIZE THIS ASPECT
Outline
• P OTTS M ODEL IN THE R ANDOM C LUSTER R EPRESENTATION
• I NTRODUCING D ISORDER
• R ESULTS AND P ERSPECTIVES
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
q -state Potts model
3
.
Z≡
X
q −βH({σ})
X
Jij δ(σi , σj )
H=−
{σ}
σi = 0, 1, · · · , q − 1
hi,ji
Jij FM random couplings
⇓
Random cluster representation
Z=
X
G⊆E
q
c(G)
Y
νij
νij = eβJij − 1
ij∈G
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
4
q=3
0
1
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
2
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
5
c(G)= 10 + 12 = 22
x
x
x
x
x
x
x
x
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
x
x
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
6
Π (e βJ e − 1)
c(G)
e
x
x
x
x
x
x
x
x
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
x
x
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
Properties of homogeneous Potts model
7
.
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
Properties of homogeneous Potts model
7
.
q < qc
⇒
q > qc
⇒
2nd order PT
1st order PT
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
Properties of homogeneous Potts model
7
.
q < qc
⇒
q > qc
⇒
2
nd
order PT
1st order PT
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
in 2D : qc
= 4 (exact result)
in the q → ∞ limit
the PT is strongly 1st order
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
Properties of homogeneous Potts model
7
.
q < qc
⇒
q > qc
⇒
2
nd
in 2D : qc
= 4 (exact result)
in the q → ∞ limit
the PT is strongly 1st order
order PT
1st order PT
Systems with Disorder
Continuous PT
1st order PT
HARRIS criterion
@ criterion
αP > 0 disorder is relevant
αP < 0 disorder is irrelevant
it is only known that DISORDER
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
will SOFTEN the transition
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
Properties of homogeneous Potts model
7
.
q < qc
⇒
q > qc
⇒
2
nd
in 2D : qc
= 4 (exact result)
in the q → ∞ limit
the PT is strongly 1st order
order PT
1st order PT
Systems with Disorder
Continuous PT
1st order PT
HARRIS criterion
@ criterion
αP > 0 disorder is relevant
αP < 0 disorder is irrelevant
it is only known that DISORDER
will SOFTEN the transition
Conventional Random FP (different values of critical exponents)
Relevant disorder
Infinite Randomness FP (IRFP) (disorder grows without limits)
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
q → ∞ limit
.
f (T 0 )
f (T ) →
ln q
0
T → T = T ln q
Z=
X
G⊆E
8
q
c(G)
Y ij∈G
q
βJij
−1
⇓q→∞
Z=
X
q φ(G)
φ(G) = c(G) + β
G⊆E
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
X
Jij
ij∈G
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
q → ∞ limit
.
f (T 0 )
f (T ) →
ln q
0
T → T = T ln q
Z=
X
G⊆E
8
q
c(G)
Y ij∈G
q
βJij
−1
⇓q→∞
Z=
X
q φ(G)
φ(G) = c(G) + β
G⊆E
φ∗
Z = n0 q (1 + . . .) where φ∗ = maxG φ(G)
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
X
Jij
ij∈G
and
φ∗ = −βN f
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
.
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
.
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
➠ T HERMAL P ROPERTIES ARE C ALCULATED FROM
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
φ∗
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
.
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
➠ T HERMAL P ROPERTIES ARE C ALCULATED FROM
φ∗
➛ free energy, internal energy, specif ic heat,...
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
.
➠ T HERMAL P ROPERTIES ARE C ALCULATED FROM
φ∗
➛ free energy, internal energy, specif ic heat,...
➠ M AGNETIZATION AND C ORRELATION F UNCTIONS ARE O BTAINED FROM
∗
THE
G EOMETRICAL S TRUCTURE
OF
G
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
.
➠ T HERMAL P ROPERTIES ARE C ALCULATED FROM
φ∗
➛ free energy, internal energy, specif ic heat,...
➠ M AGNETIZATION AND C ORRELATION F UNCTIONS ARE O BTAINED FROM
∗
THE
G EOMETRICAL S TRUCTURE
OF
G
➛ C(r), average correlation function, is related to the distribution of clusters
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
.
➠ T HERMAL P ROPERTIES ARE C ALCULATED FROM
φ∗
➛ free energy, internal energy, specif ic heat,...
➠ M AGNETIZATION AND C ORRELATION F UNCTIONS ARE O BTAINED FROM
∗
THE
G EOMETRICAL S TRUCTURE
OF
G
➛ C(r), average correlation function, is related to the distribution of clusters
➛ m, magnetization, is the fraction of sites in the infinite cluster
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
9
All information about the RBPM in the large-q limit
is contained in the OPTIMAL SET G∗
.
➠ T HERMAL P ROPERTIES ARE C ALCULATED FROM
φ∗
➛ free energy, internal energy, specif ic heat,...
➠ M AGNETIZATION AND C ORRELATION F UNCTIONS ARE O BTAINED FROM
∗
THE
G EOMETRICAL S TRUCTURE
OF
G
➛ C(r), average correlation function, is related to the distribution of clusters
➛ m, magnetization, is the fraction of sites in the infinite cluster
➛ ξ , correlation lenght, is the average size of the clusters
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
maximize φ∗
10
.
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
maximize φ∗
10
.
• One has to f ind the max over the 2|E| possible conf iguration !
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
maximize φ∗
10
.
• One has to f ind the max over the 2|E| possible conf iguration !
• φ∗ is a supermodular function ⇒
φ(A) + φ(B) ≤ φ(A ∪ B) + φ(A ∩ B) ∀A, B ∈ E
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
maximize φ∗
10
.
• One has to f ind the max over the 2|E| possible conf iguration !
• φ∗ is a supermodular function ⇒
φ(A) + φ(B) ≤ φ(A ∪ B) + φ(A ∩ B) ∀A, B ∈ E
• theorem of discrete math ⇒ ∃ a combinatorial optimization method to maximize it in polynomial time
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
maximize φ∗
10
.
• One has to f ind the max over the 2|E| possible conf iguration !
• φ∗ is a supermodular function ⇒
φ(A) + φ(B) ≤ φ(A ∪ B) + φ(A ∩ B) ∀A, B ∈ E
• theorem of discrete math ⇒ ∃ a combinatorial optimization method to maximize it in polynomial time
• for φ(G) of the Potts model a specif ic algorithm has been formulated
Angles d’Auriac et al.JPA35, 6973 (2002)
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
maximize φ∗
10
.
• One has to f ind the max over the 2|E| possible conf iguration !
• φ∗ is a supermodular function ⇒
φ(A) + φ(B) ≤ φ(A ∪ B) + φ(A ∩ B) ∀A, B ∈ E
• theorem of discrete math ⇒ ∃ a combinatorial optimization method to maximize it in polynomial time
• for φ(G) of the Potts model a specif ic algorithm has been formulated
Angles d’Auriac et al.JPA35, 6973 (2002)
d=2
L = 512
⇒
2524288 ∼ 2.6 · 10157826
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
11
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
12
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.200
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
13
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.166
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
14
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.066
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
15
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.050
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
16
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.042
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
17
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.033
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
18
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.025
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
19
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.016
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
20
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.008
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
21
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 1.000
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
22
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 0.992
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
23
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 0.983
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
24
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 0.967
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
25
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 0.9416
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
26
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 0.667
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
27
1
1
5
1
+ δ J−
P (J) = δ J −
2
6
2
6
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
T = 0.500
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.➠ I N 2D D ISORDER
DESTROY
P HASE C OEXISTENCE ⇒ it softens the 1st order PT into a 2nd order PT
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.➠ I N 2D D ISORDER
DESTROY
P HASE C OEXISTENCE ⇒ it softens the 1st order PT into a 2nd order PT
➠ I N 3D W EAK D ISORDER DOES NOT DISTROY P HASE C OEXISTENCE i.e. disorder has to be strong
enough to soften the PT into 2nd order PT
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.➠ I N 2D D ISORDER
DESTROY
P HASE C OEXISTENCE ⇒ it softens the 1st order PT into a 2nd order PT
➠ I N 3D W EAK D ISORDER DOES NOT DISTROY P HASE C OEXISTENCE i.e. disorder has to be strong
enough to soften the PT into 2nd order PT
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.➠ I N 2D D ISORDER
DESTROY
P HASE C OEXISTENCE ⇒ it softens the 1st order PT into a 2nd order PT
➠ I N 3D W EAK D ISORDER DOES NOT DISTROY P HASE C OEXISTENCE i.e. disorder has to be strong
enough to soften the PT into 2nd order PT
2D
δ
1
2nd order above
breaking lenght
ordered
phase
2nd order below
breaking lenght
1
T/d
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.➠ I N 2D D ISORDER
DESTROY
P HASE C OEXISTENCE ⇒ it softens the 1st order PT into a 2nd order PT
➠ I N 3D W EAK D ISORDER DOES NOT DISTROY P HASE C OEXISTENCE i.e. disorder has to be strong
enough to soften the PT into 2nd order PT
2D
δ
3D
δ
1
1
2nd order
2nd order above
breaking lenght
ordered
phase
ordered
phase
1st order above
breaking lenght
2nd order below
breaking lenght
1
T/d
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
1st order below
breaking lenght
1
T/d
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
28
.➠ I N 2D D ISORDER
DESTROY
P HASE C OEXISTENCE ⇒ it softens the 1st order PT into a 2nd order PT
➠ I N 3D W EAK D ISORDER DOES NOT DISTROY P HASE C OEXISTENCE i.e. disorder has to be strong
enough to soften the PT into 2nd order PT
2D
δ
3D
δ
1
1
2nd order
2nd order above
breaking lenght
ordered
phase
ordered
phase
1st order above
breaking lenght
2nd order below
breaking lenght
1
T/d
1st order below
breaking lenght
1
T/d
In a f inite size system weak disorder fluctuation could not be suff icient to break phase coexistence
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
29
Through extreme value statistics one can estimates the breaking length scale L
∼ exp[(1/δ)2 ]
[the f inite length scale L at which breaking of phase coexistence take place]
strenght of disorder δ
= ∆/J
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
29
Through extreme value statistics one can estimates the breaking length scale L
∼ exp[(1/δ)2 ]
[the f inite length scale L at which breaking of phase coexistence take place]
strenght of disorder δ
= ∆/J
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
29
Through extreme value statistics one can estimates the breaking length scale L
∼ exp[(1/δ)2 ]
[the f inite length scale L at which breaking of phase coexistence take place]
bimodal distribution
L
100
L
100
10
2.4
2.6
2.8
3
3.2
3.4
3.6
J/∆
10
6
7
8
9
10
(J/∆)
strenght of disorder δ
11
12
13
2
= ∆/J
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
30
.
Free Energy:
∗
F = c(G )T −
P
ij∈G∗
Jij
Internal Energy:
E=−
P
ij∈G∗
Jij
➠ For a given sample E is a piecewise costant function of temperature ⇒ it shows discontinuities
➠ The average over disorder generally smears out discontinuities
➠ The behavior of averaged quantities is different for the discrete and the continuous distributions
1
1.2
100∆E
1
0.8
Energy per bond
E
0.8
0.6
0.4
bimodal distribution
L=16
L=32
L=64
L=128
L=256
0.2
0
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0.6
L=128
L=64
L=32
Continuous distribution
1
0
0
0.1
0.2
1/m
0.3
0.4
0.2
1
β
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
0
0.7
0.8
0.9
1
β
1.1
1.2
1.3
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
31
a
At the critical point the largest cluster of G∗ is a fractal and its mass M
√
β
df = d −
ν
5+ 5
df =
4
∼ L df
According to scaling theory, cumulative distribution of the mass of the cluster
R(M, L) = M −τ R̃(M/Ldf )
100000
0.8
R1.8
Square L=256
Square L=128
Square L=64
Square L=32
Hexagonal L=128
Bimodal distribution
10000
0.7
L=32
L=64
L=128
L=256
24
a)
22
20
0.6
18
16
0.5
100
36
14
12
df
10
0.4
1.72
1.76
1.8
ms
mτ P(m)
ms
m(R)
1000
0.3
1.84
1.88
b)
36
28
28
0.2
20
20
10
14
14
0.1
L
32
1
64
128
L
32
256
64
128
256
0
1
10
100
1000
0
R
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
0.5
1
1.5
2
m/Ldf
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
32
Conclusions
.➠
R ESULTS
IN
2D√
3− 5
➛ α = 0, β =
,ν =1
as for the RTIM ⇒ IRFP !
4
➛ We can argue that the RTIM is the Hamiltonian version of the 2D RBPM in the large-q limit
` d’Auriac, Igloi,
´ PRE 69, 0461xx (2004);
Ref.: Mercaldo, Angles
.
` d’Auriac, Igloi
´ PRL 90, 190601 (2003)
Angles
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
32
Conclusions
.➠
R ESULTS
IN
2D√
3− 5
➛ α = 0, β =
,ν =1
as for the RTIM ⇒ IRFP !
4
➛ We can argue that the RTIM is the Hamiltonian version of the 2D RBPM in the large-q limit
` d’Auriac, Igloi,
´ PRE 69, 0461xx (2004);
Ref.: Mercaldo, Angles
.
` d’Auriac, Igloi
´ PRL 90, 190601 (2003)
Angles
➠ Q UESTIONS IN 3D
➛ is the transition line Tc /d = 1 for δ 1 ?
➛ is δ = 1/2 the tricritical point ?
➛ does the critical line depend on the disorder distribution ?
` d’Auriac, Igloi,
´ Mercaldo work in progress
Ref.: Angles
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói
32
Conclusions
.➠
R ESULTS
IN
2D√
3− 5
➛ α = 0, β =
,ν =1
as for the RTIM ⇒ IRFP !
4
➛ We can argue that the RTIM is the Hamiltonian version of the 2D RBPM in the large-q limit
` d’Auriac, Igloi,
´ PRE 69, 0461xx (2004);
Ref.: Mercaldo, Angles
.
` d’Auriac, Igloi
´ PRL 90, 190601 (2003)
Angles
➠ Q UESTIONS IN 3D
➛ is the transition line Tc /d = 1 for δ 1 ?
➛ is δ = 1/2 the tricritical point ?
➛ does the critical line depend on the disorder distribution ?
` d’Auriac, Igloi,
´ Mercaldo work in progress
Ref.: Angles
➠ R ELATED P ROBLEM
➛ Critical Properties of Quantum Potts model
Ref.: Mercaldo, De Cesare, work in progress
Congresso del Dipartimento di Fisica “E.R. Caianiello”– 19-20 Aprile 2004, Salerno.
RBPM in the large-q limit
M.T. Mercaldo, J-C. Anglès d’Auriac, F. Iglói