Renormalization Group
and critical properties of long range models
Maria Chiara Angelini
Supervisor: F. Ricci-Tersenghi
Dipartimento di Fisica, Sapienza, Università di Roma
Dottorato di ricerca - XXV ciclo
Introduction
1
Introduction
2
A new RG scheme for disordered systems
3
Long Range vs Short Range
Maria Chiara Angelini (Sapienza)
Long Range Models
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Introduction
Renormalization group (RG)
Second order phase transition
ξ ∝ (T − Tc )−ν −−−−→ ∞ :
T < Tc
T > Tc
T → Tc
important length scale
F
F
Reduction
of Degrees of freedom
P
H = (i,j) σi σj Jij
M

a→b·a






σ→σ 0 
−Mo
+Mo
M
−βH(J,{σ}) =
{σ} e
P
0
0
0
0

K =J·β→K 0 
= {σ0 } e−β (H (J ,{σ })+cost)





H(σ)→H 0 (σ 0 )
Maria Chiara Angelini (Sapienza)
Long Range Models
Z=
P
Dottorato di ricerca - XXV ciclo
3 / 18
Introduction
Renormalization group (RG)
Second order phase transition
ξ ∝ (T − Tc )−ν −−−−→ ∞ :
T < Tc
T > Tc
T → Tc
important length scale
F
F
Reduction
of Degrees of freedom
P
H = (i,j) σi σj Jij
M

a→b·a






σ→σ 0 
−Mo
+Mo
M
−βH(J,{σ}) =
{σ} e
P
0
0
0
0

K =J·β→K 0 
= {σ0 } e−β (H (J ,{σ })+cost)





H(σ)→H 0 (σ 0 )
Z=
P
fixed point: K 0 = K
K =0
w
Maria Chiara Angelini (Sapienza)
K = K∗
w
Long Range Models
K =∞
- w
Dottorato di ricerca - XXV ciclo
3 / 18
Introduction
Spin Glass (SG)
F
H=−
X
σi Jij σj
σi = ±1
<ij>
Jij extracted randomly from a given distribution P(J),
positive and negative couplings are allowed
{m}
D > 6: MF
S1
J=−1
S2
Important properties:
Disorder: double average hOi
J=+1
J=+1
Frustration
Ground state degeneracy
Computational complexity
S4
J=+1
S3
D < 6: No analytic solution, no RG analysis
Maria Chiara Angelini (Sapienza)
Long Range Models
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Introduction
The (spin glass) hierarchical model (HM)
Hn+1 (s1 , ..., s2n+1 ) = Hn (s1 , ..., s2n ) + Hn (s2n +1 , ..., s2n+1 )
+c
n+1
n+1
2X
Jij = 1 : F. Dyson,
Com. Math. Phys. 12, 91 (1969).
−
Jij si sj + cost
J2
ij
P(Jij ) ∝ e 2 : S. Franz et al.,
J. Stat. Mec., P02002 (2009).
i<j=1
Maria Chiara Angelini (Sapienza)
Long Range Models
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Introduction
The (spin glass) hierarchical model (HM)
Hn+1 (s1 , ..., s2n+1 ) = Hn (s1 , ..., s2n ) + Hn (s2n +1 , ..., s2n+1 )
+c
n+1
n+1
2X
Jij = 1 : F. Dyson,
Com. Math. Phys. 12, 91 (1969).
−
Jij si sj + cost
J2
ij
P(Jij ) ∝ e 2 : S. Franz et al.,
J. Stat. Mec., P02002 (2009).
i<j=1
It simulates a D-dimensional
short range system:
FM: c = 2−1−2/D
−1−2/D
SG: c = 2 2
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
5 / 18
Introduction
The (spin glass) hierarchical model (HM)
Hn+1 (s1 , ..., s2n+1 ) = Hn (s1 , ..., s2n ) + Hn (s2n +1 , ..., s2n+1 )
+c
n+1
n+1
2X
Jij = 1 : F. Dyson,
Com. Math. Phys. 12, 91 (1969).
−
Jij si sj + cost
J2
ij
P(Jij ) ∝ e 2 : S. Franz et al.,
J. Stat. Mec., P02002 (2009).
i<j=1
It simulates a D-dimensional
short range system:
FM: c = 2−1−2/D
−1−2/D
SG: c = 2 2
FM: P(m) =
P
mL ,mR :mL +mR =m
P(mL ) · P(mR ) · eβc
n J(m +m )2
L
R
SG: m is no more a good order parameter: no closed equation
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
5 / 18
A new RG scheme for disordered systems
1
Introduction
2
A new RG scheme for disordered systems
3
Long Range vs Short Range
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
6 / 18
A new RG scheme for disordered systems
Revised renormalization procedure
O3
O2
σ3=1
σ2=1
σ1=1
Take an ensemble , calculate n − 1 observables hOi i
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
7 / 18
A new RG scheme for disordered systems
Revised renormalization procedure
O3
σ3=1
O2 ’
O2
σ2=1
O1 ’
σ2 ’
σ1 ’
σ1=1
Take an ensemble , calculate n − 1 observables hOi i
Find new variances σ 0 of the Gaussian distribution of couplings for
a (n − 1)-level systems ensemble: hOi0 i = hOi i
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
7 / 18
A new RG scheme for disordered systems
Revised renormalization procedure
O3
σ3=1
σ3=1
O2 ’
O2
σ2=1
O1 ’
σ2 ’
σ2
σ1 ’
’
σ1
σ1=1
’
Take an ensemble , calculate n − 1 observables hOi i
Find new variances σ 0 of the Gaussian distribution of couplings for
a (n − 1)-level systems ensemble: hOi0 i = hOi i
Join two (n − 1)-level systems extracted from the new coupling
distribution with a coupling extracted from the beginning
distribution: σn = 1
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
7 / 18
A new RG scheme for disordered systems
Revised renormalization procedure
O3
σ3=1
σ3=1
O2 ’
O2
σ2=1
O1 ’
σ2 ’
σ2
σ1 ’
’
σ1
σ1=1
’
Take an ensemble , calculate n − 1 observables hOi i
Find new variances σ 0 of the Gaussian distribution of couplings for
a (n − 1)-level systems ensemble: hOi0 i = hOi i
Join two (n − 1)-level systems extracted from the new coupling
distribution with a coupling extracted from the beginning
distribution: σn = 1
Repeat the procedure from the beginning
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
7 / 18
A new RG scheme for disordered systems
Renormalization of couplings and observables
hcn i =
h
N
1 X
P
i∈L (si sj )
j∈R
0.8
1.3
0.7
2
1.4
<s1 s2>
σ1
2
ix
q P
P
N x=1
h i,j∈L (si sj )2 ix h i,j∈R (si sj )2 ix
1.2
1.1
1
0.6
0.5
0.4
0.9
1
2
3
4
5
6
7
8
1
2
x
T=0.30
T=0.40
T=0.45
3
4
5
6
7
8
x
T=0.47
T=0.50
T=0.53
T=0.55
T=0.57
T=0.60
T=0.65
T=0.70
D = 3, n = 4, Tc = 0.58(1)
Maria Chiara Angelini (Sapienza)
Long Range Models
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8 / 18
A new RG scheme for disordered systems
Critical exponent ν
J(x)
T
−
1
Tc
=
1
T
−
1
Tc
x
bν
J1(x)/T1 - J2(x)/T2
1
J1(x)/T1 - J2(x)/T2
1
D = 3 < Du
T1 = 0.47
0
1
D=3
2
3
T2 = 0.57
4
x
5
6
ν = 4.50(15)
Maria Chiara Angelini (Sapienza)
7
8
D = 8.2 > Du
T1 = 1.6
0
1
2
3
T2 = 2.1
4
x
5
6
7
8
D = 8.2 ν = 4.33(10) νT = 4.1
Long Range Models
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A new RG scheme for disordered systems
Comparison with the Edward-Anderson model
5.5
st
1 order ε-exp.
5
νD
4.5
4
3.5
3
nd
2
3
HM ERG
EA MC
D/2
order ε-exp.
4
5
6
7
8
9
10
D
-exp: Michele Castellana, Giorgio Parisi, Phys. Rev. E 83, 041134 (2011);
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
10 / 18
Long Range vs Short Range
1
Introduction
2
A new RG scheme for disordered systems
3
Long Range vs Short Range
Maria Chiara Angelini (Sapienza)
Long Range Models
Dottorato di ricerca - XXV ciclo
11 / 18
Long Range vs Short Range
Long Range systems
M. E. Fisher, S. K. Ma, B. G. Nickel, Phys. Rev. Lett. 29, 917 (1972)
H=
N
1X
Jij σi σj
2
Jij ∝ |rij |−(d+σ)
i,j=1
Z
dxL(φ) =
X
u2 (k)φ(k )φ(−k) + u
k
X
φ(k1 )φ(k2 )φ(k3 )φ(−k1 − k2 − k3 )
k1 k2 k3
u2 (k) = r + jσ k σ + j2 k 2 : crossover between k σ and k 2
MF
0
LR
σ U =d/2
ξ ∝ (T − Tc )−ν
G(x) = hσ(0)σ(x)i
T =Tc
Maria Chiara Angelini (Sapienza)
=
SR
σ L =2?
x −d+2−η
Long Range Models
σ
MF: ν = σ1
MF+LR: η = 2 − σ
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Long Range vs Short Range
Problem I : The lower critical exponent
0.5
LR
d =2
η =2−σ
Picco
Fisher et al.
Sak
0.4
η
0.3
0.2
SR
D=2
η = 0.25
0.1
0
1.5
1.6
1.7
1.8
1.9
2
σ
1
2
3
Fisher et al.: η = 2 − σ for σ < 2: σL = 2
Sak: η = max(2 − σ, ηSR ): σL = 2 − ηSR
Picco: η smoothly interpolates between 2 − σ and ηSR
Moreover we know analytically that σL = 1 if d = 1
WE WANT TO STUDY THE SYSTEM AT σL
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Correlation functions around σL in d = 2
σ = 1.75
0
G(x) = ax −η + bx −η 0
f (x) = ax −ηsak + bx −η
0
g(x) = ax −ηpicco + bx −η
T =Tc
1
1
G(x)
G(x)
128
192
256
384
512
768
1024
-0.25
x-0.48
x
0.1
1024
768
f(x)
g(x)
0.1
0.01
0.01
1
10
100
1000
1
10
100
1000
x
x
ANSWER I : It is difficult to extract η and σL
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Problem II : σ − D relations
1
LR in d = 1: from MF theory σ = D2
Problem: σL = 1 corresponds to DL = 2 while DL = 1.
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Problem II : σ − D relations
1
2
LR in d = 1: from MF theory σ = D2
Problem: σL = 1 corresponds to DL = 2 while DL = 1.
(D)
σ = 2−ηSR
D
σL = 1 corresponds to DL = 1
(D)
that implies νSR (D) = D1 νLR ( 2−ηSR
)
D
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Problem II : σ − D relations
1
2
3
LR in d = 1: from MF theory σ = D2
Problem: σL = 1 corresponds to DL = 2 while DL = 1.
(D)
σ = 2−ηSR
D
σL = 1 corresponds to DL = 1
(D)
that implies νSR (D) = D1 νLR ( 2−ηSR
)
D
Generalization in d > 1
2−ηSR (D)
σ
d =
D
superuniversality: important variable σ̂ = σ/d
(D)
νSR (D) = Dd νLR ( dσ = 2−ηSR
)
D
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Problem II : σ − D relations
1
2
3
LR in d = 1: from MF theory σ = D2
Problem: σL = 1 corresponds to DL = 2 while DL = 1.
(D)
σ = 2−ηSR
D
σL = 1 corresponds to DL = 1
(D)
that implies νSR (D) = D1 νLR ( 2−ηSR
)
D
Generalization in d > 1
2−ηSR (D)
σ
d =
D
superuniversality: important variable σ̂ = σ/d
(D)
νSR (D) = Dd νLR ( dσ = 2−ηSR
)
D
WE WANT TO VERIFY THE σ − D RELATIONS
AND SUPERUNIVERSALITY
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
MonteCarlo simulations
Cluster algorithm for LR in d = 1 and d = 2
Susceptibility χ = N/T (hm2 i) and Binder parameter B = 21 [3 −
hm4 i
]
hm2 i2
hO(L, t)i = FO (L1/ν (T − Tc )) + L−ω GO (L1/ν (T − Tc )) + . . .
0.3
0.48
0.46
0.28
0.44
0.42
Binder
0.26
χ/Lσ
8
9
10
11
12
13
14
15
16
17
18
19
20
0.24
0.22
0.4
0.38
0.36
0.34
0.32
0.2
0.3
0.18
3.188 3.189 3.19 3.191 3.192 3.193 3.194
0.28
3.182
T
3.184
3.186
3.188
3.19
3.192
3.194
T
Extraction of the critical exponent ν
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Superuniversality
νSR (D) =
d
ν (σ
D LR d
=
2−ηSR (D)
)
D
ANSWER II : σ − D relation and superuniversality hold near σU
Maria Chiara Angelini (Sapienza)
Long Range Models
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Long Range vs Short Range
Conclusions
Study of RG on disordered systems using hierarchical model:
Critical fixed point for spin glass in D < Dcu
Critical exponent ν for spin glass in D < Dcu
Non monotonic behaviour of ν with σ or D confirming -exp
Importance of ensemble RG
Study of relations between LR and SR models
Extension of the σ − D relation for d > 1
Superuniversality and σ − D relation hold near σU
Two power-laws behaviour of correlation functions near σL
Maria Chiara Angelini (Sapienza)
Long Range Models
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