critical behavior of three- dimensional disordered potts models

CRITICAL BEHAVIOR OF THREEDIMENSIONAL DISORDERED POTTS
MODELS WITH MANY STATES
Janus colaboration:
R. Alvarez Baños, A. Cruz, J. M. Gil-Narvion, J. Monforte-Garcia, D. Navarro, A. Tarancón
UNIVERSIDAD DE ZARAGOZA, BIFI
L. A. Fernandez, V. Martín-Mayor, A. Muñoz Sudupe, B. Seoane, D. Yllanes
UNIVERSIDAD COMPLUTENSE DE MADRID, BIFI
A. Gordillo-Guerrero, J. J. Ruiz-Lorenzo
UNIVERSIDAD DE EXTREMADURA, BIFI
M. Guidetti, F. Mantovani, S. F. Schifano, R. Tripiccione
UNIVERSITÀ DI FERRARA
A. Maiorano, E. Marinari, G. Parisi, S. Perez-Gaviro
UNIVERSITÀ DI ROMA “LA SAPIENZA”
BIFI2011
A.P. Young
DEPARTMENT OF PHYSICS, UNIVSERSITY OF CALIFORNIA
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
OVERVIEW
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Introduction.
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Aim of the work
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The Disordered Potts Model.
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Observables.
Results.
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Simulation details.
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Thermalization tests.
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Critical temperature and critical exponents.
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Magnetization.
Conclusions.
References.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
POTTS MODEL
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Potts models are used to describe some materials like Ar1 - x(N2)x
Different versions of Potts model are useful to describe several
phenomena like the behavior of FCC antiferromagnetic materials
with magnetic field points in the <1,1,1> direction.
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We have studied the Disorderd Potts Model which is useful to
characterize the glass transition.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
AIM OF THE STUDY
We want to characterize the spin glass phase transition by
calculating its critical temperature and critical exponents.
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The relation of the critical exponents with the number of states, p,
indicates if the transition becomes harder or not.
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We want to find if a ferromagnetic phase transition also happens.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
THE DISORDERED POTTS MODEL
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A cubic lattice of linear size L with periodic boundary conditions.
There is a scalar spin si in every site i. It takes one of the values
1,2,...,p.
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The Hamiltonian is: H =− ∑ J ij  s , s
⟨i , j ⟩
i
j
The couplings Jij are independent quenched random variables from
a bimodal distribution: Jij=±1.
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We have simulated three dimensional systems with p=4, 5, 6.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
OBSERVABLES
Simplex representation:
We rewrite the variables → vectors pointing to the corners of a (p-1)
hyper-tetrahedron.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
OBSERVABLES
Simplex representation:
We rewrite the variables → vectors pointing to the corners of a (p-1)
hyper-tetrahedron.
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The new unit vectors Si must satisfy:
p  ab −1
S a⋅S b =
where a , b∈[1, p ]
p−1
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p=2
p=3
p=4
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
OBSERVABLES
Simplex representation:
We rewrite the variables → vectors pointing to the corners of a (p-1)
hyper-tetrahedron.
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The new unit vectors Si must satisfy:
p  ab −1
S a⋅S b =
where a , b∈[1, p ]
p−1
1
q k =
V

∑i S
1 
i
S
 2 
i
e
i k⋅R i
q k ≡V ∑ ,  ⟨∣q  k ∣2 ⟩
1/ 2

0
1
=
 q
−1
2 sin k m / 2 q k m 
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1
m= ∑i S i
V
m ≡V ⟨∣m∣2 ⟩
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
SIMULATION DETAILS
Algorithm used: Metropolis and Parallel Tempering.
We have simulated L = 4, 6, 8, 16 with p = 4 and L = 4, 6, 8, 12 with
p= 5, 6.
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L = 4, 6 in PC.
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L = 8, 12, 16 in JANUS.
We have simulated 2 replicas of every sample.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
SIMULATION DETAILS
p=4
p=5
p=6
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
THERMALIZATION TESTS
Logarithmic data binning:
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The whole system's history is divided in several subsets (bins).
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The length of every subset is double that the previous one.
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One must average every observable in every bin.
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Thermalization condition: The average in three (or more) last
bins must converge.
p=6
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
CRITICAL TEMPERATURE AND CRITICAL EXPONENTS
Quotient method:
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First step: In the critical temperature there are crossing points
of the correlation length in lattice units (ξ/L).
sL , cross 
Q   L , sL≡
=s
 L , cross 
p=5
p=4
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
CRITICAL TEMPERATURE AND CRITICAL EXPONENTS
p=6
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
CRITICAL TEMPERATURE AND CRITICAL EXPONENTS
Quotient method:
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First step: In the critical temperature there are crossing points
of the correlation length in lattice units (ξ/L).
sL , cross 
Q   L , sL≡
=s
 L , cross 
Second step: Let O an observable that diverges near the critical
temperature as (β-βc)-xo.
⟨O  sL , cross ⟩ x /
=s O  L− 
⟨O  L , cross ⟩
0
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
CRITICAL TEMPERATURE AND CRITICAL EXPONENTS
We have analyzed:
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∂   to obtain 1+1/ν.
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χq to obtain 2-ηq.
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χm to obatin 2-ηm.
p
(L1,L2)
βcross
ν
ηq
ηm
4
(8,16)
4.000(48)
0.96(8)
0.12(6)
2.03(3)
5
(6,12)
5.010(40)
0.81(2)
0.16(2)
1.94(2)
6
(6,12)
6.262(71)
0.80(4)
0.16(2)
1.971(19)
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
CRITICAL TEMPERATURE AND CRITICAL EXPONENTS
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p
βcross
ν
ηq
2
1.786(6)
2.39(5)
-0.366(16)
2
1.804(16)
2.45(15)
-0.375(10)
3
2.653(35)
0.91(2)
0.16(2)
4
4.000(48)
0.96(8)
0.12(6)
5
5.010(40)
0.81(2)
0.16(2)
6
6.262(71)
0.80(4)
0.16(2)
β follows a linear behavior in p with slope close to 1.
In a disordered first order transitions we expect: ν = 2/D and
2-ηq=D/2, thus if D=3 → ν=2/3 and ηq=1/2. In our case, one can
observe that ν and ηq tend to these values.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
MAGNETIZATION
The disordered Potts model could undergo a ferromagnetic phase
transition. Therefore it could present spontaneus magnetization at
low temperature.
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Firstly as ηm≈2 there is not a ferromagnetic phase transition near
the glass one.
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If the system is in a
paramagnetic phase, χM →
const. and ⟨∣m∣⟩∝1/  V
p=4
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In a ferrormagnetic phase,
χM ∝  V and ⟨∣m∣⟩  const.≠0
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
MAGNETIZATION
p=5
p=6
In these figures one can observe that the conditions of a
paramagnetic phase are satisfied in the whole range of temperatures.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
CONCLUSIONS
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The spin glass transition is clear in p=4, 5 and 6.
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βc and critical exponents ν and ηqare computed.
βc increases like p and ν and ηq tend to the values predicted in a first
order transition. Therefore two scenarios are possible:
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The model undergoes to a first order transition for large
enough values of p.
The model presents a second order transition for all finite
values of p.
There is not any sign of a ferromagnetic phase in the range of
temperatures studied.
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
REFERENCES
The papers of this study:
1. Phys. Rev. B 79 (2009) 184408-1–6 .
2. JSTAT (2010) P05002-1–16.
Other interesting references:
3. H. G. Katzgraber, M. Körner and A. P. Young , Phys. Rev. B 73,
224432 (2006).
4. M. Hasenbusch, A. Pelissetto and E. Vicari, Phys. Rev. B 78,
(2008) 214205.
5. L. W. Lee, H. G. Katzgraber and A. P. Young, Phys Rev. B 74,
104416 (2006).
6. R. S. Andrist, D. Larson and H. G. Katzgraber, arXiv:1009.1916
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos
CRITICAL BEHAVIOR OF THREE-DIMENSIONAL DISORDERED
POTTS MODELS WITH MANY STATES
THANK YOU FOR YOUR ATTENTION
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V NATIONAL CONFERENCE Jorge Monforte García. BIFI: Instituto de Biofísica y Física de Sistemas Complejos