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A. Krawiecki
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland
DYNAMICAL PHASE TRANSITIONS IN THE ISING
MODEL ON REGULAR AND SCALE-FREE NETWORKS
Darmstadt, 2004
courtesy of Dr. P. Jóźwiak, Svalbard 2001
Dynamical phase transitions in magnetic systems
A subject of interest are magnetic systems with periodic field, e.g., the Ising model
H   J ij i j  h cos 0t  i
i, j
i
• It is known that isotropic magnets do not exhibit hysteresis loop under adiabatic
sweeps of the magnetic field. However, if the field oscillates periodically, the
oscillations of magnetization lag behind those of the field due to finite relaxation time,
and the hysteresis loop can appear (in analogy with „Lissajou figures”).
• The dynamical phase transition occurs in the magnetic systems (e.g., in the Ising
model) since, for high enough field amplitude h and low enough temperature T, even
the sign of the magnetization does not follow that of the magnetic field. Then, although
the magnetic field has zero mean, spontaneous symmetry breaking occurs in the system
and the hysteresis loop becomes asymmetric (i.e., it is constrained only to the positive
or negative ranges of the magnetization).
• The two phases in the system can be distinguished using the order parameter <|Q|>
and the hysteresis loop area <A>, where brackets denote averaging over many periods
Q
0
2
 m(t )dt ,
A   m(h)dh
[B.K. Chakrabarti, M. Acharyya, Rev. Mod. Phys. 71, 847 (1999)].
Dynamical phase transition in the mean-field Ising model
[T. Tome, M.J. de Oliveira, Phys. Rev. A 41, 4251 (1990)].
dm
 m  h cos  ot 
 m  tanh 

dt
T


Continuous (second-order) transition
Q
Discontinuous (first-order) transition
Q
In the mean-field approximation, for any frequency 0 a tricritical point exists on
the boundary between the dynamically ordered and disordered phases, which
separates regions with the first- and second-order transition.
The mean-field theory is only approximate for real systems.
What about spatially extended models?
courtesy of Dr. P. Jóźwiak, Svalbard 2001
Dynamical phase transition in the
spatially extended Ising model
Existence of the tricritical point was
confirmed by numerical simulations in
two and three dimensions.
[M. Acharyya and B.K. Chakrabarti,
Phys. Rev. B 52, 6550 (1995)].
The critical
temperature for
the second-order
dynamical phase
transition
corresponds to
maximum
fluctuations of the
dynamical order
parameter Q,
measured by dQ2,
where the
brackets denote
average over
many periods
2/0.
[M. Acharyya, Phys. Rev. E 56, 1234 (1997)].
The thermodynamic nature of the transition can be found by studying the fourthorder Binder cumulant
• For the second-order transition, UL is positive and decreases monotonically with
rising temperature from 2/3 for T=0 to 0 for T.
•For the first-order transition, UL shows a pronounced negative minimum at the
transition temperature.
[M. Acharyya, Phys. Rev. E 59, 218 (1999)].
An important question in the study of dynamical phase transitions in the
spatially extended Ising models was the existence of the discontinuous
transition and the tricritical point. Eventually, it turned out that the firstorder transition is a finite-size effect.
courtesy of Dr. P. Jóźwiak, Svalbard 2001
Simulations in spatially extended systems: the droplet nucleation theory
[G. Korniss et al., Phys. Rev E 63, 016120 (2000)]
[G. Korniss, P.A. Rikvold, M.A. Novotny, Phys. Rev E 66, 056127 (2002)]
• If the magnetic field is inverted, in small magnetic systems the magnetization changes sign via
nucleation of a single droplet of inverted spins, which grows in time and eventually comprises
the whole magnet. In large enough systems, the magnetization changes sign via nucleation of
many droplets.
• The first-order transition appears if the magnetization changes sign (in the disordered phase) or
oscillates around a non-zero value (in the ordered phase) via nucleation of a single droplet.
Hence, the occurrence of the first-order transition is a finite-size effect, and in large systems only
the second-order transition is seen.
• In large systems (in the multidroplet regime) the critical temperature for the transition is
approximately given by T0 /2=/0 =<t(T,H)>, where <t> is the average lifetime of the
metastable phase for given T, H=const.
In recent years, there has been a growing interest in the study of statistical properties of
networks with complex topologies. The Ising model on different complex networks can
exhibit nontrivial properties (magnetic ordering in „one” dimension, dependence of the
critical temperature for the ferromagnetic transition on the system size, etc.). However,
the study of the response of this kind of models to a periodic magnetic field has been so
far quite limited (mainly to the problem of stochastic resonance).
courtesy of Dr. P. Jóźwiak, Svalbard 2001
Example: scale-free networks
[A. Barabási and R. Albert, Science 286, 509 (1999)]
• Evolving networks: the networks are created by adding new nodes to the already
existing ones, one node at each step.
• Preferential attachment: a fixed number m of links from the newly added node to the
already existing nodes is created at each step, and the probability of linking the new
node to an „old” one is proportional to the number of nodes which are already linked to
the „old” node („rich get richer”).
• This results in networks with complex topology, which are called „scale-free” since
they look similar at any scale; e.g., the distribution of connectivity k (defined as the
number of connections per node) obeys a power scaling law, P(k)k-g (in the original
example by Barabási and Albert g=3).
• Examples of the scale-free networks comprise, e.g., the internet activity, the www
links, networks of cooperation (between scientists, actors, etc.), traffic networks
(airplane & railway connections, city transport schemes), biological networks (sexual
contacts, protein interactions, certain neural networks), etc.
Snapshot view of
internet connections
Imported from:
http://www.nd.edu/~networks/gallery.htm
Map of protein-protein interactions. The colour of a node signifies the
phenotypic effect of removing the corresponding protein (red, lethal;
green, non-lethal; orange, slow growth; yellow, unknown).
Imported from:
http://www.nd.edu/~networks/gallery.htm
The Ising model on the Barabasi-Albert network
• A small number m of fully connected nodes is fixed.
• New nodes are added, and each new node is linked to existing nodes with m edges
according to the „preferential attachment” probability rule: Probability of linking to a
node i is
, where ki is the actual connectivity (number of attached edges) of
node i, and
is the actual number of edges in the whole network.
• The growth process is continued until the number of nodes N is reached, when the
network structure is frozen.
• At each node a spin i with two possible orientations +1 or -1 is located, and non-zero
exchange integrals between spins linked by edges are assumed.
• The spins are subjected to thermal noise (Glauber dynamics is used in simulations)
and periodic external magnetic field.
The Hamiltonian for this model is
where Jij=1 if there is an edge between nodes i,j, and Jij=0 otherwise.
The ferromagnetic transition temperature for h=0
Tc  0.25 J ln( N )
[A. Aleksiejuk, J.A. Hołyst, D. Stauffer,
Physica A 310, 260 (2002)]
Dynamical phase transition in the Ising model on the Barabási-Albert network
(a) The dynamical fourth-order cumulant UL, (b) fluctuations of the DOP
dQ, and (c) average of the absolute value of Q vs. temperature for fixed
number of nodes N, period T, and for increasing field amplitude h (see
legend). As h rises, the DPT changes from second-order for h=0.02 to
first-order for h=0.1 and h=0.3.
(a-c) Approximate
phase boundaries
between the
dynamically ordered
(Q>0, below the
borders) and
disordered (Q=0,
above the borders)
phases for the Ising
model on the BA
network, for various
numbers of nodes N
and periods T (see
legends). (d) The
dynamical fourthorder cumulant UL for
increasing number of
nodes N; the field
amplitude h and the
period T were chosen
so that with increasing
N the DPT changes
from second-order for
N=100 and N=1000 to
first-order for
N=10000.
Conclusions
• For all N and  existence of the tricritical point is observed, separating the critical
values of h and T for which the dynamical phase transition is first- and second-order.
Hence, the phase diagram resembles that for the mean-field theory for the Ising model
on regular lattices.
• Further evidence for the existence of the tricritical point comes from the fact that the
region of critical values of h and T/Tc for which the transition is first-order increases as
N rises. I.e., if for given h,  and small N the transition is second-order, for increasing
N it finally becomes first-order. This suggests that for h>0 in the thermodynamic limit
the transition becomes discontinuous and disappears due to the divergence of the
critical temperature.
• The difference with the case of spatially extended Ising systems on periodic lattices is
probably due to lack of regularity in the Barabasi-Albert network.
• Investigation of the thermodynamic nature of the dynamical phase transition in the
Ising model on other networks with complex topology (e.g., on small-world networks)
should clarify the role of spatial disorder in this phenomenon.
• Investigation of the dynamical phase transition on other scale-free networks should
make possible to distinguish between the effect of spatial disorder and that of particular
network topology on the trasition.
Thank you for your attention...
courtesy of Dr. P. Jóźwiak, Svalbard 2001
...(and patience) :-)