A. KrawieckiFaculty 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

• 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

Dynamical phase transitions in magnetic systems

i

ijiji

ij thJH 0,

cos

A subject of interest are magnetic systems with periodic field, e.g., the Ising model

dhhmAdttmQ )(,)(2

0

[B.K. Chakrabarti, M. Acharyya, Rev. Mod. Phys. 71, 847 (1999)].

[T. Tome, M.J. de Oliveira, Phys. Rev. A 41, 4251 (1990)].

Dynamical phase transition in the mean-field Ising model

T

thmm

dtdm ocos

tanh

Q Q

Continuous (second-order) transition Discontinuous (first-order) transition

In the mean-field approximation, for any frequency 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.The mean-field theory is only approximate for real systems.What about spatially extended models?What about spatially extended models?

courtesy of Dr. P. Jóźwiak, Svalbard 2001

[M. Acharyya and B.K. Chakrabarti,Phys. Rev. B 52, 6550 (1995)].

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, Phys. Rev. E 56, 1234 (1997)].

The critical temperature for the second-order dynamical phase transition corresponds to maximum fluctuations of the dynamical order parameter Q, measured by Q2, where the brackets denote average over many periods 2/0.

[M. Acharyya, Phys. Rev. E 59, 218 (1999)].

The thermodynamic nature of the transition can be found by studying the fourth-order 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.

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 first-order transition is a finite-size effect.

courtesy of Dr. P. Jóźwiak, Svalbard 2001

[G. Korniss et al., Phys. Rev E 63, 016120 (2000)]

Simulations in spatially extended systems: the droplet nucleation theory

[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,H)>, where <> 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 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 networks with complex topologies. The Ising model on different complex networks can exhibit nontrivial properties (magnetic ordering in „one” dimension, dependence of the exhibit nontrivial properties (magnetic ordering in „one” dimension, dependence of the critical temperature for the ferromagnetic transition on the system size, etc.). However, 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 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).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- (in the original example by Barabási and Albert =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

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).

• 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

The Ising model on the Barabasi-Albert network

where Jij=1 if there is an edge between nodes i,j, and Jij=0 otherwise.

)ln(25.0 NJTc [A. Aleksiejuk, J.A. Hołyst, D. Stauffer,Physica A 310, 260 (2002)]

The ferromagnetic transition temperature for h=0

(a) The dynamical fourth-order cumulant UL, (b) fluctuations of the DOP Q, 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.

Dynamical phase transition in the Ising model on the Barabási-Albert network

(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 fourth-order 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...

...(and patience) :-)courtesy of Dr. P. Jóźwiak, Svalbard 2001