Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network

We study the critical behaviour of the $q$-state Potts model on an uncorrelated scale-free network having a power-law node degree distribution with a decay exponent $\lambda$. Previous data show that the phase diagram of the model in the $q,\lambda$ plane in the second order phase transition regime contains three regions, each being characterized by a different set of critical exponents. In this paper we complete these results by finding analytic expressions for the scaling functions and critical amplitude ratios in the above mentioned regions. Similar to the previously found critical exponents, the scaling functions and amplitude ratios appear to be $\lambda$-dependent. In this way, we give a comprehensive description of the critical behaviour in a new universality class.Comment: 10 pages, 4 figure

Phase transitions in the Potts model on complex networks

The Potts model is one of the most popular spin models of statistical physics. The prevailing majority of work done so far corresponds to the lattice version of the model. However, many natural or man-made systems are much better described by the topology of a network. We consider the q-state Potts model on an uncorrelated scale-free network for which the node-degree distribution manifests a power-law decay governed by the exponent \lambda. We work within the mean-field approximation, since for systems on random uncorrelated scale-free networks this method is known to often give asymptotically exact results. Depending on particular values of q and \lambda one observes either a first-order or a second-order phase transition or the system is ordered at any finite temperature. In a case study, we consider the limit q=1 (percolation) and find a correspondence between the magnetic exponents and those describing percolation on a scale-free network. Interestingly, logarithmic corrections to scaling appear at \lambda=4 in this case.Comment: 15 pages, 2 figure

Potts model with invisible states on a scale-free network

Different models are proposed to understand magnetic phase transitions through the prism of competition between the energy and the entropy. One of such models is a $q$-state Potts model with invisible states. This model introduces $r$ invisible states such that if spin lies in one of them, it does not interact with the rest. We consider such a model using the mean field approximation on an annealed scale-free network where the probability of a randomly chosen vertex having a degree $k$ is governed by the power-law $P(k)\propto k^{-\lambda}$. Our results confirm that $q,r$ and $\lambda$ play a role of global parameters that influence the critical behaviour of the system. Depending on their values the phase diagram is divided into three regions with different critical behaviour. However, the topological influence, presented by the marginal value of $\lambda_c(q)$, has proven to be dominant over the entropic one, governed by the number of invisible states $r$.Comment: 12 pages, 10 figure

Partition function zeros for the Ising model on complete graphs and on annealed scale-free networks

We analyze the partition function of the Ising model on graphs of two different types: complete graphs, wherein all nodes are mutually linked and annealed scale-free networks for which the degree distribution decays as $P(k)\sim k^{-\lambda}$. We are interested in zeros of the partition function in the cases of complex temperature or complex external field (Fisher and Lee-Yang zeros respectively). For the model on an annealed scale-free network, we find an integral representation for the partition function which, in the case $\lambda > 5$, reproduces the zeros for the Ising model on a complete graph. For $3<\lambda < 5$ we derive the $\lambda$-dependent angle at which the Fisher zeros impact onto the real temperature axis. This, in turn, gives access to the $\lambda$-dependent universal values of the critical exponents and critical amplitudes ratios. Our analysis of the Lee-Yang zeros reveals a difference in their behaviour for the Ising model on a complete graph and on an annealed scale-free network when $3<\lambda <5$. Whereas in the former case the zeros are purely imaginary, they have a non zero real part in latter case, so that the celebrated Lee-Yang circle theorem is violated.Comment: 36 pages, 31 figure

Classical phase transitions in a one-dimensional short-range spin model

Ising's solution of a classical spin model famously demonstrated the absence of a positive-temperature phase transition in one-dimensional equilibrium systems with short-range interactions. No-go arguments established that the energy cost to insert domain walls in such systems is outweighed by entropy excess so that symmetry cannot be spontaneously broken. An archetypal way around the no-go theorems is to augment interaction energy by increasing the range of interaction. Here we introduce new ways around the no-go theorems by investigating entropy depletion instead. We implement this for the Potts model with invisible states.Because spins in such a state do not interact with their surroundings, they contribute to the entropy but not the interaction energy of the system. Reducing the number of invisible states to a negative value decreases the entropy by an amount sufficient to induce a positive-temperature classical phase transition. This approach is complementary to the long-range interaction mechanism. Alternatively, subjecting positive numbers of invisible states to imaginary or complex fields can trigger such a phase transition. We also discuss potential physical realisability of such systems.Comment: 29 pages, 11 figure

Violation of Lee-Yang circle theorem for Ising phase transitions on complex networks

The Ising model on annealed complex networks with degree distribution decaying algebraically as $p(K)\sim K^{-\lambda}$ has a second-order phase transition at finite temperature if $\lambda> 3$. In the absence of space dimensionality, $\lambda$ controls the transition strength; mean-field theory applies for $\lambda >5$ but critical exponents are $\lambda$-dependent if $\lambda < 5$. Here we show that, as for regular lattices, the celebrated Lee-Yang circle theorem is obeyed for the former case. However, unlike on regular lattices where it is independent of dimensionality, the circle theorem fails on complex networks when $\lambda < 5$. We discuss the importance of this result for both theory and experiments on phase transitions and critical phenomena. We also investigate the finite-size scaling of Lee-Yang zeros in both regimes as well as the multiplicative logarithmic corrections which occur at $\lambda=5$.Comment: 5 pages, 5 figure

Marginal dimensions of the Potts model with invisible states

We reconsider the mean-field Potts model with $q$ interacting and $r$ non-interacting (invisible) states. The model was recently introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where the $Z_q$-symmetry is spontaneously broken. We analyse the marginal dimensions of the model, i.e., the value of $r$ at which the order of the phase transition changes. In the $q=2$ case, we determine that value to be $r_c = 3.65(5)$; there is a second-order phase transition there when $r<r_c$ and a first-order one at $r>r_c$. We also analyse the region $1 \leq q<2$ and show that the change from second to first order there is manifest through a new mechanism involving {\emph{two}} marginal values of $r$. The $q=1$ limit gives bond percolation and some intermediary values also have known physical realisations. Above the lower value $r_{c1}$, the order parameters exhibit discontinuities at temperature $\tilde{t}$ below a critical value $t_c$. But, provided $r>r_{c1}$ is small enough, this discontinuity does not appear at the phase transition, which is continuous and takes place at $t_c$. The larger value $r_{c2}$ marks the point at which the phase transition at $t_c$ changes from second to first order. Thus, for $r_{c1}< r < r_{c2}$, the transition at $t_c$ remains second order while the order parameter has a discontinuity at $\tilde{t}$. As $r$ increases further, $\tilde{t}$ increases, bringing the discontinuity closer to $t_c$. Finally, when $r$ exceeds $r_{c2}$ $\tilde{t}$ coincides with $t_c$ and the phase transition becomes first order. This new mechanism indicates how the discontinuity characteristic of first order phase transitions emerges.Comment: 15 pages, 7 figures, 2 table

On the discontinuity of the specific heat of the Ising model on a scale-free network

International audienceWe consider the Ising model on an annealed scale-free network with node-degree distribution characterized by a power-law decay P(K) similar to K-lambda. It is well established that the model is characterized by classical mean-field exponents for lambda > 5. In this note we show that the specific-heat discontinuity delta c(h) at the critical point remains lambda-dependent even for lambda > 5: delta c(h) = 3(lambda-5)(lambda-1)/[2(lambda-3)(2)] and attains its mean-field value delta c(h) = 3/2 only in the limit lambda -> infinity. We compare this behaviour with recent measurements of the d dependency of delta c(h) made for the Ising model on lattices with d > 4 [Lundow P. H., Markstrom K., Nucl. Phys. B, 2015, 895, 305]