epl draft Emergence of a non trivial fluctuating phase in the XY-rotors model on regular networks

We study an XY-rotor model on regular one dimensional lattices by varying the number of neighbours. The parameter 2 ≥ γ ≥ 1 is defined. γ = 2 corresponds to mean field and γ = 1 to nearest neighbours coupling. We find that for γ < 1.5 the system does not exhibit a phase transition, while for γ > 1.5 the mean field second order transition is recovered. For the critical value γ = γc = 1.5, the systems can be in a non trivial fluctuating phase for which the magnetisation shows important fluctuations in a given temperature range, implying an infinite susceptibilty. For all values of γ the magnetisation is computed analytically in the low temperatures range and the magnetised versus non-magnetised state which depends on the value of γ is recovered, confirming the critical value γc = 1.5. In the last two decades, systems with long-range interactions have attracted increasing attention and have been widely studied [1, 2]. In systems with short range interactions, the property of additivity allows to construct the canonical ensemble from the microcanonical, the two approaches being equivalent in the thermodynamic limit [3]. In constrast, the lack of additivity adds another layer of complexity to the picture when dealing with systems interacting via a long-range potential [4–9], giving for instance rise to possible negative specific heat in the microcanonical ensemble. Another peculiar feature corresponds to the fact that some long-range systems may dynamically keep track of their initial configuration, leading to long-lasting quasistationary states (QSSs). Peculiar of those states is their duration, which diverges with the system size [10], leading to ergodicity breaking [2,9,11,12]. Over the years the mean field rotator model (HMF), which corresponds to a mean field XY model with an added kinetic term [13], has become a paradigmatic model for the study of long range systems. In this frame, QSSs have been extensively studied and an out of equilibrium phase transition has been displayed [14, 15]. Moreover, these stationary states have been shown to display intriguing regular microscopic dynamics [16, 17] and an oscillating metastable state was observed [18], enriching the already various scenario of the HMF model. Moving one step further, a coupling constant depending on the distance r like 1/r, 0 < α < 2 was introduced, giving birth to the so called α − HMF model [19–22]. The parameter α allowed to explore the transition between the non-additive regime, for α < 1, and the additive one for α > 1: the first case, belonging to the aforementioned class of long-ranged systems, unveiled the same degree of complexity than the HMF model, displaying as well QSSs and negative specific heat [23]. Relaxing the assumption of global coupling, the XY model with just nearest neighbours interactions has been in his turn a very fertile subject for decades of numerical studies [24–30] . Among countless other remarkable features, this model in two dimensions shows a infinite order phase transition, retrieved by Kosterlitz and Thouless [31], affecting the correlation function: for low temperatures it shows a power law decay, while it switches to an exponential behaviour for high temperatures. More recently, another issue challenged the study of long-range systems: their interplay with complex network topologies inspired by real world ones [32]. Concerning the XY model, we acknowledge for instance studies on random networks [33] or on Small-World networks [34, 35]. In this Letter, we address this issue of complex networks too, investigating the transition from short-range to longrange regime from a quite different point of view than previous works. We chose as control parameter a topological condition, which is imposing the connectivity per interacting unit. We used the paradigmatic 1D-XY model p-1 ha l-0 07 21 43 7, v er si on 3 16 J an 2 01 3 Author manuscript, published in "EPL 101 (2013) 10002" DOI : 10.1209/0295-5075/101/10002