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

Rashba spin-orbit interaction enhanced by graphene in-plane deformations

Graphene consists in a single-layer carbon crystal where 2$p_z$ electrons display a linear dispersion relation in the vicinity of the Fermi level, conveniently described by a massless Dirac equation in $2+1$ spacetime. Spin-orbit effects open a gap in the band structure and offer perspectives for the manipulation of the conducting electrons spin. Ways to manipulate spin-orbit couplings in graphene have been generally assessed by proximity effects to metals that do not compromise the mobility of the unperturbed system and are likely to induce strain in the graphene layer. In this work we explore the $\rm{U(1)}\times SU(2)$ gauge fields that result from the uniform stretching of a graphene sheet under a perpendicular electric field. Considering such deformations is particularly relevant due to the counter-intuitive enhancement of the Rashba coupling between 30-50% for small bond deformations well known from tight-binding and DFT calculations. We report the accessible changes that can be operated in the band structure in the vicinity of the K points as a function of the deformation strength and direction.Comment: 10 pages, 7 figure

Network harness: bundles of routes in public transport networks

Public transport routes sharing the same grid of streets and tracks are often found to proceed in parallel along shorter or longer sequences of stations. Similar phenomena are observed in other networks built with space consuming links such as cables, vessels, pipes, neurons, etc. In the case of public transport networks (PTNs) this behavior may be easily worked out on the basis of sequences of stations serviced by each route. To quantify this behavior we use the recently introduced notion of network harness. It is described by the harness distribution P(r,s): the number of sequences of s consecutive stations that are serviced by r parallel routes. For certain PTNs that we have analyzed we observe that the harness distribution may be described by power laws. These power laws observed indicate a certain level of organization and planning which may be driven by the need to minimize the costs of infrastructure and secondly by the fact that points of interest tend to be clustered in certain locations of a city. This effect may be seen as a result of the strong interdependence of the evolutions of both the city and its PTN. To further investigate the significance of the empirical results we have studied one- and two-dimensional models of randomly placed routes modeled by different types of walks. While in one dimension an analytic treatment was successful, the two dimensional case was studied by simulations showing that the empirical results for real PTNs deviate significantly from those expected for randomly placed routes.Comment: 12 pages, 24 figures, paper presented at the Conference ``Statistical Physics: Modern Trends and Applications'' (23-25 June 2009, Lviv, Ukaine) dedicated to the 100th anniversary of Mykola Bogolyubov (1909-1992

The 2D XY model on a finite lattice with structural disorder: quasi-long-range ordering under realistic conditions

We present an analytic approach to study concurrent influence of quenched non-magnetic site-dilution and finiteness of the lattice on the 2D XY model. Two significant deeply connected features of this spin model are: a special type of ordering (quasi-long-range order) below a certain temperature and a size-dependent mean value of magnetisation in the low-temperature phase that goes to zero (according to the Mermin-Wagner-Hohenberg theorem) in the thermodynamic limit. We focus our attention on the asymptotic behaviour of the spin-spin correlation function and the probability distribution of magnetisation. The analytic approach is based on the spin-wave approximation valid for the low-temperature regime and an expansion in the parameters which characterise the deviation from completely homogeneous configuration of impurities. We further support the analytic considerations by Monte Carlo simulations performed for different concentrations of impurities and compare analytic and MC results. We present as the main quantitative result of the work the exponent of the spin-spin correlation function power law decay. It is non universal depending not only on temperature as in the pure model but also on concentration of magnetic sites. This exponent characterises also the vanishing of magnetisation with increasing lattice size.Comment: 13 pages, 7 eps figures, style files include

Green Function Simulation of Hamiltonian Lattice Models with Stochastic Reconfiguration

We apply a recently proposed Green Function Monte Carlo to the study of Hamiltonian lattice gauge theories. This class of algorithms computes quantum vacuum expectation values by averaging over a set of suitable weighted random walkers. By means of a procedure called Stochastic Reconfiguration the long standing problem of keeping fixed the walker population without a priori knowledge on the ground state is completely solved. In the $U(1)_2$ model, which we choose as our theoretical laboratory, we evaluate the mean plaquette and the vacuum energy per plaquette. We find good agreement with previous works using model dependent guiding functions for the random walkers.Comment: 14 pages, 5 PostScript Figures, RevTeX, two references adde

Gauge field theory approach to spin transport in a 2D electron gas

We discuss the Pauli Hamiltonian including the spin-orbit interaction within an U(1) x SU(2) gauge theory interpretation, where the gauge symmetry appears to be broken. This interpretation offers new insight into the problem of spin currents in the condensed matter environment, and can be extended to Rashba and Dresselhaus spin-orbit interactions. We present a few outcomes of the present formulation: i) it automatically leads to zero spin conductivity, in contrast to predictions of Gauge symmetric treatments, ii) a topological quantization condition leading to voltage quantization follows, and iii) spin interferometers can be conceived in which, starting from a arbitrary incoming unpolarized spinor, it is always possible to construct a perfect spin filtering condition.Comment: Invited contribution to Statphys conference, June 2009, Lviv (Ukraine