First- and second-order phase transitions in scale-free networks

We study first- and second-order phase transitions of ferromagnetic lattice models on scale-free networks, with a degree exponent $\gamma$. Using the example of the $q$-state Potts model we derive a general self-consistency relation within the frame of the Weiss molecular-field approximation, which presumably leads to exact critical singularities. Depending on the value of $\gamma$, we have found three different regimes of the phase diagram. As a general trend first-order transitions soften with decreasing $\gamma$ and the critical singularities at the second-order transitions are $\gamma$-dependent.Comment: 4 pages, 1 figure, published versio

Boundary critical phenomena of the random transverse Ising model in D>=2 dimensions

Using the strong disorder renormalization group method we study numerically the critical behavior of the random transverse Ising model at a free surface, at a corner and at an edge in D=2, 3 and 4-dimensional lattices. The surface magnetization exponents are found to be: x_s=1.60(2), 2.65(15) and 3.7(1) in D=2, 3 and 4, respectively, which do not depend on the form of disorder. We have also studied critical magnetization profiles in slab, pyramid and wedge geometries with fixed-free boundary conditions and analyzed their scaling behavior.Comment: 7 pages, 11 figure

Rendezetlen kvantum spinrendszerek = Disordered quantum spinsystems

Különböző sokrészecskés rendszerek (kvantum spinláncok, létrák és réteges rendszerek; klasszikusspinrendszerek és sztochasztikus folyamatok, stb.) kooperatív viselkedését vizsgáltuk rendezetlenségjelenlétében. Speciális renormálási csoport eljárást használva megállapítottuk, hogy számos vizsgáltproblémánánál a rendezetlenségi fluktuációk domináns szerepet játszanak a determinisztikus (termikus, kvantum, sztochasztikus) fluktuációkkal szemben. A kvantum spinláncoknál ismert végtelenül rendezetlenés erősen rendezetlen fixpont fogalmát kiterjesztettük klasszikus rendszerekre és sztochasztikusfolyamatokra is. Több egydimenziós probléma esetén (rendezetlen kvantum Ising modell, rendezetlen kontakt folyamat, rendezetlen aszimmetrikus kizárási folyamat, stb.) aszinguláris tulajdonságokat aszimptotikusan egzaktul meghatároztuk,mind a kritikus pontban, mind azon kívül az un. Griffiths fázisban. Vizsgáltuk a rendezetlenség erősségénekváltozásakor tapasztalható átmeneteket is. | We have studied the cooperative behaviour of different manyparticle systems (quantum spin chains, laddersand layered systems; classical spin systems and stochastic processes, etc.) in the presence of quencheddisorder. Using a special renormalization group procedure we have found that for several studied problemsthe disorder fluctuations play a dominant role over deterministic (thermal, quantum or stochastic) fluctuations. The concept of infinite disorder and strong disorder fixed points, known for random quantum spin chains hasbeen extended to classical systems and stochastic processes, too. For several one-dimensional problems(random quantum Ising model, random contact process, random asymmetric exclusion process, etc.) thesingular properties are asymptotically exactly determined, both at the critical point and outside thecritical point, in the so called Griffiths phase. We have also studied the cross-overs by varying thestrength of the disorder

Corner contribution to percolation cluster numbers

We study the number of clusters in two-dimensional (2d) critical percolation, N_Gamma, which intersect a given subset of bonds, Gamma. In the simplest case, when Gamma is a simple closed curve, N_Gamma is related to the entanglement entropy of the critical diluted quantum Ising model, in which Gamma represents the boundary between the subsystem and the environment. Due to corners in Gamma there are universal logarithmic corrections to N_Gamma, which are calculated in the continuum limit through conformal invariance, making use of the Cardy-Peschel formula. The exact formulas are confirmed by large scale Monte Carlo simulations. These results are extended to anisotropic percolation where they confirm a result of discrete holomorphicity.Comment: 7 pages, 9 figure

Random transverse-field Ising chain with long-range interactions

We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power alpha of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent z_c=alpha. Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an alpha-independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.Comment: 6 pages, 4 figure