Strongly Asymmetric Tricriticality of Quenched Random-Field Systems

In view of the recently seen dramatic effect of quenched random bonds on tricritical systems, we have conducted a renormalization-group study on the effect of quenched random fields on the tricritical phase diagram of the spin-1 Ising model in $d=3$. We find that random fields convert first-order phase transitions into second-order, in fact more effectively than random bonds. The coexistence region is extremely flat, attesting to an unusually small tricritical exponent $\beta_u$; moreover, an extreme asymmetry of the phase diagram is very striking. To accomodate this asymmetry, the second-order boundary exhibits reentrance.Comment: revtex, 4 pages, 2 figs, submitted to PR

Critical Percolation Phase and Thermal BKT Transition in a Scale-Free Network with Short-Range and Long-Range Random Bonds

Percolation in a scale-free hierarchical network is solved exactly by renormalization-group theory, in terms of the different probabilities of short-range and long-range bonds. A phase of critical percolation, with algebraic (Berezinskii-Kosterlitz-Thouless) geometric order, occurs in the phase diagram, in addition to the ordinary (compact) percolating phase and the non-percolating phase. It is found that no connection exists between, on the one hand, the onset of this geometric BKT behavior and, on the other hand, the onsets of the highly clustered small-world character of the network and of the thermal BKT transition of the Ising model on this network. Nevertheless, both geometric and thermal BKT behaviors have inverted characters, occurring where disorder is expected, namely at low bond probability and high temperature, respectively. This may be a general property of long-range networks.Comment: Added explanations and data. Published version. 4pages, 4 figure

An Analytic Equation of State for Ising-like Models

Using an Environmentally Friendly Renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, $y=f(x)$, that exhibits all desired asymptotic and analyticity properties in the three limits $x\to 0$, $x\to \infty$ and $x\to -1$. The only necessary inputs are the Wilson functions $\gamma_\lambda$, $\gamma_\phi$ and $\gamma_{\phi^2}$, associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson-Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N=1, we derive a one-loop equation of state for $2< d<4$ naturally parameterized by a ratio of non-linear scaling fields. For $d=3$ we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.Comment: 10 pages, 2 figure

On the CMAS Problem in Thermal Barrier Coatings%253A Benchmarking Thermochemical Resistance of Oxides Alternative to YSZ Through a Microscopic Standpoint

This study focuses on experimental modelling of the failure of Thermal Barrier Coatings (TBCs) due to attack of CMAS (Calcia-Magnesia-Alumina-Silicate), which is often found in harsh environments, via glassy phase infiltration. Volcanic ash and dust, sand particles, and fly ash, which contain CMAS, are imminent threats impeding predictable lifetimes of TBCs. Such incurrence directly affects the geometry and clinging to bond coat, and intrinsic material properties such as thermal conductivity and crystal structure of TBC are modified after exposure to CMAS, which ultimately results in delamination, spallation and failure of the coating material. The scope of this work is to survey the reactivity of CMAS with various oxide systems, and evaluate possible oxide systems that can be replaced and%252For used with Yttria-stabilized Zirconia (YSZ) by investigating the penetration depth and reactivity after sintering with CMAS. A cost-effective method to observe the reaction of candidate oxides with CMAS is suggested and administired%253B understanding the main mechanism that causes the failure of top coat in the wake of CMAS infiltration, and seeking solutions for the problem is performed by taking advantage of Scanning Electron Microscopy (SEM). Recently suggested ceramic oxide systems that form in pyrochlore structure, some perovskite structures in various compositions, monazite, mullite and YSZ are studied. The possible outcome consequent upon CMAS infiltration are concluded and course for designing novel material systems that are expected to withstand the CMAS attacks better than the state-of-the-art 4mole%25 YSZ is defined. 5%25 mole Yb-doped SrZrO3(5Yb-SZ) and favored pyrochlores such as Gd2r2O7 and GdYbZr2O7 are found to be better mitigating CMAS attacks

Potts-Percolation-Gauss Model of a Solid

We study a statistical mechanics model of a solid. Neighboring atoms are connected by Hookian springs. If the energy is larger than a threshold the "spring" is more likely to fail, while if the energy is lower than the threshold the spring is more likely to be alive. The phase diagram and thermodynamic quantities, such as free energy, numbers of bonds and clusters, and their fluctuations, are determined using renormalization-group and Monte-Carlo techniques.Comment: 10 pages, 12 figure

Strong Violation of Critical Phenomena Universality: Wang-Landau Study of the 2d Blume-Capel Model under Bond Randomness

We study the pure and random-bond versions of the square lattice ferromagnetic Blume-Capel model, in both the first-order and second-order phase transition regimes of the pure model. Phase transition temperatures, thermal and magnetic critical exponents are determined for lattice sizes in the range L=20-100 via a sophisticated two-stage numerical strategy of entropic sampling in dominant energy subspaces, using mainly the Wang-Landau algorithm. The second-order phase transition, emerging under random bonds from the second-order regime of the pure model, has the same values of critical exponents as the 2d Ising universality class, with the effect of the bond disorder on the specific heat being well described by double-logarithmic corrections, our findings thus supporting the marginal irrelevance of quenched bond randomness. On the other hand, the second-order transition, emerging under bond randomness from the first-order regime of the pure model, has a distinctive universality class with \nu=1.30(6) and \beta/\nu=0.128(5). This amounts to a strong violation of the universality principle of critical phenomena, since these two second-order transitions, with different sets of critical exponents, are between the same ferromagnetic and paramagnetic phases. Furthermore, the latter of these two transitions supports an extensive but weak universality, since it has the same magnetic critical exponent (but a different thermal critical exponent) as a wide variety of two-dimensional systems. In the conversion by bond randomness of the first-order transition of the pure system to second order, we detect, by introducing and evaluating connectivity spin densities, a microsegregation that also explains the increase we find in the phase transition temperature under bond randomness.Comment: Added discussion and references. 10 pages, 6 figures. Published versio

Random site dilution properties of frustrated magnets on a hierarchical lattice

We present a method to analyze magnetic properties of frustrated Ising spin models on specific hierarchical lattices with random dilution. Disorder is induced by dilution and geometrical frustration rather than randomness in the internal couplings of the original Hamiltonian. The two-dimensional model presented here possesses a macroscopic entropy at zero temperature in the large size limit, very close to the Pauling estimate for spin-ice on pyrochlore lattice, and a crossover towards a paramagnetic phase. The disorder due to dilution is taken into account by considering a replicated version of the recursion equations between partition functions at different lattice sizes. An analysis at first order in replica number allows for a systematic reorganization of the disorder configurations, leading to a recurrence scheme. This method is numerically implemented to evaluate the thermodynamical quantities such as specific heat and susceptibility in an external field.Comment: 26 pages, 11 figure

Critical Dynamics of the Contact Process with Quenched Disorder

We study critical spreading dynamics in the two-dimensional contact process (CP) with quenched disorder in the form of random dilution. In the pure model, spreading from a single particle at the critical point $\lambda_c$ is characterized by the critical exponents of directed percolation: in $2+1$ dimensions, $\delta = 0.46$, $\eta = 0.214$, and $z = 1.13$. Disorder causes a dramatic change in the critical exponents, to $\delta \simeq 0.60$, $\eta \simeq -0.42$, and $z \simeq 0.24$. These exponents govern spreading following a long crossover period. The usual hyperscaling relation, $4 \delta + 2 \eta = d z$, is violated. Our results support the conjecture by Bramson, Durrett, and Schonmann [Ann. Prob. {\bf 19}, 960 (1991)], that in two or more dimensions the disordered CP has only a single phase transition.Comment: 11 pages, REVTeX, four figures available on reques

Random walks on the Apollonian network with a single trap

Explicit determination of the mean first-passage time (MFPT) for trapping problem on complex media is a theoretical challenge. In this paper, we study random walks on the Apollonian network with a trap fixed at a given hub node (i.e. node with the highest degree), which are simultaneously scale-free and small-world. We obtain the precise analytic expression for the MFPT that is confirmed by direct numerical calculations. In the large system size limit, the MFPT approximately grows as a power-law function of the number of nodes, with the exponent much less than 1, which is significantly different from the scaling for some regular networks or fractals, such as regular lattices, Sierpinski fractals, T-graph, and complete graphs. The Apollonian network is the most efficient configuration for transport by diffusion among all previously studied structure.Comment: Definitive version accepted for publication in EPL (Europhysics Letters

Fractal and Transfractal Recursive Scale-Free Nets

We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals - possessing a finite fractal dimension - while others are small world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of "transfinite" dimension may be defined and applied to the small world nets. Concerning multiscaling, we show how first-passage time for diffusion and resistance between hub (the most connected nodes) scale differently than for other nodes. Despite the different scalings, the Einstein relation between diffusion and conductivity holds separately for hubs and nodes. The transfinite exponents of small world nets obey Einstein relations analogous to those in fractal nets.Comment: Includes small revisions and references added as result of readers' feedbac