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Sharp critical behavior for pinning model in random correlated environment  [PDF]
Quentin Berger,Hubert Lacoin
Mathematics , 2011,
Abstract: This article investigates the effect for random pinning models of long range power-law decaying correlations in the environment. For a particular type of environment based on a renewal construction, we are able to sharply describe the phase transition from the delocalized phase to the localized one, giving the critical exponent for the (quenched) free-energy, and proving that at the critical point the trajectories are fully delocalized. These results contrast with what happens both for the pure model (i.e. without disorder) and for the widely studied case of i.i.d. disorder, where the relevance or irrelevance of disorder on the critical properties is decided via the so-called Harris Criterion.
Glassy phases and driven response of the phase-field-crystal model with random pinning  [PDF]
E. Granato,J. A. P. Ramos,C. V. Achim,J. Lehikoinen,S. C. Ying,T. Ala-Nissila,K. R. Elder
Physics , 2011, DOI: 10.1103/PhysRevE.84.031102
Abstract: We study the structural correlations and the nonlinear response to a driving force of a two-dimensional phase-field-crystal model with random pinning. The model provides an effective continuous description of lattice systems in the presence of disordered external pinning centers, allowing for both elastic and plastic deformations. We find that the phase-field crystal with disorder assumes an amorphous glassy ground state, with only short-ranged positional and orientational correlations even in the limit of weak disorder. Under increasing driving force, the pinned amorphous-glass phase evolves into a moving plastic-flow phase and then finally a moving smectic phase. The transverse response of the moving smectic phase shows a vanishing transverse critical force for increasing system sizes.
The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics  [PDF]
Quentin Berger,Hubert Lacoin
Mathematics , 2010, DOI: 10.1007/s10955-010-0110-x
Abstract: We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on Z^d with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t \ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta L_t(X,Y)), beta in R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta_c, such that if beta>beta_c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d larger or equal to three 3, the presence of disorder makes the phase transition at least of second order. This, in dimension larger or equal to 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.
Path properties of the disordered pinning model in the delocalized regime  [PDF]
Kenneth S. Alexander,Nikos Zygouras
Mathematics , 2012, DOI: 10.1214/13-AAP930
Abstract: We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense "tight in probability" as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.
Copolymer with pinning: variational characterization of the phase diagram  [PDF]
Frank den Hollander,Alex A. Opoku
Mathematics , 2012, DOI: 10.1007/s10955-013-0747-3
Abstract: This paper studies a polymer chain in the vicinity of a linear interface separating two immiscible solvents. The polymer consists of random monomer types, while the interface carries random charges. Both the monomer types and the charges are given by i.i.d. sequences of random variables. The configurations of the polymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The Hamiltonian has two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface interaction ("pinning"). The quenched and the annealed version of the model each undergo a transition from a localized phase (where the polymer stays close to the interface) to a delocalized phase (where the polymer wanders away from the interface). We exploit the approach developed in [5] and [3] to derive variational formulas for the quenched and the annealed free energy per monomer. These variational formulas are analyzed to obtain detailed information on the critical curves separating the two phases and on the typical behavior of the polymer in each of the two phases. Our main results settle a number of open questions.
Vortex glass transition in a random pinning model  [PDF]
Anders Vestergren,Jack Lidmar,Mats Wallin
Physics , 2001, DOI: 10.1103/PhysRevLett.88.117004
Abstract: We study the vortex glass transition in disordered high temperature superconductors using Monte Carlo simulations. We use a random pinning model with strong point-correlated quenched disorder, a net applied magnetic field, longrange vortex interactions, and periodic boundary conditions. From a finite size scaling study of the helicity modulus, the RMS current, and the resistivity, we obtain critical exponents at the phase transition. The new exponents differ substantially from those of the gauge glass model, but are consistent with those of the pure three-dimensional XY model.
Scaling limits of $(1+1)$-dimensional pinning models with Laplacian interaction  [PDF]
Francesco Caravenna,Jean-Dominique Deuschel
Mathematics , 2008, DOI: 10.1214/08-AOP424
Abstract: We consider a random field $\varphi:\{1,...,N\}\to \mathbb{R}$ with Laplacian interaction of the form $\sum_iV(\Delta\varphi_i)$, where $\Delta$ is the discrete Laplacian and the potential $V(\cdot)$ is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime $(\varepsilon<\varepsilon_c)$ and a localized one $(\varepsilon>\varepsilon_c)$, where $0<\varepsilon_c<\infty$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance $N^{3/2}$, while in the localized regime the distance is just $O((\log N)^2)$. A subtle scenario shows up in the critical regime $(\varepsilon=\varepsilon_c)$, where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable L\'evy process of index 2/5. Our approach is based on Markov renewal theory.
Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction  [PDF]
Francesco Caravenna,Jean-Dominique Deuschel
Mathematics , 2007, DOI: 10.1214/08-AOP395
Abstract: We consider a random field $\varphi:\{1,...,N\}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $\varphi=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum_iV(\Delta\varphi_i))$ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $\Delta$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\varepsilon$ of the pinning reward varies: both in the pinning ($a=\mathrm{p}$) and in the wetting ($a=\mathrm{w}$) case, there exists a critical value $\varepsilon_c^a$ such that when $\varepsilon>\varepsilon_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\varepsilon<\varepsilon_c^a$ (delocalization). The two critical values are nontrivial and distinct: $0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty$, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\varepsilon=\varepsilon_c^{\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\varepsilon=\varepsilon_c^{\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.
Random Pinning Glass Model  [PDF]
Smarajit Karmakar,Giorgio Parisi
Physics , 2012, DOI: 10.1073/pnas.1222848110
Abstract: Glass transition where viscosity of liquids increases dramatically upon decrease of temperature without any major change in structural properties, remains one of the most challenging problems in condensed matter physics (Cavagna, 2009; Berthier and Biroli, 2011) in spite of tremendous research efforts in last decades. On the other hand disordered freezing of spins in a magnetic materials with decreasing temperature, the so-called spin glass transition, is relatively better understood (Mezard, Parisi and Virasoro, 1987; Castellani and Cavagna, 2005). Previously found similarity between some spin glass models with the structural glasses (Kirkpatrick and Thirumalai, 1987; Kirkpatrick and Wolynes, 1987; Kirkpatrick and Wolynes, 1987; Franz and Parisi, 1999; Moore and Drossel, 2002) inspired development of theories of structural glasses (Kirkpatrick, Thirumalai and Wolynes, 1989; Barrat, Franz and Parisi, 1997; M\'ezard and Parisi, 1999; Lubchenko and Wolynes, 2007; Biroli and Bouchaud, 2012) based on the scenario of spin glass transition. This scenario though looks very appealing is still far from being well established. One of the main differences between standard spin systems to molecular systems is the absence of quenched disorder and the presence of translational invariance: it often assumed that this difference is not relevant, but this conjecture is still far from being established. The quantities, which are well defined and characterized for spin models, are not easily calculable for molecular glasses due to the lack of quenched disorder which breaks the translational invariance in the system and the characterization of the similarity between the spin and the structural glass transition remained an elusive subject still now. In this study we introduced a model structural glass with built in quenched disorder which alleviates this main difference between the spin and molecular glasses thereby helping us to compare these two systems: the possibility of producing a good thermalization at rather low temperatures is one of the advantages of this model.
Quantum phase transition in the delocalized regime of the spin-boson model  [PDF]
Qing-Jun Tong,Jun-Hong An,Hong-Gang Luo,C. H. Oh
Physics , 2011, DOI: 10.1103/PhysRevB.84.174301
Abstract: The existence of the delocalized-localized quantum phase transition (QPT) in the ohmic spin-boson model has been commonly recognized. While the physics in the localized regime is relatively simple, the delocalized regime shows many interesting behaviors. Here we reveal that in this regime there exists a novel QPT: namely, from a phase without a bound state to a phase with a bound state, which leads to completely different dynamical behaviors in these two phases. If the reservoir is initially in the displaced vacuum state (i.e., the coherent state), the spin dynamics exhibits lossless oscillation when the bound state exists; otherwise, the oscillation will decay completely. This result is compatible with the coherence-incoherence transition occurring in the small-tunneling limit. Our work indicates that the QPT physics in the spin-boson model needs further exploration.
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