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Register a new userNucleation for one-dimensional long-range Ising models
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In this note we study metastability phenomena for a class of long-range Ising models in one-dimension. We prove that, under suitable general conditions, the configuration -1 is the only metastable state and we estimate the mean exit time. Moreover, we illustrate the theory with two examples (exponentially and polynomially decaying interaction) and we show that the critical droplet can be macroscopic or mesoscopic, according to the value of the external magnetic field.
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In this paper we study bond percolation on a one-dimensional chain with power-law bond probability $C/ r^{1+sigma}$, where $r$ is the distance length between distinct sites. We introduce and test an order $N$ Monte Carlo algorithm and we determine as a function of $sigma$ the critical value $C_{c}$ at which percolation occurs. The critical exponents in the range $0<sigma<1$ are reported and compared with mean-field and $varepsilon$-expansion results. Our analysis is in agreement, up to a numerical precision $approx 10^{-3}$, with the mean field result for the anomalous dimension $eta=2-sigma$, showing that there is no correction to $eta$ due to correlation effects.
We study the phase diagram and critical properties of quantum Ising chains with long-range ferromagnetic interactions decaying in a power-law fashion with exponent $alpha$, in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzero-temperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents $alpha > 1$ up to small deviations in some critical exponents. We also address the elusive regime $alpha < 1$, where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at $alpha = 0$. Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.
We investigate the nonequilibrium dynamics following a quench to zero temperature of the non-conserved Ising model with power-law decaying long-range interactions $propto 1/r^{d+sigma}$ in $d=2$ spatial dimensions. The zero-temperature coarsening is always of special interest among nonequilibrium processes, because often peculiar behavior is observed. We provide estimates of the nonequilibrium exponents, viz., the growth exponent $alpha$, the persistence exponent $theta$, and the fractal dimension $d_f$. It is found that the growth exponent $alphaapprox 3/4$ is independent of $sigma$ and different from $alpha=1/2$ as expected for nearest-neighbor models. In the large $sigma$ regime of the tunable interactions only the fractal dimension $d_f$ of the nearest-neighbor Ising model is recovered, while the other exponents differ significantly. For the persistence exponent $theta$ this is a direct consequence of the different growth exponents $alpha$ as can be understood from the relation $d-d_f=theta/alpha$; they just differ by the ratio of the growth exponents $approx 3/2$. This relation has been proposed for annihilation processes and later numerically tested for the $d=2$ nearest-neighbor Ising model. We confirm this relation for all $sigma$ studied, reinforcing its general validity.
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature $T_c$ for a one-dimensional Ising spin glass model with diluted long-range interactions. In this model, the probability $P_{ij}$ that an edge ${i,j}$ has nonvanishing interaction falls as a power-law with chord distance, $P_{ij}propto1/R_{ij}^{2sigma}$, and we study a range of values of $sigma$ with $1/2<sigma<1$. We consider a correlation function $C_{4}(r,t)$. A dynamic correlation length that shows power-law growth with time $xi(t)propto t^{1/z}$ can be identified in the data and, for large time $t$, $C_{4}(r,t)$ decays as a power law $r^{-alpha_d}$ with distance $r$ when $rll xi(t)$. The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent $alpha_d$ differentiates between a trivial MMAS ($alpha_d=0$), as expected in the droplet picture of spin glasses, and a nontrivial MMAS ($alpha_d e 0$), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero $alpha_d$ even in the regime $sigma >2/3$ which corresponds to short-range systems below six dimensions. For $sigma < 2/3$, the decay exponent $alpha_d$ follows the RSB prediction for the decay exponent $alpha_s = 3 - 4 sigma$ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches $alpha_d=1-sigma$ in the regime $2/3<sigma<1$; however, it deviates from both lines in the vicinity of $sigma=2/3$.
We study persistence in one-dimensional ferromagnetic and anti-ferromagnetic nearest-neighbor Ising models with parallel dynamics. The probability P(t) that a given spin has not flipped up to time t, when the system evolves from an initial random configuration, decays as P(t) sim 1/t^theta_p with theta_p simeq 0.75 numerically. A mapping to the dynamics of two decoupled A+A to 0 models yields theta_p = 3/4 exactly. A finite size scaling analysis clarifies the nature of dynamical scaling in the distribution of persistent sites obtained under this dynamics.
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