Dual time scales in simulated annealing of a two-dimensional Ising spin glass

We apply a generalized Kibble-Zurek out-of-equilibrium scaling ansatz to simulated annealing when approaching the spin-glass transition at temperature $T=0$ of the two-dimensional Ising model with random $J= \pm 1$ couplings. Analyzing the spin-glass order parameter and the excess energy as functions of the system size and the annealing velocity in Monte Carlo simulations with Metropolis dynamics, we find scaling where the energy relaxes slower than the spin-glass order parameter, i.e., there are two different dynamic exponents. The values of the exponents relating the relaxation time scales to the system length, $\tau \sim L^z$, are $z=8.28 \pm 0.03$ for the relaxation of the order parameter and $z=10.31 \pm 0.04$ for the energy relaxation. We argue that the behavior with dual time scales arises as a consequence of the entropy-driven ordering mechanism within droplet theory. We point out that the dynamic exponents found here for $T \to 0$ simulated annealing are different from the temperature-dependent equilibrium dynamic exponent $z_{\rm eq}(T)$, for which previous studies have found a divergent behavior; $z_{\rm eq}(T\to 0) \to \infty$. Thus, our study shows that, within Metropolis dynamics, it is easier to relax the system to one of its degenerate ground states than to migrate at low temperatures between regions of the configuration space surrounding different ground states. In a more general context of optimization, our study provides an example of robust dense-region solutions for which the excess energy (the conventional cost function) may not be the best measure of success.Comment: 13 pages, 16 figure

Nonequilibrium dynamics in the O(N) model to next-to-next-to-leading order in the 1/N expansion

Nonequilibrium dynamics in quantum field theory has been studied extensively using truncations of the 2PI effective action. Both 1/N and loop expansions beyond leading order show remarkable improvement when compared to mean-field approximations. However, in truncations used so far, only the leading-order parts of the self energy responsible for memory loss, damping and equilibration are included, which makes it difficult to discuss convergence systematically. For that reason we derive the real and causal evolution equations for an O(N) model to next-to-next-to-leading order in the 2PI-1/N expansion. Due to the appearance of internal vertices the resulting equations appear intractable for a full-fledged 3+1 dimensional field theory. Instead, we solve the closely related three-loop approximation in the auxiliary-field formalism numerically in 0+1 dimensions (quantum mechanics) and compare to previous approximations and the exact numerical solution of the Schroedinger equation.Comment: 29 pages, minor changes, references added; to appear in PR

Meteoritic material on the moon

Three types of meteoritic material are found on the moon: micrometeorites, ancient planetesimal debris from the "early intense bombardment," and debris of recent, craterforming projectiles. Their amounts and compositions have been determined from trace element studies. The micrometeorite component is uniformly distributed over the entire lunar surface, but is seen most clearly in mare soils. It has a primitive, C1-chondrite-like composition, and comprises 1 to 1.5 percent of mature soils. Apparently it represents cometary debris. The ancient component is seen in highland breccias and soils. Six varieties have been recognized, differing in their proportions of refractories (Ir, Re), volatiles (Ge, Sb), and Au. All have a fractionated composition, with volatiles depleted relative to siderophiles. The abundance patterns do not match those of the known meteorite classes. These ancient meteoritic components seem to represent the debris of an extinct population of bodies (planetisimals, moonlets) that produced the mare basins during the first 700 Myr of the moon's history. On the basis of their stratigraphy and geographic distribution, five of the six groups are tentatively assigned to specific mare basins: Imbrium, Serenitatis, Crisium, Nectaris, and Humorum or Nubium

Meteoritic material on the moon

Micrometeorites, ancient planetesimal debris from the early intense bombardment, and debris of recent, crater-forming projectiles are discussed and their amounts and compositions have been determined from trace element studies. The micrometeorite component is uniformly distrubuted over the entire lunar surface, but is seen most clearly in mare soils whereas, the ancient component is seen in highland breccias and soils. A few properties of the basin-forming objects are inferred from the trace element data. An attempt is made to reconstruct the bombardment history of the moon from the observation that only basin-forming objects fell on the moon after crustal differentiation. The apparent half-life of basin-forming bodies is close to the calculated value for earth-crossing planetesimals. It is shown that a gap in radiometric ages is expected between the Imbrium and Nectaris impacts, because all 7 basins formed in this interval lie on the farside or east limb

Quantum phase transitions in disordered dimerized quantum spin models and the Harris criterion

We use quantum Monte Carlo simulations to study effects of disorder on the quantum phase transition occurring versus the ratio g=J/J' in square-lattice dimerized S=1/2 Heisenberg antiferromagnets with intra- and inter-dimer couplings J and J'. The dimers are either randomly distributed (as in the classical dimer model), or come in parallel pairs with horizontal or vertical orientation. In both cases the transition violates the Harris criterion, according to which the correlation-length exponent should satisfy nu >= 1. We do not detect any deviations from the three-dimensional O(3) universality class obtaining in the absence of disorder (where nu = 0.71). We discuss special circumstances which allow nu<1 for the type of disorder considered here.Comment: 4+ pages, 3 figure

Neutron activation analysis for trace elements. Elements depleted on lunar surface - Implications for origin of moon and meteorite influx rate Final report, 1 Aug. 1969 - 31 Jan. 1971

Neutron activation analysis for trace elements depleted on lunar surface with implications for origin of moon and meteorite influx rat

Dynamic scaling in the 2D Ising spin glass with Gaussian couplings

We carry out simulated annealing and employ a generalized Kibble-Zurek scaling hypothesis to study the 2D Ising spin glass with normal-distributed couplings. The system has an equilibrium glass transition at temperature $T=0$. From a scaling analysis when $T\rightarrow 0$ at different annealing velocities, we extract the dynamic critical exponent $z$, i.e., the exponent relating the relaxation time $\tau$ to the system length $L$; $\tau\sim L^z$. We find $z=13.6 \pm 0.4$ for both the Edwards-Anderson spin-glass order parameter and the excess energy. This is different from a previous study of the system with bimodal couplings [S. J. Rubin, N. Xu, and A. W. Sandvik, Phys. Rev. E {\bf 95}, 052133 (2017)] where the dynamics is faster and the above two quantities relax with different exponents (and that of the energy is larger). We here argue that the different behaviors arise as a consequence of the different low-energy landscapes---for normal-distributed couplings the ground state is unique (up to a spin reflection) while the system with bimodal couplings is massively degenerate. Our results reinforce the conclusion of anomalous entropy-driven relaxation behavior in the bimodal Ising glass. In the case of a continuous coupling distribution, our results presented here indicate that, although Kibble-Zurek scaling holds, the perturbative behavior normally applying in the slow limit breaks down, likely due to quasi-degenerate states, and the scaling function takes a different form.Comment: 10 pages, 5 figure

Variational ground states of 2D antiferromagnets in the valence bond basis

We study a variational wave function for the ground state of the two-dimensional S=1/2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes h(x,y) for valence bonds connecting spins separated by (x,y) lattice spacings. In contrast to previous studies, in which a functional form for h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to 32*32 spins. We use two different schemes for optimizing the amplitudes; a Newton/conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only approx. 0.06% from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling approx. 2% below the exact ones at long distances. The amplitudes h(r) for valence bonds of long length r decay as 1/r^3. We also discuss some results for small frustrated lattices.Comment: v2: 8 pages, 5 figures, significantly expanded, new optimization method, improved result

Dynamical mean field solution of the Bose-Hubbard model

We present the effective action and self-consistency equations for the bosonic dynamical mean field (B-DMFT) approximation to the bosonic Hubbard model and show that it provides remarkably accurate phase diagrams and correlation functions. To solve the bosonic dynamical mean field equations we use a continuous-time Monte Carlo method for bosonic impurity models based on a diagrammatic expansion in the hybridization and condensate coupling. This method is readily generalized to bosonic mixtures, spinful bosons, and Bose-Fermi mixtures.Comment: 10 pages, 3 figures. includes supplementary materia

Ground state phases of the Half-Filled One-Dimensional Extended Hubbard Model

Using quantum Monte Carlo simulations, results of a strong-coupling expansion, and Luttinger liquid theory, we determine quantitatively the ground state phase diagram of the one-dimensional extended Hubbard model with on-site and nearest-neighbor repulsions U and V. We show that spin frustration stabilizes a bond-ordered (dimerized) state for U appr. V/2 up to U/t appr. 9, where t is the nearest-neighbor hopping. The transition from the dimerized state to the staggered charge-density-wave state for large V/U is continuous for U up to appr. 5.5 and first-order for higher U.Comment: 4 pages, 4 figure