Coulomb energy of uniformly charged spheroidal shell systems.

We provide exact expressions for the electrostatic energy of uniformly charged prolate and oblate spheroidal shells. We find that uniformly charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shell's aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.

Electrostatics-driven shape transitions in soft shells.

Manipulating the shape of nanoscale objects in a controllable fashion is at the heart of designing materials that act as building blocks for self-assembly or serve as targeted drug delivery carriers. Inducing shape deformations by controlling external parameters is also an important way of designing biomimetic membranes. In this paper, we demonstrate that electrostatics can be used as a tool to manipulate the shape of soft, closed membranes by tuning environmental conditions such as the electrolyte concentration in the medium. Using a molecular dynamics-based simulated annealing procedure, we investigate charged elastic shells that do not exchange material with their environment, such as elastic membranes formed in emulsions or synthetic nanocontainers. We find that by decreasing the salt concentration or increasing the total charge on the shell's surface, the spherical symmetry is broken, leading to the formation of ellipsoids, discs, and bowls. Shape changes are accompanied by a significant lowering of the electrostatic energy and a rise in the surface area of the shell. To substantiate our simulation findings, we show analytically that a uniformly charged disc has a lower Coulomb energy than a sphere of the same volume. Further, we test the robustness of our results by including the effects of charge renormalization in the analysis of the shape transitions and find the latter to be feasible for a wide range of shell volume fractions.

Novel disordering mechanism in ferromagnetic systems with competing interactions.

Ferromagnetic Ising systems with competing interactions are considered in the presence of a random field. We find that in three space dimensions the ferromagnetic phase is disordered by a random field which is considerably smaller than the typical interaction strength between the spins. This is the result of a novel disordering mechanism triggered by an underlying spin-glass phase. Calculations for the specific case of the long-range dipolar LiHo(x)Y(1-x)F(4) compound suggest that the above mechanism is responsible for the peculiar dependence of the critical temperature on the strength of the random field and the broadening of the susceptibility peaks as temperature is decreased, as found in recent experiments by Silevitch et al.. [Nature (London) 448, 567 (2007)]. Our results thus emphasize the need to go beyond the standard Imry-Ma argument when studying general random-field systems.

Numerically exact correlations and sampling in the two-dimensional Ising spin glass.

A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest-neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.

Boolean decision problems with competing interactions on scale-free networks: critical thermodynamics.

We study the critical behavior of Boolean variables on scale-free networks with competing interactions (Ising spin glasses). Our analytical results for the disorder-network-decay-exponent phase diagram are verified using Monte Carlo simulations. When the probability of positive (ferromagnetic) and negative (antiferromagnetic) interactions is the same, the system undergoes a finite-temperature spin-glass transition if the exponent that describes the decay of the interaction degree in the scale-free graph is strictly larger than 3. However, when the exponent is equal to or less than 3, a spin-glass phase is stable for all temperatures. The robustness of both the ferromagnetic and spin-glass phases suggests that Boolean decision problems on scale-free networks are quite stable to local perturbations. Finally, we show that for a given decay exponent spin glasses on scale-free networks seem to obey universality. Furthermore, when the decay exponent of the interaction degree is larger than 4 in the spin-glass sector, the universality class is the same as for the mean-field Sherrington-Kirkpatrick Ising spin glass.

Simplest model to study reentrance in physical systems.

We numerically investigate the necessary ingredients for reentrant behavior in the phase diagram of physical systems. Studies on the possibly simplest model that exhibits reentrance, the two-dimensional random-bond Ising model, show that reentrant behavior is generic whenever frustration is present in the model. For both discrete and continuous disorder distributions, the phase diagram in the disorder-temperature plane is found to be reentrant, where for some disorder strengths a paramagnetic phase exists at both high and low temperatures, but an ordered ferromagnetic phase exists for intermediate temperatures.

Zero- and low-temperature behavior of the two-dimensional ±J Ising spin glass.

Scaling arguments and precise simulations are used to study the square lattice ±J Ising spin glass, a prototypical model for glassy systems. Droplet theory explains, and our numerical results show, entropically stabilized long-range spin-glass order at zero temperature, which resembles the energetic stabilization of long-range order in higher-dimensional models at finite temperature. At low temperature, a temperature-dependent crossover length scale is used to predict the power-law dependence on temperature of the heat capacity and clarify the importance of disorder distributions.

Optimizing glassy p-spin models.

Computing the ground state of Ising spin-glass models with p-spin interactions is, in general, an NP-hard problem. In this work we show that unlike in the case of the standard Ising spin glass with two-spin interactions, computing ground states with p=3 is an NP-hard problem even in two space dimensions. Furthermore, we present generic exact and heuristic algorithms for finding ground states of p-spin models with high confidence for systems of up to several thousand spins.

Exact algorithm for sampling the two-dimensional Ising spin glass.

A sampling algorithm is presented that generates spin-glass configurations of the two-dimensional Edwards-Anderson Ising spin glass at finite temperature with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proceedings of the Eighth Symposium on Discrete Algorithms (SIAM, Philadelphia, 1997), p 258] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin-glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination rather than Gaussian elimination for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n(3/2)); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128(2), with fixed and periodic boundary conditions.

Bubble-raft model for a paraboloidal crystal.

We investigate crystalline order on a two-dimensional paraboloid of revolution by assembling a single layer of millimeter-sized soap bubbles on the surface of a rotating liquid, thus extending the classic work of Bragg and Nye on planar soap bubble rafts. Topological constraints require crystalline configurations to contain a certain minimum number of topological defects such as disclinations or grain boundary scars whose structure is analyzed as a function of the aspect ratio of the paraboloid. We find the defect structure to agree with theoretical predictions and propose a mechanism for scar nucleation in the presence of large Gaussian curvature.

Irrational mode locking in quasiperiodic systems.

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous-time models with irrational mode locking and possible experimental realizations are discussed.