Universal properties of Ising clusters and droplets near criticality

Clusters and droplets of positive spins in the two-dimensional Ising model percolate at the Curie temperature in absence of external field. The percolative exponents coincide with the magnetic ones for droplets but not for clusters. We use integrable field theory to determine amplitude ratios which characterize the approach to criticality within these two universality classes of percolative critical behavior.     

Universal topological properties of two-dimensional trivalent cellular patterns

Universal topological properties of two-dimensional trivalent cellular patterns are found from shell analysis of soap froth and computer-generated Voronoi diagrams. We introduce a cluster analysis based on the shell model and find the universal relation ln(a/mu(2)) = A+Bln(mu(2)), with the generalized Aboav parameter a and second moment of the number of cell edge distribution mu(2). For the second, third, and fourth shells of the cluster, A and B are the same for all samples. Furthermore, A is increasing with shell number while B is a universal number, -0.90. For the first shell, the slope B is the same for soap froths, but slightly different from Voronoi graphs.     

Universal properties of self-avoiding walks from two-dimensional field theory

We use the recently conjectured exact S-matrix of the massive O(n) model to derive its form factors and ground state energy. This information is then used in the limit n → 0 to obtain quantitative results for various universal properties of self-avoiding chains and loops. In particular, we give the first theoretical prediction of the amplitude ratio C/D which relates the mean square end-to-end distance of chains to the mean square radius of gyration of closed loops. This agrees with the results from lattice enumeration studies to within their errors, and gives strong support for the various assumptions which enter into the field theoretic derivation. In addition, we obtain results for the scaling function of the structure factor of long loops, and for various amplitude ratios measuring the shape of self-avoiding chains. These quantities are all related to moments of correlation functions which are evaluated as a sum over m-particle intermediate states in the corresponding field theory. We show that in almost all cases, the restriction to m 2 gives results which are accurate to at least one part in 10³. This remarkable fact is traced to a softening of the m > 2 branch cuts relative to their behaviour based on phase space arguments alone, a result which follows from the threshold behaviour of the two-body S-matrix, S(O) = −1. Since this is a general property of interacting 2D field theories, it suggests that similar approximation may well hold for other models. However, we also study the moments of the area of self-avoiding loops, and show that, in this case, the two-particle approximation is not valid.     

Universal periods in quantum Hall droplets

Using the hierarchy picture of the fractional quantum Hall effect, we study the ground-state periodicity of a finite size quantum Hall droplet in a quantum Hall fluid of a different filling factor. The droplet edge charge is periodically modulated with flux through the droplet and will lead to a periodic variation in the conductance of a nearby point contact, such as occurs in some quantum Hall interferometers. Our model is consistent with experiment and predicts that superperiods can be observed in geometries where no interfering trajectories occur. The model may also provide an experimentally feasible method of detecting elusive neutral modes and otherwise obtaining information about the microscopic edge structure in fractional quantum Hall states.     

Isoperimetric phase transitions of two-dimensional droplets

We consider two-dimensional assemblies of particles governed by hamiltonians depending on the area and the perimeter of their convex hull. Provided the hamiltonian is quadratically homogeneous in the coordinates, we find an exact formula for the free energy. Phase transitions resulting from the competition between area and perimeter can easily be produced and explicitly dealt with. We illustrate those features by a simple example undergoing a second-order transition.     

Ground states in two-dimensional boson quantum field-theory

We consider λ:P(?):₂ models of Euclidean quantum field theory, P(?) = Σ_k=1²ⁿa_k?^k and define for them translation-invariant ground states. It appears that the number of them is not more than two provided $\text{λ}\text{m}{}_0{}^2$ is large.     

Continuum models of two-dimensional random droplets

We study the statistical mechanical properties of a two-dimensional assembly of free particles coupled to a mechanical reservoir. The particles-reservoir interaction is modelised by an Hamiltonian depending on the convex hull of the particles only. We concentrate on models whose energy is the sum of an area-term, a perimeter term and possibly a term preventing the particles occupying the interior of the convex hull. The range of coupling constants insuring a thermodynamic behaviour, as well as the associated free energy per particle are exactly determined.Work partially supported by the Swiss National Foundation for Scientific Research.     

Quantum hall effect of two-dimensional interacting boson systems

While non-interacting two-dimensional (2-D) boson systems cannot have the quantum Hall effect (QHE), we show that 2-D interacting boson systems can have the QHE if there is a downward cusp in the ground state energy. A necessary condition to have the QHE at filling factor v = p/q with σ_H = (p/q)e²/h is that p and q must satisfy e^ipqπ= 1, so the possible candidates are v = 2n₁/(2n₂ + 1) and (2n₁ + 1)/2n₂ where n₁ and n₂ are integers.     

Quantum droplets in two-dimensional optical lattices

We study the stability of zero-vorticity and vortex lattice quantum droplets (LQDs), which are described by a two-dimensional (2D) Gross–Pitaevskii (GP) equation with a periodic potential and Lee– Huang–Yang (LHY) term. The LQDs are divided in two types: onsite-centered and offsite-centered LQDs, the centers of which are located at the minimum and the maximum of the potential, respectively. The stability areas of these two types of LQDs with different number of sites for zero-vorticity and vorticity with S = 1 are given. We found that the μ–N relationship of the stable LQDs with a fixed number of sites can violate the Vakhitov–Kolokolov (VK) criterion, which is a necessary stability condition for nonlinear modes with an attractive interaction. Moreover, the μ–N relationship shows that two types of vortex LQDs with the same number of sites are degenerated, while the zero-vorticity LQDs are not degenerated. It is worth mentioning that the offsite-centered LQDs with zero-vorticity and vortex LQDs with S = 1 are heterogeneous.     

Collisional dynamics of symmetric two-dimensional quantum droplets

The collisional dynamics of two symmetric droplets with equal intraspecies scattering lengths and particle number density for each component is studied by solving the corresponding extended Gross−Pitaevskii equation in two dimensions by including a logarithmic correction term in the usual contact interaction. We find the merging droplet after collision experiences a quadrupole oscillation in its shape and the oscillation period is found to be independent of the incidental momentum for small droplets. With increasing collision momentum the colliding droplets may separate into two, or even more, and finally into small pieces of droplets. For these dynamical phases we manage to present boundaries determined by the remnant particle number in the central area and the damped oscillation of the quadrupole mode. A stability peak for the existence of droplets emerges at the critical particle numberN_c ≃ 48 for the quasi-Gaussian and flat-top shapes of the droplets.     

Electromagnetic properties of theZ boson

Direct (nonresonant) 3-body decays of charmed mesons are calculated in chiral perturbation theory. The magnitudes of the 20 and 84 representations of the effective chiralSU(4)×SU(4) Lagrangian are determined from the measured 2-bodyD→Kπ rate. For decay modes which do not contain nonspectator contributions, the agreement of theoretical predictions with the data is satisfactory. Nevertheless, the large discrepancy between theory and experiment for decays which can proceed through theW-exchange orW-annihilation might imply the importance of nonperturbative corrections to the nonspectator diagram and the existence of final-state interactions.     

Polarization properties of boson radiation

The polarization properties of boson radiation in a magnetic field have been studied, and expressions obtained for the integral intensity of boson and fermion radiation in a magnetic field which are suitable for random energies. The investigation of boson radiation in an external field is of limited practical interest; however, from the methodological point of view it is very instructive and has already been investigated by several authors [1, 2, 4], Also of undoubted interest is the comparison of boson and fermion (electron) radiation and hence the explanation of the role of spin in radiation.In conclusion the author expresses his deep appreciation to Prof. I, M. Ternov for assistance with the work and discussion of the results.     

Finite-temperature phase transitions in a two-dimensional boson Hubbard model

We study finite-temperature phase transitions in a two-dimensional boson Hubbard model with zero-point quantum fluctuations via Monte Carlo simulations of a quantum rotor model and construct the corresponding phase diagram. Compressibility shows a thermally activated gapped behavior in the insulating regime. Finite-size scaling of the superfluid stiffness clearly shows the nature of the Kosterlitz-Thouless transition. The transition temperature T(c) confirms a scaling relation T(c) proportional, rho(0)(x), with x=1.0. Some evidence of anomalous quantum behavior at low temperatures is presented.     

Universal distribution functions in two-dimensional localized systems

We find the conductance distribution function of the two-dimensional Anderson model in the strongly localized limit. The fluctuations of lng grow with lateral size as L1/3 and follow a universal distribution that depends on the type of leads. For narrow leads, it is the Tracy-Widom distribution, which appears in the problem of the largest eigenvalue of random matrices from the Gaussian unitary ensemble and in many other problems like the longest increasing subsequence of a permutation, directed polymers, or polynuclear growth. We also show that for wide leads the conductance follows a related, but different, distribution.