Spin in the worldline path integral in 2 + 1 dimensions

We present a constructive derivation of a worldline path integral for the effective action and the propagator of a Dirac field in 2 + 1 dimensions, in terms of spacetime and SU(2) paths. After studying some general properties of this representation, we show that the auxiliary gauge-group variable can be integrated, deriving a worldline action depending only on x(τ), the spacetime paths. We then show that the functional integral automatically imposes the constraint , while there is a spin action, which agrees with the one one should expect for a spin- field.     

Highly improved lattice field-strength tensor

We derive an -improved lattice version of the continuum field-strength tensor. Discretization errors are reduced via the combination of several clover terms of various sizes, complemented by tadpole improvement. The resulting improved field-strength tensor is used to construct -improved topological charge and action operators. As a test of our improved field-strength tensor, we compare the values attained by these operators as we cool several configurations to self-duality with a previously defined highly improved action and assess the relative scale of the remaining discretization errors. We demonstrate accuracy to better than one part in 10,000.     

On superdiffusive behavior of a passive tracer in a random flow

In this note we consider a passive tracer model describing particle dispersion in a turbulent flow. The trajectory of the particle is given by the solution of an ordinary differential equation , , where is a divergence-free, random vector field that is spatially homogeneous and isotropic. We show that trajectories of the tracer display superdiffusive behavior when the random velocity decorrelates, at large distances, but does it not rapidly but rather at some moderate rate. The main tools used in the proofs are variational principles and Tauberian-type theorems.     

Euler equations with non-homogeneous Navier slip boundary conditions

We consider the flow of an ideal fluid in a 2D bounded domain, admitting flows through the boundary of this domain. The flow is described by Euler equations with non-homogeneous Navier slip boundary conditions. These conditions can be written in the form , , where the tensor is the rate of strain of the fluid’s velocity and is the pair formed by the normal and tangent vectors to the boundary. We establish the solvability of this problem for the class of solutions with L_p-bounded vorticity, p∈(2,∞]. To prove the solvability we realize the passage to the limit in Navier-Stokes equations with vanishing viscosity.     

Comparison of |Q|=1 and |Q|=2 gauge-field configurations on the lattice four-torus

It is known that exactly self-dual gauge-field configurations with topological charge |Q|=1 cannot exist on the untwisted continuum four-torus. We explore the manifestation of this remarkable fact on the lattice four-torus for SU(3) using advanced techniques for controlling lattice discretization errors, extending earlier work of De Forcrand et al. for SU(2). We identify three distinct signals for the instability of |Q|=1 configurations, and show that these signals manifest themselves early in the cooling process, long before the would-be instanton has shrunk to a size comparable to the lattice discretization threshold. These signals do not appear for the individual instantons which make up our |Q|=2 configurations. This indicates that these signals reflect the truly global nature of the instability, rather than the local discretization effects which cause the eventual disappearance of the would-be single instanton. Monte-Carlo generated SU(3) gauge-field configurations are cooled to the self-dual limit using an -improved gauge action chosen to have small but positive errors. This choice prevents lattice discretization errors from destroying instantons provided their size exceeds the dislocation threshold of the cooling algorithm. Lattice discretization errors are evaluated by comparing the -improved gauge-field action with an -improved action constructed from the square of an -improved lattice field-strength tensor, thus having different discretization errors. The number of action-density peaks, the instanton size, and the topological charge of configurations is monitored. We observe a fluctuation in the total topological charge of |Q|=1 configurations, and demonstrate that the onset of this unusual behavior corresponds with the disappearance of multiple-peaks in the action density. At the same time discretization errors are minimal.     

Schrieffer–Wolff transformation for quantum many-body systems

The Schrieffer–Wolff (SW) method is a version of degenerate perturbation theory in which the low-energy effective Hamiltonian is obtained from the exact Hamiltonian by a unitary transformation decoupling the low-energy and high-energy subspaces. We give a self-contained summary of the SW method with a focus on rigorous results. We begin with an exact definition of the SW transformation in terms of the so-called direct rotation between linear subspaces. From this we obtain elementary proofs of several important properties of such as the linked cluster theorem. We then study the perturbative version of the SW transformation obtained from a Taylor series representation of the direct rotation. Our perturbative approach provides a systematic diagram technique for computing high-order corrections to . We then specialize the SW method to quantum spin lattices with short-range interactions. We establish unitary equivalence between effective low-energy Hamiltonians obtained using two different versions of the SW method studied in the literature. Finally, we derive an upper bound on the precision up to which the ground state energy of the nth-order effective Hamiltonian approximates the exact ground state energy.     

Fluctuation scaling of quotation activities in the foreign exchange market

We study the scaling behavior of quotation activities for various currency pairs in the foreign exchange market. The components’ centrality is estimated from multiple time series and visualized as a currency pair network. The power-law relationship between a mean of quotation activity and its standard deviation for each currency pair is found. The scaling exponent α and the ratio between common and specific fluctuations η increase with the length of the observation time window . The result means that although for , the market dynamics are governed by specific processes, and at a longer time scale the common information flow becomes more important. We point out that quotation activities are not independently Poissonian for , and temporally or mutually correlated activities of quotations can happen even at this time scale. A stochastic model for the foreign exchange market based on a bipartite graph representation is proposed.     

Quantum critical behaviour in doped Y bB12 studied by ab initio calculations

The Kondo insulator Y bB₁₂ is known to undergo a transition to the metallic state with doping or under an external magnetic field. Within the virtual crystal approximation (VCA), we calculated the occupation of the Yb 4f and 5d shells, and , as a function of doping of Y bB₁₂ with the rare earths Tm and Lu. We found that exhibits an anomalous change at the critical concentration of the dopant, in agreement with experiment ( for Y b_1−xLu_xB₁₂ and for Y b_1−xTm_xB₁₂). We suggest that the critical behaviour seems to be strictly connected with the change of and in consequence the change of the Yb valency.     

A ring of instantons inducing a monopole loop

We consider the superposition of infinitely many instantons on a circle in . The construction yields a self-dual solution of the Yang-Mills equations with action density concentrated on the ring. We show that this configuration is reducible in which case magnetic charge can be defined in a gauge invariant way. Indeed, we find a unit charge monopole (worldline) on the ring. This is an analytic example of the correlation between monopoles and action/topological density, however with infinite action. We show that both the Maximal Abelian Gauge and the Laplacian Abelian Gauge detect the monopole, while the Polyakov gauge does not. We discuss the implications of this configuration.