Abstract:
In many interesting physical settings, such as the vulcanization of rubber, the introduction of permanent random constraints between the constituents of a homogeneous fluid can cause a phase transition to a random solid state. In this random solid state, particles are permanently but randomly localized in space, and a rigidity to shear deformations emerges. Owing to the permanence of the random constraints, this phase transition is an equilibrium transition, which confers on it a simplicity (at least relative to the conventional glass transition) in the sense that it is amenable to established techniques of equilibrium statistical mechanics. In this Paper I shall review recent developments in the theory of random solidification for systems obeying permanent random constraints, with the aim of bringing to the fore the similarities and differences between such systems and those exhibiting the conventional glass transition. I shall also report new results, obtained in collaboration with Weiqun Peng, on equilibrium correlations and susceptibilities that signal the approach of the random solidification transition, discussing the physical interpretation and values of these quantities both at the Gaussian level of approximation and, via a renormalization-group approach, beyond.

Abstract:
The structure and energetics of superflow around quantized vortices, and the motion inherited by these vortices from this superflow, are explored in the general setting of the superfluidity of helium-four in arbitrary dimensions. The vortices may be idealized as objects of co-dimension two, such as one-dimensional loops and two-dimensional closed surfaces, respectively, in the cases of three- and four-dimensional superfluidity. By using the analogy between vorticial superflow and Ampere-Maxwell magnetostatics, the equilibrium superflow containing any specified collection of vortices is constructed. The energy of the superflow is found to take on a simple form for vortices that are smooth and asymptotically large, compared with the vortex core size. The motion of vortices is analyzed in general, as well as for the special cases of hyper-spherical and weakly distorted hyper-planar vortices. In all dimensions, vortex motion reflects vortex geometry. In dimension four and higher, this includes not only extrinsic but also intrinsic aspects of the vortex shape, which enter via the first and second fundamental forms of classical geometry. For hyper-spherical vortices, which generalize the vortex rings of three dimensional superfluidity, the energy-momentum relation is determined. Simple scaling arguments recover the essential features of these results, up to numerical and logarithmic factors.

Abstract:
The scattering of light at large wave-vector magnitudes k in nematic systems containing topological defects is investigated theoretically. At large k the structure factor S(k) is dominated by power-law contributions originating from singular order-parameter variations associated with topological defects and from transverse thermal fluctuations of the nematic director. These defects (nematic disclinations and hedgehogs) lead to contributions of the form rho A k^{-xi} (``the Porod tail''), where rho is the number density of a given type of defect, A is a dimensionless Porod amplitude, and xi is an integer-valued Porod exponent. The Porod amplitudes and exponents are calculated for all types of topologically stable defects occurring in uniaxial and biaxial nematics in two or three spatial dimensions. The range of wave-vectors in which the contributions to the scattering intensity due to defects dominate the contribution due to thermal fluctuations is estimated, and it is concluded that for experimentally accessible defect densities the range of observability of the Porod tail extends over one to three decades in scattering wave-vector magnitude k. Available experimental results on phase ordering in uniaxial nematics are analyzed, and applications of our results are suggested for light-scattering studies of other nematic systems containing numerous defects.

Abstract:
Correlators describing the vulcanization transition are constructed and explored via a renormalization group approach. This approach is based on a minimal model that accounts for the thermal motion of constituents and the quenched random constraints imposed on their motion by crosslinks. Critical exponents associated with the correlators are obtained near six dimensions, and found to equal those governing analogous entities in percolation theory. Some expectations for how the vulcanization transition is realized in two dimensions, developed with H. E. Castillo, are discussed.

Abstract:
Andreev billiards are finite, arbitrarily-shaped, normal-state regions, surrounded by superconductor. At energies below the superconducting gap, single-quasiparticle excitations are confined to the normal region and its vicinity, the essential mechanism for this confinement being Andreev reflection. This Paper develops and implements a theoretical framework for the investigation of the short-wave quantal properties of these single-quasiparticle excitations. The focus is primarily on the relationship between the quasiparticle energy eigenvalue spectrum and the geometrical shape of the normal-state region, i.e., the question of spectral geometry in the novel setting of excitations confined by a superconducting pair-potential. Among the central results of this investigation are two semiclassical trace formulas for the density of states. The first, a lower-resolution formula, corresponds to the well-known quasiclassical approximation, conventionally invoked in settings involving superconductivity. The second, a higher-resolution formula, allows the density of states to be expressed in terms of: (i) An explicit formula for the level density, valid in the short-wave limit, for billiards of arbitrary shape and dimensionality. This level density depends on the billiard shape only through the set of stationary-length chords of the billiard and the curvature of the boundary at the endpoints of these chords; and (ii) Higher-resolution corrections to the level density, expressed as a sum over periodic orbits that creep around the billiard boundary. Owing to the fact that these creeping orbits are much longer than the stationary chords, one can, inter alia, hear the stationary chords of Andreev billiards.

Abstract:
Interest in the protein folding problem has motivated a wide range of theoretical and experimental studies of the kinetics of the collapse of flexible homopolymers. In this Paper a phenomenological model is proposed for the kinetics of the early stages of homopolymer collapse following a quench from temperatures above to below the theta temperature. In the first stage, nascent droplets of the dense phase are formed, with little effect on the configurations of the bridges that join them. The droplets then grow by accreting monomers from the bridges, thus causing the bridges to stretch. During these two stages the overall dimensions of the chain decrease only weakly. Further growth of the droplets is accomplished by the shortening of the bridges, which causes the shrinking of the overall dimensions of the chain. The characteristic times of the three stages respectively scale as the zeroth, 1/5 and 6/5 power of the the degree of polymerization of the chain.

Abstract:
Andreev billiards are finite, arbitrarily-shaped, normal-state regions, surrounded by superconductor. At energies below the superconducting energy gap, single-quasiparticle excitations are confined to the normal region and its vicinity, the mechanism for confinement being Andreev reflection. Short-wave quantal properties of these excitations, such as the connection between the density of states and the geometrical shape of the billiard, are addressed via a multiple scattering approach. It is shown that one can, inter alia, hear the stationary chords of Andreev billiards.

Abstract:
Certain density correlators, measurable via various experimental techniques, are studied in the context of the vulcanization transition. It is shown that these correlators contain essential information about both the vulcanization transition and the emergent amorphous solid state. Contact is made with various physical ingredients that have featured in experimental studies of amorphous colloidal and gel systems and in theoretical studies of the glassy state.

Abstract:
We consider the quantum-control task of obtaining a target unitary operation via control fields that couple to the quantum system and are chosen to best mitigate errors resulting from time-dependent noise. We allow for two sources of noise: fluctuations in the control fields and those arising from the environment. We address the issue of error mitigation by means of a formulation rooted in the Martin-Siggia-Rose (MSR) approach to noisy, classical statistical-mechanical systems. We express the noisy control problem in terms of a path integral, and integrate out the noise to arrive at an effective, noise-free description. We characterize the degree of success in error mitigation via a fidelity, which characterizes the proximity of the sought-after evolution to ones achievable in the presence of noise. Error mitigation is then accomplished by applying the optimal control fields, i.e., those that maximize the fidelity subject to any constraints obeyed by the control fields. To make connection with MSR, we reformulate the fidelity in terms of a Schwinger-Keldysh (SK) path integral, with the added twist that the `forward' and `backward' branches of the time-contour are inequivalent with respect to the noise. The present approach naturally allows the incorporation of constraints on the control fields; a useful feature in practice, given that they feature in real experiments. We illustrate this MSR-SK approach by considering a system consisting of a single spin $s$ freedom (with $s$ arbitrary), focusing on the case of $1/f$ noise. We discover that optimal error-mitigation is accomplished via a universal control field protocol that is valid for all $s$, from the qubit (i.e., $s=1/2$) case to the classical (i.e., $s \to \infty$) limit. In principle, this MSR-SK approach provides a framework for addressing quantum control in the presence of noise for systems of arbitrary complexity.

Abstract:
We consider the quantum-control task of obtaining a target unitary operation via control fields that couple to the quantum system and are chosen to best mitigate errors resulting from time-dependent noise. We allow for two sources of noise: fluctuations in the control fields and those arising from the environment. We address the issue of error mitigation by means of a formulation rooted in the Martin-Siggia-Rose (MSR) approach to noisy, classical statistical-mechanical systems. We express the noisy control problem in terms of a path integral, and integrate out the noise to arrive at an effective, noise-free description. We characterize the degree of success in error mitigation via a fidelity, which characterizes the proximity of the sought-after evolution to ones achievable in the presence of noise. Error mitigation is then accomplished by applying the optimal control fields, i.e., those that maximize the fidelity subject to any constraints obeyed by the control fields. To make connection with MSR, we reformulate the fidelity in terms of a Schwinger-Keldysh (SK) path integral, with the added twist that the `forward' and `backward' branches of the time-contour are inequivalent with respect to the noise. The present approach naturally allows the incorporation of constraints on the control fields; a useful feature in practice, given that they feature in real experiments. We illustrate this MSR-SK approach by considering a system consisting of a single spin $s$ freedom (with $s$ arbitrary), focusing on the case of $1/f$ noise. We discover that optimal error-mitigation is accomplished via a universal control field protocol that is valid for all $s$, from the qubit (i.e., $s=1/2$) case to the classical (i.e., $s \to \infty$) limit. In principle, this MSR-SK approach provides a framework for addressing quantum control in the presence of noise for systems of arbitrary complexity.