30 research outputs found
Renormalization-group Method for Reduction of Evolution Equations; invariant manifolds and envelopes
The renormalization group (RG) method as a powerful tool for reduction of
evolution equations is formulated in terms of the notion of invariant
manifolds. We start with derivation of an exact RG equation which is analogous
to the Wilsonian RG equations in statistical physics and quantum field theory.
It is clarified that the perturbative RG method constructs invariant manifolds
successively as the initial value of evolution equations, thereby the meaning
to set is naturally understood where is the arbitrary initial
time. We show that the integral constants in the unperturbative solution
constitutes natural coordinates of the invariant manifold when the linear
operator in the evolution equation has no Jordan cell; when has a
Jordan cell, a slight modification is necessary because the dimension of the
invariant manifold is increased by the perturbation. The RG equation determines
the slow motion of the would-be integral constants in the unperturbative
solution on the invariant manifold. We present the mechanical procedure to
construct the perturbative solutions hence the initial values with which the RG
equation gives meaningful results. The underlying structure of the reduction by
the RG method as formulated in the present work turns out to completely fit to
the universal one elucidated by Kuramoto some years ago. We indicate that the
reduction procedure of evolution equations has a good correspondence with the
renormalization procedure in quantum field theory; the counter part of the
universal structure of reduction elucidated by Kuramoto may be the Polchinski's
theorem for renormalizable field theories. We apply the method to interface
dynamics such as kink-anti-kink and soliton-soliton interactions in the latter
of which a linear operator having a Jordan-cell structure appears.Comment: 67 pages. No figures. v2: Additional discussions on the unstable
motion in the the double-well potential are given in the text and the
appendix added. Some references are also added. Introduction is somewhat
reshape
Emergent Phenomena Induced by Spin-Orbit Coupling at Surfaces and Interfaces
Spin-orbit coupling (SOC) describes the relativistic interaction between the
spin and momentum degrees of freedom of electrons, and is central to the rich
phenomena observed in condensed matter systems. In recent years, new phases of
matter have emerged from the interplay between SOC and low dimensionality, such
as chiral spin textures and spin-polarized surface and interface states. These
low-dimensional SOC-based realizations are typically robust and can be
exploited at room temperature. Here we discuss SOC as a means of producing such
fundamentally new physical phenomena in thin films and heterostructures. We put
into context the technological promise of these material classes for developing
spin-based device applications at room temperature