Role of Broken Gauge Symmetry in Transport Phenomena Involving Neutral and Charged Particle

This paper formalizes and generalizes a number of intuitive ideas associated with electron band theory in solid state physics, in the idealized limit of infinitely-repeating, periodic lattices, to situations involving finite structures, with real boundaries. The arguments are general: they can be applied to all ordered or partially ordered solids, in the limit of low, but finite temperature T. In particular, in this paper, the terms order and periodic order are used inter-changeably. A solid that possesses partial order (or partial, periodic order) refers to the requirement that the particles in some finite volume, within the solid, occupy one or more many-body states Ψ possessing some degree of periodic symmetry, for some finite period of time. This last condition is defined by the requirement that one or more Bragg diffraction peaks (Ashcroft & Mermin 1976A), beyond the zeroeth order peak, be observable in the diffraction pattern that results when
Role of Broken Gauge Symmetry in Transport Phenomena Involving Neattices
Scott R. Chubb Remote Sensing Division Naval Research Laboratory, Washington, DC 20375, USA
particular particles (x-rays, neutrons, etc. ) scatter elastically off of the solid. An ordered, periodic, or periodically ordered solid refers to the limit in which the resolution between different diffraction peaks and the number of peaks asymptotically approach the resolution and number that one would conventionally associate with a diffraction pattern involving a situation in which a macroscopic number of scattering centers can be involved with the scattering processes that are responsible for the pattern. For the purpose of determining critical time- and length- scales associated with experiments involving finite size optical lattices, a number of the arguments involving finite size effects (and the associated overlap features) in partially ordered solids can also be generalized to optical lattices. (This topic is addressed in the final section of the paper.) The underlying logic behind the analysis follows from a number of general results that are required in order to insure that the reaction rate be minimized (through minimal overlap with potential perturbations) for a potential process that can couple the ground state (GS) to the lowest-lying excited states, in a solid or lattice. In particular, in general, the GS of a many-body system is required to have minimal overlap with external perturbations and with the lowest-lying excited states. But in the idealized limit in which a set of measurements can be performed, in such a way that the measurement process does not affect the energy of any of a number of potential many-body states (including the GS), locally, in a particular region of space, each measurement can be viewed as a form of “symmetry operation”. When each of these symmetry operations can be related to the others through one or more continuously-varying parameters (for example, through measurements of a continuously-varying angle or displacement, performed outside the particular region of space), the set of symmetry operations is referred to as a gauge symmetry. But because no symmetry operation is perfect, residual perturbations, tied to the measurement process exist. In the presence of these perturbations, the overlap of the GS with low-lying excited states is minimized through processes that minimize the GS energy. The resulting changes in wave function overlap and energy occur through a process that is referred to as broken gauge symmetry. Provided the particular region (which will be referred to as the bulk region) occurs within the solid (or within a lattice), where the numbers of neutral and charged particles are conserved (relative to a particular, well-defined boundary or boundary region), the rate of change of any process involving the GS and its overlap with low-lying excited states (through residual perturbations) is dominated by a universal broken gauge symmetry that occurs in ordered, finite lattices (the loss of translation symmetry at the boundaries of a lattice). In this particular limit: 1. In the absence of the residual perturbations, the energy remains invariant with respect to the gauge symmetry operation associated with performing a rigid shift of the bulk region, in which the coordinate of each particle in the bulk is shifted by the same, constant amount; 2. Since, by construction, the associated coordinate transformation (which is referred to as a Galilean, coordinate transformation) preserves the separation between each particle and the remaining particles in the bulk, a large degeneracy exists, and (in the presence of broken gauge symmetry) the lowest-lying excited states are all related to the GS through one of the possible (Galilean) transformations; and 3. In the presence of residual interactions, the lowest-lying excited states conserve particle number within the bulk region. Since the associated transformations do not alter the relative separations between any of the particles, effectively, the resulting change in position of the particles can be viewed as occurring through a form of “perfect” rigid body motion, in which the entire collection of particles moves with a common velocity, that is similar to the nearly-perfect, lattice recoil that