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Journal of Physics A Mathematical and General

A new concept of geometric phase in parameter space: coupling as a parameter

Lei Xing

Published Jul 12, 2006• 6 citations
Publisher

Abstract

We have obtained the geometric phase of the tripartite spin-1/2 system with spin-spin coupling and discussed the influence of this coupling on the geometric phase. We have discovered the coupling-independent geometric phase and explained it. Based on this work, we propose the concept of quasimagnetic field and assimilate the coupling into the parameter space. By this theory, the relation between the geometric phase and the solid angle in parameter space still holds for a system with spin-spin coupling if the concept of the parameter is properly revised in the way we have elucidated.

DOI
10.1088/0305-4470/39/30/010

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We have obtained the geometric phase of the tripartite spin-1/2 system with spin-spin coupling and discussed the influence of this coupling on the geometric phase. We have discovered the coupling-independent geometric phase and explained it. Based on this work, we propose the concept of quasimagnetic field and assimilate the coupling into the parameter space. By this theory, the relation between the geometric phase and the solid angle in parameter space still holds for a system with spin-spin coupling if the concept of the parameter is properly revised in the way we have elucidated.
The geometric phase of the bipartite Heisenberg spin-1/2 system with one spin driven by rotating magnetic field is investigated. It is found that in the one-site drive case, the intersubsystem coupling can be equivalent to a static quasi-magnetic field in the parameter space. This perspective has satisfactorily explained the irregular asymptote effect of geometric phase. We discuss the property of the two-site magnetic drive spin system and discover that a stationary state with no geometric phase shift is generated.
The geometric phase of the bipartite Heisenberg spin-1/2 system with one spin driven by rotating magnetic field is investigated. It is found that in the one-site drive case, the intersubsystem coupling can be equivalent to a static quasi-magnetic field in the parameter space. This perspective has satisfactorily explained the irregular asymptote effect of geometric phase. We discuss the property of the two-site magnetic drive spin system and discover that a stationary state with no geometric phase shift is generated.
An open note connecting Lei Xing's foundational 2006 insight — that coupling belongs in parameter space (J. Phys. A: Math. Gen., 39(30), doi:10.1088/0305-4470/39/30/010) — to Geometric Coupling Theory and its application in radiation oncology. Xing proved that the Berry phase is preserved when spin-spin coupling is assimilated into parameter space via a quasimagnetic field. GCT makes the identical architectural move across 40+ domains, treating coupling as a coordinate with geometric features (sign-flip boundary, irrational attractor, convergent ladder). Five open problems in Xing's current Stanford radiation oncology lab — auto-contouring in dense tissue, radiation field precision, tissue heterogeneity, density-independent screening, and pre-treatment density reduction — are each explained by the coupling geometry he identified twenty years ago. The bridge between quantum spin systems and tissue coupling is structural.
It is well known that any cyclic solution of a spin 1/2 neutral particle moving in an arbitrary magnetic field has a nonadiabatic geometric phase proportional to the solid angle subtended by the trace of the spin. For neutral particles with higher spin, this is true for cyclic solutions with special initial conditions. For more general cyclic solutions, however, this does not hold. As an example, we consider the most general solutions of such particles moving in a rotating magnetic field. If the parameters of the system are appropriately chosen, all solutions are cyclic. The nonadiabatic geometric phase and the solid angle are both calculated explicitly. It turns out that the nonadiabatic geometric phase contains an extra term in addition to the one proportional to the solid angle. The extra term vanishes automatically for spin 1/2. For higher spin, however, it depends on the initial condition. We also consider the valence electron of an alkaline atom. For cyclic solutions with special initial conditions in an arbitrary strong magnetic field, we prove that the nonadiabatic geometric phase is a linear combination of the two solid angles subtended by the traces of the orbit and spin angular momenta. For more general cyclic solutions in a strong rotating magnetic field, the nonadiabatic geometric phase also contains extra terms in addition to the linear combination.
The influence of intersubsystem coupling on the cyclic adiabatic geometric phase in bipartite systems is investigated. We examine the geometric phase effects for two uniaxially coupled spin-12\frac{1}{2}21​ particles, both driven by a slowly rotating magnetic field. It is demonstrated that the relation between the geometric phase and the solid angle enclosed by the magnetic field is broken by the spin-spin coupling, in particular leading to a quenching effect on the geometric phase in the strong coupling limit.