Non-Abelian geometric phases carried by the spin fluctuation tensor

Abstract

The expectation values of the first and second moments of the quantum mechanical spin operator can be used to define a spin vector and spin fluctuation tensor, respectively. The former is a vector inside the unit ball in three space, while the latter is represented by an ellipsoid in three space. They are both experimentally accessible in many physical systems. By considering transport of the spin vector along loops in the unit ball, it is shown that the spin fluctuation tensor picks up geometric phase information. For the physically important case of spin one, the geometric phase is formulated in terms of an SO(3) operator. Loops defined in the unit ball fall into two classes: those which do not pass through the origin and those which pass through the origin. The former class of loops subtend a well-defined solid angle at the origin, while the latter do not, and the corresponding geometric phase is non-Abelian. To deal with both classes, a notion of generalized solid angle is introduced, which helps to clarify the interpretation of the geometric phase information. The experimental systems that can be used to observe this geometric phase are also discussed.

Understanding temperature-dependent SU(3) spin dynamics in the S = 1 antiferromagnet Ba2FeSi2O7

“…Since the AFM to QPM transition is driven by an enhancement of the local quadrupolar moment at the expense of the magnitude of the local dipolar moment, a proper classical description must allow for the coexistence of local dipolar and quadrupolar fluctuations, leading to transverse and longitudinal collective modes 26 . SU(3) coherent states fulfill this condition because an SU(3) spin has 8 = 3 + 5 components that include the three components of the dipole moment and the five components of the quadupolar moment (trace-less symmetric tensor) 16,17,[28][29][30] .…”