Quantum Geometric Phase in Majorana’s Stellar Representation: Mapping onto a Many-Body Aharonov-Bohm Phase

The (Berry-Aharonov-Anandan) geometric phase acquired during a cyclic quantum evolution of finite-dimensional quantum systems is studied. It is shown that a pure quantum state in a ($2J+1$)-dimensional Hilbert space (or, equivalently, of a spin-$J$ system) can be mapped onto the partition function of a gas of independent Dirac strings moving on a sphere and subject to the Coulomb repulsion of $2J$ fixed test charges (the Majorana stars) characterizing the quantum state. The geometric phase may be viewed as the Aharonov-Bohm phase acquired by the Majorana stars as they move through the gas of Dirac strings. Expressions for the geometric connection and curvature, for the metric tensor, as well as for the multipole moments (dipole, quadrupole, etc.), are given in terms of the Majorana stars. Finally, the geometric formulation of the quantum dynamics is presented and its application to systems with exotic ordering such as spin nematics is outlined.

Figure 1

Typical diagrams used to compute $D_{\mathbf{U}}^{(J,n)}$ (a), $D_{\mathbf{U}\mu }^{(J,n)}$ (b), and $D_{\mathbf{U}\mu \nu }^{(J,n)}$ (c). The solid dots represent the MSs (labeled from 1 to $2J$), the open dots (labeled $\mu ,\nu ,\dots$) represent the auxiliary stars used to compute the multipole moments. The rules are: (i) draw all possible distinct diagrams with $n$ pairing links (each dot, solid or open, may be linked only once in a given diagram); (ii) calculate the contribution of each diagram as indicated below, and then sum over all diagrams; (iii) an unlinked solid dot yields a factor 1; (iv) an unlinked open dot yields a factor 0; (v) a link between 2 solid dots $i$ and $j$ yields a factor $d_{ij}$; (vi) a link between a solid dot $i$ and an open dot $\mu$ yields a factor ${\widehat{u}}_{i\mu }$; (vii) a link between 2 open dots $\mu$ and $\nu$ yields a factor $-2{\delta }_{\mu \nu }$. The diagrams (a), (b), (c) shown here yield the contributions $d_{12}{d}_{45}$, ${\widehat{u}}_{5\mu }{d}_{23}{d}_{46}$, and ${\widehat{u}}_{3\mu }{\widehat{u}}_{6\nu }$ to $D_{\mathbf{U}}^{(3,2)}$, $D_{\mathbf{U}\mu }^{(3,3)}$, and $D_{\mathbf{U}\mu \nu }^{(3,2)}$, respectively.