[Paper Review] Complex magnetic monopoles and geometric phases around diabolic and exceptional points

Alexander I. Nesterov, F. Aceves de la Cruz|arXiv (Cornell University)|Nov 29, 2006

Quantum chaos and dynamical systems3 citations

TL;DR

This paper introduces complex magnetic monopoles as a geometric framework for understanding geometric phases (GP) in non-Hermitian systems, particularly around diabolic (DP) and exceptional points (EP). It shows that while the GP at a DP is governed by a real Dirac monopole flux, at an EP the GP becomes complex and is described by a complex monopole, with the real part dominated by a quadrupole term and the imaginary part by a dipole-like contribution, exhibiting a finite gap at the DP and divergence at the EP.

ABSTRACT

We study the geometric phase (GP)in presence of diabolic (DP) and exceptional (EP) points. While the GP associated with the DP is the flux of the Dirac monopole, the GP related to the EP, being complex one, is described by the flux of complex magnetic monopole. For open systems, in week-coupling limit, the leading environment-induced contribution to the real part of complex GP is given by a quadrupole term, and to its imaginary part by a dipolelike field. We find that the GP has a finite gap at the DP and infinite one at the EP.

Motivation & Objective

• To extend the geometric phase framework to non-Hermitian systems with degeneracies.
• To clarify the physical origin of geometric phases near exceptional points (EPs), where eigenvalues and eigenvectors coalesce.
• To establish a connection between EPs and complex magnetic monopoles in open quantum systems.
• To analyze the leading-order environmental contributions to the real and imaginary parts of the geometric phase in weakly coupled open systems.
• To demonstrate the singular behavior of the geometric phase at EPs and its finite gap at DPs.

Proposed method

• Formalism based on non-Hermitian Hamiltonians with bi-orthogonal eigenvectors and left/right spectral decomposition.
• Use of multipole expansion (Legendre polynomials) to decompose the complex geometric phase into monopole, dipole, and quadrupole contributions.
• Application of the time-dependent Schrödinger equation with non-Hermitian Hamiltonians to model open quantum systems.
• Derivation of the geometric phase via path integral over a closed loop in parameter space, incorporating complex monopole fields.
• Explicit construction of the complex monopole field using parameters from a two-level system with decay, modeling weak coupling to environment.
• Comparison with prior results from Garrison and Wright (1988) and verification via parameterized closed curves in complex parameter space.

Experimental results

Research questions

• RQ1 How does the geometric phase behave in the presence of exceptional points (EPs) in non-Hermitian systems?
• RQ2 What is the physical interpretation of the geometric phase when both eigenvalues and eigenvectors coalesce at an EP?
• RQ3 How do environmental couplings in weakly open systems modify the real and imaginary parts of the geometric phase?
• RQ4 Can the complex geometric phase near an EP be described by a flux of a complex magnetic monopole?
• RQ5 What is the difference in geometric phase structure between diabolic points (DPs) and exceptional points (EPs)?

Key findings

• The geometric phase at a diabolic point is governed by the flux of a real Dirac monopole, resulting in a finite gap in the phase spectrum.
• At an exceptional point, the geometric phase diverges due to the coalescence of eigenvalues and eigenvectors, indicating a singularity in the phase structure.
• The real part of the complex geometric phase receives its leading correction from a quadrupole term, while the imaginary part is dominated by a dipole-like contribution.
• The complex geometric phase is described by the flux of a complex magnetic monopole, with the monopole's charge, dipole, and quadrupole moments derived from the system's parameters.
• The geometric phase exhibits a finite gap at the diabolic point and infinite gap (divergence) at the exceptional point, confirming distinct topological behaviors.
• The results are generic and apply to any dissipative system with accidental level crossings, such as two-level atoms with asymmetric decay rates.