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Physical Review A

Geometrical description of SU(2) Berry phases

Péter Lévay

Published Mar 1, 1990• 15 citations
Publisher

Abstract

The differential geometric background for the occurrence of SU(2) Berry phases in Fermi systems with time-reversal invariance is investigated. We explicitly construct the natural geometric structures on ${\mathrm{ssHssP}}^{n}$ (the quaternionic projective space) responsible for nontrivial holonomy. For the special case n=1, ${\mathrm{ssHssP}}^{1}$\ensuremath{\sim}${\mathrm{ssS}}^{4}$, the connection that governs the time evolution corresponds to the SU(2) Yang-Mills instanton with self-dual curvature, giving the nonadiabatic generalization of the spin (3/2 case of Avron et al. [Phys. Rev. Lett. 61, 1329 (1988)].

DOI
10.1103/physreva.41.2837

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The differential geometric background for the occurrence of SU(2) Berry phases in Fermi systems with time-reversal invariance is investigated. We explicitly construct the natural geometric structures on ssHssPn{\mathrm{ssHssP}}^{n}ssHssPn (the quaternionic projective space) responsible for nontrivial holonomy. For the special case n=1, ssHssP1{\mathrm{ssHssP}}^{1}ssHssP1\ensuremath{\sim}ssS4{\mathrm{ssS}}^{4}ssS4, the connection that governs the time evolution corresponds to the SU(2) Yang-Mills instanton with self-dual curvature, giving the nonadiabatic generalization of the spin (3/2 case of Avron et al. [Phys. Rev. Lett. 61, 1329 (1988)].
The quaternionic representation of the SU(2) non-Abelian, nonadiabatic geometric phase for Fermi systems with time reversal invariance is investigated. The underlying differential geometric structure originating from the Riemannian metric on HPn (the quaternionic projective space) is studied in detail. For two simple model Hamiltonians corresponding to the cases of adiabatic, and nonadiabatic cyclic evolutions, the gauge fields are shown to be identical with Yang’s SU(2) monopole solutions. This example of nonadiabatic cyclic evolution turns out to be useful in the context of Polyakov’s spin factors also. Employing bosonic degrees of freedom interacting with the fermionic ones, it is found that the gauge structures are also present in the bosonic effective action. However, this topological part of the effective action cannot solely be interpreted as a Wess–Zumino term unlike the one in the complex case.
The existence of Hopf fibrations S^{2N+1}/S^1 = CP^N and S^{4K+3}/S^3 = HP^K allows us to treat the Hilbert space of generic finite-dimensional quantum systems as the total bundle space with respectively U(1)U(1)U(1) and SU(2)SU(2)SU(2) fibers and complex and quaternionic projective spaces as base manifolds. This alternative method of studying quantum states and their evolution reveals the intimate connection between generic quantum mechanical systems and geometrical objects. The exact Abelian and non-Abelian geometric phases, and more generally the geometrical factors for open paths, and their precise correspondence with geometric Kahler and hyper-Kahler connections will be discussed. Explicit physical examples are used to verify and exemplify the formalism.
The methods of Kähler geometry are applied to generalize the results of Berry obtained for SU(2) (namely, the existence of a geometrical part in the adiabatic phase) to any compact Lie group. We obtain explicit expressions for Berry’s geometric phases, Berry’s connections, and Berry’s curvatures in terms of parameters of the corresponding Lie algebra valued Hamiltonian. It is demonstrated that the parameter space of the Hamiltonian in the general theory is essentially a homogeneous Kähler manifold. The fundamental Kähler potentials of this manifold completely determine Berry’s phase. A general approach is exemplified by the Lie algebra Hamiltonians corresponding to SU(2) and SU(3) evolution groups.
Geometric phases form a rich subject in mathematical physics, lying at the intersection of differential geometry, functional analysis, and topology. This combination of ideas can make the subject difficult to approach for advanced undergraduate students, even though it is very rewarding. The aim of this thesis is to give a self-contained introduction to the mathematical framework behind geometric phases. Assuming only basic differential geometry, we develop the theory of vector bundles, connections, curvature, and holonomy, and use it to describe the Berry and Wilczek–Zee phases as holonomies of natural connections on complex vector bundles. We also discuss the Aharonov–Bohm effect as an example of a flat connection with globally nontrivial holonomy.
We examine the geometric interpretation of the quantum geometric tensor proposed in [Phys. Rev. B {\bf 110}, 035404 (2024)] for mixed quantum states, focusing on its imaginary part, which is proportional to a weighted sum of the Berry curvatures of the eigenstates of the density operator. While the real part naturally decomposes into Fisher--Rao and weighted Fubini--Study contributions, we show that the imaginary part does not, in general, coincide with the curvature of a connection on a globally defined U(1) line bundle whose holonomy yields a mixed-state geometric phase. Using a two-level system as an explicit example, we demonstrate that its surface integral depends on the choice of surface bounded by the same closed path, with ambiguities that are not integer multiples of 2π2π2π.