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 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 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 (the quaternionic projective space) responsible for nontrivial holonomy. For the special case n=1, ssHssP1\ensuremath{\sim}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)].
This thesis examines the quantum-mechanical geometric phase with a view toward time reversal symmetry considerations. The idea of time reversal in quantum mechanics is investigated, disagreements and inconsistencies in the literature are examined, and the action of the time reversal operator is extended to time-dependent Hamiltonians. With this background, and using a definition of time reversal symmetry based on the evolution operator, I demonstrate that the existence of a non-zero geometric phase can in all cases be attributed to a breakdown of time reversal symmetry in some form. This result holds for both adiabatic and nonadiabatic evolutions, and for arbitrary dimensional parameter spaces. I explore the role of the geometric phase in a two-level Kramers system described by a parameter-dependent Hamiltonian such that the two levels can become degenerate for some value of the parameters, and discuss, from a mathematical point of view, the monopole geometric potential that results. I then extend this analysis by considering a pair of Kramers doublets, each doublet degenerate due to time reversal symmetry, where the parameters can be chosen so that each of the pair of doublets becomes degenerate with the other. I find the explicit functional forms for the two resulting nonabelian geometric gauge potentials and show that they can be identified exactly with the only two gauge-inequivalent SU(2) monopole potentials of Yang. Furthermore, following a conformal transformation these potentials can be mapped to those of the SU(2) instanton/anti-instanton pair. Finally I examine the relevance of the geometric phase to the molecular physics of time-odd systems. Time-odd coupling in molecular physics is a much under-studied area, with many potentially interesting results. Specifically I study time-odd coupling in Jahn-Teller systems under the Born-Oppenheimer approximation, where the electronic position states are coupled to the lattice momentum rather than the usual time-even lattice position. As an example I solve the E ⊗ (b₁ ⊕ b₂ ⊕ a₂) Jahn-Teller system exactly, showing that once again monopole-like geometric potentials arise, and comment on how this affects the angular momentum of the lattice subsystem.
Geometric phase is an interesting topic that is germane to numerous and varied research areas: molecules, optics, quantum computing, quantum Hall effect, graphene, and so on. It exists only when the system of interest interacts with something it perceives as exterior. An isolated system cannot display geometric phase. This article addresses geometric phase in polyatomic molecules from a gauge field theory perspective. Gauge field theory was introduced in electrodynamics by Fock and examined assiduously by Weyl. It yields the gauge field A(μ), particle-field couplings, and the Aharonov-Bohm phase, while Yang-Mills theory, the cornerstone of the standard model of physics, is a template for non-Abelian gauge symmetries. Electronic structure theory, including nonadiabaticity, is a non-Abelian gauge field theory with matrix-valued covariant derivative. Because the wave function of an isolated molecule must be single-valued, its global U(1) symmetry cannot be gauged, i.e., products of nuclear and electron functions such as χ(n)ψ(n) are forbidden from undergoing local phase transformation on R, where R denotes nuclear degrees of freedom. On the other hand, the synchronous transformations (first noted by Mead and Truhlar): ψ(n)→ψ(n)e(iζ) and simultaneously χ(n)→χ(n)e(-iζ), preserve single-valuedness and enable wave functions in each subspace to undergo phase transformation on R. Thus, each subspace is compatible with a U(1) gauge field theory. The central mathematical object is Berry's adiabatic connection i<n|∇n>, which serves as a communication link between the two subsystems. It is shown that additions to the connection according to the gauge principle are, in fact, manifestations of the synchronous (e(iζ)/e(-iζ)) nature of the ψ(n) and χ(n) phase transformations. Two important U(1) connections are reviewed: qA(μ) from electrodynamics and Berry's connection. The gauging of SU(2) and SU(3) is reviewed and then used with molecules. The largest gauge group applicable in the immediate vicinity of a two-state intersection is U(2), which factors to U(1) × SU(2). Gauging SU(2) yields three fields, whereas U(1) is not gauged, as the result cannot be brought into registry with electronic structure theory, and there are other problems as well. A parallel with spontaneous symmetry breaking in electroweak theory is noted. Loss of SU(2) symmetry as the energy gap between adiabats increases yields the inter-related U(1) symmetries of the upper and lower adiabats, with spinor character imprinted in the vicinity of the degeneracy.
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) and 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.
Adiabatic U(2) geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians and their linear combinations for spin j = 1 systems. In particular it is shown that the nuclear quadrupole Hamiltonian does actually lead to non-Abelian geometric phases for j = 1. This system, being bosonic, is time-reversal invariant. Therefore, it cannot support Abelian adiabatic geometrical phases.
We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-1∕2 particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.