Chern numbers for fermionic quadrupole systems

Abstract

Abstract. We analyse families of quantum quadrupole Hamiltonians H = Z,, QnpJuJp for half-odd-integer spin, and calculate the second Chern numbers of the energy levels. Each non-zero integer occurs only a finite number of times. The adiabatic time evolution, the non-Abelian generalisation of Berry's phase, is different for each system, in contrast to Berry's example. The j = $ and j = f cases previously analysed are the only ones with self-dual curvatures and SO(5) symmetry.Geometrical and topological techniques applied to the study of time-dependent quantum Hamiltonians have recently generated much interest [l]. Berry's examples of a family of Hamiltonians of the form B . J display the diversity of phenomena. The degenerate, or non-Abelian, case has received much attention [2]. In the class of time-reversal-invariant fermionic Hamiltonians, which have Kramers degeneracy [3], the quadrupole systems Zap QapJaJp for half-odd-integer spin are in many ways the analogues of Berry's examples [4]. The relevant topological invariants are the first Chern number over a 2-sphere for Berry's examples, and the second Chern number over a 4-sphere for the quadrupoles. Chern numbers are defined for energy levels which have a fixed degree of degeneracy for all Hamiltonians in the family. The Chern numbers for quadrupoles with j s f are defined and have been previously computed [4].In this paper, we will calculate the second Chern numbers for all quadrupole systems with half-odd-integer spin. In fact, every topological invariant of twodimensional complex vector bundles over S4 is a function of the second Chern number, i.e. these bundles are classified by the second Chern number up to topological equivalence. It will be shown in [5] that the second Chern numbers are indeed well defined for all half-odd-integer j .An energy level can be specified by the eigenvalue of a particular Hamiltonian. It is convenient to take the quadrupole Hamiltonian Qo = J : -$J2 which commutes with J 3 . The energy level can then be labelled by ( j , mT), where j is the total angular momentum and m: is the eigenvalue of J:. We shall refer to this Hamiltonian as the north pole, and to minus this Hamilitonian as the south pole. The level can alternately be labelled by ( j , mB), where m', is the eigenvalue of J: at the south pole, with m B = j ++ -mT.Second Chern numbers over the 4-sphere will be calculated as the integral of the 4-form o = -Tr(fl A fl)/87r2, where fl is the curvature of the connection on the eigenstate bundle given by adiabatic time evolution [ 6 ] . As a first step we will reduce the integral of a general rotationally invariant 4-form over the 4-sphere of unit quadrupoles

The geometry of entanglement: metrics, connections and the geometric phase

“…For the details of these orbits see Ref. [23], here we merely give the (skew-Hermitian) SO (3) generators needed to generate such loops…”

Non-self-dual Yang-Mills connections with nonzero Chern number

“…The study of the YM equations on the quadrupole bundles of [ASSS1,SS,ASSS2] was initiated by Gil Bor and Richard Montgomery [BoMo]. Some of the ideas presented in this paper were developed in collaboration with Bor and Montgomery.…”

Rearrangement of Energy Levels Between Energy Super-Bands Characterized by Second Chern Class

“…The eigenvalues of the semi-quantum hyperhermitian quaternionic Hamiltonian have codimension 5 degeneracies and consequently, the simplest model with qualitative modifications of the super-band structure could appear for at least four fast states (two Kramers degenerate pairs), a slow subsystem of two degrees of freedom (four classical variables), and one control parameter. The topological invariant associated with the formation of the codimention-5 degeneracy is now the second Chern class [31]. We conjecture again that this class corresponds to the number of quantum energy levels redistributed between the super-bands under the variation of the control parameter.…”

“…with v S = (2S + 1)/2 the number of superbands, calculated in [8,31] for the parametric spin-quadrupole system, which we used as the initial point of our dynamical construction. 7, both types appear in the example system with Hamiltonian (10) and compact slow classical phase space P = S 2 × S 2 .…”