ar X iv : c ha o - dy n / 96 08 01 5 v 1 2 4 A ug 1 99 6 Partial Dynamical Symmetry and Mixed Dynamics

Partial dynamical symmetry describes a situation in which some eigenstates have a symmetry which the quantum Hamiltonian does not share. This property is shown to have a classical analogue in which some tori in phase space are associated with a symmetry which the classical Hamiltonian does not share. A local analysis in the vicinity of these special tori reveals a neighbourhood of phase space foliated by tori. This clarifies the suppression of classical chaos associated with partial dynamical symmetry. The results are used to divide the states of a mixed system into “chaotic” and “regular” classes. PACS numbers: 05.45.+b, 03.65.Fd, 03.65.Sq, 03.20.+i Typeset using REVTEX 1 Symmetry plays a central role in affecting the character of classical and quantum dynamics. Constants of motion associated with a symmetry govern the integrability of the system. Often the symmetry in question is not obeyed uniformly, i.e. only a subset of quantum states fulfill the symmetry while other states do not. In such circumstances, referred to as partial symmetries, the symmetry of these “special” (at times solvable) eigenstates does not arise from invariance properties of the Hamiltonian. Selected examples in this category are adiabatic regular states in the stadium billiard [1]; regular quasi-Landau resonances of a hydrogen atom in strong magnetic fields, which coexist in a region of otherwise chaotic dynamics [2]; discrete nuclear states embedded in a continuum of decay channels [3]; partial SU(3) symmetry found in deformed nuclei [4]. Hamiltonians with partial symmetries are not completely integrable hence can exhibit stochastic behavior. As such they are relevant to the study of mixed systems with coexisting regularity and chaos [5], which are the most generic. The effect of discrete symmetries on a mixed-phase-space system was examined in [6], while the consequences of breaking a discrete symmetry on the spectral statistics of a chaotic Hamiltonian were discussed in [7]. In the present paper we focus on continuous symmetries (associated with Lie groups) which can be conveniently studied in the framework of algebraic models. Such symmetry-based models are amenable to both quantum and classical treatments and have been used extensively in nuclear and molecular physics [8]. Their integrable limits are associated with dynamical symmetries in which the Hamiltonian is written in terms of Casimir operators of a chain of nested algebras. The eigenstates are labeled by the irreducible representations (irreps) of the algebras in the chain and the eigenvalues and wave functions are known analytically. A Hamiltonian with dynamical symmetry is completely solvable quantum mechanically and classically [9]. Non-integrability is obtained by breaking the dynamical symmetry and may lead to chaotic dynamics [9,10]. To address situations of mixed dynamics, an algorithm was developed [11] for constructing algebraic Hamiltonians with partial dynamical symmetries. Such Hamiltonians are not invariant under a symmetry group and yet possess a subset of “special” solvable states which do respect the symmetry. 2 In the context of a nuclear physics model involving five quadrupole degrees of freedom, it was shown that partial dynamical symmetry induced a strong suppression of classical chaos [12]. This was true even though the fraction of special states vanished as h̄, so one might have expected no classical effect. In order to better understand this effect, we consider a simpler model and use its partial dynamical symmetry to infer relationships between the classical and quantum dynamics of a Hamiltonian in a mixed KAM régime. As a simple test-case we consider a model based on a U(3) algebra. This algebra (with fermionic operators) was considered previously in the context of chaos, but not regarding partial dynamical symmetry [10]. Here we employ a realization in terms of three bosons a, b, c satisfying the usual commutation relations (different types of bosons commute). The nine number-conserving bilinear products of creation and destruction operators comprise the U(3) algebra. The conservation of the total boson-number N̂ = n̂a + n̂b + n̂c (n̂a = aa with eigenvalue na etc.) ensures that the model describes a system with only two independent degrees of freedom. All states of the model are assigned to the totally symmetric representation [N] of U(3). One of the dynamical symmetries of the model is associated with the following chain of algebras U(3) ⊃ U(2) ⊃ U(1) (1) Here U(2) ≡ SU(2)×Uab(1) with a linear Casimir n̂ab = n̂a+ n̂b (which is also the generator of Uab(1) ). The generators of SU(2) are Ĵ+ = b a, Ĵ− = a b, Ĵz = (n̂b − n̂a)/2 and its Casimir ~ J = n̂ab(n̂ab + 2)/4. The subalgebra U(1) in Eq. (1) is composed of the operator Ĵz. A choice of Hamiltonian with a U(2) dynamical symmetry is H0 = (1 + A) a a+ (1−A) bb = ωa(n̂ab − Ĵz) + ωb(n̂ab + Ĵz) (2) where ωa,b = 1 ± A, and A is introduced to break degeneracies. Diagonalization of this Hamiltonian is trivial and leads to eigenenergies Ena,nb = ωana + ωbnb and eigenstates |na, nb, nc〉 or equivalently |N, J, Jz〉 where the label J = nab/2 identifies the SU(2) irrep. These are states with well defined na, nb and nc = N−na−nb. To create a partial dynamical symmetry we add the term 3 H1 = b (ba+ bc+ ab+ cb)b , (3) which preserves the total boson number but not the individual boson numbers, so it breaks the dynamical symmetry. However states of the form |na, nb = 0, nc〉 (or equivalently |N, J = na/2, Jz = −J〉 ) with na = 0, 1, 2, . . .N are annihilated by H1 and therefore remain eigenstates of H0 + BH1. The latter Hamiltonian is not an SU(2) scalar yet has a subset of (N + 1) “special” solvable states with SU(2) symmetry, and therefore has partial dynamical symmetry. There is one special state per SU(2) irrep J = na/2 (the lowest weight state in each case) with energy ωana independent of the parameter B. Other eigenstates are mixed. Although H0 and H1 do not commute, when acting on the “special” states they satisfy