Compact quantum systems

Robert Fawcett

This thesis is primarily concerned with compact quantum systems, their relationship to non-compact quantum systems and the classical limit of these systems.A compact quantum system is a quantum mechanical system whose underlying kinematical Lie algebra is a compact algebra instead of the usual non-compact Weyl-Heisenberg algebra w n (of the canonical hermitian position and momentum operators) which can be recovered by a contraction of the compact algebra.It is shown that compact quantum systems can be described in terms of the special orthogonal, unitary and compact sympletic algebras and their representations.Boson realisations of the compact Lie algebras ao{n + 2), u(n + 1) and sp(2n + 2) are given, modified slightly from realisations given by Exner, Havlicek and Lassner (Czechoslovak J. Phys.B 26,1213,1976) in terms of canonical coordinate and momentun operators.These boson realisations provide a natural generalisation of the Dyson boson realisation of JO(3).They also have a recurrent structure which in the cases of so(n + 2) and u(n + 1) allows the explicit construction of any desired irreducible representation entirely in terms of bosons.Using these boson realisations, a general formulation for compact quantum systems with n degrees of freedom based on so{n + 2) and u(n + 1) is given in terms of n compact coordinate and momentum operators (2n + 1 degrees of freedom and 2n + 1 coordinate and position operators in the case of sp(2n + 2)).The dynamics of such a system is assumed to be determined in the usual way by a Hamiltonian operator, taken here to be a polynomial in the coordinate and momentum operators.Such compact quantum systems can be contracted to corresponding non-compact quantum systems by means of a contraction of the underlying compact kinematical Lie algebra.A new definition of the contraction of a Lie algebra and its representations by the method of sequences of representations is given, suitable to the purposes of this application.The consequences of the definition are explored at length, in particular, with regard to the reducibility and decomposability of the resultant representations of the contracted

Philipp O. J. Scherer

L. E. Reichl

Stefan Aimet, Philipp Schmoll, Jens Eisert +2 more

Zero modes, understood here as degrees of freedom with vanishing confining frequency, play a central role in the nonequilibrium dynamics of bosonic systems. In Gaussian models, however, they lead to an unbounded, logarithmic growth of entanglement entropy. We show that this divergence is not an intrinsic property of zero modes themselves, but arises specifically for non-compact zero modes. Their non-compact configuration space allows unbounded spreading in position space, while their continuous spectra enable indefinite dephasing in momentum space. By contrast, compact zero modes in compact bosonic systems behave fundamentally differently: Spreading and dephasing are eventually halted, so that compactness caps the entanglement entropy at a finite value, making its dynamical role most transparent in the presence of a zero mode. We demonstrate this mechanism in a minimal setting by comparing two coupled harmonic oscillators with two coupled quantum rotors. We then show that the same physics persists in many-body systems by contrasting an N-site compact rotor chain with the non-compact harmonic chain. Finally, we relate these insights to ultra-cold-atom realizations of compact quantum field theories. In particular, we clarify when a compact free-boson (Tomonaga-Luttinger liquid) description is required and when the commonly used non-compact massless Klein-Gordon model breaks down. Even when the initial state is accurately captured by a non-compact Gaussian description, compactness ultimately governs the late-time quench dynamics, curbing entanglement growth rather than allowing a dynamical divergence.

Robert Fawcett, Anthony J. Bracken

Compact quantum dynamics, based on the Lie algebra so(n), is known to be relevant to the description of the internal motion of the electron and neutrino. It is shown here that a compact dynamics can also be based on u(n). In both cases there exists a contraction to a noncompact algebra containing a Weyl–Heisenberg subalgebra and consequently, there also exists an approximation scheme defined entirely by the contraction process, in which a sequence of compact systems described in terms of finite square matrices of increasing size, approaches a noncompact system of the usual type, described in terms of operators on a Hilbert space. Particular boson realizations of so(n) and u(n) are shown to be especially convenient for the discussion of the contractions. Some simple illustrative examples of compact systems are treated algebraically and numerically in order to show how their properties approach those of the corresponding noncompact systems.

Philipp O. J. Scherer