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arXiv:1607.00560 · WORLD SCIENTIFIC (EUROPE) eBooks

Symmetry and Natural Quantum Structures for Three-Particles in One-Dimension

N. L. Harshman

Published Nov 14, 2016• 4 citations
arXivHTMLPDF

Abstract

How do symmetries induce natural and useful quantum structures? This question is investigated in the context of models of three interacting particles in one-dimension. Such models display a wide spectrum of possibilities for dynamical systems, from integrability to hard chaos. This article demonstrates that the related but distinct notions of integrability, separability, and solvability identify meaningful collective observables for Hamiltonians with sufficient symmetry. In turn, these observables induce tensor product structures on the Hilbert space that are especially useful for storing and processing quantum information and provide guidance to interpretation and phenomenology of quantum few-body physics.

DOI
10.1142/9781786341419_0011

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How do symmetries induce natural and useful quantum structures? This question is investigated in the context of models of three interacting particles in one-dimension. Such models display a wide spectrum of possibilities for dynamical systems, from integrability to hard chaos. This article demonstrates that the related but distinct notions of integrability, separability, and solvability identify meaningful collective observables for Hamiltonians with sufficient symmetry. In turn, these observables induce tensor product structures on the Hilbert space that are especially useful for storing and processing quantum information and provide guidance to interpretation and phenomenology of quantum few-body physics.
Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of interpretation. For models with symmetry, the properties of irreducible representations constrain the possibilities of Hilbert space arithmetic, i.e.\ how a Hilbert space can be decomposed into sums of subspaces and factored into products of subspaces. Partitioning the Hilbert space is equivalent to parsing the system into subsystems, and these emergent subsystems provide insight into the kinematics, dynamics, and informatics of a quantum model. This article provides examples of how complex models can be built up from irreducible representations that correspond to `natural' ontological units like spins and particles. It also gives examples of the reverse process in which complex models are partitioned into subsystems that are selected by the representations of the symmetries and require no underlying ontological commitments. These techniques are applied to a few-body model in one-dimension with a Hamiltonian depending on an interaction strength parameter. As this parameter is tuned, the Hamiltonian runs dynamical spectrum from integrable to chaotic, and the subsystems relevant for analyzing and interpreting the dynamics shift accordingly.
Using the quantum two-body system as a familiar model, this talk will describe how entanglement can be used to select preferred observables for interrogating a physical system. The symmetries and dynamics of the quantum two-body system provide a backdrop for testing the relativity of entanglement with respect to observable-induced tensor product structures. We believe this exploration leads us to a general statement: the physically-meaningful observable subalgebras are the ones that minimize entanglement in typical states.
We investigate relations between the group of symmetries of a subspace of the symmetric tensor product of a separable Hilbert space and a representation of the partition function of a quantum entangled family of particles associated with this subspace. Consider a system of NNN noninteracting identical bosons. In general, the system is described by the NNN-fold symmetric tensor product E⊙N,\mathcal{E}^{\odot N},E⊙N, where E\mathcal{E}E is the Hilbert space that describes the one-particle system. In some cases, the system can be described by some subspace of E⊙N.\mathcal{E}^{\odot N}.E⊙N. We consider the case in which this subspace is the closure of the linear span of some subset of the eigenbasis of the Hamiltonian of the system. We describe the group of symmetries SSS such that the partition function of the system is SSS-symmetric.
Quantum mechanical systems with some degree of complexity due to multiple scattering behave as if their Hamiltonians were random matrices. Such behavior, while originally surmised for the interacting many-body system of highly excited atomic nuclei, was later discovered in a variety of situations including single-particle systems with disorder or chaos. A fascinating theme in this context is the emergence of universal laws for the fluctuations of energy spectra and transport observables. After an introduction to the basic phenomenology, the talk highlights the role of symmetries for universality, in particular the correspondence between symmetry classes and symmetric spaces that led to a classification scheme dubbed the 'Tenfold Way'. Perhaps surprisingly, the same scheme has turned out to organize also the world of topological insulators.
It is known that the development and deep understanding of modern interaction theory and classical mechanics are made through geometry and symmetry. Yet, quantum mechanics which was regarded to be the microscopic theory of classical mechanics and achieved the crowning success in interpreting the entire microscopic world was developed purely from algebraic methods. In this thesis, we will study the geometry and dynamical symmetry in quantum systems, from which the question of integrability and chaos are explicitly addressed. First of all, the quantum dynamical degrees of freedom and quantum integrability are precisely defined and the inherent geometrical structure of quantum systems is explored from the fundamental structure of quantum theory. Such a geometrical structure can provide a framework to simultaneously build quantum and classical mechanics. The quantum-classical correspondence is then explicitly deduced. The dynamics of quantum system before it reaches the classical limit is formulated. Thus, the classical chaos is proven to be a special limiting phenomena of quantum systems and the dynamics before the system reaches its classical chaos is explored. The latter is the first step to seek the quantum manifestation of chaos. The relationship between integrability and dynamical symmetry are studied and some universal properties are discovered: a dynamical system (both quantum and classical) is integrable if it possesses a dynamical symmetry. Chaos will occur if the system undergoes a dynamical symmetry breaking and is accompanied by a structural phase transition. Thus, the concept of dynamical symmetry can be used to predict the general behaviors of a system. The theoretical underpinnings developed in this thesis are verified by many basic quantum mechanical examples. In the second part, we have applied the theory to construct the geometrical description for fermionic systems and then use this general formulation to study the geometrical properties, integrability and chaos in nuclear collective motions. The study of geometry for the fermion systems clearly show how the pure quantum properties (e.g. Pauli principle) are exhibited in geometry and how these geometrical manifestations can interpret experimental facts in nuclear collective motion (e.g. the satiation of nuclear deformation and narrow oblate window). The dynamical symmetry and dynamical symmetry breaking in nuclear collective motion are extensively studied.
In quantum many-body systems, complex dynamics delocalize the physical degrees of freedom. This spreading of information throughout the system has been extensively studied in relation to quantum thermalization, scrambling, and chaos. Locality is typically defined with respect to a tensor product structure (TPS) which identifies the local subsystems of the quantum system. In this paper, we investigate a simple geometric measure of operator spreading by quantifying the distance of the space of local operators from itself evolved under a unitary channel. We show that this TPS distance is related to the scrambling properties of the dynamics between the local subsystems and coincides with the entangling power of the dynamics in the case of a symmetric bipartition. Additionally, we provide sufficient conditions for the maximization of the TPS distance and show that the class of 2-unitaries provides examples of dynamics that achieve this maximal value. For Hamiltonian evolutions at short times, the characteristic timescale of the TPS distance depends on scrambling rates determined by the strength of interactions between the local subsystems. Beyond this short-time regime, the behavior of the TPS distance is explored through numerical simulations of prototypical models exhibiting distinct ergodic properties, ranging from quantum chaos and integrability to Hilbert space fragmentation and localization.