Identical Wells, Symmetry Breaking, and the Near-Unitary Limit

Harshman, N. L.

doi:10.1007/s00601-017-1214-5

Cited by 6 publications

(7 citation statements)

References 33 publications

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“…ordering sectors is identical and the permutation group of ordering sectors S N ! is a symmetry, and not just S N [23]. For equal mass particles, this is a symmetry of the Hamiltonian for any trap shape.…”

Section: Topology Of Configuration Space With the Coincidence Structumentioning

confidence: 99%

Coincidence Structures and Hard-Core Few-Body Interactions

Harshman

Knapp

2018

Few-Body Syst

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The symmetry and topology of the coincidence structure, i.e. the locus of points in configuration space corresponding to particles in the same position, plays a critical role in extracting universal properties for few-body models with hard-core interactions. The coincidence structure is a scale-invariant union of manifolds possessing rich symmetry. When there are zero-range hardcore two-body interactions, the coincidence structure forms a nodal surface for finite-energy wave functions in configuration space. More generally, it acts like a defect that changes the topology of configuration space in a way that depends on the dimension of the underlying space, the total number of particles, and the number of particles in the hard-core interaction. We show that for the specific case of three-body hard-core interactions in one-dimension, the configuration space is no longer simply-connected, providing a topological explanation for several models that exhibit anyonic behavior.

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Section: Topology Of Configuration Space With the Coincidence Structumentioning

confidence: 99%

Coincidence Structures and Hard-Core Few-Body Interactions

Harshman

Knapp

2018

Few-Body Syst

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“…If there are M identical wells, then there is a symmetry isomorphic to S M T t that corresponds to permuting the the individual, disconnected wells. Tunneling breaks this local symmetry in a manner that depends on the global structure of the wells [44].…”

Section: B Kinematic Symmetriesmentioning

confidence: 99%

“…identical N -dimensional wells corresponding to each dynamically-stable possible ordering and the symmetry has the form T t × S N ! [44]. In the near unitary limit, this symmetry is broken in a way that can be calculated exactly for contact interactions [9,10,43].…”

Section: B Kinematic Symmetriesmentioning

confidence: 99%

“…Because the spectrum of sectors A and C shares energy levels with the spectrum of sectors B and D, the energy levels are either two-fold or eight-fold degenerate. Note that for more than two particles, ordering permutation symmetry has a richer structure [43,44] because the number of orders N ! is greater than the number of particles N .…”

Section: Ordering Permutation Symmetrymentioning

confidence: 99%

“…identical N -dimensional wells corresponding to each dynamically-stable possible ordering and the symmetry has the form T t × S N ! [44].…”

Section: B Kinematic Symmetriesmentioning

confidence: 99%

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Infinite barriers and symmetries for a few trapped particles in one dimension

Harshman

2017

Phys. Rev. A

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This article investigates the properties of a few interacting particles trapped in a few wells and how these properties change under adiabatic tuning of interaction strength and inter-well tunneling. While some system properties are dependent on the specific shapes of the traps and the interactions, this article applies symmetry analysis to identify generic features in the spectrum of stationary states of few-particle, few-well systems. Extended attention is given to a simple but flexible threeparameter model of two particles in two wells in one dimension. A key insight is that two limiting cases, hard-core repulsion and no inter-well tunneling, can both be treated as emergent symmetries of the few-particle Hamiltonian. These symmetries are the mathematical consequences of infinite barriers in configuration space. They are necessary to explain the pattern of degeneracies in the energy spectrum, to understand how degeneracies are broken for models away from limiting cases, and to explain separability and integrability. These symmetry methods are extendable to more complicated models and the results have practical consequences for stable state control in fewparticle, few-well systems with ultracold atoms in optical traps.

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Symmetry, Structure, and Emergent Subsystems

Harshman¹

2019

Quantum Worlds

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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.

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Identical Wells, Symmetry Breaking, and the Near-Unitary Limit

Cited by 6 publications

References 33 publications

Coincidence Structures and Hard-Core Few-Body Interactions

Coincidence Structures and Hard-Core Few-Body Interactions

Infinite barriers and symmetries for a few trapped particles in one dimension

Symmetry, Structure, and Emergent Subsystems

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