Translation-invariant quantum systems with effectively broken symmetry
Brooks M. 2022. Translation-invariant quantum systems with effectively broken symmetry. Institute of Science and Technology Austria.
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Thesis
| PhD
| Published
| English
Author
Supervisor
Department
Series Title
ISTA Thesis
Abstract
The scope of this thesis is to study quantum systems exhibiting a continuous symmetry that
is broken on the level of the corresponding effective theory. In particular we are going to
investigate translation-invariant Bose gases in the mean field limit, effectively described by
the Hartree functional, and the Fröhlich Polaron in the regime of strong coupling, effectively
described by the Pekar functional. The latter is a model describing the interaction between a
charged particle and the optical modes of a polar crystal. Regarding the former, we assume in
addition that the particles in the gas are unconfined, and typically we will consider particles
that are subject to an attractive interaction. In both cases the ground state energy of the
Hamiltonian is not a proper eigenvalue due to the underlying translation-invariance, while on
the contrary there exists a whole invariant orbit of minimizers for the corresponding effective
functionals. Both, the absence of proper eigenstates and the broken symmetry of the effective
theory, make the study significantly more involved and it is the content of this thesis to
develop a frameworks which allows for a systematic way to circumvent these issues.
It is a well-established result that the ground state energy of Bose gases in the mean field limit,
as well as the ground state energy of the Fröhlich Polaron in the regime of strong coupling, is
to leading order given by the minimal energy of the corresponding effective theory. As part
of this thesis we identify the sub-leading term in the expansion of the ground state energy,
which can be interpreted as the quantum correction to the classical energy, since the effective
theories under consideration can be seen as classical counterparts.
We are further going to establish an asymptotic expression for the energy-momentum relation
of the Fröhlich Polaron in the strong coupling limit. In the regime of suitably small momenta,
this asymptotic expression agrees with the energy-momentum relation of a free particle having
an effectively increased mass, and we find that this effectively increased mass agrees with the
conjectured value in the physics literature.
In addition we will discuss two unrelated papers written by the author during his stay at ISTA
in the appendix. The first one concerns the realization of anyons, which are quasi-particles
acquiring a non-trivial phase under the exchange of two particles, as molecular impurities.
The second one provides a classification of those vector fields defined on a given manifold
that can be written as the gradient of a given functional with respect to a suitable metric,
provided that some mild smoothness assumptions hold. This classification is subsequently
used to identify those quantum Markov semigroups that can be written as a gradient flow of
the relative entropy.
Publishing Year
Date Published
2022-12-15
Publisher
Institute of Science and Technology Austria
Page
196
ISSN
IST-REx-ID
Cite this
Brooks M. Translation-invariant quantum systems with effectively broken symmetry. 2022. doi:10.15479/at:ista:12390
Brooks, M. (2022). Translation-invariant quantum systems with effectively broken symmetry. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:12390
Brooks, Morris. “Translation-Invariant Quantum Systems with Effectively Broken Symmetry.” Institute of Science and Technology Austria, 2022. https://doi.org/10.15479/at:ista:12390.
M. Brooks, “Translation-invariant quantum systems with effectively broken symmetry,” Institute of Science and Technology Austria, 2022.
Brooks M. 2022. Translation-invariant quantum systems with effectively broken symmetry. Institute of Science and Technology Austria.
Brooks, Morris. Translation-Invariant Quantum Systems with Effectively Broken Symmetry. Institute of Science and Technology Austria, 2022, doi:10.15479/at:ista:12390.
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Material in ISTA:
Part of this Dissertation