Polarons in Bose gases and polar crystals: Some rigorous energy estimates
The polaron model is a basic model of quantum field theory describing a single particle
interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned
with models applicable in the context of an impurity atom in a Bose-Einstein condensate as
well as the problem of electrons moving in polar crystals.
The model has a simple structure in which the interaction of the particle with the field is given
by a term linear in the field’s creation and annihilation operators. In this work, we investigate
the properties of this model by providing rigorous estimates on various energies relevant to the
problem. The estimates are obtained, for the most part, by suitable operator techniques which
constitute the principal mathematical substance of the thesis.
The first application of these techniques is to derive the polaron model rigorously from first
principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an
impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas
in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak
interactions as a low-energy effective theory for this problem.
In the second part, we investigate rigorously the ground state of the model at fixed momentum
and for large values of the coupling constant. Qualitatively, the system is expected to display
a transition from the quasi-particle behavior at small momenta, where the dispersion relation
is parabolic and the particle moves through the medium dragging along a cloud of phonons, to
the radiative behavior at larger momenta where the polaron decelerates and emits free phonons.
At the same time, in the strong coupling regime, the bosonic field is expected to behave purely
classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to
be asymptotically equal to the one obtained from the semiclassical counterpart of the problem,
first studied by Landau and Pekar in the 1940s. For polaron models with regularized form
factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear
function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove
that for a large window of momenta below the radiation threshold, the energy-momentum
relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the
Landau–Pekar effective mass, as expected.
For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is
of the optical type and the form factor is formally UV–singular due to the nature of the point
charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to
the regular case, this requires the inclusion of the quantum fluctuations of the phonon field,
which makes the problem considerably more difficult.
The results are supplemented by studies on the absolute ground-state energy at strong coupling,
a proof of the divergence of the effective mass with the coupling constant for a wide class of
polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model
and the application of the techniques used for its removal for the energy estimates.
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Institute of Science and Technology Austria (ISTA)
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