Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extremal Black Holes, Stueckelberg Scalars and Phase Transitions   We calculate the entropy of a static extremal black hole in 4D gravity,
non-linearly coupled to a massive Stueckelberg scalar. We find that the scalar
field does not allow the black hole to be magnetically charged. We also show
that the system can exhibit a phase transition due to electric charge
variations. For spherical and hyperbolic horizons, the critical point exists
only in presence of a cosmological constant, and if the scalar is massive and
non-linearly coupled to electromagnetic field. On one side of the critical
point, two extremal solutions coexist: Reissner-Nordstr\"{o}m (A)dS black hole
and the charged hairy (A)dS black hole, while on the other side of the critical
point the black hole does not have hair. A near-critical analysis reveals that
the hairy black hole has larger entropy, thus giving rise to a zero temperature
phase transition. This is characterized by a discontinuous second derivative of
the entropy with respect to the electric charge at the critical point. The
results obtained here are analytical and based on the entropy function
formalism and the second law of thermodynamics.

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The categories of this paper are: patt-sol
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The categories of this paper are: math
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th