AbstractWe consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane $${{\mathbb {R}}}^2$$
R
2
perpendicular to an external constant magnetic field of strength $$B>0$$
B
>
0
. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential $$\mu \ge B$$
μ
≥
B
(in suitable physical units). For this (pure) state we define its local entropy $$S(\Lambda )$$
S
(
Λ
)
associated with a bounded (sub)region $$\Lambda \subset {{\mathbb {R}}}^2$$
Λ
⊂
R
2
as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region $$\Lambda $$
Λ
of finite area $$|\Lambda |$$
|
Λ
|
. In this setting we prove that the leading asymptotic growth of $$S(L\Lambda )$$
S
(
L
Λ
)
, as the dimensionless scaling parameter $$L>0$$
L
>
0
tends to infinity, has the form $$L\sqrt{B}|\partial \Lambda |$$
L
B
|
∂
Λ
|
up to a precisely given (positive multiplicative) coefficient which is independent of $$\Lambda $$
Λ
and dependent on B and $$\mu $$
μ
only through the integer part of $$(\mu /B-1)/2$$
(
μ
/
B
-
1
)
/
2
. Here we have assumed the boundary curve $$\partial \Lambda $$
∂
Λ
of $$\Lambda $$
Λ
to be sufficiently smooth which, in particular, ensures that its arc length $$|\partial \Lambda |$$
|
∂
Λ
|
is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case $$B=0$$
B
=
0
, where an additional logarithmic factor $$\ln (L)$$
ln
(
L
)
is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space $$\text{ L}^2({{\mathbb {R}}}^2)$$
L
2
(
R
2
)
to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case $$B=0$$
B
=
0
, the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.
A proof of the Biswas–Mitra–Bhattacharyya conjecture for the ideal quantum gas trapped under the generic power law potential $U=\sum\nolimits_{i=1}^{d}{{c}_{i}}\left\vert \frac{{{x}_{i}}}{{{a}_{i}}}\right\vert^{{{n}_{i}}}$ inddimensions
