Leschke, H;
Sobolev, A;
Spitzer, W;
(2021)
Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field.
Communications in Mathematical Physics
, 381
(2)
pp. 673-705.
10.1007/s00220-020-03907-w.
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Abstract
We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane R2 perpendicular to an external constant magnetic field of strength 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 μ≥B (in suitable physical units). For this (pure) state we define its local entropy S(Λ) associated with a bounded (sub)region Λ⊂R2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Λ of finite area |Λ|. In this setting we prove that the leading asymptotic growth of S(LΛ), as the dimensionless scaling parameter L>0 tends to infinity, has the form LB−−√|∂Λ| up to a precisely given (positive multiplicative) coefficient which is independent of Λ and dependent on B and μ only through the integer part of (μ/B−1)/2. Here we have assumed the boundary curve ∂Λ of Λ to be sufficiently smooth which, in particular, ensures that its arc length |∂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, where an additional logarithmic factor 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 L2(R2) 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, the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.
| Type: | Article |
|---|---|
| Title: | Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s00220-020-03907-w |
| Publisher version: | https://doi.org/10.1007/s00220-020-03907-w |
| Language: | English |
| Additional information: | This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10126706 |
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