Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

M. A. Ivanov, S. S. Lipovetskiǐ, V. S. Sekerzhitski

Yu. A. Bychkov, A. V. Kolesnikov

Hajo Leschke, Alexander V. Sobolev, Wolfgang Spitzer

Abstract We 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 $$B=0$$ <m

Tomislav Prokopec, Michael G. Schmidt, Jan Weenink

P. T. Landsberg, J. S. Shiner

Paul E. Pfeiffer

We consider the two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane $\mathbb R^2$ and whose strength $B(x)$ at $x\in\mathbb R^2$ converges to some $B_0>0$ as $\|x\|\to\infty$. Furthermore, we allow for an electric potential $V_\varepsilon$ which vanishes at infinity. They define the single-particle Landau Hamiltonian of our Fermi gas (up to gauge fixing). Starting from the ground state of this Fermi gas with chemical potential $μ\ge B_0$ we study the asymptotic growth of its bipartite entanglement entropy associated to $LΛ$ as $L\to\infty$ for some fixed bounded region $Λ\subset\mathbb R^2$. We show that its leading order in $L$ does not depend on the perturbations $B_\varepsilon := B_0 - B$ and $V_\varepsilon$ if they satisfy some mild decay assumptions. Our result holds for all $α$-R\' enyi entropies $α>1/3$; for $α\le 1/3$, we have to assume in addition some differentiability of the perturbations $B_\varepsilon$ and $V_\varepsilon$. The case of a constant magnetic field $B_\varepsilon = 0$ and with $V_\varepsilon= 0$ was treated recently for general $μ$ by Leschke, Sobolev and Spitzer. Our result thus proves the stability of that area law under the same regularity assumptions on the boundary $\partial Λ$.