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

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 $$B=0$$ B = 0 , the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.

On the stability of the area law for the entanglement entropy of the Landau Hamiltonian

“…The following corollary is our main result. It combines Theorem 8 in [10], which can be stated as the corollary for the case B ε = V ε = 0, with our Theorem 3.5.…”

“…This allows us to focus on bounding the error term that arises, as we introduce a perturbation to H 0 . Our main result is Corollary 3.6 and relies on the exact calculations of the leading term for H = H 0 , see [10], and the estimates we will prove in this paper.…”

“…This quantity is the local entropy or entanglement entropy of the ground state restricted to LΛ. Under the assumption that Λ has C 2 boundary, the leading term of order L for the operator H = H 0 has been calculated by Leschke, Sobolev and Spitzer in [10]. This allows us to focus on bounding the error term that arises, as we introduce a perturbation to H 0 .…”

“…If we assume V ε > 0 pointwise, then all eigenvalues of H are strictly larger than B 0 and hence 1 I (H) = 0. But Theorem 8 in [10], which we will elaborate on shortly, states, that the leading order asymptotic expansion of tr h α (1 LΛ 1 I (H 0 )1 LΛ ) for large L is of order O(L) and does not vanish. On the other hand, if we assume that V ε < 0 pointwise, there is a spectral gap of the form (B 0 , B 0 + δ] in the spectrum of H. Hence, we can move b to B 0 + δ without changing the operators.…”

“…Entanglement entropy of the ground state of the latter Landau Hamiltonian (for the ground state with chemical potential µ = B 0 ) has been studied in [11,18,19] with some additional assumptions on the region Λ. The case of µ = B 0 has been solved by Charles and Estienne in [3] and then for arbitrary µ by Leschke, Sobolev and Spitzer in [10], both under some regularity assumptions on the boundary ∂Λ. Our main result is Corollary 3.6.…”