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 $\partial Λ$.
We consider the two-dimensional ideal Fermi gas subject to a magnetic field which is perpendicular to the Euclidean plane R2 and whose strength B(x) at x∈R2 converges to some B0>0 as ∥x∥→∞. Furthermore, we allow for an electric potential Vε 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≥B0 we study the asymptotic growth of its bipartite entanglement entropy associated to LΛ as L→∞ for some fixed bounded region Λ⊂R2. We show that its leading order in L does not depend on the perturbations Bε:=B0−B and Vε if they satisfy some mild decay assumptions. Our result holds for all α-R' enyi entropies α>1/3; for α≤1/3, we have to assume in addition some differentiability of the perturbations Bε and Vε. The case of a constant magnetic field Bε=0 and with Vε=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 ∂Λ.
Abstract We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane R2 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 μ≥B μ ≥ B (in suitable physical units). For this (pure) state we define its local entropy S(Λ) S ( Λ ) associated with a bounded (sub)region Λ⊂R2 Λ ⊂ R 2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Λ Λ of finite area ∣Λ∣ | Λ | . In this setting we prove that the leading asymptotic growth of S(LΛ) S ( L Λ ) , as the dimensionless scaling parameter L>0 L > 0 tends to infinity, has the form LB∣∂Λ∣ L B | ∂ Λ | up to a precisely given (positive multiplicative) coefficient which is independent of Λ Λ and dependent on B and μ μ only through the integer part of (μ/B−1)/2 ( μ / B - 1 ) / 2 . Here we have assumed the boundary curve ∂Λ ∂ Λ of Λ Λ to be sufficiently smooth which, in particular, ensures that its arc length ∣∂Λ∣ | ∂ 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 L2(R2) 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
Abstract We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3 R 3 subject to a constant magnetic field perpendicular to a plane. As for the case with no magnetic field we find, to leading order L2ln(L) L 2 ln ( L ) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3 L Λ ⊂ R 3 as the scaling parameter L tends to infinity. This is in contrast to the two-dimensional case since particles can now move freely in the direction of the magnetic field, which causes the extra ln(L) ln ( L ) factor. The explicit expression for the coefficient of the leading order contains a surface integral similar to the Widom–Sobolev formula in the non-magnetic case. It differs, however, in the sense that the dependence on the boundary, ∂Λ ∂ Λ , is not solely on its area but on the “surface perpendicular to the direction of the magnetic field”. We utilize a two-term asymptotic expansion by Widom (up to an error term of order one) of certain traces of one-dimensional Wiener–Hopf operators with a discontinuous symbol. This leads to an improved error term of the order L2 L 2 of the relevant trace for piecewise C1,α C 1 , α smooth surfaces ∂Λ ∂ Λ .
Abstract We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schrödinger operators.
The leading asymptotic large-scale behavior of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensionsal Euclidean space at zero absolute temperature, T=0, is by now well understood. Here, we announce and discuss the first rigorous results for the corresponding EE of thermal equilibrium states at T>0. The leading large-scale term of this thermal EE turns out to be twice the leading finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the boundary surface of the bipartition. However, it is given by a rather complicated analytical expression derived from semiclassical functional calculus and differs, at least at high temperature, from simpler expressions previously obtained by arguments based on conformal field theory. In the zero-temperature limit, the leading large-scale term of the thermal EE considerably simplifies and displays a ln(1/T)-singularity which one may identify with the known logarithmic correction at T=0 to the so-called area-law scaling. Our results extend to the whole one-parameter family of (quantum) R\'enyi entropies.
We consider the fermionic entanglement entropy for the free Dirac field in a bounded spatial region of Minkowski spacetime. In order to make the system ultraviolet finite, a regularization is introduced. An area law is proven in the limiting cases where the volume tends to infinity and/or the regularization length tends to zero. The technical core of the paper is to generalize a theorem of Harold Widom to pseudo-differential operators whose principal symbols develop a specific discontinuity at a single point.
We study the entanglement entropy of ground states of a Hamiltonian defined on a domain with a boundary. Surprisingly, boundary conditions can change the spectrum and the nature of the spectrum drastically but not the leading behaviour of the entanglement entropy. As is well-known, the Landau Hamiltonian on the full plane has pure point spectrum (the infinitely degenerate Landau levels) and ground states display a so-called strict area law. On the other hand, the Landau Hamiltonian on the half-plane has purely absolutely continuous spectrum and yet we prove a strict area law for its ground states. We raise the question of what extra or finer conditions on the absolutely continuous spectrum are necessary to guarantee a logarithmically enhanced area-law as we have for the Laplace operator.