Abstract We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in $${{\mathbb {R}}}^3$$ 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 $$L^2\ln (L)$$ L 2 ln ( L ) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region $$L\Lambda \subset {{\mathbb {R}}}^3$$ 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, $$\partial \Lambda $$ ∂ 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 $$L^2$$ L 2 of the relevant trace for piecewise $$\textsf{C}^{1,\alpha }$$ C 1 , α smooth surfaces $$\partial \Lambda $$ ∂ Λ .
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 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
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.
We prove a special case of a conjecture in asymptotic analysis by Harold Widom. More precisely, we establish the leading and next-to-leading term of a semi-classical expansion of the trace of the square of certain integral operators on the Hilbert space L2(Rd). As already observed by Gioev and Klich, this implies that the bi-partite entanglement entropy of the free Fermi gas in its ground state grows at least as fast as the surface area of the spatially bounded part times a logarithmic enhancement.
We prove a logarithmically enhanced area law for all Rényi entanglement entropies of the ground state of a free gas of relativistic Dirac fermions. Such asymptotics occur in any dimension if the modulus of the Fermi energy is larger than the mass of the particles and in the massless case at Fermi energy zero in one space dimension. In all other cases of mass, Fermi energy and dimension, the entanglement entropy grows no faster than the area of the involved spatial region. The result is established for a general class of test functions which includes the ones corresponding to Rényi entropies and relies on a recently proved extension of the Widom-Sobolev formula to matrix-valued symbols by the authors.
In a remarkable paper [Phys. Rev. Lett. 96, 100503 (2006)], Gioev and Klich conjectured an explicit formula for the leading asymptotic growth of the spatially bipartite von Neumann entanglement entropy of noninteracting fermions in multidimensional Euclidean space at zero temperature. Based on recent progress by one of us (A. V. S.) in semiclassical functional calculus for pseudodifferential operators with discontinuous symbols, we provide here a complete proof of that formula and of its generalization to Rényi entropies of all orders α>0. The special case α=1/2 is also known under the name logarithmic negativity and often considered to be a particularly useful quantification of entanglement. These formulas exhibiting a "logarithmically enhanced area law" have been used already in many publications.
Abstract The leading asymptotic large-scale behaviour of the spatially bipartite entanglement entropy (EE) of the free Fermi gas infinitely extended in multidimensional Euclidean space at zero absolute temperature, T = 0, is by now well understood. Here, we present 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 first-order finite-size correction to the infinite-volume thermal entropy (density). Not surprisingly, this correction is just the thermal entropy on the interface of the bipartition. However, it is given by a rather complicated integral derived from a semiclassical trace formula for a certain operator on the underlying one-particle Hilbert space. But in the zero-temperature limit T ↓ 0 , 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 enhancement at T = 0 of the so-called area-law scaling.