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arXiv:0906.4946 · International Mathematics Research Notices

A Special Case of a Conjecture by Widom with Implications to Fermionic Entanglement Entropy

Robert C. Helling, Hajo Leschke, W. Spitzer

Published Jun 21, 2010• 46 citations
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Abstract

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 $L^2(\R^d)$. 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.

DOI
10.1093/imrn/rnq085

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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)L^2(\R^d)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.
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 We study the asymptotic growth of the entanglement entropy of ground states of non-interacting (spinless) fermions in R3{{\mathbb {R}}}^3R3 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)L2ln(L) L 2 ln ( L ) , a logarithmically enhanced area law of this entropy for a bounded, piecewise Lipschitz region LΛ⊂R3L\Lambda \subset {{\mathbb {R}}}^3LΛ⊂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)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\partial \Lambda ∂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 L2L^2L2 L 2 of the relevant trace for piecewise C1,α\textsf{C}^{1,\alpha }C1,α C 1 , α smooth surfaces ∂Λ\partial \Lambda ∂Λ ∂ Λ .
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 Tgt0T\\gt 0Tgt0. 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 Tdownarrow0T\\downarrow 0Tdownarrow0, the leading large-scale term of the thermal EE considerably simplifies and displays a mathrmln(1/T)\\mathrm{ln}(1/T)mathrmln(1/T)-singularity which one may identify with the known logarithmic enhancement at T = 0 of the so-called area-law scaling.
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.
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.