The von Neumann entropy asymptotics in multidimensional fermionic systems

Farkas, Szilárd; Zimborás, Zoltán

doi:10.1063/1.2800167

Cited by 35 publications

(38 citation statements)

References 30 publications

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“…with a known prefactor [22][23][24][25][26][27]. Except for d = 1 this entanglement entropy does not even have the same scaling form as the CFT result, hence no direct comparison is possible.…”

Section: Fermi Surfacementioning

confidence: 99%

Entanglement does not generally decrease under renormalization

Swingle¹

2014

J. Stat. Mech.

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Renormalization is often described as the removal or "integrating out" of high energy degrees of freedom. In the context of quantum matter, one might suspect that quantum entanglement provides a sharp way to characterize such a loss of degrees of freedom. Indeed, for quantum many-body systems with Lorentz invariance, such entanglement monotones have been proven to exist in one, two, and three spatial dimensions. In each dimension d, a certain term in the entanglement entropy of a d-ball decreases along renormalization group (RG) flows. Given that most quantum many-body systems available in the laboratory are not Lorentz invariant, it is important to generalize these results if possible. In this work we demonstrate the impossibility of a wide variety of such generalizations. We do this by exhibiting a series of counterexamples with understood renormalization group flows which violate entanglement RG monotonicity. We discuss bosons at finite density, fermions at finite density, and majorization in Lorentz invariant theories, among other results.

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Section: Fermi Surfacementioning

confidence: 99%

Entanglement does not generally decrease under renormalization

Swingle¹

2014

J. Stat. Mech.

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“…We recommend [6] for a recent review of such area laws, with an emphasis on rigorous results for many-particle systems on the lattice Z d .It is somewhat surprising that an area law is not quite valid for the simple system of a free Fermi gas in R d or Z d . Indeed, various studies [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] have suggested that the EE of its ground state grows (at least as fast) as L d−1 ln L. Physically, this slight logarithmic enhancement is due to the effective long-range correlations of the particles by the Fermi-Dirac statistics, the algebraic statement of Pauli's exclusion principle. Mathematically, the resulting "sharp" Fermi surface modifies the asymptotic analysis as L → ∞ in such a way that the extra factor ln L emerges.The theorem.…”

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confidence: 99%

“…if t ∈ [0, 1] and by h α (t) (1) with α = 1 has appeared meanwhile in many publications [21][22][23][24][25][26][27] and with general α > 0 (and d ≥ 2) besides in [24] also in [28,30].…”

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confidence: 99%

Scaling of Rényi Entanglement Entropies of the Free Fermi-Gas Ground State: A Rigorous Proof

2014

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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.

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“…[3]). For a system with short-ranged interactions in d spatial dimensions, the area law states that the entanglement entropy of a region A of linear size L grows like L d−1 , that is like the area |∂A| of the boundary ∂A of A. Fermi liquids are extremely interesting from an entanglement perspective because they possesses long-range entanglement that manifests as a violation of the area law [4][5][6][7][8][9][10][11][12]. Indeed, entanglement entropy in a Fermi liquid ground state scales like L d−1 ln (L) hence showing a logarithmic violation of the area law.…”

Section: Introductionmentioning

confidence: 99%

Oscillating terms in the Renyi entropy of Fermi gases and liquids

Swingle

McMinis

Tubman³

2013

Phys. Rev. B

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In this work we compute subleading oscillating terms in the Renyi entropy of Fermi gases and liquids corresponding to 2kF -like oscillations. Our theoretical tools are the one dimensional formulation of Fermi liquid entanglement familiar from discussions of the logarithmic violation of the area law and quantum Monte Carlo calculations. The main result is a formula for the oscillating term for any region geometry and a spherical Fermi surface in any dimension. Specializing to two dimensions, we compare this term to numerical calculations of Renyi entropies using the correlation function method and find excellent agreement. We also compare with quantum Monte Carlo data on interacting Fermi liquids where we also find agreement up to moderate interaction strengths.

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The von Neumann entropy asymptotics in multidimensional fermionic systems

Cited by 35 publications

References 30 publications

Entanglement does not generally decrease under renormalization

Entanglement does not generally decrease under renormalization

Scaling of Rényi Entanglement Entropies of the Free Fermi-Gas Ground State: A Rigorous Proof

Oscillating terms in the Renyi entropy of Fermi gases and liquids

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