Finite-temperature entanglement negativity of free fermions

Shapourian, Hassan; Ryu, Shinsei

doi:10.1088/1742-5468/ab11e0

Cited by 66 publications

(77 citation statements)

References 109 publications

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“…As shown in Ref. [108], the negativity of codimension-1 free fermions obey a similar form in terms of the one dimensional negativity. So, for finite temperature negativity we can write…”

Section: Fermi Surface Systemssupporting

confidence: 58%

The negativity contour: a quasi-local measure of entanglement for mixed states

2020

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In this paper, we study the entanglement structure of mixed states in quantum many-body systems using the negativity contour, a local measure of entanglement that determines which real-space degrees of freedom in a subregion are contributing to the logarithmic negativity and with what magnitude. We construct an explicit contour function for Gaussian states using the fermionic partial-transpose. We generalize this contour function to generic many-body systems using a natural combination of derivatives of the logarithmic negativity. Though the latter negativity contour function is not strictly positive for all quantum systems, it is simple to compute and produces reasonable and interesting results. In particular, it rigorously satisfies the positivity condition for all holographic states and those obeying the quasi-particle picture. We apply this formalism to quantum field theories with a Fermi surface, contrasting the entanglement structure of Fermi liquids and holographic (hyperscale violating) non-Fermi liquids. The analysis of non-Fermi liquids show anomalous temperature dependence of the negativity depending on the dynamical critical exponent. We further compute the negativity contour following a quantum quench and discuss how this may clarify certain aspects of thermalization.1

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“…As shown in Ref. [108], the negativity of codimension-1 free fermions obey a similar form in terms of the one dimensional negativity. So, for finite temperature negativity we can write…”

Section: Fermi Surface Systemssupporting

confidence: 58%

The negativity contour: a quasi-local measure of entanglement for mixed states

2020

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“…which was studied by some of us in [112] and was shown to obey the same expressions as the bosonic negativity [70] for both even and odd values of n. In this appendix, we briefly report the results for various geometries. A technical point is that the monodromy of the field around T k,n for the susy trace is given by ψ k → e ±i(2πk/n−ϕ n ) ψ k where ϕ n = π or π(n − 1)/n for n even or odd, respectively [110,133].…”

Section: Partial Transpose With Supersymmetric Tracementioning

confidence: 99%

Twisted and untwisted negativity spectrum of free fermions

et al. 2019

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A basic diagnostic of entanglement in mixed quantum states is known as the positive partial transpose (PT) criterion. Such criterion is based on the observation that the spectrum of the partially transposed density matrix of an entangled state contains negative eigenvalues, in turn, used to define an entanglement measure called the logarithmic negativity. Despite the great success of logarithmic negativity in characterizing bosonic many-body systems, generalizing the operation of PT to fermionic systems remained a technical challenge until recently when a more natural definition of PT for fermions that accounts for the Fermi statistics has been put forward. In this paper, we study the many-body spectrum of the reduced density matrix of two adjacent intervals for one-dimensional free fermions after applying the fermionic PT. We show that in general there is a freedom in the definition of such operation which leads to two different definitions of PT: the resulting density matrix is Hermitian in one case, while it becomes pseudo-Hermitian in the other case. Using the path-integral formalism, we analytically compute the leading order term of the moments in both cases and derive the distribution of the corresponding eigenvalues over the complex plane. We further verify our analytical findings by checking them against numerical lattice calculations.

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“…This is not possible for free-fermion models [80][81][82][83][84][85][86][87]. An alternative entanglement measure, which is effectively calculable using free-fermion techniques, has been introduced [15,85,86,[88][89][90], and it is also an upper bound for the negativity [87]. Very recently, much attention has been focused to study the behaviour of the negativity at a finite-temperature phase transition.…”

Section: Entanglement Entropies Mutual Information and Logarithmicmentioning

confidence: 99%

“…(7)) at equilibrium. They can be readily obtained by first expressing s n , p n in terms of b k , b † k that diagonalise the model, and using (15). One obtains [96] ⟨s n s m ⟩ = 1 2V k e i(n−m)⋅k α 2 k coth(βE k 2) (16)…”

Section: A Two-point Correlation Functionsmentioning

confidence: 99%

“…For systems described by Conformal Field Theory, its behaviour has been fully characterized, both at zero temperature [11], and at finite temperature [12]. Similar to the mutual information, for any temperature the negativity exhibits the area-law scaling, which has been checked in several systems [13][14][15][16][17]. An important question, which is key in this paper, is how quantum and classical fluctuations are intertwined at a finite temperature critical point.…”

Section: Introductionmentioning

confidence: 99%

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Entanglement and classical fluctuations at finite-temperature critical points

Wald¹,

Arias²,

Alba³

2020

J. Stat. Mech.

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We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature T and of the quantum parameter g, which measures the strength of quantum fluctuations. While the von Neumann and the Rényi entropies exhibit the volume-law for any g and T , the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of g, T , the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any g, it exhibits the so-called sudden death, i.e., it is exactly zero for large enough T . The vanishing of the negativity defines a "death line", which we characterise analytically at small g. Importantly, for any finite T the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit T → 0. Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.

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Finite-temperature entanglement negativity of free fermions

Cited by 66 publications

References 109 publications

The negativity contour: a quasi-local measure of entanglement for mixed states

The negativity contour: a quasi-local measure of entanglement for mixed states

Twisted and untwisted negativity spectrum of free fermions

Entanglement and classical fluctuations at finite-temperature critical points

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