SciPost Phys. 8, 063 (2020) ·
published 20 April 2020
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In this paper, we study the entanglement structure of mixed states in quantum
many-body systems using the $\textit{negativity contour}$, 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.
SciPost Phys. 7, 037 (2019) ·
published 25 September 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.
