SciPost Phys. 7, 037 (2019) ·
published 25 September 2019

· pdf
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 manybody 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 manybody spectrum of the reduced density matrix of two adjacent intervals
for onedimensional 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 pseudoHermitian in the other case. Using the
pathintegral 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.
SciPost Phys. 6, 051 (2019) ·
published 30 April 2019

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The recent results of [J. Dubail, J.M. St\'ephan, J. Viti, P. Calabrese,
Scipost Phys. 2, 002 (2017)], which aim at providing access to large scale
correlation functions of inhomogeneous critical onedimensional quantum systems
 e.g. a gas of hard core bosons in a trapping potential  are extended to a
dynamical situation: a breathing gas in a timedependent harmonic trap. Hard
core bosons in a timedependent harmonic potential are well known to be exactly
solvable, and can thus be used as a benchmark for the approach. An extensive
discussion of the approach and of its relation with classical and quantum
hydrodynamics in one dimension is given, and new formulas for correlation
functions, not easily obtainable by other methods, are derived. In particular,
a remarkable formula for the large scale asymptotics of the bosonic
$n$particle function $\left< \Psi^\dagger (x_1,t_1) \dots \Psi^\dagger
(x_n,t_n) \Psi(x_1',t_1') \dots \Psi(x_n',t_n') \right>$ is obtained. Numerical
checks of the approach are carried out for the fermionic twopoint function 
easier to access numerically in the microscopic model than the bosonic one 
with perfect agreement.
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in Report on Twisted and untwisted negativity spectrum of free fermions