470 research outputs found
Evolution of entanglement after a local quench
We study free electrons on an infinite half-filled chain, starting in the
ground state with a bond defect. We find a logarithmic increase of the
entanglement entropy after the defect is removed, followed by a slow relaxation
towards the value of the homogeneous chain. The coefficients depend
continuously on the defect strength.Comment: 14 pages, 9 figures, final versio
Entanglement entropy and quantum phase transitions in quantum dots coupled to Luttinger liquid wires
We study a quantum phase transition which occurs in a system composed of two
impurities (or quantum dots) each coupled to a different interacting
(Luttinger-liquid) lead. While the impurities are coupled electrostatically,
there is no tunneling between them. Using a mapping of this system onto a Kondo
model, we show analytically that the system undergoes a
Berezinskii-Kosterlitz-Thouless quantum phase transition as function of the
Luttinger liquid parameter in the leads and the dot-lead interaction. The phase
with low values of the Luttinger-liquid parameter is characterized by an abrupt
switch of the population between the impurities as function of a common applied
gate voltage. However, this behavior is hard to verify numerically since one
would have to study extremely long systems. Interestingly though, at the
transition the entanglement entropy drops from a finite value of to
zero. The drop becomes sharp for infinite systems. One can employ finite size
scaling to extrapolate the transition point and the behavior in its vicinity
from the behavior of the entanglement entropy in moderate size samples. We
employ the density matrix renormalization group numerical procedure to
calculate the entanglement entropy of systems with lead lengths of up to 480
sites. Using finite size scaling we extract the transition value and show it to
be in good agreement with the analytical prediction.Comment: 12 pages, 9 figure
Non-zero entropy density in the XY chain out of equilibrium
The von Neumann entropy density of a block of n spins is proved to be
non-zero for large n in the non-equilibrium steady state of the XY chain
constructed by coupling a finite cutout of the chain to the two infinite parts
to its left and right which act as thermal reservoirs at different
temperatures. Moreover, the non-equilibrium density is shown to be strictly
greater than the density in thermal equilibrium
Entanglement entropy of two disjoint blocks in XY chains
We study the Renyi entanglement entropies of two disjoint intervals in XY
chains. We exploit the exact solution of the model in terms of free Majorana
fermions and we show how to construct the reduced density matrix in the spin
variables by taking properly into account the Jordan-Wigner string between the
two blocks. From this we can evaluate any Renyi entropy of finite integer
order. We study in details critical XX and Ising chains and we show that the
asymptotic results for large blocks agree with recent conformal field theory
predictions if corrections to the scaling are included in the analysis
correctly. We also report results in the gapped phase and after a quantum
quench.Comment: 34 pages, 11 figure
A contour for the entanglement entropies in harmonic lattices
We construct a contour function for the entanglement entropies in generic harmonic lattices. In one spatial dimension, numerical analysis are performed by considering harmonic chains with either periodic or Dirichlet boundary conditions. In the massless regime and for some configurations where the subsystem is a single interval, the numerical results for the contour function are compared to the inverse of the local weight function which multiplies the energy-momentum tensor in the corresponding entanglement hamiltonian, found through conformal field theory methods, and a good agreement is observed. A numerical analysis of the contour function for the entanglement entropy is performed also in a massless harmonic chain for a subsystem made by two disjoint intervals
Entanglement Hamiltonians in 1D free lattice models after a global quantum quench
We study the temporal evolution of the entanglement Hamiltonian of an interval after a global quantum quench in free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous Hamiltonian. We focus on critical evolution Hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement Hamiltonians in these models are characterised by matrices that provide also contours for the entanglement entropies. The temporal evolution of these contours for the entanglement entropy is studied, also by employing existing conformal field theory results for the semi-infinite line and the quasi-particle picture for the global quench
Entanglement and correlation functions following a local quench: a conformal field theory approach
We show that the dynamics resulting from preparing a one-dimensional quantum
system in the ground state of two decoupled parts, then joined together and
left to evolve unitarily with a translational invariant Hamiltonian (a local
quench), can be described by means of quantum field theory. In the case when
the corresponding theory is conformal, we study the evolution of the
entanglement entropy for different bi-partitions of the line. We also consider
the behavior of one- and two-point correlation functions. All our findings may
be explained in terms of a picture, that we believe to be valid more generally,
whereby quasiparticles emitted from the joining point at the initial time
propagate semiclassically through the system.Comment: 19 pages, 4 figures, v2 typos corrected and refs adde
Universal corrections to scaling for block entanglement in spin-1/2 XX chains
We consider the R\'enyi entropies in the one dimensional spin-1/2
Heisenberg XX chain in a magnetic field. The case n=1 corresponds to the von
Neumann ``entanglement'' entropy. Using a combination of methods based on the
generalized Fisher-Hartwig conjecture and a recurrence relation connected to
the Painlev\'e VI differential equation we obtain the asymptotic behaviour,
accurate to order , of the R\'enyi entropies
for large block lengths . For n=1,2,3,10 this constitutes the 3,6,10,48
leading terms respectively. The o(1) contributions are found to exhibit a rich
structure of oscillatory behaviour, which we analyze in some detail both for
finite and in the limit .Comment: 25 pages, 5 figure
Entanglement spectrum degeneracy and the Cardy formula in 1+1 dimensional conformal field theories
We investigate the effect of a global degeneracy in the distribution of the entanglement spectrum in conformal field theories in one spatial dimension. We relate the recently found universal expression for the entanglement Hamiltonian to the distribution of the entanglement spectrum. The main tool to establish this connection is the Cardy formula. It turns out that the Affleck-Ludwig non-integer degeneracy, appearing because of the boundary conditions induced at the entangling surface, can be directly read from the entanglement spectrum distribution. We also clarify the effect of the noninteger degeneracy on the spectrum of the partial transpose, which is the central object for quantifying the entanglement in mixed states. We show that the exact knowledge of the entanglement spectrum in some integrable spinchains provides strong analytical evidences corroborating our results
Universal scaling of the logarithmic negativity in massive quantum field theory
We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1 + 1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length r and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance r. We show that the former saturates to a finite value, and that the latter tends to zero, as r -> â. We show that in both cases, the leading corrections are exponential decays in r (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy (EE), the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the EE, a large-r analysis of the negativity allows for the detection of bound states
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