Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions

We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidence that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the R\ue9nyi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the R\ue9nyi entropy of order 1/2

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 two-dimensional conformal field theory

We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O(1) terms in the entanglement entropy. ArXI

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

Spin structures and entanglement of two disjoint intervals in conformal field theories

We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We find that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal field theory on the underlying Riemann surface with a given spin structure. The analytical findings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point

Partial transpose of two disjoint blocks in XY spin chains

We consider the partial transpose of the spin reduced density matrix of two disjoint blocks in spin chains admitting a representation in terms of free fermions, such as XY chains. We exploit the solution of the model in terms of Majorana fermions and show that such partial transpose in the spin variables is a linear combination of four Gaussian fermionic operators. This representation allows to explicitly construct and evaluate the integer moments of the partial transpose. We numerically study critical XX and Ising chains and we show that the asymptotic results for large blocks agree with conformal field theory predictions if corrections to the scaling are properly taken into account

Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion

We study the moments of the partial transpose of the reduced density matrix of two intervals for the free massless Dirac fermion. By means of a direct calculation based on a coherent state path integral, we find an analytic form for these moments in terms of the Riemann theta function. We show that moments of arbitrary order are equal to the same quantities for the compactified boson at the self-dual point. These equalities also imply the nontrivial result that the negativity of the free fermion and the self-dual boson are equal

HD CAGnome: A Search Tool for Huntingtin CAG Repeat Length-Correlated Genes

Background: The length of the huntingtin (HTT) CAG repeat is strongly correlated with both age at onset of Huntington’s disease (HD) symptoms and age at death of HD patients. Dichotomous analysis comparing HD to controls is widely used to study the effects of HTT CAG repeat expansion. However, a potentially more powerful approach is a continuous analysis strategy that takes advantage of all of the different CAG lengths, to capture effects that are expected to be critical to HD pathogenesis. Methodology/Principal Findings We used continuous and dichotomous approaches to analyze microarray gene expression data from 107 human control and HD lymphoblastoid cell lines. Of all probes found to be significant in a continuous analysis by CAG length, only 21.4% were so identified by a dichotomous comparison of HD versus controls. Moreover, of probes significant by dichotomous analysis, only 33.2% were also significant in the continuous analysis. Simulations revealed that the dichotomous approach would require substantially more than 107 samples to either detect 80% of the CAG-length correlated changes revealed by continuous analysis or to reduce the rate of significant differences that are not CAG length-correlated to 20% (n = 133 or n = 206, respectively). Given the superior power of the continuous approach, we calculated the correlation structure between HTT CAG repeat lengths and gene expression levels and created a freely available searchable website, “HD CAGnome,” that allows users to examine continuous relationships between HTT CAG and expression levels of ∼20,000 human genes. Conclusions/Significance: Our results reveal limitations of dichotomous approaches compared to the power of continuous analysis to study a disease where human genotype-phenotype relationships strongly support a role for a continuum of CAG length-dependent changes. The compendium of HTT CAG length-gene expression level relationships found at the HD CAGnome now provides convenient routes for discovery of candidates influenced by the HD mutation

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

Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems

Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement