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

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 -> ∞. 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

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

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

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

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

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 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 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

Hipertensão arterial em funcionários de um Hospital Universitário

PURPOSE: To find out the prevalence of hypertension in employees of the Hospital and relate it to social demographic variables. METHODS: Blood pressure measurement was performed with a mercury sphygmomanometer, using an appropriate cuff size for arm circumference, weight, and height in a population sample of 864 individuals out of the 9,905 employees of a University General Hospital stratified by gender, age, and job position. RESULTS: Hypertension prevalence was 26% (62% of these reported being aware of their hypertension and 38% were unaware but had systolic/diastolic blood pressures of >;140 and/or >;90 mm Hg at the moment of the measurement). Of those who were aware of having hypertension, 51% were found to be hypertensive at the moment of the measurement. The prevalence was found to be 17%, 23%, and 29% (P ;50 years, work unit being the Institute of Radiology and the Administration Building, educational level ;10 years, and body mass index >;30 kg/m². The multivariate logistic regression model revealed a statistically significant association of hypertension with the following variables: gender, age, skin color, family income, and body mass index. CONCLUSIONS: Hypertension prevalence was high, mainly in those who were not physicians or members of the nursing staff. High-risk groups (obese, non-white, men, low family income) should be better advised of prevention and early diagnosis of hypertension by means of special programs.OBJETIVO: Conhecer a prevalência de hipertensão arterial em funcionários de um complexo hospitalar e relacionar com variáveis sócio demográficas. MÉTODOS: Foi medida a pressão arterial com aparelho de coluna de mercúrio e manguito adequado à circunferência do braço, o peso e a altura em amostra de 864 dos 9.905 funcionários do Hospital Universitário estratificada de acordo com sexo, idade e ocupação. RESULTADOS: A prevalência de hipertensão foi de 26% (hipertensão referida = 62% ou pressão sistólica >; 140 e/ou >; 90 mm Hg no momento da medida = 38%). Dos que referiram 51% estavam hipertensos no momento da medida. A prevalência foi 17, 23 e 29% (p ; 50 anos, unidade de trabalho para o Instituto de Radiologia e Prédio da Administração, escolaridade ; 10 anos e índice de massa corporal (IMC) maior ou igual a 30 kg/m². O modelo de regressão logística com procedimento "stepwise" mostrou associação estatisticamente significante com hipertensão arterial para as variáveis: sexo, idade, cor da pele, renda familiar e IMC. CONCLUSÃO: A prevalência de hipertensão foi alta em funcionários do Complexo Hospital das Clínicas, principalmente nos de ocupação diferente de médico e enfermagem. Os grupos de maior risco (homens, cor preta, baixa renda familiar, obesos) precisam ser orientados quanto a prevenção e diagnóstico precoce da doença através de programas especiais