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Abstract. We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power α of the density matrix of the block. We calculate the asymptotic for L → ∞ analytically in terms of Klein's elliptic λ -function. We study the limiting entropy as a function of its parameter α. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map α → α −1 and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when α is a integer power of 2, this includes the purity trρ 2 . We also analyze the behavior of the entropy as α → 0 and ∞ and at the critical magnetic field and in the isotropic limit [XX model].

We study an asymptotic behavior of a special correlator known as the Emptiness Formation Probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field. This correlator is essentially the probability of formation of a ferromagnetic string of length n in the antiferromagnetic ground state of the chain and plays an important role in the theory of integrable models. For the XY Spin Chain, the correlator can be expressed as the determinant of a Toeplitz matrix and its asymptotical behaviors for n → ∞ throughout the phase diagram are obtained using known theorems and conjectures on Toeplitz determinants. We find that the decay is exponential everywhere in the phase diagram of the XY model except on the critical lines, i.e. where the spectrum is gapless. In these cases, a power-law prefactor with a universal exponent arises in addition to an exponential or Gaussian decay. The latter Gaussian behavior holds on the critical line corresponding to the isotropic XY model, while at the critical value of the magnetic field the EFP decays exponentially. At small anisotropy one has a crossover from the Gaussian to the exponential behavior. We study this crossover using the bosonization approach.

For systems consisting of distinguishable particles, there exists an agreed upon notion of entanglement which is fundamentally based on the possibility of addressing individually each one of the constituent parties. Instead, the indistinguishability of identical particles hinders their individual addressability and has prompted diverse, sometimes discordant definitions of entanglement. In the present review, we provide a comparative analysis of the relevant existing approaches, which is based on the characterization of bipartite entanglement in terms of the behaviour of correlation functions. Such a a point of view provides a fairly general setting where to discuss the presence of non-local effects; it is performed in the light of the following general consistency criteria: i) entanglement corresponds to non-local correlations and cannot be generated by local operations; ii) when, by "freezing" suitable degrees of freedom, identical particles can be effectively distinguished, their entanglement must reduce to the one that holds for distinguishable particles; iii) in absence of other quantum resources, only entanglement can outperform classical information protocols. These three requests provide a setting that allows to evaluate strengths and weaknesses of the existing approaches to indistinguishable particle entanglement and to contribute to the current understanding of such a crucial issue. Indeed, they can be classified into five different classes: four hinging on the notion of particle and one based on that of physical modes. We show that only the latter approach is consistent with all three criteria, each of the others indeed violating at least one of them.

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighboring spins.We study a double scaling limit: the size of the block is much larger than 1 but much smaller than the length of the whole chain. The entropy of the block has an asymptotic limit. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas and that they all meet at one point (essential critical point).Depending on the approach to the essential critical point the entropy can take any value between 0 and ∞. In the vicinity of this point small changes in the parameters cause large change of the entropy.

We study an asymptotic behavior of the probability of formation of a ferromagnetic string (referred to as EFP) of length n in a ground state of the one-dimensional anisotropic XY model in a transversal magnetic field as n → ∞. We find that it is exponential everywhere in the phase diagram of the XY model except at the critical lines where the spectrum is gapless. One of those lines corresponds to the isotropic XY model where EFP decays in a Gaussian way, as was shown in Ref. [1]. The other lines are at the critical value of the magnetic field. There, we show that EFP is still exponential but acquires a non-trivial power-law prefactor with a universal exponent.

We study the Renyi entropy of the one-dimensional XY Z spin-1/2 chain in the entirety of its phase diagram. The model has several quantum critical lines corresponding to rotated XXZ chains in their paramagnetic phase, and four tricritical points where these phases join. Two of these points are described by a conformal field theory and close to them the entropy scales as the logarithm of its mass gap. The other two points are not conformal and the entropy has a peculiar singular behavior in their neighbors, characteristic of an essential singularity. At these nonconformal points the model undergoes a discontinuous transition, with a level crossing in the ground state and a quadratic excitation spectrum. We propose the entropy as an efficient tool to determine the discontinuous or continuous nature of a phase transition also in more complicated models. In the past several years there has been a constantly increasing interest in quantifying and studying the entanglement of virtually every physical system [1][2][3]. This interest is not surprising, as entanglement provides valuable insights from many different perspectives.As it is often measured as entanglement entropy, that is, the Von Neumann entropy of the reduced density matrix of a system S = −Trρ lnρ, its origin lies in the context of quantum information theory [4,5]. Essentially, it quantifies the "quantumness" of a state and therefore it provides a measure of its suitability for efficient quantum algorithms, in the ongoing quest for quantum computing.In the physics of strongly interacting systems, entanglement has been welcomed as a new interesting correlation function, different in nature compared to the traditional ones, due to its nonlocal structure. In particular, it has provided a new challenge for the integrable model community on one side [6][7][8] and has led to new, more efficient approaches for numerical simulations (tensor network states) on the other [9][10][11]. For statistical physics, the entanglement entropy has also been proposed as a numerically efficient way to characterize a system, due to its diverging behavior across a phase transition [12,13].If we concentrate on the bipartite entanglement, that is, the entanglement entropy between two complementary regions, a lot is understood about its qualitative behavior. In general, it satisfies the so-called area law [2], that is, it is proportional to the area of the boundary dividing the two regions. This is naively understood considering that the entanglement is carried by correlations, that, in general, decay exponentially with the distance with a rate given by the correlation length ξ. If the volume of the two regions is much bigger than ξ, entanglement is localized at the boundary. Hence the area law. This behavior is modified when correlations decay more slowly, like close to phase transitions, where ξ → ∞.The one-dimensional (1D) case is most interesting.As the boundary between regions consists of individual points, in a massive phase the entropy saturates to a finite value for sufficien...

The fluorescence detection of ultra high energy (≳1018 eV) cosmic rays requires a detailed knowledge of the fluorescence light emission from nitrogen molecules, which are excited by the cosmic ray shower particles along their path in the atmosphere. We have made a precise measurement of the fluorescence light spectrum excited by MeV electrons in dry air. We measured the relative intensities of 34 fluorescence bands in the wavelength range from 284 to 429 nm with a high resolution spectrograph. The pressure dependence of the fluorescence spectrum was also measured from a few hPa up to atmospheric pressure. Relative intensities and collisional quenching reference pressures for bands due to transitions from a common upper level were found in agreement with theoretical expectations. The presence of argon in air was found to have a negligible effect on the fluorescence yield. We estimated that the systematic uncertainty on the cosmic ray shower energy due to the pressure dependence of the fluorescence spectrum is reduced to a level of 1% by the AIRFLY results presented in this paper