In a quantum critical chain, the scaling regime of the energy and momentum of the ground state and low lying excitations are described by conformal field theory (CFT). The same holds true for the von Neumann and Rényi entropies of the ground state, which display a universal logarithmic behaviour depending on the central charge. In this letter we generalize this result to those excited states of the chain that correspond to primary fields in CFT. It is shown that the n-th Rényi entropy is related to a 2n-point correlator of primary fields. We verify this statement for the critical XX and XXZ chains. This result uncovers a new link between quantum information theory and CFT.Entanglement is one of the central concepts in quantum physics since Schroedinger used the term in an answer to the Einstein-Podolsky-Rosen article in 1935. A particularly active line of research is concerned with the role played by entanglement in the physics of many-body systems [1]. One is typically interested in the amount of entanglement between two spatial partitions, say A and B, of a many-body system in its ground state. For a pure ground state the amount of entanglement is usually quantified with the entanglement entropy, or the von Neumann entropy of the reduced density matrix ρ A :n , the entanglement entropy being lim n→1 S n . One of the most important results in this topic is the celebrated area law [2-4], which, roughly speaking, states that ground states of gapped many-body systems with short-range interactions have an entanglement entropy proportional to the area of the hypersurface separating both partitions. The area law restricts the fraction of the Hilbert space accessible to ground states of local Hamiltonians in an essential way, allowing for their efficient numerical simulation [4].Violations of the area law occur in gapless (critical) systems. In one dimension most of critical systems, as well as being gapless, are also conformal invariant. The attention to the entanglement properties on these systems came after the seminal result of Holzhey, Larsen and Wilczek [5], who showed that the leading behavior of the ground state entropies S gs n is proportional to the central charge of the underlying conformal field theory (CFT) governing the long-distance physics of the discrete quantum chain. If ℓ and N are the lengths of the partition A and of the total system, both measured in lattice spacing units, then the Rényi entropy of the ground state, with periodic boundary conditions, is [5-7]where c is the central charge of the CFT and γ n is a nonuniversal constant. In a critical model, the finite-size scaling of the energy of excitations is given by the scaling dimension of the corresponding conformal operators [8]. This fact suggests that also the entanglement entropy could be related to properties of these operators. Entanglement of excited states has been considered previously. In [9] it was shown that the negativity of the excited states in the XXZ critical model shows a universal scaling. In [10] it was shown that a violat...

Rényi and von Neumann entropies quantifying the amount of entanglement in ground states of critical spin chains are known to satisfy a universal law which is given by the Conformal Field Theory (CFT) describing their scaling regime. This law can be generalized to excitations described by primary fields in CFT, as was done in reference [1], of which this work is a completion. An alternative derivation is presented, together with numerical verifications of our results in different models belonging to the c = 1, 1/2 universality classes. Oscillations of the Rényi entropy in excited states and descendant fields are also discussed.

The origin and meaning of facial beauty represent a longstanding puzzle. Despite the profuse literature devoted to facial attractiveness, its very nature, its determinants and the nature of inter-person differences remain controversial issues. Here we tackle such questions proposing a novel experimental approach in which human subjects, instead of rating natural faces, are allowed to efficiently explore the face-space and “sculpt” their favorite variation of a reference facial image. The results reveal that different subjects prefer distinguishable regions of the face-space, highlighting the essential subjectivity of the phenomenon. The different sculpted facial vectors exhibit strong correlations among pairs of facial distances, characterising the underlying universality and complexity of the cognitive processes, and the relative relevance and robustness of the different facial distances.

The critical behavior of the O(2) model on dilute Lévy graphs built on a 2D square lattice is analyzed. Different qualitative cases are probed, varying the exponent ρ governing the dependence on the distance of the connectivity probability distribution. The mean-field regime, as well as the long-range and short-range non-mean-field regimes are investigated by means of high-performance parallel Monte-Carlo numerical simulations running on GPU's. The relationship between the longrange ρ exponent and the effective dimension of an equivalent short-range system with the same critical behavior is investigated. Evidence is provided for the effective short-range dimension to coincide with the spectral dimension of the Lévy graph for the XY model in the mean-field regime.

We review recent statistical mechanical approaches to multimode laser theory. The theory has proved very effective to describe standard lasers. We refer of the mean field theory for passive mode locking and developments based on Monte Carlo simulations and cavity method to study the role of the frequency matching condition. The status for a complete theory of multimode lasing in open and disordered cavities is discussed and the derivation of the general statistical models in this framework is presented. When light is propagating in a disordered medium, the system can be analyzed via the replica method. For high degrees of disorder and nonlinearity, a glassy behavior is expected at the lasing threshold, providing a suggestive link between glasses and photonics. We describe in details the results for the general Hamiltonian model in mean field approximation and mention an available test for replica symmetry breaking from intensity spectra measurements. Finally, we summary some perspectives still opened for such approaches.Comment: 16 pages, 7 figure