On the construction of evolution equations admitting a master symmetry

A method for constructing evolution equations admitting a master symmetry is proposed. Several examples illustrating the method are presented. It is also noted that for certain evolution equations master symmetries can be useful for obtaining new conservation laws from a given one

Global properties of the spectrum of the Haldane-Shastry spin chain

We derive an exact expression for the partition function of the su(m) Haldane-Shastry spin chain, which we use to study the density of levels and the distribution of the spacing between consecutive levels. Our computations show that when the number of sites N is large enough the level density is Gaussian to a very high degree of approximation. More surprisingly, we also find that the nearest-neighbor spacing distribution is not Poissonian, so that this model departs from the typical behavior for an integrable system. We show that the cumulative spacing distribution of the model can be well approximated by a simple functional law involving only three parameters.Comment: RevTeX 4, 7 pages, 4 figures. To appear in Phys. Rev.

Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions

We introduce a family of inhomogeneous XX spin chains whose squared couplings are a polynomial of degree at most four in the site index. We show how to obtain an asymptotic approximation for the Rényi entanglement entropy of all such chains in a constant magnetic field at half filling by exploiting their connection with the conformal field theory of a massless Dirac fermion in a suitably curved static background. We study the above approximation for three particular chains in the family, two of them related to well-known quasi-exactly solvable quantum models on the line and the third one to classical Krawtchouk polynomials, finding an excellent agreement with the exact value obtained numerically when the Rényi parameter α is less than one. When α ≥ 1 we find parity oscillations, as expected from the homogeneous case, and show that they are very accurately reproduced by a modification of the Fagotti-Calabrese formula. We have also analyzed the asymptotic behavior of the Rényi entanglement entropy in the non-standard situation of arbitrary filling and/or inhomogeneous magnetic field. Our numerical results show that in this case a block of spins at each end of the chain becomes disentangled from the rest. Moreover, the asymptotic approximation for the case of half filling and constant magnetic field, when suitably rescaled to the region of non-vanishing entropy, provides a rough approximation to the entanglement entropy also in this general case

A duality principle for the multi-block entanglement entropy of free fermion systems

The analysis of the entanglement entropy of a subsystem of a one-dimensional quantum system is a powerful tool for unravelling its critical nature. For instance, the scaling behaviour of the entanglement entropy determines the central charge of the associated Virasoro algebra. For a free fermion system, the entanglement entropy depends essentially on two sets, namely the set A of sites of the subsystem considered and the set K of excited momentum modes. In this work we make use of a general duality principle establishing the invariance of the entanglement entropy under exchange of the sets A and K to tackle complex problems by studying their dual counterparts. The duality principle is also a key ingredient in the formulation of a novel conjecture for the asymptotic behavior of the entanglement entropy of a free fermion system in the general case in which both sets A and K consist of an arbitrary number of blocks. We have verified that this conjecture reproduces the numerical results with excellent precision for all the configurations analyzed. We have also applied the conjecture to deduce several asymptotic formulas for the mutual and r-partite information generalizing the known ones for the single block case

Las marcas y sus estrategias de inclusión de género en las redes sociales

En los países industriales el 80% del consumo total es realizado por mujeres, si bien esta cifra contrasta con la escasa identificación que éstas tienen con las marcas. Este fenómeno no es ajeno a las redes sociales. El presente trabajo se basa en una investigación sobre el nivel de inclusión de género de grandes marcas comerciales en las principales redes sociales en el contexto español. Partimos de la constatación de que las mujeres en su mayoría siguen las marcas en lugar de identificarse y participar activamente en las mismas. En un primer momento identificamos las estrategias de marketing definitorias de cada una de las marcas analizadas en función de su sector comercial, así como sus distintas estrategias de comunicación y marketing en medios sociales. Seguidamente analizamos en qué medida estas marcas, a partir de las dimensiones analizadas, promueven la inclusión de las mujeres y el tipo de inclusión que fomentan. Por último, a partir de los análisis realizados, identificamos algunas claves operativas que permitan distinguir, evaluar y diseñar estrategias más sensibles a la diferencias de género en las campañas de comunicación y marketing en medios sociales

Supersymmetric spin chains with nonmonotonic dispersion relation: criticality and entanglement entropy

We study the critical behavior and the ground-state entanglement of a large class of su(1|1) supersymmetric spin chains with a general (not necessarily monotonic) dispersion relation. We show that this class includes several relevant models, with both short-and long-range interactions of a simple form. We determine the low temperature behavior of the free energy per spin, and deduce that the models considered have a critical phase in the same universality class as a (1 + 1)-dimensional conformal field theory (CFT) with central charge equal to the number of connected components of the Fermi sea. We also study the Renyi entanglement entropy of the ground state, deriving its asymptotic behavior as the block size tends to infinity. In particular, we show that this entropy exhibits the logarithmic growth characteristic of (1 + 1)-dimensional CFTs and one-dimensional (fermionic) critical lattice models, with a central charge consistent with the low-temperature behavior of the free energy. Our results confirm the widely believed conjecture that the critical behavior of fermionic lattice models is completely determined by the topology of their Fermi surface

Solvable scalar and spin models with near-neighbors interactions

We construct new solvable rational and trigonometric spin models with near-neighbors interactions by an extension of the Dunkl operator formalism. In the trigonometric case we obtain a finite number of energy levels in the center of mass frame, while the rational models are shown to possess an equally spaced infinite algebraic spectrum. For the trigonometric and one of the rational models, the corresponding eigenfunctions are explicitly computed. We also study the scalar reductions of the models, some of which had already appeared in the literature, and compute their algebraic eigenfunctions in closed form. In the rational cases, for which only partial results were available, we give concise expressions of the eigenfunctions in terms of generalized Laguerre and Jacobi polynomials

Thermodynamics and criticality of supersymmetric spin chains with long-range interactions

We study the thermodynamics and critical behavior of su(m|n) supersymmetric spin chains of Haldane-Shastry type with a chemical potential term. We obtain a closed-form expression for the partition function and deduce a description of the spectrum in terms of the supersymmetric version of Haldane's motifs, which we apply to obtain an analytic expression for the free energy per site in the thermodynamic limit. By studying the low-temperature behavior of the free energy, we characterize the critical behavior of the chains with 1 <= m, n <= 2, determining the critical regions and the corresponding central charge. We also show that in the su(2|1), su(1| 2) and su(2| 2) chains the bosonic or fermionic densities can undergo first-order ( discontinuous) phase transitions at T = 0, in contrast with the previously studied su(2) case

Quasi-exactly solvable models in nonlinear optics

We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases nth harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigenspaces of the integrals of motion and show that the action of the Hamiltonian on these common eigenspaces can be represented by a quasiexactly solvable reduced Hamiltonian, whose expression in terms of the usual generators of sl_2 is computed explicitly