Person:
Tempesta, Piergiulio

First Name

Piergiulio

Last Name

Tempesta

URI

https://hdl.handle.net/20.500.14352/74979

Affiliation

Universidad Complutense de Madrid

Faculty / Institute

Ciencias Físicas

Department

Física Teórica

Area

Matemática Aplicada

Identifiers

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Now showing 1 - 10 of 16

Group entropies: from phase space geometry to entropy functionals via Group Theory
(Entropy, 2018) Jensen, Henrik Jeldtoft; Tempesta, Piergiulio
The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective.
Critical behavior of su(1/1) supersymmetric spin chains with long-range interactions
(Physical review E, 2016) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We introduce a general class of su(1 / 1) supersymmetric spin chains with long-range interactions which includes as particular cases the su (1 / 1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1 / 1 ) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low- energy excitations and the low-temperature behavior of the free energy, hich coincides with that of a (1 + 1)-dimensional conformal field theory (CFT) with central charge c = 1 when the chemical potential lies in the critical interval (0, ε (π)), ε (p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1 + 1) –dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c = 1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1 / 1) elliptic chain.
Generalized isotropic Lipkin-Meshkov-Glick models: ground state entanglement and quantum entropies
(Journal of statistical mechanics : theory and experiment, 2016) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We introduce a new class of generalized isotropic Lipkin–Meshkov–Glick models with su(m+1) spin and long-range non-constant interactions, whose non-degenerate ground state is a Dicke state of su(m+1) type. We evaluate in closed form the reduced density matrix of a block of Lspins when the whole system is in its ground state, and study the corresponding von Neumann and Rényi entanglement entropies in the thermodynamic limit. We show that both of these entropies scale as a log L when L tends to infinity, where the coefficient a is equal to (m − k)/2 in the ground state phase with k vanishing magnon densities. In particular, our results show that none of these generalized Lipkin–Meshkov–Glick models are critical, since when L-->∞ their Rényi entropy R_q becomes independent of the parameter q. We have also computed the Tsallis entanglement entropy of the ground state of these generalized su(m+1) Lipkin–Meshkov–Glick models, finding that it can be made extensive by an appropriate choice of its parameter only when m-k≥3. Finally, in the su(3) case we construct in detail the phase diagram of the ground state in parameter space, showing that it is determined in a simple way by the weights of the fundamental representation of su(3). This is also true in the su(m+1) case; for instance, we prove that the region for which all the magnon densities are non-vanishing is an (m + 1)-simplex in R^m whose vertices are the weights of the fundamental representation of su(m+1).
A duality principle for the multi-block entanglement entropy of free fermion systems
(Scientific reports, 2017) Carrasco, J. A.; Finkel Morgenstern, Federico; González López, Artemio; Tempesta, Piergiulio
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.
Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions
(Physical review E, 2016) Carrasco, José A.; Finkel Morgenstern, Federico; González López, Artemio; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
Nilpotent integrability, reduction of dynamical systems and a third-order Calogero-Moser system
(Annali di matematica pura ed applicata, 2019) Ibort, A.; Marmo, G.; Rodríguez González, Miguel Ángel; Tempesta, Piergiulio
We present an algebraic formulation of the notion of integrability of dynamical systems, based on a nilpotency property of its flow: It can be explicitly described as a polynomial on its evolution parameter. Such a property is established in a purely geometric-algebraic language, in terms both of the algebra of all higher-order constants of the motion (named the nilpotent algebra of the dynamics) and of a maximal Abelian algebra of symmetries (called a Cartan subalgebra of the dynamics). It is shown that this notion of integrability amounts to the annihilator of the nilpotent algebra being contained in a Cartan subalgebra of the dynamics. Systems exhibiting this property will be said to be nilpotent-integrable. Our notion of nilpotent integrability offers a new insight into the intrinsic dynamical properties of a system, which is independent of any auxiliary geometric structure defined on its phase space. At the same time, it extends in a natural way the classical concept of integrability for Hamiltonian systems. An algebraic reduction procedure valid for nilpotent-integrable systems, generalizing the well-known reduction procedures for symplectic and/or Poisson systems on appropriate quotient spaces, is also discussed. In particular, it is shown that a large class of nilpotent-integrable systems can be obtained by reduction of higher-order free systems. The case of the third-order free system is analyzed and a non-trivial set of third-order Calogero-Moser-like nilpotent-integrable equations is obtained.
Groups, information theory, and Einstein's likelihood principle
(Physical review E, 2016) Sicuro, Gabriele; Tempesta, Piergiulio
We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts.
Beyond the Shannon-Khinchin formulation: The composability axiom and the universal-group entropy.
(Annals of physics, 2016) Tempesta, Piergiulio
The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann-Gibbs (BC) entropy and could be applicable in thermodynamics, quantum mechanics and information theory. In Khinchin (1957), by extending previous ideas of Shannon (1948) and Shannon and Weaver (1949), Khinchin proposed a characterization of the BG entropy, based on four requirements, nowadays known as the Shannon-Khinchin (SK) axioms. The purpose of this paper is twofold. First, we show that there exists an intrinsic group-theoretical structure behind the notion of entropy. It comes from the requirement of composability of an entropy with respect to the union of two statistically independent systems, that we propose in an axiomatic formulation. Second, we show that there exists a simple universal family of trace form entropies. This class contains many well known examples of entropies and infinitely many new ones, a priori multi-parametric. Due to its specific relation with Lazard's universal formal group of algebraic topology, the new general entropy introduced in this work will be called the universal-group entropy. A new example of multi-parametric entropy is explicitly constructed.
A parallel space saving algorithm for frequent items and the Hurwitz zeta distribution
(Information sciences, 2016) Cafaro, Massimo; Pulimeno, Marco; Tempesta, Piergiulio
We present a message-passing based parallel version of the Space Saving algorithm designed to solve the k-majority problem. The algorithm determines in parallel frequent items, i.e., those whose frequency is greater than a given threshold, and is therefore useful for iceberg queries and many other different contexts. We apply our algorithm to the detection of frequent items in both real and synthetic datasets whose probability distribution functions are a Hurwitz and a Zipf distribution respectively. Also, we compare its parallel performances and accuracy against a parallel algorithm recently proposed for merging summaries derived by the Space Saving or Frequent algorithms.
Singularity confinement for matrix discrete Painleve equations
(Nonlinearity, 2014) Cassatella-Contra, Giovanni A; Mañas Baena, Manuel; Tempesta, Piergiulio
We study the analytic properties of a matrix discrete system introduced by Cassatella and Manas (2012 Stud. Appl. Math. 128 252-74). The singularity confinement for this system is shown to hold generically, i.e. in the whole space of parameters except possibly for algebraic subvarieties. This paves the way to a generalization of Painleve analysis to discrete matrix models.